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Electrodynamic response of mesoscopic objects carbon nanotubes and superconducting nanowires


Electrodynamic response of mesoscopic objects carbon nanotubes and superconducting nanowires

Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector MagniďŹ cus prof. ir. K.Ch.A.M. Luyben, voorzitter van het College voor Promoties, in het openbaar te verdedigen op woensdag 26 februari 2014 om 10:00 uur door

Hendrik Lude HORTENSIUS natuurkundig ingenieur geboren te Zeist.


Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. T. M. Klapwijk Samenstelling van de promotiecommissie: Rector MagniďŹ cus, Prof. dr. ir. T. M. Klapwijk Prof. dr. K. K. Berggren Prof. dr. ir. T. H. Oosterkamp Prof. dr. Y. M. Blanter Prof. dr. J. M. van Ruitenbeek Prof. dr. H. W. Zandbergen Dr. ir. E. F. C. Driessen Prof. dr. A. Neto

voorzitter Technische Universiteit Delft, promotor Massachusetts Institute of Technology, USA Universiteit Leiden Technische Universiteit Delft Universiteit Leiden Technische Universiteit Delft CEA Grenoble, France Technische Universiteit Delft, reservelid

Published by: H.L. Hortensius Printed by: GVO printers & designers | Ponsen & Looijen, Ede, The Netherlands An electronic version of this thesis is available at: http://repository.tudelft.nl c 2014 by H.L. Hortensius. All rights reserved. Copyright  Casimir PhD Series, Delft-Leiden, 2014-4 ISBN 978-90-8593-179-9


Contents

1 Shining a new light on mesoscale physics 1.1 Introduction . . . . . . . . . . . . . . . . . . 1.2 Electron interaction in mesoscopic systems . 1.3 The link to the real world . . . . . . . . . . 1.4 Thesis outline . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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1 2 2 4 5 7

2 Carbon nanotubes and superconducting nanowires 2.1 Carbon nanotubes . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Atomic structure . . . . . . . . . . . . . . . . . . . . 2.1.2 Low-dimensional transport . . . . . . . . . . . . . . 2.1.3 Excitation dynamics of a Tomonaga-Luttinger liquid 2.2 Superconducting nanowires . . . . . . . . . . . . . . . . . . 2.2.1 Superconductivity . . . . . . . . . . . . . . . . . . . 2.2.2 Critical current . . . . . . . . . . . . . . . . . . . . . 2.2.3 Single-photon detection . . . . . . . . . . . . . . . . 2.2.4 Superconductivity in strongly disordered materials . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 10 10 11 16 18 19 21 22 26 29

3 Experimental methods 3.1 Materials and sample-development . . . . . . . . . . . . . 3.1.1 Microwave-coupled, suspended, carbon nanotubes . 3.1.2 Nanogeometries in thin disordered superconductors 3.2 Cryogenics and radiation sources . . . . . . . . . . . . . . 3.2.1 Measurement setups . . . . . . . . . . . . . . . . .

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35 36 36 38 39 39

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v


3.2.2 Microwave and terahertz sources . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40 42

4 Microwave-induced nonequilibrium temperature in a suspended carbon nanotube 43 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Suspended carbon nanotubes under microwave radiation . . . . 44 4.3 Luttinger liquid behavior . . . . . . . . . . . . . . . . . . . . . 47 4.4 Microwave heating . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5 Electron temperature proďŹ le in a suspended carbon nanotube . 50 4.6 Thermovoltage of microwave-heated carbon nanotube . . . . . 52 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5 Frequency-dependent response of suspended carbon nanotubes 5.1 Collective modes in a Luttinger liquid . . . . . . . . . . . . . . 5.2 DC characterization of used suspended carbon nanotubes . . . 5.3 Free-electron laser for 1.5 to 3 terahertz . . . . . . . . . . . . . 5.4 Radiation-induced changes in DC transport . . . . . . . . . . . 5.5 Frequency dependence of the carbon nanotube response . . . . 5.6 Analysis of the observed frequency dependence . . . . . . . . . 5.7 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 58 59 61 62 65 69 72 74

6 Critical-Current Reduction in Thin Superconducting Wires Due to Current Crowding 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 NbTiN geometries . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Reduction of the critical current due to current crowding . . . . 6.4 Temperature dependence of the critical current distribution . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 78 79 81 84 85 86

7 Possible Indications of Electronic Inhomogeneities in Superconducting Nanowire Detectors 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Steplike transitions . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Thermal model with randomly uctuating parameters . . . . . 7.5 Simulated characteristics . . . . . . . . . . . . . . . . . . . . . . 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 90 90 92 93 96 97 98

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8 Eect of spatial variations of the superconducting properties on common observables. 101 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.2 Atomic-layer-deposited-TiN nanowires . . . . . . . . . . . . . . 102 8.3 The resistive transition . . . . . . . . . . . . . . . . . . . . . . . 104 8.4 The critical current . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.5 The voltage-carrying state . . . . . . . . . . . . . . . . . . . . . 113 8.6 Qualitative model of inhomogeneity in a voltage-biased self-heating hotspot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Summary

127

Samenvatting

131

Curriculum Vitae

135

List of publications

137

Acknowledgements

139

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viii


Chapter 1 Shining a new light on mesoscale physics

1


1. Shining a new light on mesoscale physics

1.1

Introduction

The interaction of (far-infrared) light with mesoscopic objects is interesting from two points of view. On the one hand, it allows us to detect the light with single-photon sensitivity, with applications in for instance astronomy and quantum information technology. On the other hand, it provides a probe of the intrinsic physics. In this thesis, we choose to study two distinct mesoscale samples: suspended carbon nanotubes and disordered superconducting nanowires. In both these examples, it is the interaction between electrons confined in small devices, which give rise to unique physical processes.

1.2

Electron interaction in mesoscopic systems

Mesoscopic literally means ”that which is seen in the middle”. Figure 1.1 shows ’Bridge over a Pond of Water Lilies’ by the impressionist painter Claude Monet. If we look on the smallest scale at the single brushstrokes of the painter, such as in Fig. 1.1(a), we can learn something about the technique, the choice of paint, etc., but it is only when we look on a larger scale, as in Fig. 1.1(b), that the vision of the painter emerges. On the other hand, visibly building the painting up of discrete brushstrokes, allows the impressionists to give a new dimension to their paintings compared to classical techniques. In a similar way, mesoscopic physics is the area of condensed matter physics between the physics of single particles and the classic ’macroscopic’ physics we experience in our daily live. If we approach from the world of single particles, we enter the mesoscopic regime when the number of atoms or electrons becomes so large that collective interactions lead to the emergence of new physics. For example, superconductivity exists as a consequence of a large number of electrons interacting through lattice vibrations. This leads to amazing material properties such as the ability to carry dissipationless currents and expulsion of magnetic fields [1]. One could never start from the quantum mechanical description of single electrons and come up with superconductivity, it is through the interaction of a large ensemble that new physics emerges [2]. While ’conventional’ superconductivity is well understood, the complex interactions in more exotic superconductors give rise to a wide scala of phenomena, such as a pseudogap or transitions into different states of matter [3,4]. This is an active and fruitful field of research from a fundamental physics point of view, with wide potential applications in for instance energy technology. Recent experiments [5–7] show that in strongly disordered superconductors, such as those typically used for single-photon detectors, the superconducting properties can become position dependent. Disorder takes the form of a random spatial distribution of atoms and defects on which electrons can scatter in the 2


1.2 Electron interaction in mesoscopic systems

Figure 1.1. Bridge over a Pond of Water Lilies by Claude Monet (1899). (a) is a detail of the painting, showing the individual brushstrokes. Just as the vision of a bridge emerges from the interplay of a number of individual strokes, mesoscopic physics emerges from the interaction of a large number of individual particles.

material. Superconducting single-photon detectors have promising applications in for instance quantum information technology [8]. For eďŹƒcient device operation, the homogeneity of the superconducting current-carrying state may be crucial. In this thesis, we will see that geometrical and intrinsic inhomogeneities can arise from device architecture [9] and disordered superconductivity [5] respectively. If we approach from the macroscopic world, we enter the mesoscopic regime 3


1. Shining a new light on mesoscale physics

when sample dimensions become smaller than characteristic length scales, such as the scattering length or wavelength of the electrons. At a certain point, the classical mechanics that govern our daily lives do not describe the complete picture. For example, if we confine electrons in a one-dimensional structure such as a carbon nanotube, quantum mechanics plays an important role in the nature of the charge transport [10]. This leads to exotic states of the complete electron system, fundamentally changing the charge transport properties [11].

1.3

The link to the real world

Mesoscopic physics occurs on a length scale too small to observe without the use of powerful microscopes. To study them, these samples need to be connected to our world of measurement electronics and computers. In order to perform charge transport measurements, such as those discussed throughout this thesis, at least two electrodes need to be connected to the sample to inject and extract electrons.

Figure 1.2. The connection between a mesoscopic sample displaying interesting physics (blue) and measurement devices in our macroscopic world, is often not deeply considered. The way that this connection is made can however have significant influence on the physics of the mesoscopic sample or lead to altogether new phenomena.

Figure 1.2 shows the way we intuitively tend to imagine an ’ideal’, ’welldefined’ contact between a sample and the outside world. Our object of interest (blue) forms a bridge between the reservoirs (gray). Typically, we assume these contacts to be an infinite reservoir of electrons at a fixed temperature and chemical potential. There is a sharp transition point where the sample 4


1.4 Thesis outline

is connected to the reservoirs (dotted line). This line can be given certain properties (resistance, temperature, etc.) that describe the contact. The mesoscopic sample and the reservoirs are generally not made of the same material. This can lead to a variety of effects in which the contacts play a crucial role in the physics of the device. For example current-conversion processes at the interface between normal metal reservoirs and a superconducting wire can lead to complex phenomena such as a non-equilibrium energy distribution of the charge carriers [12]. Typically the contact can in reality not be considered to occur at one single location; its effect is present over a region extending both into the reservoirs and into the sample. In this thesis, we find that even when the contacts and the mesoscopic device consist of the same material, a sudden change in dimensions can have profound effects. The presence of sharp corners in a contact scheme such as shown in Fig. 1.2 leads to a reduced ability to carry a supercurrent through the device. Ironically, we find that due to efforts to create a ’nice’, ’ideal’, ’well-defined’ system, we tend to create unexpected new problems. The nature of thermal transport in nanoscale devices is one of the main current open questions in quantum transport. For instance, if electrons pass through an object faster than they exchange their energy with their surroundings, Joule heating occurs non-locally from the associated resistance. This can lead to asymmetric heating in the two reservoirs, depending on the direction of the current [13]. In the case of carbon nanotubes, we will find that under microwave or terahertz irradiation a strong gradient in local electron temperature can form at the contacts, which induces the development of a thermovoltage. By studying charge transport under the influence of incident radiation, we can extract new information about the physical processes occurring in mesoscopic systems. One can for instance study time-dependent processes by varying the frequency or add energy to the system without going to large bias voltages.

1.4

Thesis outline

This thesis describes experiments on mesoscale samples in relation to their interaction with light. In the case of carbon nanotubes, the microwave and terahertz radiation enables selective heating of the carbon nanotube. In the case of the superconducting nanowires, the application of the devices lies in the detection of single photons. The influence of geometrical and intrinsic inhomogeneities on important device characteristics is studied. Chapter 2 describes the important concepts and context for the experiments discussed in this thesis. Chapter 3 discusses the experimental techniques, sample fabrication, and available sources of terahertz radiation. In Chapter 4 we report observations of the increase of the local electron 5


1. Shining a new light on mesoscale physics

temperature in suspended carbon nanotubes exposed to 108 GHz radiation. The DC current-voltage characteristics are studied, showing both an increase in conductivity and the development of a thermovoltage under microwave irradiation. Chapter 5 describes experiments at FELIX, the Free-Electron Laser for Infrared eXperiments, where we studied the current-voltage characteristics of suspended carbon nanotubes under continuously tunable radiation from 1.5 to 3 THz. We again ďŹ nd the increase in conductivity and development of a DC voltage, and report on the frequency dependence of the nanotube response. In Chapter 6 we change our focus to disordered superconducting nanowires. We study the inuence of geometrical inhomogeneities, in particular the presence of sharp corners, on the critical current of these devices. The critical current is an important characteristic in the operation of superconducting nanowire single photon detectors. We ďŹ nd a strong reduction of the critical current and show that it is avoidable by the use of optimally rounded corners. Chapter 7 describes the observation of a step-like transition from the normal to the superconducting state in current-biased disordered superconducting nanowires. Geometrical inhomogeneities do not dominate these features. We develop a thermal model based on a variation of the critical temperature along the wires, which reproduces the observed current-voltage characteristics. In Chapter 8 we discuss a variety of typical DC measurements on titanium nitride nanowires of varying thickness. We measure the resistive transition, the critical current, and both the current- and voltage-biased transition from the normal state to the superconducting state. We discuss the results of these measurements in the context of electronic inhomogeneities in strongly disordered superconductors.

6


1.4 References

References [1] M. Tinkham, Introduction to superconductivity, McGraw-Hill, New York, (1975). [2] P. W. Anderson, More is different, Science 177, 393 (1972). [3] M. R. Norman, The challenge of unconventional superconductivity, Science 332, 196 (2011). [4] J. C. Davis and D.-H. Lee, Concepts relating magnetic interactions, intertwined electronic orders, and strongly correlated superconductivity, PNAS 110, 17623 (2013). [5] B. Sac´ep´e, C. Chapelier, T. I. Baturina, V. M. Vinokur, M. R. Baklanov, and M. Sanquer, Disorder-induced inhomogeneities of the superconducting state close to the superconductor-insulator transition, Phys. Rev. Lett. 101, 157006 (2008). [6] M. Chand, G. Saraswat, A. Kamlapure, M. Mondal, S. Kumar, J. Jesudasan, V. Bagwe, L. Benfatto, V. Tripathi, and P. Raychaudhuri, Phase diagram of the strongly disordered s-wave superconductor NbN close to the metal-insulator transition, Phys. Rev. B 85, 014508 (2012). [7] P. C. J. J. Coumou, E. F. C. Driessen, J. Bueno, C. Chapelier, and T. M. Klapwijk, Electrodynamic response and local tunneling spectroscopy of strongly disordered superconducting TiN films, Phys. Rev. B 88, 180505 (2013). [8] C. M. Natarajan, M. G. Tanner, and R. H. Hadfield, Superconducting nanowire single-photon detectors: physics and applications, Supercond. Sci. Technol. 25, 063001 (2012). [9] J. R. Clem and K. K. Berggren, Geometry-dependent critical currents in superconducting nanocircuits, Phys. Rev. B 84, 174510 (2011). [10] E. Thune and C. Strunk, Quantum transport in carbon nanotubes, Lect. Notes Phys. 680, 351 (2005). [11] V. V. Deshpande, M. Bockrath, L. I. Glazman, and A. Yacoby, Electron liquids and solids in one dimension, Nature 464, 209 (2010). [12] N. Vercruyssen, T. G. A. Verhagen, M. G. Flokstra, J. P. Pekola, and T. M. Klapwijk, Evanescent states and nonequilibrium in driven superconducting nanowires, Phys. Rev. B 85, 224503 (2012). [13] W. Lee, K. Kim, W. Jeong, L. A. Zotti, F. Pauly, J. C. Cuevas, and P. Reddy, Heat dissipation in atomic-scale junctions, Nature 498, 209 (2013).

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1. Shining a new light on mesoscale physics

8


Chapter 2 Carbon nanotubes and superconducting nanowires We discuss the theoretical concepts and experimental context of two types of nanowires: suspended carbon nanotubes showing collective behavior of the electron system and superconducting nanowires made of strongly disordered superconductors. The interaction with light is important for both. In carbon nanotubes, incident terahertz radiation may provide a probe for their intrinsic excitations. In the case of superconducting nanowires, the superconducting state allows them to be used as sensitive single photon detectors. Here structural and intrinsic inhomogeneities play an important role.

9


2. Carbon nanotubes and superconducting nanowires

2.1 2.1.1

Carbon nanotubes Atomic structure

The fullerenes are a special class of carbon-based materials, which consist of sheets of carbon atoms in a (mainly) hexagonal lattice. Since the discovery of the fullerenes in 1985 by Kroto et al. [1], a variety of realizations with different dimensionality were found. Figure 2.1 shows the most famous examples: (a) C60 (0D) [1], (b) carbon nanotubes (1D) [2], and (c) graphene (2D) [3].

(a)

(b)

(c)

Figure 2.1. Structure of fullerenes with different dimensionality: (a) Buckminster fullerene (C60 ) (0D), (b) carbon nanotube (1D), (c) graphene (2D) [4].

We will focus our attention on carbon nanotubes, which were discovered in 1991 by Iijima [2]. As shown in Fig. 2.1(b), a carbon nanotube consists of a ’rolled up’ sheet of graphene. Depending on the growth mechanism and particular realization, they can either be made of a single graphene layer: single-wall carbon nanotubes (SWCNT), or multiple sheets of graphene wrapped around each other: multi-walled carbon nanotubes (MWCNT). Carbon nanotubes have extraordinary physical properties. For instance, they are extraordinarily stiff, their Young’s modulus can be as high as ∼1 TPa [5]. This has led to their integration in other materials for strengthening [6]. Due to their small mass, the mechanical vibrations of suspended carbon nanotubes are extremely sensitive to the adhesion of small particles [7, 8]. Because of the high frequency and low losses of the mechanical vibrations, suspended carbon nanotubes are a promising candidate for the study of vibrational modes of a system cooled to the quantum groundstate [9]. However, in this work we will focus on the electrodynamic properties. Fig. 2.2(a) shows the band diagram of a graphene sheet [10]. The conduction and valence band touch at the Dirac points. In the absence of doping, the Fermi energy lies exactly between the conduction and valence band. The linear dispersion relation in the cones around these points leads to many of the 10


2.1 Carbon nanotubes

(a)

(b)

E k

E

(c) ky

E k

kx

Figure 2.2. (a) Band structure of graphene [10]. The points where the valence (dark gray) and conduction band (light transparent gray) touch are the Dirac points. The intersecting planes correspond to a metallic (red) and semiconducting (blue) carbon nanotube. (b) Band diagram of a metallic carbon nanotube. (c) Band diagram of a semiconducting carbon nanotube.

electronic properties of graphene. The band diagram of a carbon nanotube can be found by making a planar cut-through of the graphene band diagram, as indicated in Fig. 2.2(a). Depending on the orientation of the vector along which the graphene sheet is ’rolled up’ (the chirality), the plane can cross through a Dirac point or not. By changing the diameter of the tube the plane can be shifted. By changing the chirality the plane can be rotated. If a Dirac point is crossed, the density of states of the nanotube is finite around the Fermi energy and the nanotube is metallic (Fig. 2.2(b)). If no Dirac point is crossed, there is a gap in the density of states around the Fermi energy and the nanotube is semi-conducting (Fig. 2.2(c)). In practice, 1/3 of randomly grown carbon nanotubes are metallic and 2/3 are semi-conducting [11].

2.1.2

Low-dimensional transport

In a bulk three-dimensional metal, a very large number of electrons is present. All of these electrons interact through the strong Coulomb repulsion. This Coulomb interaction can be screened by the movement of nearby electrons to short distances. Effectively, every electron interacts with a cloud of electrons around them. Still, the system can be surprisingly adequatly described as a system of free noninteracting electron-like particles through the Fermi liquid 11


2. Carbon nanotubes and superconducting nanowires

theory. The elementary excitations of the system can be described as fermionic quasiparticles. The effects of the interactions are contained in the properties of the quasiparticles, such as an effective mass or lifetime [12]. In this thesis, we focus on systems in which the interactions between electrons changes in such a way that the Fermi liquid description breaks down. This can happen for instance due to a reduced screening of the Coulomb interaction, when the electrons are confined to move in a low-dimensional system or due to the presence of an effective attractive interaction in a superconductor. The electronic transport through carbon nanotubes can show, depending on the details of the experiment, aspects of electrons as waves (ballistic transport), electrons as single charged particles (Coulomb blockade), and collective modes of the complete electron system (Luttinger liquid). All of these effects stem from aspects of the confined electron system. In reality, the crossover between these different aspects can be continuous, see for instance Jezouin et al. [13]. Ballistic transport Clean carbon nanotubes have a very low scattering probability for electrons, because of their structural uniformity and one-dimensional character [14]. They can therefore be treated as perfect ballistic transmission channels in the LandauerB¨ uttiker framework [15]. This theory describes the electronic transport properties of a nano-object in terms of a number of transmission channels, each of which have a certain transmission probability T . A single channel with perfect transmission can carry the quantum conductance e2 /h. Due to the degeneracy of the states at the two Dirac points in a unit cell and the degeneracy of the two spin states, there are four one-dimensional channels in a clean carbon nanotube, each of which can have a near-perfect transmission. The maximum total conductance of a carbon nanotube in the ballistic limit G = 4e2 /h ≈ (6.4kΩ)−1 [11]. To measure the electrical transport characteristics of any system, contact has to be made to the outside environment. Depending on the fabrication method, contact resistances up to several hundreds of kilo-Ohms can occur. Contact resistances are lower for samples on which the metal contacts are deposited on top of the carbon nanotube, as opposed to the reverse procedure [11]. Coulomb blockade If tunnel contacts occur between the metal reservoirs and the carbon nanotube, the nanotube effectively forms an island for the electrons. Due to the Coulomb interaction, a certain energy has to be paid to add an additional electron. This charging energy is given by EC = e2 /2C, where C is the total capacitance from the island to the environment. Typically, the environment consists of the metal 12


2.1 Carbon nanotubes

(a)

(b)

(c)

Figure 2.3. (a) Schematic representation of a carbon nanotube with two tunnel contacts with bias voltage Vsd and a gate electrode with the gate potential UG and gate capacitance CG . (b) Energy diagram of a carbon nanotube quantum dot. The energy levels on the carbon nanotube are separated by the charging energy EC and can be shifted by the gate voltage UG . (c) Coulomb diamonds indicating the regions in which the amount of electrons on the carbon nanotube is constant and no current can flow through the carbon nanotube (gray diamonds) as a function of gate voltage (horizontal) and bias voltage (vertical) [11].

contacts and a gate electrode, at a potential UG . This system is schematically represented in Fig. 2.3(a). Fig. 2.3(b) shows the energy diagram for a carbon nanotube quantum dot. The vertical axis represents the energy of the electron states. The horizontal bars in the middle represent the energy levels of electrons on the carbon nanotube, spaced by the charging energy EC . If the charging energy is larger than the thermal energy: EC  kB T , the energy levels of the electrons on the nanotube are sharply defined. The energy levels can be shifted up and down by the gate voltage UG . The gray regions to the left and right represent the filled electron states of the left and right contact electrodes with chemical potential μL and μR respectively. The energy difference is given by the bias voltage VSD . If an energy level of the carbon nanotube lies in the window of width VSD between μL and μR , an electron can tunnel from the left electrode to the carbon nanotube and from there into an empty state on the right electrode and a net current will flow. If no energy levels lie in the bias window, no net current can flow. Fig. 2.3(c) shows the resultant diamond pattern in a two-dimensional plot of the differential conductance as a function of bias voltage (vertical) and gate voltage (horizontal). In the dark regions no current can flow, since there is no energy level of the carbon nanotube in the bias window. The conducting state can be reached either by opening up the bias window by increasing the bias voltage, or by shifting the energy levels on the carbon nanotube by the gate 13


2. Carbon nanotubes and superconducting nanowires

voltage. The asymmetry in the diamonds results from an asymmetry in the capacitances at the two contacts. Tomonaga-Luttinger liquid Because of the screening of the long range electron-electron interaction in bulk metals, those systems can be described as a system of single particle states with a finite lifetime, a Fermi liquid [16]. Due to the one-dimensional nature of a carbon nanotube, Coulomb screening of the electrons is largely ineffective compared to a bulk three-dimensional metal. In the absence of screening of the Coulomb interaction in one-dimensional systems, the Fermi liquid theory breaks down. The system has to be described as a collective system of strongly interacting particles. For instance, when an electron is added to the wire, it affects the position of all the electrons on the wire. It is therefore no longer possible to consider fermionic single charged particles as the elementary low energy excitations of the system. Instead, the system is typically described by bosonic modes of the collective electron system in the Tomonaga-Luttinger liquid theory [17, 18]. Experimental evidence for the Tomonaga-Luttinger liquid theory can be found in the tunneling density of states, which is suppressed around zero bias energy. The addition of an electron to the wire causes all other electrons on the wire to be displaced, as illustrated in Fig. 2.4(a). Therefore, additional energy is required for the addition of an electron, which can come either from the thermal energy of the electron system or the bias voltage. In a TomonagaLuttinger liquid therefore, the tunneling density of states is suppressed with a power-law dependence on energy around zero bias. This in contrast to a bulk metal, in which only a cloud of electrons immediately around an added charge is shifted, resulting in a feature-less density of states around zero bias energy. The tunneling density of states can be probed by measuring the differential conductance across a tunnel junction between a normal metal and the object under study. The tunneling current between two systems, labeled 1 and 2, is given by Fermi’s golden rule:  ∞ 2 I(V ) ∝ |T | N1 (E)N2 (E + eV )[f (E) − f (E + eV )]dE (2.1) ∞

with T the tunneling probability, Ni (E) the density of states of system i and f (E) the Fermi distribution. If system 1 is a normal metal with constant density, this leads to the following equation for the differential conductance:  ∞ ∂f (E + eV ) dI ∝ ]dE (2.2) N2 (E)[− dV ∂(eV ) ∞ At zero temperature, when the Fermi distribution is a step function, the dif14


2.1 Carbon nanotubes

(a)

E~

kBT eV

(b)

Figure 2.4. (a) Schematic representation of electrons on a carbon nanotube. Due to the one-dimensional nature, the addition of an electron requires the shift of all the electrons on the carbon nanotube. The thermal energy or the bias voltage can provide the energy required for this. (b) The differential conductance (vertical axis) is scaled by T α . The data then collapse onto a universal curve, showing the predicted power-law scaling of conductance with bias-voltage and temperature of the Tomonaga-Luttinger liquid [19].

ferential tunneling conductance directly measures the density of states: dI (V ) ∝ N2 (e|V |) dV

(2.3)

Figure 2.4 shows experimental evidence of the suppression of the tunneling density of states for tunneling into a carbon nanotube, measured by Bockrath et al. in 1999 [19]. For eV > kB T , the differential conductance increases with increasing bias voltage with a power-law scaling: dI/dV ∼ V α . For eV < kB T , the differential conductance increases with increasing temperature with the 15


2. Carbon nanotubes and superconducting nanowires

same power-law scaling: dI/dV ∼ T α . The conductance in Fig. 2.4(b) has been scaled by T α . Due to the strong interactions in a Tomonaga-Luttinger liquid, the density of states NE itself changes when an electron is added to the system or the temperature of the Tomonaga-Luttinger liquid is increased. This leads to the characteristic power-law behavior of tunneling into a TomonagaLuttinger liquid [20]. Although the suppression of the tunneling density of states around zero bias energy strongly hints at the expected Tomonaga-Luttinger liquid behavior in a one-dimensional system, it is not a fully conclusive evidence of the unique nature of such a system. Other mechanisms, such as Coulomb blockade in a dissipative electromagnetic environment, are known to cause a similar powerlaw suppression in for instance strongly disordered metals [21]. As we will see, the exotic nature of the Tomonaga-Luttinger liquid is more strongly present in the electrodynamics of the system.

2.1.3

Excitation dynamics of a Tomonaga-Luttinger liquid

To have a more complete experimental confirmation of the Tomonaga-Luttinger liquid theory, one would like to directly probe the elementary excitations. Tomonaga-Luttinger liquid theory predicts the existence of two kinds of bosonic excitations: spin-density waves moving at the Fermi velocity vF and charge density waves moving at an increased velocity vF /g, with g < 1 a parameter describing the strength of the long-range Coulomb interaction [22]. A measurement of the velocity of the excitations allows a direct determination of the nature of these excitations. A carbon nanotube connected to metallic leads with tunnel barriers forms a resonator for electronic excitations. Due to the typically low transmission of the tunnel barriers and the low scattering probability in the nanotube, it can act as a Fabry-Perot-like cavity for electron waves. If the length is known, this allows a direct measurement of the velocity of excitations in the system. Liang et al. have measured the Fermi velocity vF = 8 · 105 m/s for electron waves in a nanotube. Figure 2.5(b) shows a color plot of the modeled and measured differential conductance as a function of gate- and bias-voltage [23]. A striking consequence of electron transport in the Tomonaga-Luttinger liquid theory is the separation of spin and charge into spin- and charge-density waves. In a Fermi liquid, quasiparticles carry both spin and charge and travel at the Fermi-velocity. In a Tomonaga-Luttinger liquid however, the longrange Coulomb interaction gives rise to collective modes of the electron system. Waves of spin density can propagate along the one-dimensional wire, independent of charge flowing along the wire. Because no charges are dislocated in the spin-density waves, the unscreened long-range Coulomb interaction does 16


2.1 Carbon nanotubes

(a)

Charge mode: v > vF Spin mode:

(b)

v = vF

(c)

Figure 2.5. (a) Schematic representation of a charge density wave (top) and spin density wave (bottom) in a carbon nanotube TomonagaLuttinger liquid. The velocity of the charge density wave is greater than the Fermi velocity, due to the unscreened Coulomb interaction between the electrons. (b) Fabry-Perot oscillations in the diďŹ&#x20AC;erential conductance of a carbon nanotube as a function of bias voltage and gate voltage due to interference of electron waves, calculated (red) and measured (blue), Fermi velocity vF = 8 ¡ 105 m/s for single electron excitations [23]. (c) Velocity of excitations in a GaAs/AlGaAs onedimensional nanowires showing the increased velocity of the TomonagaLuttinger liquid charge excitations [24].

not inďŹ&#x201A;uence their propagation velocity, they travel at the Fermi velocity vF . Charge density waves however travel faster due to the long-range Coulomb interaction, as indicated schematically in Fig. 2.5(a). Their velocity is given by v = vF /g, where g is a parameter indicating the strength of the Coulomb interaction. g = 1 for the non-interaction particles in a Fermi liquid, while g < 1 for stronger interactions [22]. Momentum-resolved tunneling [24] from a one-dimensional wire provides experimental evidence of the spin-charge separation in a Tomonaga-Luttinger 17


2. Carbon nanotubes and superconducting nanowires

liquid. By studying the electron transfer as a function of bias voltage and magnetic field between two parallel ballistic one-dimensional channels in a GaAs/AlGaAs-heterostructure, the velocity of the collective modes in the wires can be determined. Using momentum-resolved tunneling the separation of spinand charge-density waves was experimentally confirmed, as shown in Fig. 2.5. As predicted, the spin-density waves propagate at the Fermi velocity. The charge-density waves were found to have an increased velocity of up to twice the Fermi velocity depending on the strength of the electron-electron interaction [24]. A measurement of the frequency dependence of the input impedance of a carbon nanotube is predicted to provide a direct measurement of the propagation velocity of the excitations in the system [25]. Also collective chargedensity waves are expected to give a frequency dependent response, as long as they couple to the incident radiation. For typical carbon nanotube lengths of a few micrometer and charge-density wave velocities of a few times the Fermivelocity, resonances are expected to occur in the terahertz frequency range. Measurement of the electrodynamics at these frequencies could thus provide the first direct measurement of the increased velocity of charge-density waves in carbon nanotube Tomonaga-Luttinger liquids.

2.2

Superconducting nanowires

In the preceding Section, the wavelike nature of light allows us to study resonances in carbon nanotubes. The application of superconducting nanowires on the other hand, is based on the particle nature of light; they are used to detect single photons. The ability to reliably detect single photons is a key technique in many experiments and applications such as quantum information science, quantum key distribution, optical quantum computing, characterization of quantum emitters, space-to-ground communications, integrated circuit testing, fiber sensing, and time-of-flight ranging [26]. Most commonly used single-photon detectors are currently based on silicon semiconductor technology [27]. These devices can not operate at frequencies below the bandgap of silicon. There is growing interest in technologies providing single-photon detection capabilities in the infrared range. Partially due to their much smaller energy gap, superconductors are commonly used for the detection of light in the infrared to terahertz range. Photons with insufficient energy to create excitations over a semiconductor gap, can still be detected in a superconducting system. Devices such as transition edge sensors [28], hot electron bolometers [29], superconducting tunnel junctions [30], and kinetic inductance detectors [31], have enabled detection of low-energy photons. 18


2.2 Superconducting nanowires

For applications in quantum information science however, other criteria determine the feasibility of a detector design. The main demands are the ability to detect single incident photons with a detection efficiency close to 100%, fast reset times, low timing jitter, and low dark count rate. In the last decade, the development of the superconducting nanowire single-photon detector (SNSPD), has quickly started to fulfill all these requirements.

2.2.1

Superconductivity

In Section 2.1.2, the Fermi liquid theory of independent quasiparticles in bulk normal metals with Coulomb screening was discussed, as well as the deviations that occur when the dimensionality of the system is reduced and the Coulomb screening becomes ineffective. Another class of systems in which the Fermi liquid theory breaks down, is those in which there exists an attractive potential between the charge carriers. At low temperatures, such systems undergo a remarkable phase-transition to a state where a dissipationless current can flow through the material: the superconducting state. The superconducting state was first observed by Heike Kamerlingh Onnes in 1911, when the successful liquefaction of helium allowed access to temperatures close to absolute zero [32]. Below a material-specific critical temperature TC , it was observed that the resistance of a metal suddenly dropped sharply to zero. At the time of these experiments, this result was completely unexpected; the experiment was designed to distinguish between theoretical predictions that the resistance of pure metals would gradually approach zero as the temperature goes to zero Kelvin, or would start to increase at sufficiently low temperature and tend to infinity as the temperature goes to zero. Although many properties of the superconducting state could be described by phenomenological theories such as that of Ginzburg and Landau [33] and the London equations [34], it was not until over four decades after the first experimental observations of superconductivity, that a microscopic theory was developed by Bardeen, Cooper, and Schrieffer [35]. Dynamic lattice deformations due to the presence of electrons, can lead to an attractive interaction between electrons, overpowering the (screened) repulsive Coulomb interaction and leading to an effective attractive interaction. This attractive interaction between electrons leads them to form so-called Cooper-pairs. Because Cooperpairs are made up of pairs of spin-1/2 particles (with opposite spin and momentum), they are boson-like and at low temperature condense into a single macroscopic groundstate, described by the Ginzburg-Landau order parameter: ψ(r) = |ψ|eiθ(r) ,

(2.4)

where |ψ(r)|2 = ns /2 is the Cooper pair density. Note here that, apart from the phase, the order parameter is taken to be independent of position. This 19


2. Carbon nanotubes and superconducting nanowires

is a general property of conventional superconductivity; the superconducting state is insensitive to underlying disorder in the material, as originally discussed by P.W. Anderson [36]. This leads to homogeneous superconducting properties. As we will see later however, in the case of very strong disorder, the superconducting properties become strongly dependent on the disorder and inhomogeneities emerge. The nature of the underlying mechanism of the attractive pairing interaction has no influence on many of the predictions of BCS-theory, as long as there is an effective attractive interaction between electrons. There is in fact evidence that in certain high-temperature superconductors, the attractive interaction does not result from lattice deformations. The formation of Cooper pairs under any effective attractive potential, is a consequence of the properties of the Fermi sea. As more and more Cooper pairs are formed from electrons close to the Fermi energy, the density of states around the Fermi energy becomes more and more altered. In the end, this leads to a maximum pairing energy Δ around the Fermi energy EF over which electrons form into Cooper pairs. This Δ is the gap energy of the superconductor and plays a crucial role in most of its characteristics. Due to the formation of Cooper pairs at the Fermi energy from electrons at energies up to Δ around the Fermi energy, the density of states of the superconductor is modified from its form in the normal metal state. The density of states of the excitations is given by:  √ E , if (E > Δ) Ns (E) E 2 −Δ2 = (2.5) N (0) 0, if (E < Δ) with N (0) the density of states in the normal state, which is usually taken constant, since Δ  EF [37]. Since the density of states is zero up to the gap energy Δ and a Cooper pair is made up of two electrons, the minimum energy required to break a Cooper pair is 2Δ. This sets the minimum detectable photon energy in pair-breaking detector applications of superconductors. The full microscopic BCS-theory becomes difficult to apply to problems dealing with the macroscopic state of the superconductor. It has been shown [38], that the Ginzburg-Landau description [33], which was originally derived as an empirical model, is a limiting case of BCS-theory, valid close to the critical temperature. The Ginzburg-Landau theory describes the superconducting order parameter ψ as given in Eq. 2.4. Two characteristic lengths play a role in the Ginzburg-Landau theory. The first is the coherence length ξGL , which is the minimum range over which the order parameter ψ can vary, and in the dirty limit (l  ξ0 ), applicable to the materials in this thesis, is given by: √ ξ0 l ξGL (T ) = 0.855 √ (2.6) 1−t 20


2.2 Superconducting nanowires

ξ0 =

vF πΔ(0)

(2.7)

with t = TTC , vF the Fermi velocity, and l the mean free path. The second important length scale is a magnetic penetration depth λ, which gives a measure of the thickness of the layer at the surface of a superconductor in which the currents screening external magnetic fields flow. This length scale is not a pure material property, but also depends on the sample geometry and field orientation. For perpendicular magnetic field and a bulk superconductor in the dirty limit, the magnetic penetration depth is given by:  ξ0 λeff = λL (T ) (2.8) 1.33l with λL (T ) the London penetration depth at temperature T . In the case of a thin film of thickness d  λ, the screening length for perpendicular field is instead given by the Pearl length [39]: Λ = 2λ2 /d

2.2.2

(2.9)

Critical current

Although in a superconductor a current can flow without any dissipation, there is a maximum current that any superconducting wire can carry. This is the critical current IC , which will turn out to be important for applications such as the superconducting nanowire single-photon detector. In wires with transverse dimension smaller than the Pearl length (w, d  Λ), the current density and the amplitude of the order parameter are constant over the width and thickness of the wire. The Ginzburg-Landau equations then lead to the following equation for the supercurrent velocity vs as a function of current density Js : 2 Js = 2eψ∞ (1 −

m∗ vs2 )vs 2|α|

(2.10)

with ψ∞ the amplitude of the order parameter in a bulk sample. Equation 2.10 is plotted in Fig. 2.6. There is a maximum current density as a function of supercurrent velocity; this maximum current density is the critical current of the wire above which no supercurrent can flow. The temperature dependence of this critical current is given by: Jc ∝ (1 −

T 3/2 ) TC

(2.11)

As these results follow from Ginzburg-Landau theory, they are valid in a limited temperature range close to the critical temperature [37]. 21


2. Carbon nanotubes and superconducting nanowires

(a)

(b)

Js

Jc

vs Figure 2.6. (a)Supercurrent density as a function of supercurrent velocity from Ginzburg-Landau theory as given by Equation 2.10. The maximum current is the critical current density of the wire. (b) Critical current of an aluminum wire over a wide temperature range up to the critical temperature from Romijn et al. [40]. The data is compared to the Ginzburg-Landau temperature dependence from Eq. 2.11 valid close to TC (straight line), Kupriyanov-Lukichev (solid curve) [41], and Bardeen (dashed curve) [42].

To find the critical current over the full temperature range, one has to solve the Eilenberger or Usadel equations [43, 44]. Kupriyanov and Lukichev numerically solved these to calculate the temperature dependence of the critical current [41]. The temperature dependence of the critical current of a wire is described fully by a critical temperature TC,IC , and a scaling factor IC,0 : IC,0

√  8π 2 2π (kB TC )3 w = , 21ζ(3)e 3D Rsq

(2.12)

with ζ Riemann’s zeta function, e the electron charge, kB Boltzmann’s constant, D the diffusion constant, and Rsq the normal-state square resistance of the wire. In Fig. 2.6 the temperature dependence of the critical current of an aluminum wire is compared to the various theoretical predictions [40].

2.2.3

Single-photon detection

The operational principle of superconducting nanowire single-photon detectors is based on the breaking of the superconducting state by the absorption of a single photon. In essence, the device consists of a long and narrow superconducting wire. This wire is biased close to its critical current, so that when 22


2.2 Superconducting nanowires

a photon is absorbed it switches to a dissipative state and a voltage pulse is detected.

(a)

(b)

(c)

Figure 2.7. (a) Scanning tunneling microscope image of current stateof-the-art superconducting nanowire single-photon detector. The upper panel shows the 130 nm wide WSi nanowire in blue meandering over a circular area with diameter 15 μm. The surrounding area is patterned to improve fabrication yield. The bottom panel shows one of the bends in detail [45]. (b) Circuit diagram of the typical device architecture of a superconducting nanowire single-photon detector [46]. (c) Voltage response on the load resistor Rshunt . The rise and restore time are determined by the resistance and inductance of the circuit elements [46].

Figure 2.7(b) shows a schematic representation of the typical device architecture of a superconducting nanowire single-photon detector. The superconducting nanowire is shunted by a resistor Rshunt  RN , with RN the resistance of the normal region in the nanowire. The wire has a kinetic inductance LK related to the energy stored in the kinetic energy of the Cooper pairs. When a photon is absorbed, the nanowire switches to the normal state. The bias current is then redistributed to the shunt resistor, leading to a detectable voltage pulse over the shunt resistor. The rise time of this pulse is determined by: τrise = LK /(Rshunt + RN,wire ). The nanowire relaxes back to the superconducting state on a timescale related to the electron-phonon scattering time τe−ph and the phonon escape time τesc of the superconducting film. When the nanowire is again in the superconducting state, the supercurrent is restored on a timescale τrestore = LK /Rshunt . The device is then ready to register another 23


2. Carbon nanotubes and superconducting nanowires

photon. Since Rshunt  RN,wire , typically τrise  τrestore . If the photon energy would not have left the nanowire by the time the current is restored, there is a risk of ’latching’ to a self-heating normal state. Therefore, material choices and device design are typically such that τrestore is the limiting timescale for the device: τrestore  τe−ph , τesc [26, 46]. The first superconducting nanowire single-photon detector was made of a NbN wire by Gol’tsman et al. in 2001 [47]. Figure 2.7(a) shows a current stateof-the-art superconducting nanowire single-photon detector, made by Marsili et al. [45, 48]. It shows a 130 nm wide WSi nanowire, meandering over a circular area with a diameter of 15 μm. Important advances made over the first decade of superconducting nanowire single-photon detectors include for instance placing the device in an optical cavity and increasing the sensitive area by meandering the nanowire [49, 50]. The important device characteristics for a superconducting nanowire singlephoton detector are the following: • The detection efficiency: The chance that an incident photon is detected is given by the detection efficiency η. There are several contributions to the overall detection efficiency; the coupling efficiency of the optics to the detector ηcoupling , the absorption efficiency of the material and geometry ηabsorption , and the chance that an absorbed photon leads to a detectable pulse ηregistering . The total system detection efficiency is then: ηsde = ηcoupling · ηabsorption · ηregistering . • The dark count rate: Noise or stray light can lead to false detection events when no signal photon is present. Biasing the detector closer to the critical current typically increases the detection efficiency, but also increases the dark count rate. • The timing jitter: For many applications of single photon detection the determination of the arrival time of the photon is crucial. For instance, certain proposed communication schemes rely on the photon arrival times [51]. Superconducting nanowire single-photon detectors have an intrinsic variation in the time between photon incidence and the output of a voltage pulse. This is the timing jitter. • The reset time: The reset time determines how fast the detector is ready to detect another photon, and therefore determines the rate with which information can be transferred in a single photon communication scheme. Both the intrinsic detection efficiency ηregistering and the dark count rate strongly depend on the bias current. Figure 2.8 shows the system detection efficiency and dark count rate as a function of bias current for a NbTiN SNSPD at 24


2.2 Superconducting nanowires

Figure 2.8. Detection efficiency and dark count rate as a function of bias current at two temperatures. The detection efficiency depends on the absolute value of the bias current, while the dark count rate depends on IB /IC [52].

two different temperatures, as measured by Dorenbos et al. [52]. As discussed in Section 2.2.2, the critical current of a superconducting nanowire decreases with increasing temperature. The data in Fig. 2.8 suggest that the dark count rate is related to IB /IC , while the detection efficiency is related to the absolute bias current. An increase of the critical current of a device may therefore directly lead to better device performance, either through an increased efficiency or lower dark count rate depending on the choice of bias current. The critical current of a wire with inhomogeneities, such as structural defects, will be determined by its weakest link. It has been shown that the device performance of identically fabricated superconducting nanowire singlephoton detectors depends on localized constrictions which limit the device current [53, 54]. Figure 2.9 shows the measured detection rate as a function of position for a high detection efficiency NbN SNSPD (a) and a NbN SNSPD with a low detection efficiency (b). These measurements were performed using a focused incident photon beam with a spotsize of 300-600 nm and an x,ytranslation stage. The detection efficiency of the sample with lower overall detection efficiency is clearly non-uniform. The device only functions when the photons are incident at the defect location [55]. Similar measurements were performed by Rosticher et al. [56] using a scanning electron beam as the excitation. In this case the detector responds to single electrons instead of single photons. With this method a high spatial resolution detection efficiency map of a sample can be obtained. 25


2. Carbon nanotubes and superconducting nanowires

(a)

(b)

Figure 2.9. Detection rate as function of position on two superconducting nanowire single-photon detectors: one with a high-detection efficiency (a) and one with a low detection efficiency (b). The low detection efficiency device clearly has a strongly inhomogeneous response [55].

Inhomogeneities in the device design and fabrication can lead to decreased device performance. However, the materials of choice can also exhibit intrinsic inhomogeneity.

2.2.4

Superconductivity in strongly disordered materials

The first superconducting nanowire single-photon detector was made of a thin film of NbN [47]. Ever since, NbN and NbTiN have remained the main materials of choice. This is in part due to historical preference and partially due to the relatively high critical temperature of these materials, which allows device performance at liquid helium temperature. Another important consideration in the choice of material of a superconducting nanowire single-photon detector, are the thermal relaxation properties. A fast electron-phonon scattering time τe−ph and a good thermal conductivity from the film to the substrate, allow for faster device operation and therefore better timing resolution and bandwidth. As we have seen in the previous section, inhomogeneities in superconducting nanowire single-photon detectors can directly influence the device performance. Recently, experimental evidence has been found that strong disorder in a superconductor can lead to the emergence of intrinsic electronic inhomogeneity [57, 58]. Thin films of NbN, NbTiN, and TiN are in this family of strongly disordered superconductors. Measurements of the electrodynamic response and local tunneling density of states of TiN films of varying disorder, show that for increasing disorder there is a growing discrepancy between the experimental data and models based on homogeneous material properties [59]. 26


2.2 Superconducting nanowires

Figure 2.10. Map of the superconducting gap in a thin TiN-film close to the superconductor-insulator transition measured by STM spectroscopy. Disorder-induced inhomogeneity of the superconducting state is clearly visible [57].

Figure 2.10 shows a map of the superconducting gap of a TiN-film, determined from a tunneling spectroscopy measurement at each point [57]. The superconducting gap varies significantly with position, on a length-scale unrelated to the grain size of the material. Similar results have been obtained on films of thin, strongly disordered, NbN films [58]. These materials are on the limit of conventional superconductivity theory. The mean free path is on the order of the Fermi-wavelength: kF l ∼ 1. The average distance between Cooper pairs approaches the size of a Cooper pair: dCP ∼ ξ, with dCP the average distance between Cooper pairs determined from the Cooper pair density, and ξ the coherence length. This means that there is no longer an overlap of a large amount of Cooper pairs. This is reminiscent of the BCS-BEC transition as studied in cold atom gasses [60]. The robustness of the superconducting state to underlying disorder, as originally discussed by P.W. Anderson [36], starts to break down in these limits. When the disorder is increased even further, a quantum phase transition into an insulating state is observed [61]. A quantum phase transition is a phase transition that occurs at zero temperature by varying a physical parameter, in this case the disorder. The phase transition from a strongly-, but homogeneously disordered superconductor to an insulator and the spontaneous 27


2. Carbon nanotubes and superconducting nanowires

emergence of electronic inhomogeneity in superconducting films close to this transition, is an active field of theoretical and experimental study. Disorder is homogeneous if, on the smallest length scale of physical interest, the statistical distribution of the disorder potential is the same, independent of position in the sample [62]. In the case of a superconductor, the coherence length is the shortest distance over which the superconducting state can vary. In strongly disordered superconductors, the coherence length will typically be of the order of ∼5 nm [63]. A cubic region of dimension 5x5x5 nm contains on the order of 1 · 104 atoms, which is enough to consider it homogeneously disordered, if the scattering length due to disorder is on the lengthscale of the interatomic distance. Two theoretical frameworks exist for the superconducting-insulator transition. One can think of the disordered superconducting film as a network of superconducting islands coupled to each other. The macroscopic superconducting state can then either be destroyed by letting the amplitude of the pairing potential on the islands go to zero, or by decreasing the coupling between the islands until no macroscopic phase-coherence remains. These are the fermionic and bosonic mechanism respectively. In strongly disordered materials, the electrons are unable to effectively screen the Coulomb interaction, because of their short mean free path. At some point the repulsive Coulomb interaction completely counteracts the attractive interaction forming the Cooper pairs. At this point, the superconducting order parameter Δ goes to zero. This means that at the transition, all Cooper pairs are split into single fermionic quasiparticles. Therefore, this is called the fermionic mechanism of the superconductor-insulator transition. The reduction of the order parameter can be found by taking into account the Coulomb interaction between electrons together with the disorder. In the dirty limit, the Coulomb interaction leads to a reduction of the critical temperature TC [64]. In the case of strong disorder, the quasiparticle wavefunction is localized and the material is an insulator. In the bosonic mechanism, the binding energy does not go to zero at the superconductor-insulator transition. This means that Cooper pairs are present also on the insulating side of the transition. They are condensed in superconducting ’islands’, the transport between which is suppressed due to disorderinduced effects. The phase-coherence between the islands of the superconducting film is lost and the macroscopic superconductivity disappears. Experimentally this results in the measurement of a ’pseudogap’ in the normal state [57,65] and the observation of a strong negative magnetoresistance [61]. The exact mechanism remains unclear at present, and a combination of the different effects is likely to play a role. It is clear however that one has to consider the consequences of the choice for materials close to the superconductorinsulator transition for applications in for instance single photon detectors.

28


2.2 References

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[34] F. London and H. London, The electromagnetic equations of the supraconductor, Proc. Roy. Soc. 149, 71 (1935). [35] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev. 108, 1175 (1957). [36] P. W. Anderson, Theory of dirty superconductors, J. Phys. Chem. Solids 11, 26 (1959). [37] M. Tinkham, Introduction to superconductivity, McGraw-Hill, New York, (1975). [38] L. P. Gor’kov, Zh. Eksperim. i Teor. Fiz. 36, 1918 (1959). [39] J. Pearl, Current distribution in superconducting films carrying quantized fluxoids, Appl. Phys. Lett. 5, 65 (1964). [40] J. Romijn, T. M. Klapwijk, M. J. Renne, and J. E. Mooij, Critical pairbreaking current in superconducting aluminum strips far below Tc , Phys. Rev. B 26, 3648 (1982). [41] M. Y. Kupriyanov and V. F. Lukichev, Temperature dependence of pairbreaking current in superconductors, Sov. J. Low Temp. Phys. 210, 210 (1980). [42] J. Bardeen, Critical fields and currents in superconductors, Rev. Mod. Phys. 34, 667 (1962). [43] G. Eilenberger, Transformation of Gorkovs equation for type II superconductors into transport-like equations, Z. Phys. 214, 195 (1968). [44] K. D. Usadel, Generalized diffusion equation for superconducting alloys, Phys. Rev. Lett. 25, 507 (1970). [45] E. F. C. Driessen, Fast and efficient, Nature photonics 7, 168 (2013). [46] A. J. Kerman, E. A. Dauler, W. E. Keicher, J. K. W. Yang, K. K. Berggren, G. Gol’tsman, and B. Voronov, Kinetic-inductance-limited reset time of superconducting nanowire photon counters, Appl. Phys. Lett. 88, 111116 (2006). [47] G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, Picosecond superconducting single-photon optical detector, Appl. Phys. Lett. 79, 705 (2001). [48] F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, Detecting single infrared photons with 93 percent system efficiency, Nature photonics 7, 210 (2013). [49] K. M. Rosfjord, J. K. W. Yang, E. A. Dauler, A. J. Kerman, V. Anant, B. M. Voronov, G. N. Gol’tsman, and K. K. Berggren, Nanowire singlephoton detector with an integrated optical cavity and anti-reflection coating, Opt. Express 14, 527 (2006). [50] A. Verevkin, J. Zhang, R. Sobolewski, A. Lipatov, O. Okunev, G. Chulkova, A. Korneev, K. Smirnov, G. N. Gol’tsman, and A. Semenov, 31


2. Carbon nanotubes and superconducting nanowires

[51] [52]

[53]

[54]

[55]

[56]

[57]

[58]

[59]

[60] [61] [62]

32

Detection efficiency of large-active-area NbN single-photon superconducting detectors in the ultraviolet to near-infrared range, Appl. Phys. Lett. 80, 4687 (2002). R. H. Hadfield, Single-photon detectors for optical quantum information applications, Nature Photon. 3, 696 (2009). S. N. Dorenbos, R. W. Heeres, E. F. C. Driessen, and V. Zwiller, Efficient and robust fiber coupling of superconducting single photon detectors, arXiv:1109.5809 (2011). F. Mattioli, R. Leoni, A. Gaggero, M. G. Castellano, P. Carelli, F. Marsili, and A. Fiore, Electrical characterization of superconducting single-photon detectors, J. of Appl. Phys. 101, 054302 (2007). A. J. Kerman, E. A. Dauler, J. K. W. Yang, K. M. Rosfjord, V. Anant, K. K. Berggren, G. N. Gol’tsman, and B. M. Voronov, Constriction-limited detection efficiency of superconducting nanowire single-photon detectors, Appl. Phys. Lett. 90, 101110 (2007). R. H. Hadfield, P. A. Dalgarno, J. A. O’Connor, E. Ramsay, R. J. Warburton, E. J. Gansen, B. Baek, M. J. Stevens, R. P. Mirin, and S. W. Nam, Submicrometer photoresponse mapping of nanowire superconducting single-photon detectors, Appl. Phys. Lett. 91, 241108 (2007). M. Rosticher, F. R. Ladan, J. P. Maneval, S. N. Dorenbos, T. Zijlstra, T. M. Klapwijk, V. Zwiller, A. Lupa¸scu, and G. Nogues, A high efficiency superconducting nanowire single electron detector, Appl. Phys. Lett. 97, 183106 (2010). B. Sac´ep´e, C. Chapelier, T. I. Baturina, V. M. Vinokur, M. R. Baklanov, and M. Sanquer, Disorder-induced inhomogeneities of the superconducting state close to the superconductor-insulator transition, Phys. Rev. Lett. 101, 157006 (2008). M. Chand, G. Saraswat, A. Kamlapure, M. Mondal, S. Kumar, J. Jesudasan, V. Bagwe, L. Benfatto, V. Tripathi, and P. Raychaudhuri, Phase diagram of the strongly disordered s-wave superconductor NbN close to the metal-insulator transition, Phys. Rev. B 85, 014508 (2012). P. C. J. J. Coumou, E. F. C. Driessen, J. Bueno, C. Chapelier, and T. M. Klapwijk, Electrodynamic response and local tunneling spectroscopy of strongly disordered superconducting TiN films, Phys. Rev. B 88, 180505 (2013). V. M. Loktev, R. M. Quick, and S. G. Sharapov, Phase fluctuations and pseudogap phenomena, Physics Reports 349, 1 (2001). V. F. Gantmakher and V. T. Dolgopolov, Superconductor-insulator quantum phase transition, Physics-Uspekhi 53, 1 (2010). J. M. Ziman, Models of disorder; the theoretical physics of homogeneously disordered systems, Cambridge University Press, (1982).


2.2 References

[63] E. F. C. Driessen, P. C. J. J. Coumou, R. R. Tromp, P. J. de Visser, and T. M. Klapwijk, Strongly disordered TiN and NbTiN s-wave superconductors probed by microwave electrodynamics, Phys. Rev. Lett. 109, 107003 (2012). [64] A. M. Finkelstein, Superconductings transition-temperature in amorphous ďŹ lms, JETP Lett. 45, 46 (1987). [65] S. P. Chockalingam, M. Chand, A. Kamlapure, J. Jesudasan, A. Mishra, V. Tripathi, and P. Raychaudhuri, Tunneling studies in a homogeneously disordered s-wave superconductor: NbN, Phys. Rev. B 79, 094509 (2009).

33


2. Carbon nanotubes and superconducting nanowires

34


Chapter 3 Experimental methods We discuss the experimental realization of the objects used to identify and study the concepts described in this thesis. The nanofabrication techniques used are outlined for both suspended carbon nanotubes and superconducting nanowires. We brieďŹ&#x201A;y discuss the experimental setup, including the terahertz radiation sources used in the experiments on carbon nanotubes.

35


3. Experimental methods

3.1 3.1.1

Materials and sample-development Microwave-coupled, suspended, carbon nanotubes

There are experimental and theoretical indications that interactions with the substrate lead to a quick decay of the charge modes in a carbon nanotube Luttinger liquid [1, 2]. Therefore, we believe these modes could be significantly longer lived in suspended carbon nanotubes. Furthermore, suspended nanotubes are thermally isolated from the substrate, leading to a stronger bolometric response. We thus want to fabricate clean, suspended carbon nanotubes, at a specific location, namely at the focus of a bowtie-antenna. The carbon nanotube fabrication process developed by Dai et al. and Steele et al., allows the growth of suspended carbon nanotubes from specifically located iron-molybdenum catalyst particles [3, 4]. Since the nanotube growth occurs in the final step of fabrication, this process leads to high-quality, clean carbon nanotubes, free from surface interactions. Figure 3.1 shows a schematic representation of the fabrication process of suspended carbon nanotubes as used in our experiments. In order to study the electrical transport properties of nanotubes under terahertz illumination (100 GHz up to several THz), the carbon nanotube is fabricated in the center of a bowtie-antenna, whose two halves also function as the electrodes in the electrical transport measurement. We start with a silicon wafer, covered by a ∼500 nm thick silicon oxide layer. This is in turn covered by a ∼40 nm thick silicon nitride layer, using plasma-enhanced chemical vapor deposition. In the first lithography step, we define the gate and its electrodes, these are situated in a trench etched into the substrate. The patterning is done using electron beam lithography on MMA/PMMA bilayer resist. First the silicon nitride layer is etched by CHF3 /O2 reactive-ion etching. Subsequently, the exposed silicon oxide layer is etched in buffered hydrofluoric acid for 60 seconds to create a ∼250 nm deep trench. Finally, the tungsten/platinum gate electrode is deposited using electron beam evaporation. The etch mask is now used as a lift-off resist. In the second lithography step the antenna-electrodes and contact wires are defined. For this step we use a single layer of PMMA. There is no etching of the silicon nitride and silicon oxide in this step. The tungsten/platinum electrodes and contact wires are thus deposited directly on the silicon nitride substrate. In the third and final lithography step, the ∼1 μm2 iron-molybdenum catalyst particles are defined 500 nm from the edge of one the two electrodes. The catalyst is prepared by mixing 40 mg of ferric nitrate nonahydrate, 30 mg of fumed aluminum oxide, and 8 mg of molybdenum dioxydiacetylacetonate in a 30 ml methanol solution. After sonication, it is applied to the sample using a syringe. The methanol is then evaporated on a hot-plate. We again use the 36


3.1 Materials and sample-development

38 nm SiNx 490 nm SiOx Si 30 nm Pt 10 nm W

30 nm Pt 10 nm W

Catalyst

CNT

Figure 3.1. Schematic representation of the fabrication process of the suspended carbon nanotubes. On the left we show a vertical cutaway, on the right a top view. The bottom-right picture is a scanning electron microscope picture of an actual fabricated device.

same lithography and lift-off recipe as in the previous step. The carbon nanotubes are grown in a flow-oven at 900 degree Celsius under a continuous flow of CH4 and H2 . Since this process happens at the final step of fabrication, it places stringent requirements on the material choice for the electrodes. For our experiments, we choose platinum with a tungsten adhesion layer, because of their high melting temperatures. In thin film patterned 37


3. Experimental methods

structures however, we find that the carbon nanotube growth process can still lead to significant deterioration of the films due to the formation of droplets. We found the optimal values for the thicknesses to be 10 nm of tungsten and 30 nm of platinum, leading to a high-quality film even after carbon nanotube growth.

3.1.2

Nanogeometries in thin disordered superconductors

Thin films of high resistivity superconductors, such as TiN and NbTiN, show interesting new physics, due to their proximity to the superconductor-insulator transition, as discussed in Chapter 2. Accurate control of the thickness of the superconducting films, allows us to tune the resistivity of the film and thereby its ’disorder’. Control of the growth of the film and its parameters is therefore crucial for experiments on these disordered films. For the work in this thesis, two deposition techniques of superconducting films are used: sputter deposition and Atomic Layer Deposition (ALD). In sputter deposition, a plasma is created in the deposition chamber. Ions from the plasma are accelerated towards the material to be deposited. These accelerated ions ’knock loose’ atoms in the target material. These atoms will then be deposited on the sample, situated opposite to the target material in the deposition chamber. In principle different target atoms will have different sputtering yields. However, due to the low penetration length of the incident ions in the target material, sputtering allows the deposition of films with the same stoichiometry as the target, except for the nitrogen which is drawn from the plasma [5]. Atomic layer deposition is a chemical deposition method. It is based on sequential self-terminating gas-solid reactions at the film surface. This allows the controlled growth of a film by a single atomic layer at a time. In our deposition process for TiN, the growth rate is approximately 0.5 ˚ A/cycle. The TiN layer is polycrystalline, the typical grain size is 5 nm, as revealed by crosssectional transmission electron microscopy [6]. To measure the electronic transport characteristics of the thin superconducting films, they need to be patterned into wires. In order to achieve measurable values of, for instance, resistance and critical current, the wire dimensions are typically in the 100 nm to 1 μm scale. Furthermore, high precision patterning is essential for the study of the influence of geometrical features, such as corners, on the transport characteristics. To transfer patterns into our superconducting films, we spin-coat hydrogen silsesquioxane resist (HSQ) on the sample and selectively expose the desired pattern using electron beam lithography. The unexposed parts are then dissolved in tetra-methyl ammonium hydroxide solution (TMAH), leaving the exposed resist as an etch mask. The pattern can then be transferred into the superconducting film by reactive-ion etching. 38


3.2 Cryogenics and radiation sources

In order to improve the selectivity of the etching process, the patterned and developed resist can be post-exposed [7]. The greater etch resistance of the resist then allows the use of a thinner resist layer, which results in a higher resolution. We post-exposed the critical parts of our geometries in a scanning electron microscope, with a dose of â&#x2C6;ź100 mC/cm2 .

3.2 3.2.1

Cryogenics and radiation sources Measurement setups

All measurements in this thesis are performed at cryogenic temperatures. We use one of three cryogenic setups depending on the demands of the particular experiment. For the study of the response of mesoscopic objects to terahertz radiation, we have developed a helium cryostat with optical access. The cryostat consists of an inner reservoir of liquid helium at 4.2 K, a shield at liquid nitrogen temperature (77 K), and an outside wall. Each of these are isolated from each other by a vacuum space. The sample stage is located in a copper housing attached to the 4.2 K stage. The housing is coated on the inside with a mixture R and silicon carbide grains to form a blackbody, in order to absorb of Stycast stray radiation [8]. In the outside wall, the 77 K shield, and the 4.2 K sample housing, aligned high-density polyethylene windows are mounted. The sample is glued on a copper sample stage, which is attached directly to a copper block in the sample housing. This block can be weakly thermally isolated from the cold-plate of the cryostat by washer rings. A heater element and thermometer are integrated in the copper block. 24 twisted pairs of constantan wires run from room temperature to the 4.2 K stage in two ribbon loom cables. The wiring from the 4.2 K stage to the sample consists of twisted pairs of copper wire. All measurement wires are thermalized at the copper block. This allows for controlled temperature dependent measurements in a range from 4.2 K to approximately 20 K. For a quick characterization that does not require incident radiation, we have developed a dipstick for use in a helium vessel. This allows for a fast sample cycling time. The samples are again located on a copper block with an integrated temperature sensor and heater element, to allow temperature dependent measurements. For the study of thin TiN ďŹ lms, which have a critical temperature below the temperature of liquid helium, we use a He-3 sorption cooler mounted in a similar liquid helium cryostat as described before, except now without optical access [9]. This allows us to reach a base temperature of approximately 310 mK. A superconducting magnetic shield surrounds the sample stage. 39


3. Experimental methods

3.2.2

Microwave and terahertz sources

Sources of terahertz radiation are limited. Typically one either tries to extend microwave techniques to higher frequencies or convert optical techniques, such as gas lasers, to lower frequencies. Advances have been made in the development of dedicated terahertz sources, such as quantum cascade lasers [10]. There are only a few high power, continuously tunable terahertz sources in the world however: the free-electron lasers. In this thesis we will use an active multiplier chain with an output frequency of 108 GHz, and a free electron laser in the frequency range from 1.5 to 3 THz. Active multiplier chain In an active multiplier chain, a base frequency signal is converted to a higher frequency. In our case we use a YIG oscillator to generate the base frequency signal at 18 GHz. In a YIG oscillator a sphere of single crystalline yttrium iron garnet is used to generate an oscillating signal. The output frequency is tunable by varying an external magnetic field. This signal is then converted using an active multiplier chain containing Schottky diodes for multiplication and amplification of the input signal. We use a single multiplication step to reach a maximum frequency of 108 GHz. Further multiplication to higher frequencies is possible, but comes at the cost of significantly reduced output power [11]. The output frequency is tunable by changing the input frequency, however the tuning range is typically limited to about 10% of the output frequency. Free-electron laser The only high-power, continuously-tunable terahertz radiation sources are the free-electron lasers. Figure 3.2(a) shows the basic operational principle of freeelectron lasers [12]. A relativistic electron beam is moved trough an alternating set of magnetic undulators. The electron beam wiggles due to the periodic undulators, and will therefore emit synchrotron radiation. By tuning the distance between the undulators, the frequency of the emitted radiation can be tuned. The electromagnetic field in the cavity is amplified by stimulated emission of the synchrotron radiation, therefore the FEL does indeed function as a laser [12]. FELIX provides pulsed light in a wavelength range from 3 to 250 μm. The instantaneous tuning range can be up to a factor of three. The pulse repetition rate is 10 Hz. In this thesis, we will discuss some of the last measurements done using the Free Electron Laser for Infrared eXperiments (FELIX) in Nieuwegein before its move to Nijmegen. FELIX uses an optical cavity with copper mirrors as in Fig. 3.2(a). Figure 3.2(b) shows the basic layout of FELIX [13]. An electron 40


3.2 Cryogenics and radiation sources

photon beam

(a)

mirror electron beam mirror

(b) electron gun buncher linac prebuncher

undulator

SEM linac undulator

(c)

Figure 3.2. (a) Operation principle of a free-electron laser. An electron beam is guided through an undulator, which consists of a series of magnets with opposite polarity. The emitted synchrotron radiation is confined in an optical cavity. The frequency of the emitted radiation can be tuned by controlling the distance between the magnets [12]. (b) Schematic overview of the free-electron laser FELIX in Nieuwegein [13]. For the experiments in this thesis, we use the first (long frequency) undulator. (c) Photograph taken at FELIX showing the first linear accelerator on the left.

pulse originates from an electron gun and buncher and is then accelerated to energies from 15 to 25 MeV. At this point the electron beam can be guided to the first undulator stage for long-wavelength experiments or accelerated further for shorter wavelength experiments. In our experiments, we use the first undulator stage. Figure 3.2(c) shows a photograph taken at the electron beam chamber at FELIX, showing the scale of the device.

41


3. Experimental methods

References [1] W. Chen, A. V. Andreev, E. G. Mishchenko, and L. I. Glazman, Decay of a plasmon into neutral modes in a carbon nanotube, Phys. Rev. B 82, 115444 (2010). [2] M. Steiner, M. Freitag, V. Perebeinos, J. C. Tsang, J. P. Small, M. Kinoshita, D. Yuan, J. Liu, and P. Avouris, Phonon populations and electrical power dissipation in carbon nanotube transistors, Nat. Nanotechnol. 4, 320 (2009). [3] H. Cao, Q. Wang, D. W. Wang, and H. J. Dai, Suspended carbon nanotube quantum wires with two gates, Small 1, 138 (2005). [4] G. A. Steele, A. K. Huttel, B. Witkamp, M. Poot, H. B. Meerwaldt, L. P. Kouwenhoven, and H. S. J. van der Zant, Strong coupling between singleelectron tunneling and nanomechanical motion, Science 325, 1103 (2009). [5] R. Waser, Nanoelectronics and information technology, Wiley-VCH, (2003). [6] P. C. J. J. Coumou, E. F. C. Driessen, J. Bueno, C. Chapelier, and T. M. Klapwijk, Electrodynamic response and local tunneling spectroscopy of strongly disordered superconducting TiN films, Phys. Rev. B 88, 180505 (2013). [7] J. K. W. Yang, V. Anant, and K. K. Berggren, Enhancing etch resistance of hydrogen slisesquioxane via postdevelop electron curing, J. Vac. Sci. Tech. B 24, 3157 (2006). [8] T. O. Klaassen, J. H. Blok, N. J. Hovenier, G. Jakob, D. Rosenthal, and K. J. Wildeman, Absorbing coatings and diffuse reflectors for the Herschel platform sub-millimeter spectrometers HIFI and PACS, in Terahertz Electronics Proceedings, 2002. IEEE Tenth International Conference on, page 32, 2002. [9] R. Barends, Photon-detecting superconducting resonators, PhD thesis, Delft University of Technology, 2009. [10] J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Quantum Cascade Laser, Science 264, 553 (1994). [11] J. Hesler, D. Porterfield, W. Bishop, T. Crowe, A. Baryshev, R. Hesper, and J. J. A. Baselmans, Development and characterization of an easyto-use THz source, in 16th International Symposium on Space Terahertz Technology, page 378, 2005. [12] S. Khan, Free-electron lasers, Journal of Modern Optics 55, 3469 (2008). [13] P. W. van Amersfoort et al., First lasing with FELIX, Nucl. Instrum. Meth. A 318, 42 (1992).

42


Chapter 4 Microwave-induced nonequilibrium temperature in a suspended carbon nanotube

Antenna-coupled suspended single carbon nanotubes exposed to 108 GHz microwave radiation are shown to be selectively heated with respect to their metal contacts. This leads to an increase in the conductance as well as to the development of a power-dependent DC voltage. The increased conductance stems from the temperature dependence of tunneling into a one-dimensional electron system. The DC voltage is interpreted as a thermovoltage, due to the increased temperature of the electron liquid compared to the equilibrium temperature in the leads.

This chapter is based on H.L. Hortensius, A. Ozturk, P. Zeng, E.F.C. Driessen, and T.M. Klapwijk, Applied Physics Letters 100, 223112 (2012)

43


4. Microwave-induced nonequilibrium temperature in a suspended carbon nanotube

4.1

Introduction

The temperature response of carbon nanotubes exposed to microwave and terahertz radiation is of great interest both from a practical and a fundamental point of view. One of the many potential applications of carbon nanotubes lies in their response to far-infrared radiation. The bolometric response of either bundles or films of carbon nanotubes has been shown in the far- to mid- infrared range [1, 2]. Recently the bolometric and rectifying response of a single carbon nanotube was studied at 77 K and at 4.2 K [3]. Also, detectors based on thermoelectric properties of suspended films of carbon nanotubes have been proposed for the terahertz range and have been demonstrated to work at optical frequencies [4]. The unique one-dimensional electronic states of carbon nanotubes play an important role in the description of their DC electrical response [5]. Due to the strong interactions between the electrons, the response is a collective effect of the entire electron system, as expressed in the Tomonaga-Luttinger-liquid theory [6]. In this theory, these interactions induce a reduced tunneling density of states around the Fermi energy, which leads to a power-law scaling of the conductance with bias voltage and temperature [7–10]. In studying transport through a carbon nanotube, an important consideration is the energy relaxation. In recent experimental work, indications of weak energy relaxation were reported [11]. In subsequent theoretical work, the intrinsic relation between weak energy relaxation and one-dimensional physics was emphasized [12, 13]. It is to be expected that the conductivity of a carbon nanotube will also depend on this weak energy relaxation, when exposed to radiation. In previous work, carbon nanotubes in direct contact with a substrate were studied. This contact leads to interaction with charges in surface dielectrics, an increase in the scattering probability, and phonon-exchange with the substrate [14–16]. In order to avoid these potential complications and increase the thermal response, we have developed suspended carbon nanotubes in broadband antennas. We describe the realization of these suspended carbon nanotubes and their response to 108 GHz radiation. Two main effects are observed and analyzed: the differential conductance of the nanotube is described by an increased effective temperature, and a thermoelectric offset voltage develops due to the strong gradients in the electron temperature profile at the nanotube-lead interfaces.

4.2

Suspended carbon nanotubes under microwave radiation

Fig. 4.1 shows a schematic layout of the sample used. An antenna pattern is made with a catalyst particle, which is used as a seed to locally grow the 44


4.2 Suspended carbon nanotubes under microwave radiation

Figure 4.1. Scanning electron microscope image of a suspended carbon nanotube sample. The Pt/W layer of the antenna and gate is shown in yellow (light gray), while the silicon nitride substrate is shown in green (dark gray). The iron-molybdenum catalyst particle is indicated by the arrow. The gate is located in a 250 nm deep trench. The inset shows a schematic representation of a cutthrough at the nanotube location.

nanotubes. One of these can bridge the trench between the antenna pads, a methodology pioneered by Cao et al. [17] and Steele et al. [18]. The substrate is highly resistive silicon with a 490 nm silicon oxide layer and a 38 nm silicon nitride layer. First, a window is opened in the silicon nitride by CHF3 /O2 reactive ion etching, followed by a buffered hydrofluoric acid etch to create a 250 nm deep, 2 μm wide trench in the silicon oxide. At the bottom of the trench, a metal gate was realized, but this was not usable in the present experiment. On top of the silicon nitride, a metallic pattern of 30 nm platinum with a 10 nm tungsten adhesion layer is evaporated, which serves both as an antenna and as DC electrodes. The nanotubes are grown in the final step of fabrication by chemical vapor deposition from the iron-molybdenum catalyst seed, which is positioned at 1 μm from the trench using a lift-off technique. In one fabrication run 9 samples are made with 30 antenna pairs each. About 5% of the antenna pairs yield a finite resistance, indicating the presence of a nanotube. The results presented here all originate from one specific sample. Similar results were obtained for five samples. The measurement setup consists of a helium cryostat with optical access, as shown in the bottom-right inset of Fig. 4.2. The sample is mounted on a copper block, allowing a variation of the temperature from 5 K to 20 K by heating of a resistor. Microwave radiation is generated by a YIG-oscillator at 18 GHz and an active multiplier chain, which multiplies the frequency to 45


4. Microwave-induced nonequilibrium temperature in a suspended carbon nanotube

I (nA)

108 GHz. The power is varied by changing the bias voltage of an amplifier, which is part of the multiplier chain. The maximum power incident on the sample is estimated to be 83 nW/mm2 , based on a calibration with a pyroelectric detector placed at the position of the cryostat. The radiation passes through three high-density poly-ethylene windows at room temperature, 77 K, and 4.2 K, respectively. These windows are transmissive up to ∼ 90% at the frequencies used in these experiments. The total area covered by one bowtie antenna is about 0.1 mm2 , leading to the estimated maximum power incident on the area of the bowtie antenna of 8.3 nW. In the remainder of the article, we will refer to this estimate as the incident microwave power PMW . Due to the large impedance mismatch between free space and the carbon nanotube, only a small fraction of the incident power is absorbed in the nanotube.

-1

3) PMW = 8.3 nW 2) TR = 15K 1) TR = 5K

20

0

0

VB (mV)

1

LN

-20

108 GHz YIG x6

LHe

18 GHz Sample

Figure 4.2. Zero-current offset voltage as a function of the temperature increase in the nanotube under microwave irradiation (bottom axis) and the incident power on the antenna (top axis). The top-left inset shows the low-bias current-voltage characteristics. Curve 1 (black) is for the suspended carbon nanotube at 5 K, curve 2 (blue) is taken under heating of the entire substrate-lead-nanotube system to 15 K, and curve 3 (red) is taken under microwave irradiation with the bath temperature 5 K. The bottom-left inset shows a schematic representation of the experimental setup with microwave radiation incident on the sample.

The top left inset of Fig. 4.2 shows the small-signal current-voltage characteristics of the nanotube for two different bath temperatures as well as one in the presence of radiation. Clearly, the resistances of the nanotube are constant over this bias voltage range. Curve 1 (black) is measured at a bath temperature of 5 K, without applied radiation. Curve 2 (blue) is measured with the system heated to 15 K, which leads to an increase in conductance. Curve 3 (red) is a typical trace measured with the nanotube exposed to microwave radiation, 46


4.3 Luttinger liquid behavior

while the substrate is kept at 5 K. Two effects are clearly visible: an increase in slope under both conventional heating and applied microwave radiation, and, strikingly, a strong zero-current offset voltage appearing only with applied radiation. This zero-current offset voltage increases with increasing microwave power (upper scale in Fig. 4.2). We will argue below that these two effects observed with radiation are due to an increase of the electron temperature in the suspended carbon nanotube.

4.3

Luttinger liquid behavior

Figure 4.3. Power-law scaling of the differential conductance with bias voltage and temperature, indicative of a Luttinger liquid. The different curves correspond to an increase in the sample temperature from 6 K (bottom, blue) to 15 K (top, red). The thick black line shows the powerlaw scaling with bias voltage at high bias, with a scaling exponent of α = 0.28. The black curves are fits from Eq. 4.3. The inset shows the zero-bias differential conductance as a function of temperature, where the red line is a power law fit to the high temperature data, giving α = 0.28.

First, we study the DC differential conductance, as a function of voltage and at different temperatures (Fig. 4.3), to determine the nature of the electronic states in the sample. The temperature is varied by a heating resistor on the copper block on which the sample is mounted. The colored curves correspond to increasing temperatures from bottom (blue) to top (red) with steps of 1 K between each curve. The conductance at low voltages increases with 47


4. Microwave-induced nonequilibrium temperature in a suspended carbon nanotube

temperature. Similarly, the conductance increases with voltage beyond a voltage of about 1 mV. We interpret these data in the context of tunneling into a Luttinger liquid. In this theory, both dependencies will show scaling with a power law with identical exponents α: at low voltages (kB T  eV ) we expect dI dI dI α and at high voltages (eV  kB T ) we expect dV ∝ V α , where dV is dV ∝ T the differential conductance, V the bias voltage, T the temperature and α an exponent, which is determined by the strength of the interaction between the electrons in the one-dimensional wire [7, 8]. Applied to the data of Fig. 4.3, we find for kB T  eV power law scaling of the zero-bias differential conductance with temperature, with an exponent α = 0.28 ± 0.05 (red line in inset). The solid black line in the main figure shows power law scaling with bias voltage with the same exponent α = 0.28. As shown by Bockrath et al. [9] for tunneling from a metallic lead at zero temperature to a Luttinger liquid at finite temperature TLL , the full set of differential conductance curves is given by  dI  (4.1) f (V ) = dV  TR =0

ieV eV 1+α +γ )|Γ( )|2 2kB TLL 2 2πkB TLL where A is a constant, and Γ is the complex gamma function. The parameter γ is the ratio of the resistances of the two tunnel contacts of the lead-nanotubelead system, a measure of an experimentally unavoidable asymmetry. In the limits of low and high bias Eq. 4.1 reduces to the limiting values discussed above. In reality, the reservoirs also have a finite temperature TR , leading to a broadening in the differential conductance. Therefore, Eq. 4.1 should be convoluted with the derivative of the Fermi distribution in the reservoirs, which is a bell-shaped curve of width kB TR , given by α cosh(γ = ATLL

g(V ) = (1/4kB TR )sech2 (γeV /kB TR )

(4.2)

The experimentally measurable differential conductance is then given by:  ∞ dI (VB ) = f (VB − V )g(V )dV (4.3) dV −∞ where VB is the applied bias voltage. In equilibrium, the temperature of the reservoirs and the temperature in the nanotube are identical. However, since we will argue below that our experimental results under microwave radiation can be understood by assuming differences in these temperatures, we explicitly use different temperatures in Eqs. 4.1 and 4.2. In Fig. 4.3 the calculated differential conductance, for different temperatures of the entire substrate-lead-nanotube system, is shown (black curves), with the assumption that TLL = TR . The average asymmetry parameter γ from these fits is 0.48±0.09, and A is adjusted to match the data at VB = 0. 48


4.4 Microwave heating

Microwave heating

15 10

dI/dV ( S)

T

LL

20

(K )

4.4

5 0

5

PMW (nW)

10

15 PMW= 8.3 nW PMW= 0 nW

0.1

1

VB (mV)

10

Figure 4.4. Differential conductance of the suspended carbon nanotube as a function of bias voltage under increasing microwave power. The different curves correspond to an increase in the incident 108 GHz power from 0 nW (bottom, blue) to 8.3 nW (top, red). The black curves are fits from Eq. 4.3, with TLL = TR The inset shows the effective temperature determined from the zero-bias conductance, as a function of incident microwave power.

We now turn to the response to microwave radiation. Fig. 4.4 shows the differential conductance as a function of bias voltage for increasing incident power at 108 GHz. The colored curves correspond to increasing microwave power from bottom (blue) to top (red). The evolution of these conductance curves is analogous to that shown in Fig. 4.3, although the bath temperature is kept at a fixed value of 4.85 K. Given the macroscopic dimensions of the metallic contacts, we estimate that they will stay within 1 μK of the bath temperature given the microwave power used. However, the thermal conductance out of the suspended nanotube is small, since energy can only flow out through the tunnel junctions at the ends. Therefore, we assume that the increase in conductance is due to an increase in electron temperature TLL of the nanotube, and that the temperature TR in the reservoirs stays at 4.85 K. With these assumptions the data are well described by Eq. 4.3. From the fits we find an average value of γ = 0.46 ± 0.06 and α = 0.24 ± 0.02. The inset shows TLL determined from the zero-bias conductance. It increases from 5 K to 14 K with increasing incident power. However, as opposed to the data in Fig. 4.3, we now have a 49


4. Microwave-induced nonequilibrium temperature in a suspended carbon nanotube

non-equilibrium situation where the electron temperature in the nanotube is substantially higher than the temperature in the contacts. The main temperature drop occurs over the tunnel barriers at the contacts of the nanotube and the leads, because of the relatively good thermal conductance along a nanotube and the high resistance of the tunnel barriers. In the nanotube growth process we obtain nanotubes with total resistances typically varying from 50 kΩ to 300 kΩ. Because of the random nature of the growth process, these resistances will typically be distributed differently over the two different ends of the nanotube.

4.5

Electron temperature profile in a suspended carbon nanotube

We have modeled the temperature profile in a suspended carbon nanotube with a total contact resistance of 100 kΩ, under the absorption of microwave radiation up to a power of 80 pW. The asymmetry is included in the calculations by keeping the total contact resistance at 100 kΩ and only varying the distribution of the total resistance between the two contacts, i.e. the parameter γ in Eqs. 4.1 and 4.2. The absorption of microwave power by the electrons is assumed to be uniform over the length of the nanotube. We follow the commonly used approach, as used for example by Skocpol et al. in superconducting microbridges [19] and recently applied by Santavicca et al. to carbon nanotubes [20]. The temperature profile is determined by a heat-balance equation and a suitable boundary condition:

gth

dTe d (gth ) + PMW = 0 dx dx

(4.4)

dTe = GC [Te (x = 0) − TR ] dx x=0

(4.5)

where gth is the thermal conductivity along the nanotube, Te (x) is the electron temperature at position x in the nanotube, PMW is the absorbed microwave power, GC is the thermal conductance through the contacts to the electrodes and TR is the temperature of the electrodes. The thermal conductance GC across a contact is given by the contact resistance RC , via the WiedemannFranz law, GC = LTavg /RC , with L the Lorenz number, Tavg the average temperature between the lead and the nanotube, and RC the electrical contact resistance. We take the resistance along the length of the suspended carbon nanotube to be 1 kΩ/μm [20] and again apply the Wiedemann-Franz law to find the thermal conductance along the tube gth . Since we are interested in a qualitative picture we ignore possible deviations from the Wiedemann-Franz law originating in the Luttinger-liquid properties. 50


4.5 Electron temperature profile in a suspended carbon nanotube

Figure 4.5. Simulation of the electron temperature profile in a suspended carbon nanotube of length L under microwave irradiation. The total contact resistance of 100 kΩ is split between the right and left contact 10 kΩ:90 kΩ (γ = 0.9, blue), 20 kΩ:80 kΩ (γ = 0.8), 30 kΩ:70 kΩ (γ = 0.7), 40 kΩ:60 kΩ (γ = 0.6) and 50 kΩ:50 kΩ (γ = 0.5, red) respectively. The inset shows the temperature in the center of the nanotube as a function of absorbed microwave power for γ = 0.5.

Typical examples of the calculated temperature profile across the carbon nanotube, under the absorption of 50 pW of microwave radiation, are shown in 4.5. For the given parameters one gets an increase in temperature to about 15 K. The total contact resistance of 100 kΩ is distributed over the two contacts from 50 kΩ:50 kΩ (γ = 0.5, red line) to 10 kΩ:90 kΩ (γ = 0.9, blue line) in steps of 10 kΩ. The average temperature in the carbon nanotube decreases when the asymmetry increases, since more energy can flow out through the low resistance contact. The temperature profile itself remains quite flat, which justifies describing it with a single temperature. We obtain a maximum temperature gradient over the nanotube of 350 mK, for an asymmetry of 10 kΩ:90 kΩ, under absorption of 50 pW of microwave radiation. The inset shows the predicted effective temperature at the center of the suspended carbon nanotube for the symmetric case as a function of absorbed power.

51


4. Microwave-induced nonequilibrium temperature in a suspended carbon nanotube

4.6

Thermovoltage of microwave-heated carbon nanotube

Having established that a consistent interpretation of the data is possible by assuming a different temperature for the electrons in the nanotube from those in the leads, we return to the offset voltage under microwave radiation, which was shown in Fig. 4.2. In principle, an offset-voltage can be generated by rectification of the microwave field by the non-linearity of the carbon nanotube current-voltage characteristic [3,21]. However, given the measured nonlinearity, we estimate that an AC voltage of more than 12 mV is needed to obtain results on the order of the observed effect. This corresponds to an incoming microwave power of about 90 nW, given the microwave impedance of the system of 12.5 kΩ and irradiation from free space. This is an order of magnitude larger than the maximum incident microwave power of 8.3 nW. Instead, we argue that the offset-voltage arises from the strong temperature difference across the barriers between the nanotube and the contacts. Under a temperature difference hot charge carriers diffuse from the nanotube to the colder reservoirs. This current will build up an electric field over the region where the temperature gradient is present, leading to an offset voltage at zero current. The voltage is assumed to be given by V0 = −SΔT

(4.6)

where ΔT is the temperature difference and S is the thermopower (Seebeckcoefficient). A strong temperature gradient occurs primarily at the nanotubecontact barriers. In a perfectly symmetric sample, a thermovoltage would develop at both ends with opposite sign, and the net detected voltage would be zero. However, with an asymmetry in the contact resistances, the two thermovoltages will be different as well and a net voltage remains. It has been predicted that S is enhanced in a Luttinger-liquid system coupled to leads of non-interacting electrons [22]: SLL (T ) = CS (g)S0 (T )

(4.7)

where S0 (T ) is the thermopower for non-interacting electrons. The thermopower is enhanced by a factor CS (g) > 1 for g < 1. Here g is the Luttinger-liquid parameter measuring the strength of the interactions, which is directly related to the power law scaling exponent α. For an end-contacted nanotube α = (g −1 − 1)/4 while for a bulk-contacted nanotube α = (g −1 + g − 2)/8 [8]. The increase in offset voltage as a function of microwave power, shown in Fig. 4.2, reflects the increase in temperature difference between the nanotube and the metallic leads. The lower axis in the inset gives the temperature difference determined from the differential-conductance measurements. In our 52


4.7 Conclusion

experiments, the effective thermopower SLL is constant over the temperature range studied. It is found to be 43 ± 1 μV/K, which is an order of magnitude larger than the values reported for a carbon nanotube on a substrate [23].

4.7

Conclusion

In conclusion, we have shown that we can selectively raise the temperature of the electron system in a single suspended carbon nanotube by microwave irradiation, without heating the electrodes. This results in both an increase of the differential conductance and the development of a strong thermovoltage due to the temperature gradients occurring locally at the contacts between the nanotube and the leads.

53


4. Microwave-induced nonequilibrium temperature in a suspended carbon nanotube

References [1] K. Fu, R. Zannoni, C. Chan, S. H. Adams, J. Nicholson, E. Polizzi, and K. S. Yngvesson, Terahertz detection in single wall carbon nanotubes, Appl. Phys. Lett. 92, 033105 (2008). [2] M. E. Itkis, F. Borondics, A. Yu, and R. C. Haddon, Bolometric infrared photoresponse of suspended single-walled carbon nanotube films, Science 312, 413 (2006). [3] D. F. Santavicca, J. D. Chudow, D. E. Prober, M. S. Purewal, and K. P., Bolometric and nonbolometric radio frequency detection in a metallic single-walled carbon nanotube, Applied Physics Letters 98, 223503 (2011). [4] B. C. St-Antoine, D. M´enard, and R. Martel, Single-walled carbon nanotube thermopile for broadband light detection, Nano Lett. 11, 609 (2011). [5] V. V. Deshpande, M. Bockrath, L. I. Glazman, and A. Yacoby, Electron liquids and solids in one dimension, Nature 464, 209 (2010). [6] J. Luttinger, An exactly soluble model of a many-fermion system, J. Mat. Phys. 4, 1154 (1963). [7] K. A. Matveev and L. I. Glazman, Coulomb blockade of tunneling into a quasi-one-dimensional wire, Phys. Rev. Lett. 70, 990 (1993). [8] C. Kane, L. Balents, and M. P. A. Fisher, Coulomb interactions and mesoscopic effects in carbon nanotubes, Phys. Rev. Lett. 79, 5086 (1997). [9] M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E. Smalley, L. Balents, and P. L. McEuen, Luttinger-liquid behaviour in carbon nanotubes, Nature 397, 598 (1999). [10] Z. Yao, H. W. C. Postma, L. Balents, and C. Dekker, Carbon nanotube intramolecular junctions, Nature 402, 273 (1999). [11] Y.-F. Chen, T. Dirks, G. Al-Zoubi, N. O. Birge, and N. Mason, Nonequilibrium tunneling spectroscopy in carbon nanotubes, Phys. Rev. Lett. 102, 036804 (2009). [12] D. A. Bagrets, I. V. Gornyi, and D. G. Polyakov, Nonequilibrium kinetics of a disordered Luttinger liquid, Phys. Rev. B 80, 113403 (2009). [13] T. Karzig, G. Refael, L. I. Glazman, and F. von Oppen, Energy partitioning of tunneling currents into luttinger liquids, Phys. Rev. Lett. 107, 176403 (2011). [14] Z. Zhong, N. M. Gabor, J. E. Sharping, A. L. Gaeta, and P. L. McEuen, Terahertz time-domain measurement of ballistic electron resonance in a single-walled carbon nanotube, Nat. Nanotechnol. 3, 201 (2008). [15] W. Chen, A. V. Andreev, E. G. Mishchenko, and L. I. Glazman, Decay of a plasmon into neutral modes in a carbon nanotube, Phys. Rev. B 82, 115444 (2010). [16] M. Steiner, M. Freitag, V. Perebeinos, J. C. Tsang, J. P. Small, M. Kinoshita, D. Yuan, J. Liu, and P. Avouris, Phonon populations and electrical 54


4.7 References

[17] [18]

[19] [20]

[21]

[22]

[23]

power dissipation in carbon nanotube transistors, Nat. Nanotechnol. 4, 320 (2009). H. Cao, Q. Wang, D. W. Wang, and H. J. Dai, Suspended carbon nanotube quantum wires with two gates, Small 1, 138 (2005). G. A. Steele, A. K. Huttel, B. Witkamp, M. Poot, H. B. Meerwaldt, L. P. Kouwenhoven, and H. S. J. van der Zant, Strong coupling between singleelectron tunneling and nanomechanical motion, Science 325, 1103 (2009). W. J. Skocpol, M. R. Beasly, and M. Tinkham, Self-heating hotspots in superconducting thin-ďŹ lm microbridges, J. Appl. Phys. 45, 4054 (1974). D. F. Santavicca, J. D. Chudow, D. E. Prober, M. S. Purewal, and P. Kim, Energy loss of the electron system in individual single-walled carbon nanotubes, Nano Lett. 10, 4538 (2010). F. Rodriguez-Morales, R. Zannoni, J. Nicholson, M. Fischetti, K. S. Yngvesson, and J. Appenzeller, Direct and heterodyne detection of microwaves in a metallic single wall carbon nanotube, Appl. Phys. Lett. 89, 083502 (2006). I. V. Krive, I. A. Romanovsky, E. N. Bogachek, A. G. Scherbakov, and U. Landman, Thermoelectric eďŹ&#x20AC;ects in a Luttinger liquid, Low Temp. Phys. 27, 821 (2001). J. P. Small, L. Shi, and P. Kim, Mesoscopic thermal and thermoelectric measurements of individual carbon nanotubes, Solid State Comm. 127, 181 (2003).

55


4. Microwave-induced nonequilibrium temperature in a suspended carbon nanotube

56


Chapter 5 Frequency-dependent response of suspended carbon nanotubes Suspended carbon nanotubes are exposed to radiation from a freeelectron laser in the frequency range from 1.5 to 3 terahertz. The Free Electron Laser for Infrared eXperiments, FELIX, located at the FOM institute Rijnhuizen, has been used as a tunable source. We ďŹ nd that the response of a suspended carbon nanotube is a heating of the electron system in the nanotube, analogous to the response to 100 GHz radiation as described in Chapter 4. An increase in conductance is accompanied by the development of an oďŹ&#x20AC;set voltage. This result is interpreted as an increase of the temperature in the carbon nanotube. The nanotube response is strongly dependent on the frequency of the incident radiation. Further analysis of the frequency dependence could provide a direct measurement of the charge-density modes in a Luttinger liquid.

57


5. Frequency-dependent response of suspended carbon nanotubes

5.1

Collective modes in a Luttinger liquid

The exotic nature of the Luttinger liquid system in one-dimensional samples, such as carbon nanotubes, is found primarily in their electrodynamics, as discussed in Section 2.1.3. Although the measurement of the tunneling density of states through DC measurements does give an indication of the presence of a Luttinger liquid, the measurement of the velocity of the fundamental excitations in the system would provide a more direct probe of the Luttinger liquid properties. So far, experiments looking to directly probe these collective bosonic excitations in carbon nanotubes have not been successful. An often-used approach is based on optical techniques such as photoluminescence, Raman scattering, and ultrafast optical spectroscopy. In these cases however, it is a challenge to perform measurements on single isolated metallic carbon nanotubes [1]. An example of a measurement on isolated single nanotubes is the work by Zhong et al. [2], where the velocity of the excitations is determined using a new time-domain technique. The transfer time of a short pulse signal along the nanotube is measured, from which the velocity of the elementary excitations is deduced. Zhong et al. find that the velocity of the excitation is equal to the Fermi velocity vF ∼ 8 · 105 m/s typical for single electron excitations, while the collective charge excitations are expected to travel at a significantly higher velocity. Similar results are obtained by Liang et al. by observing Fabry-Perot resonances in the transport through a carbon nanotube [3]. A potential reason for the difficulty in observing the collective charge modes in carbon nanotubes is the decay into neutral modes [4]. Scattering on substrate states is known to play a role in carbon nanotubes in direct contact with a dielectric substrate [5]. Also in this case, suspended carbon nanotubes grown at the final step of device fabrication will provide cleaner quantum systems [6]. Furthermore, the energy outflow from a carbon nanotube in direct contact with a substrate is dominated by heat flowing into the substrate [7], leading to a strongly increased thermal response for suspended nanotubes. We aim for a study of the elementary excitations in clean, suspended carbon nanotubes. In this chapter, we will describe our measurements following the measurement scheme proposed by Burke [8], in suspended carbon nanotubes using a tunable free-electron laser as the radiation source. A possible frequency dependence relies on the following reasoning. The input impedance of antenna-coupled carbon nanotubes varies with frequency, due to the Fabry-Perot resonances in the nanotube [8, 9]. The power coupled into the carbon nanotube is maximum at half-wave resonances: fres = n 58

vF , 2gL

(5.1)


5.2 DC characterization of used suspended carbon nanotubes

with n a positive integer, vF the single-particle Fermi velocity, L the nanotube length, and g the Luttinger liquid parameter. For an end-contacted carbon nanotube, relevant in our experiments, the Luttinger liquid parameter g is given by [10]: 1 , (5.2) g= 4α + 1 with α the scaling-exponent for the differential conductance with bias voltage and temperature: dI/dV ∝ VBα , T α . This impedance modulation would lead to a frequency dependent signal in the photoresponse measurements.

5.2

DC characterization of used suspended carbon nanotubes

We report the study of terahertz absorption of two suspended carbon nanotube samples. The samples are fabricated in the same way as those discussed in Chapter 4, following the process developed by Cao et al. [6] and Steele et al. [11]. Figure 4.1 shows a scanning electron micrograph of a similar sample. One of the samples (CNT-A) has a trench width between source and drain electrode of 2 μm, the other (CNT-B) has a trench width of 4 μm. On the 2 μm sample we had a functional gate electrode, we could therefore also study the gatevoltage dependence of the electronic transport and the terahertz absorption. The source and drain electrodes also serve as the arms of a bowtie antenna, as described in the previous chapter. The suspended carbon nanotubes are grown in the final step of fabrication and have no contact with a gate oxide or substrate. The tunneling density of states is suppressed around zero energy in a Luttinger liquid, as discussed in Chapter 4. This leads to a power-law increase of the differential conductance with increasing bias voltage and temperature. Figure 5.1(a) shows the bias voltage dependence of the differential conductance of the 2 μm sample, for temperatures from 5 K (blue) to 14 K (red) at zero gate voltage. The inset gives the temperature dependence of the zero-bias conductance. We do not see a single clear power-law dependence over the entire bias range, for the higher temperatures however the dependence is close to linear. Figure 5.1(b) shows the differential conductance of the 4 μm sample for the same temperature range. The inset again shows the zero-bias differential conductance as a function of temperature, which more clearly follows a power law dI/dV ∼ T 0.35 . The differential conductance increases with increasing bias voltage, with a clear asymmetry between negative and positive bias voltages. Figure 5.2 shows the dependence of the differential conductance (color scale) on gate voltage (horizontal) and on bias voltage (vertical). Figure 5.1 is a cutthrough of Fig. 5.2 at VG = 0. We observe diamond-shaped regions of 59


5. Frequency-dependent response of suspended carbon nanotubes

CNT-A, L = 2 ȝm

(a)

5 2E-6

α = 0.99

dI/dV (μS)

1.5E-6

dI/dV (μS)

4

3

1E-6

5E-7

5

10

15

T (K)

2

1

0 -10

-5

0

5

10

VB (mV)

CNT-B, L = 4 ȝm

(b)

10 α = 0.35

dI/dV (μS)

12

8

dI/dV (μS)

6

5

10

15

T (K)

10

8

6 -10

-5

0

5

10

VB (mV)

Figure 5.1. Differential conductance dI/dV as a function of bias voltage for the 2 μm (a) and 4 μm (b) carbon nanotubes for increasing temperature from blue (5 K) to red (14 K). The insets show the zerobias conductance as a function of temperature, with a power-law fit with an exponent of 0.99 for the 2 μm- and 0.35 for the 4 μm- carbon nanotube.

low differential conductance (blue) and a general increase of the differential conductance with increasing bias voltage. As usual, we attribute these patterns to the onset of Coulomb blockade in the 2 μm long carbon nanotube [12]. The conductance of the carbon nanotubes along their length is usually high. For these samples, we find a large (∼ 100 kΩ resistance between the nanotube and the electrodes). Therefore, the carbon nanotube forms a Coulomb blockaded island, as discussed in Section 2.1.2. 60


5.3 Free-electron laser for 1.5 to 3 terahertz

Figure 5.2. DiďŹ&#x20AC;erential conductance (color) as a function of gate voltage (horizontal) and bias voltage (vertical). Diamond-like shapes of lower conductance can be seen. In general an increased bias voltage leads to increased diďŹ&#x20AC;erential conductance.

We conclude that these samples show features reminiscent of Coulomb blockade and Luttinger liquid behavior and are therefore suitable for a study of the response to terahertz radiation.

5.3

Free-electron laser for 1.5 to 3 terahertz

We use the Free Electron Laser for Infrared eXperiments, FELIX, located at the FOM institute Rijnhuizen, just prior to its move to the university of Nijmegen. As described in Section 3.2.2, FELIX is a free-electron laser, which allows continuous tunability of the wavelength over a wide frequency range and a large maximum output power with variable attenuation. In view of the length of our nanotube, we have chosen to study the response in the range from 1.5 to 3 THz, which is the lowest available frequency range. Water vapor is a notoriously strong absorber of terahertz radiation, with a large number of strong absorption lines in the terahertz frequency range. To avoid absorption of radiation, the beam is guided from the free-electron laser to the setup through a vacuum tube. From the output of the vacuum tube to the cryostat window, the beam travels through a box containing the beamsplitter, 61


5. Frequency-dependent response of suspended carbon nanotubes

pyroelectric detector and lens. This box is continuously purged with nitrogen gas, to avoid water absorption. A small remnant water vapor can not be excluded, and might still lead to an unwanted contribution to the spectrum. This is minimized by choosing equal path lengths from the output of the vacuum tube to the pyroelectric detector and to the cryostat. Therefore, any frequency dependence resulting from water absorption should be cancelled out by the power-measurement of the pyroelectric detector, carried out simultaneously to the measurements. The experimental setup is shown in Fig. 5.3. The suspended carbon nanotubes, located in the bowtie antenna, are mounted in the same cryostat with optical access as described in Chapter 4. The radiation bundle enters from the top-right in Fig. 5.3. A part of the bundle is reflected to a pyro-electric detector using a beam-splitter (20 μm Mylar), to monitor the changes in incident power during the frequency scan. The remainder of the radiation is coupled into the cryostat, through the windows at room temperature, 77 K, and 4.2 K, using a high-density polyethylene lens. As shown in Fig. 5.3(a), the output of FELIX is pulsed with a repetition rate of 10 Hz. The pulseprofile is a blockwave containing a large number of micropulses. The total pulse length is 10 μs. From here on, we will use the word pulse to refer to the complete burst of micropulses. An electrical trigger signal sent ∼1 ms prior to the pulse, allows us to reliably measure the time-domain signal of the nanotube as the pulse arrives. Figure 5.4 shows the temporal evolution of the current through the 4 μm carbon nanotube, for three different bias voltages (0, 2, and 5 mV). These data have been taken as follows. We continuously measure the current through the nanotube at a given bias voltage. The 10 μs duration of the FELIX pulse is indicated in green in Fig. 5.4. The observed rise- and decay-times are limited by the 100 kHz filter at the output of our measurement electronics. We measure every time-trace thirty times to improve the signal-to-noise ratio. We observe a clear peak in the current with the same delay to the trigger signal for every pulse. Furthermore, we do not see any response when the polarization of the incident FELIX pulse is orthogonal to the antenna polarization. We therefore exclude a response due to effects bypassing the carbon nanotube, such as a heating of the entire sample or electronic cross-talk. We conclude that the peak current seen in the temporal response of the current through the carbon nanotube is due to the response of the nanotube itself.

5.4

Radiation-induced changes in DC transport

In the current-voltage characteristics of a nanotube under continuous 108 GHz radiation, described in Chapter 4, we find an increase of the conductance and the development of an offset voltage, both of which we attribute to an increase 62


5.4 Radiation-induced changes in DC transport

(a) 1 pulse = 10.000 micropulses

Electronic trigger

t=0 1 ms

(b)

10 μs

FELIX output Beamsplitter

Cryostat

300K 77K 4.2K

Pyroelectric detector

Figure 5.3. Schematic representation of the experimental setup. (a) A ’pulse’ from FELIX consists of 10.000 micropulses. The pulse arrives 1 ms after an electronic trigger signal, which we take as t = 0. (b) The FELIX beam enters from the top-right and is partially coupled to a pyroelectric detector for power monitoring. The main part of the beam is coupled to the carbon nanotube in the cryostat using a lens. The beam travels through a continuously flushed nitrogen environment to avoid water absorption.

of the temperature in the electron system in the suspended carbon nanotube. Since the free-electron laser has a pulsed output, we need to construct the current-voltage characteristics from data such as those shown in Fig. 5.4. We construct current-voltage characteristics (Fig. 5.5) by measuring timetraces such as those shown in Fig. 5.4 for a range of DC bias voltages from -10 to 10 mV. In Fig. 5.5 (red dots), we plot the highest value of the current, defined as IP , occurring just before the pulse ends. The black dots are the 63


5. Frequency-dependent response of suspended carbon nanotubes

100

FELIX pulse

I (nA)

5 mV bias

50

2 mV bias

0 1000

0 mV bias

1020

1040

1060

1080

1100

t (μs) Figure 5.4. Averaged current through the nanotube for three bias voltages, as the pulse arrives. The time is measured from the electrical trigger signal. The duration of the pulse is indicated in green. The rise and decay times of the response are limited by the measurement electronics.

average values of the current taken in the time-domain from 0 to 200 μs after the electronic trigger signal (800 μs before the terahertz pulse), which we define as I0 . The green dots represent the current-voltage characteristics measured in a conventional way without any applied radiation, these measurements were performed with the windows blocked when the free-electron laser was switched off. The agreement between the black and green dots clearly shows that in the pulsed measurements we measure the ’dark’ current outside the pulse duration. The measured peak current value, IP , is not the final equilibrium value for the given incident power. Since the rise-time in the response is known to be due to the filter at the output of the measurement electronics, with a time-constant of 10 μs, we can estimate that the measured peak current is approximately ∼ 85% of the steady state value at the given incident power. The current-voltage characteristics of the peak current, IP (red dots in Fig. 5.5), show an increase in conductance and an offset voltage compared to the current measured outside the FELIX pulses, I0 (black dots). The currentvoltage characteristics are similar to the DC current-voltage characteristics measured under continuous 108 GHz radiation in Chapter 4. We conclude that within 10 μs the heating process has already been established. In addition, we 64


5.5 Frequency dependence of the carbon nanotube response

300 IP

I (nA)

I0 Dark

200

100

0 -10

-5

0 -100

5

10

VB (mV)

-200

-300

Figure 5.5. Bias-voltage dependence of the current through the 4 μm carbon nanotube measured at the peak of the FELIX pulse (red), average current outside of the FELIX pulse (black), and a dark measurement (green). The resultant current-voltage characteristics in the FELIX pulse show an increased conductance and offset-voltage.

conclude that there is no significant qualitative difference between the heating effect of 108 GHz and 1.5 to 3 THz radiation.

5.5

Frequency dependence of the carbon nanotube response

We measure the response, ΔI, of the nanotubes to radiation over a frequency range from 1.5 to 3 THz to determine the frequency dependence of the terahertz absorption of our suspended carbon nanotubes. The response of the carbon nanotube is defined as ΔI = IP − I0 , with IP the current at the peak of the FELIX pulse and I0 the out-of-pulse current. Fig. 5.6 shows the measured response at zero bias voltage for both nanotubes. CNT-A was measured over two different sessions. The main advantage of the use of a free-electron laser is that it allows continuous tunability of the frequency over a large range. This is achieved by varying the electron path in the free-electron laser, as discussed in Chapter 3. Therefore, unavoidably, we get a strong power-variation as a function of frequency. The response ΔI is strongly affected by this variation in power. In 65


5. Frequency-dependent response of suspended carbon nanotubes

15

CNT-A: 2 μm CNT-A CNT-B: 4 μm

ΔI (nA)

10

5

0 1.5

2.0

2.5

3.0

f (THz)

Figure 5.6. Raw data of the response to the incident radiation, ΔI, as a function of frequency, not normalized for the output power.

order to be able to interpret the data we need to correct for this variation. Figure 5.7 shows the measured incident power during one of the frequency scans. We constantly monitor the incident power with the pyroelectric detector, as shown in Fig. 5.3. In a separate measurement, we find that ΔI responds to the input power in a non-linear way, as shown in Fig. 5.8. We use this observation as the basis to correct the measured frequency dependence of the response ΔI (Fig. 5.6) for the variation in the output power (Fig. 5.7). Fig. 5.8 shows the response of the 4 μm carbon nanotube at zero bias voltage, as a function of the incident power for fixed frequencies at 1.84 and 2 THz. We observe an increase of the response of the nanotube with increasing incident power. The exact power-dependence of the response is difficult to predict, since it involves both the thermovoltage and the change of conductance with temperature, and the dependence of the carbon nanotube temperature on the incident power. To enable a proper rescaling, we parameterize the data of Fig. 5.8 as a function of power by (5.3) ΔI = B(f ) · (Pin )α We expect α to be independent of frequency, since the response of the nanotube depends on the temperature increase, which only depends on the amount of power absorbed and not on the original frequency of that incident radiation. We indeed consistently find only a variation smaller than the fitting uncertainty 66


5.5 Frequency dependence of the carbon nanotube response

150

Ppyro (μW)

100

50

0 1.5

2.0

2.5

3.0

f (THz)

Figure 5.7. Output power of the FELIX beam measured with the pyroelectric detector during a frequency scan.

1.84 THz 2 THz

14

α = 0.29

12

α = 0.27

ΔI (nA)

10 8 α

ΔI = B * P

6 4 2 0

0

5

10

15

20

25

Pantenna (mW)

Figure 5.8. Carbon nanotube response ΔI as a function of incident power on the antenna (measured with the pyroelectric detector) at zero bias voltage for the 4 μm carbon nanotube. The power-dependence of the response is parameterized with a power-law.

67


5. Frequency-dependent response of suspended carbon nanotubes

in our measurements. B(f ) is a frequency dependent transfer function, which contains the complete frequency dependence of the nanotube response. From Eq. 5.3 and the measured data in Figs. 5.6 and 5.7, we can determine the frequency dependent transfer function: B(f ) =

ΔI(f ) . Pin (f )α

(5.4)

B (A.U.)

CNT-A: 2 μm CNT-A CNT-B: 4 μm

1.5

2.0

2.5

3.0

f (THz)

Figure 5.9. Frequency dependence of the response of the suspended carbon nanotubes, corrected for variations in the input power.

Figure 5.9 shows the corrected frequency dependence of the carbon nanotube response from 1.5 to 3 THz for both the 2 μm (green/blue) and 4 μm (red) carbon nanotube. The dashed line in Fig. 5.7 indicates the minimum power for which we determined the corrected frequency dependence. We can now compare the spectrum in Fig. 5.9 to the measured response, ΔI, in Fig. 5.6 and the variation in incident power in Fig. 5.7. We see that in the frequency range above ∼2.2 THz the frequency dependence of the response is mainly due to the variation in incident power, giving a flat spectrum in the corrected data in Fig. 5.9. In the frequency range from 1.5 to 2 THz however, there is a strong frequency dependence also in the corrected data. Figure 5.10 shows the data of Fig. 5.9 in a smaller frequency range from 1.6 to 2 THz. We see a strong frequency dependence, especially in the 2 μm nanotube (green). The arrows in Fig. 5.10 indicate local maxima in the spectrum. The average spacing between the peaks is 73 GHz. The frequency dependence 68


5.6 Analysis of the observed frequency dependence

B (A.U.)

CNT-A: 2 μm CNT-A CNT-B: 4 μm

1.6

1.7

1.8

1.9

2.0

f (THz)

Figure 5.10. Smaller frequency range of the frequency dependence of the corrected response of the nanotubes shown in Fig. 5.9, highlighting periodic oscillations of the response with a spacing of 67, 98, and 54 GHz respectively.

of the 4 μm nanotube (red) is significantly weaker, but there do appear to be variations on a similar frequency scale.

5.6

Analysis of the observed frequency dependence

There are a number of possible sources for the observed frequency dependence of the absorbed terahertz power. Apart from intrinsic electronic effects in the nanotubes, as discussed in Section 5.1, there may also be contributions from elements of the experiment. We simultaneously measure the frequency dependence of the FELIX output power with the pyro-electric detector. As indicated in Fig. 5.3, the light travels the same distance from the beamsplitter to the pyro-electric detector and the cryostat. Therefore we can rule out any frequency dependencies that would show up in both, such as for instance spectral contributions from water absorption. We are therefore left with the elements in the path from the beam splitter to the nanotubes: the beam splitter, the cryostat windows, the antenna, and the nanotubes themselves. 69


5. Frequency-dependent response of suspended carbon nanotubes

Beamsplitter We use a 20 μm thick mylar (polyethylene terephthalate) film as a beamsplitter, in order to separate the incoming terahertz radiation into two beams. One beam goes to the cryostat containing the carbon nanotube, and the other to the pyroelectric detector. Due to the finite thickness of the beamsplitter, multiple reflections inside the film lead to interference between the radiation coupled out after various number of reflections. Depending on the wavelength of the incident light λ0 , the thickness of the film d, the dielectric constant of the beamsplitter material n, and the angle between the propagation direction and the beamsplitter θ0 , the ratio between transmitted and reflected power will differ. The transmission of the radiation is given by: Et 1 − Rp = E0 1 − Rp eiδ

(5.5)

with Rp ≈ 0.27 the reflection constant [13], and δ the phase difference between internally reflected and directly transmitted beams: δ=

n 4πd (d tan(θ) − tan(θ) sin(θ0 )) λ0 n0

(5.6)

with d the thickness of the beamsplitter, λ0 the wavelength of the radiation, θ the angle of the light in the beamsplitter, θ0 the angle of the incident radiation, n the refractive index of the beamsplitter, and n0 the refractive index of the surrounding medium. For the used material and thickness of the beamsplitter, this leads to a flat transmission spectrum with a maximum variation from 90% to 95% over the band from 1.5 to 3 THz [14, 15]. Windows The radiation enters the cryostat through a window of high-density polyethylene (HDPE) at room temperature. There are also HDPE windows at the 77K and 4K stages of the cryostat. HDPE has a frequency dependent transmission in the terahertz range, as shown in Fig. 5.11. The total thickness of HDPE the terahertz beam passes through before reaching the sample is 6 mm. The dip in transmission between 120 and 150 μm (2.5 and 2 THz), could explain the low measured response in this region in Fig. 5.9. Antenna In order to couple the incident terahertz radiation to the nanotube, we use a bowtie antenna. The frequency dependence of the bowtie antenna is similar to a broadband dipole antenna [17]. The resonance frequencies are given by: ca (5.7) fres = 2nL 70


5.6 Analysis of the observed frequency dependence

Frequency (THz) 6

3

2

1.5

1

0.5

Figure 5.11. Transmission of 2 mm thick high-density polyethylene in the terahertz frequency range [16].

with c the speed of light, a an odd integer, n the effective refractive index, and L = 308 μm the antenna length. For the given parameters, we come to a periodicity for our antennae of ∼650 GHz. In order to test the frequency dependence of the bowtie-antenna design in the terahertz range, we performed near-field time-domain spectroscopy measurements on the bare antennas. Near-field time-domain spectroscopy is a technique using an optical laser pulse as a probe and a femtosecond pulse as an excitation on an electro-optic substrate (GaP) in order to locally measure the electric fields on a subwavelength scale [18]. By Fourier-transform of the time-domain signal, the frequency components at each position can be determined. We find the expected broadband resonances according to Eq. 5.7. It is however unclear how the presence of the gate, carbon nanotube and various substrate layers will change the frequency-dependence of the antenna behavior at this point. Carbon nanotube The observed frequency dependence in Figs. 5.9 and 5.10 can not be explained to originate from any of the experimental elements. We therefore conclude that it is likely to be due to processes in the nanotubes themselves. If we use the average frequency spacing of the peaks in Fig. 5.10 in Eq. 5.1, we can determine 71


5. Frequency-dependent response of suspended carbon nanotubes

the corresponding g-parameter. We find g ≈ 2.7 for the 2 μm carbon nanotube and g ≈ 1.3 for the 4 μm carbon nanotube. We recall that g = 1 for a noninteracting excitation and g < 1 for a Luttinger liquid charge excitation [8, 19]. While the current data allows only for an order of magnitude approximation, we do find that there are features in the frequency dependence in the correct order of magnitude for electronic resonances. Although the trench width is precisely controlled to be 2 μm or 4 μm, the length of the carbon nanotube itself is more uncertain. The angle with which the nanotube crosses the trench might cause the suspended part to be slightly longer than the trench width. Furthermore, it is not necessarily clear whether the effective length of the nanotube for the electronic excitations ends exactly where the suspended part ends, or continues where the nanotube is in contact with the electrodes. In conclusion, due to the many contributing factors to the frequency dependence of the terahertz absorption, it is difficult to fully distinguish between intrinsic effects in the nanotube and experimental contributions to the frequency dependence. Although we can not at present draw any conclusion about the propagation velocity of the fundamental excitations in the carbon nanotubes from the measurements shown in Fig. 5.9 and 5.10, we can conclude that the applied measurement scheme allows for the observation of intrinsic spectral effects in the nanotubes.

5.7

Conclusions and outlook

Under terahertz radiation from 1.5 to 3 THz, we observe an increase in conductance and the development of an offset voltage in the bias voltage dependence of the current measured through our suspended carbon nanotubes. The response appears to be dominated by the increased temperature in the nanotube and the temperature difference with its surroundings, similar to the response to 108 GHz radiation discussed in Chapter 4. The frequency dependence of the current increase under irradiation, ΔI, provides a measure of the relative absorption efficiency of terahertz power in the nanotube, if corrected for variations in the input power. There is a strong dependence of this corrected response on the frequency of the incident radiation. Both in the 2 μm and 4 μm nanotube, the frequency dependence of the response is stronger in the 1.5 to 2 THz range and relatively flat in the 2 to 3 THz range. Especially in the 2 μm nanotube there is a regular set of peaks, with a spacing of ∼70 GHz. These measurements form a first step towards the measurement of Luttinger liquid bosonic modes in single suspended carbon nanotubes. Future experiments should attempt to more accurately determine the other possible contributions to the frequency dependence in the experiment by using a known 72


5.7 Conclusions and outlook

detector element (such as a hot-electron bolometer) in the same sample-layout and measurement setup. One could also measure a larger range of carbon nanotube lengths, in order to isolate the intrinsic, length dependent, contributions.

73


5. Frequency-dependent response of suspended carbon nanotubes

References [1] E. H. Haroz, J. G. Duque, X. Tu, M. Zheng, A. R. Hight Walker, R. H. Hauge, S. K. Doorn, and J. Kono, Fundamental optical processes in armchair carbon nanotubes, Nanoscale 5, 1411 (2013). [2] Z. Zhong, N. M. Gabor, J. E. Sharping, A. L. Gaeta, and P. L. McEuen, Terahertz time-domain measurement of ballistic electron resonance in a single-walled carbon nanotube, Nat. Nanotechnol. 3, 201 (2008). [3] W. Liang, M. Bockrath, D. Bozovic, J. H. Hafner, M. Tinkham, and H. Park, Fabry-Perot interference in a nanotube waveguide, Nature 411, 665 (2001). [4] W. Chen, A. V. Andreev, E. G. Mishchenko, and L. I. Glazman, Decay of a plasmon into neutral modes in a carbon nanotube, Phys. Rev. B 82, 115444 (2010). [5] M. Steiner, M. Freitag, V. Perebeinos, J. C. Tsang, J. P. Small, M. Kinoshita, D. Yuan, J. Liu, and P. Avouris, Phonon populations and electrical power dissipation in carbon nanotube transistors, Nat. Nanotechnol. 4, 320 (2009). [6] H. Cao, Q. Wang, D. W. Wang, and H. J. Dai, Suspended carbon nanotube quantum wires with two gates, Small 1, 138 (2005). [7] D. F. Santavicca, J. D. Chudow, D. E. Prober, M. S. Purewal, and P. Kim, Energy loss of the electron system in individual single-walled carbon nanotubes, Nano Lett. 10, 4538 (2010). [8] P. J. Burke, Luttinger liquid theory as a model of the gigahertz electrical properties of carbon nanotubes, IEEE Trans. Nanotechnol. 1, 129 (2002). [9] D. F. Santavicca and D. E. Prober, Terahertz resonances and bolometric response of a single-walled carbon nanotube, 33rd international conference on infrared, millimeter and terahertz waves Vols. 1 and 2, 727 (2008). [10] C. Kane, L. Balents, and M. P. A. Fisher, Coulomb interactions and mesoscopic eďŹ&#x20AC;ects in carbon nanotubes, Phys. Rev. Lett. 79, 5086 (1997). [11] G. A. Steele, A. K. Huttel, B. Witkamp, M. Poot, H. B. Meerwaldt, L. P. Kouwenhoven, and H. S. J. van der Zant, Strong coupling between singleelectron tunneling and nanomechanical motion, Science 325, 1103 (2009). [12] E. Thune and C. Strunk, Quantum transport in carbon nanotubes, Lect. Notes Phys. 680, 351 (2005). [13] H. Lihn, P. Kung, C. Settakorn, H. Wiedemann, and D. Bocek, Measurement of subpicosecond electron pulses, Phys. Rev. E 53, 6413 (1996). [14] E. Hecht, Optics, Addison-Wesley, (2001). [15] T. Baar, Masterâ&#x20AC;&#x2122;s thesis, TU Delft, 2009. [16] TYDEX Optics, THz Materials, Technical report, http://www.tydexoptics.com/pdf/THz Materials.pdf. [17] C. A. Balanis, Antenna theory, Wiley-Interscience, (2005). 74


5.7 References

[18] K. J. Ahn, K. G. Lee, K. W. Kihm, M. A. Seo, A. J. L. Adam, P. C. M. Planken, and D. S. Kim, Optical and terahertz near-ďŹ eld studies of surface plasmons in subwavelength metallic slits, New Journal of Physics 10, 105003 (2008). [19] M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E. Smalley, L. Balents, and P. L. McEuen, Luttinger-liquid behaviour in carbon nanotubes, Nature 397, 598 (1999).

75


5. Frequency-dependent response of suspended carbon nanotubes

76


Chapter 6 Critical-Current Reduction in Thin Superconducting Wires Due to Current Crowding

We demonstrate experimentally that the critical current in superconducting NbTiN wires is dependent on their geometrical shape, due to current-crowding effects. Geometric patterns such as 90◦ corners and sudden expansions of wire width are shown to result in the reduction of critical currents. The results are relevant for single-photon detectors as well as parametric amplifiers.

This chapter is based on H.L. Hortensius, E.F.C. Driessen, T.M. Klapwijk, K.K. Berggren, and J.R. Clem, Applied Physics Letters 100, 182602 (2012)

77


6. Critical-Current Reduction in Thin Superconducting Wires Due to Current Crowding

6.1

Introduction

Superconducting wires made of strongly disordered superconducting materials such as NbN and NbTiN are used for single-photon detection [1–3], singleelectron detection [4] and parametric amplification [5]. In all cases, for optimal performance, the devices are biased at as high currents as possible, without exceeding the critical current. In principle, for wires smaller than both the Pearl length Λ = 2λ2 /d (λ is the dirty London penetration depth and d is the film thickness) [6] and the dirty-limit coherence length ξ, the critical current is determined by the critical pair-breaking current, which has been theoretically calculated over the full temperature range by Kupriyanov and Lukichev [7]. These predictions have been tested experimentally in aluminum by Romijn et al. [8] and Anthore et al. [9]. In many practical cases, it is found that the critical current varies from device to device and is significantly lower than this intrinsic maximum value. This reduction is usually attributed to defects in the films and slight variations in width. In addition, for strongly disordered superconductors electronic inhomogeneity may develop, even for homogeneously disordered materials [10]. However, Clem and Berggren [11], responding to the observed dependence of the critical currents of superconducting single-photon detectors on the fill factor of the pattern [12], explained that the critical current may depend on geometric factors in the wires, such as bends. In their model analysis, the superconducting wires were narrower than the Pearl length, but wider than the coherence length. Consequently, the current is not necessarily uniform and the critical current is reached when the current density locally exceeds the critical pair-breaking current. At this current a vortex enters the superconducting wire, causing the transition to a resistive state [13]. In this letter, we present an explicit comparison of critical currents of superconducting NbTiN nanowires with different geometrical shapes and confirm that the observed critical current depends on the geometry. In the analysis of Clem and Berggren [11], the sheet-current distribution is calculated for a given geometry using conformal mapping. Then, the critical current is defined as the current at which the Gibbs-free-energy barrier for a vortex to enter the film is reduced to zero. Any geometry where the current is led around a sharp feature, even when spreading into a suddenly widening portion of the wire, is predicted to lead to an increase in local current density at the inner curve of the corner, and a correspondingly reduced critical current. This behavior is similar to that in a normal-metal current divider, in which the shorter, lower-resistance, path around the inner corner carries a higher proportion of the current than the outer, higher-resistance, path. Fig. 6.1 shows the current streamlines in a superconducting thin-film wire for two cases: (a) a sharp 90◦ turn and (b) a sudden widening or stub. Note the increased current density in the inner corner. The dashed curve in (a) follows a contour where the current density equals the current density of an infinite straight wire. 78


6.2 NbTiN geometries

Figure 6.1. Calculated current stream lines for the cases of (a) a 90◦ bend and (b) a suddenly widening “stub” region. The increased density of stream lines near the inner corners shows the current crowding. The dashed curve in (a) represents the median contour: half the integrated current flows on one side of the contour, and half flows on the other side. An inner corner shaped according to this contour will exhibit no current crowding, and is said to be optimally rounded.

Outside this contour, the current never exceeds the average current density in the straight wire segment. Therefore, a corner conforming to this geometry will always be able to carry the current supplied to it, without exhibiting a reduction in the critical current. The shape of this optimally rounded corner is given in Eqs. (112) and (113) of Ref. [11]. Even though in the structure in Fig. 6.1(b) material is added with respect to a straight wire of width W , the critical current is still predicted to be reduced.

6.2

NbTiN geometries

Fig. 6.2 shows examples of the four patterns we used to test the theoretical predictions of Ref. [11]: (a) straight wires (3 devices); (b) corners with optimally designed inner curves (2 devices); (c) corners with sharp inner curves (3 devices); and (d) a straight line that included a wider region (a stub) extending 1 μm to one side (1 device). All geometries consisted of a 1-μm-wide wire with a length of 20 μm. The devices were contacted at their terminals by using a gradually widening region, to avoid potential current crowding at the contacts. To quantify the expected effect, we define the reduction factor R to be a number between zero and one, where the critical current of the geometry is given by Ic = R · Ic,straight , with Ic,straight the critical current of an infinitely long wire of width W . R = 1 for the straight wire and for the optimally rounded corner, indicating an unsuppressed critical current. For the sharp corner and 79


6. Critical-Current Reduction in Thin Superconducting Wires Due to Current Crowding

(a)

(b)

981 nm 967 nm

(c)

(d)

972 nm 60 nm 1970 nm

Figure 6.2. SEM images of test structures measured in these experiments: (a) a straight line segment; (b) an optimally rounded 90◦ bend; (c) a 90◦ bend; and (d) a suddenly widening “stub” region. The minimum inner radius of curvature achievable in our fabrication process was approximately 60 nm. The light spot visible on (d) is due to charging after zooming in on the wire-contact transition region.

the wire with a stub, R ∝ (ξ/W )1/3 , with ξ the coherence length and W the width of the wire. The suppression is thus larger for shorter coherence length and larger wire width. In order to satisfy the condition ξ  W  Λ, we fabricated the wires of width W = 1 μm in a 8 nm thick NbTiN film with a resistivity of√170 μΩcm, implying a mean free path l ≈ 0.6 nm. The coherence length ξ ≈ ξ0 l ≈ 7 nm, with ξ0 the BCS coherence length, and the Pearl length Λ ≈ 20 μm. The NbTiN film was grown on a high-resistivity (ρ > 1 kΩcm), hydrogenpassivated silicon substrate using DC magnetron sputtering from a 70% Nb, 30% Ti target, at a pressure of 8 mTorr, a DC power of 300 W, a nitrogen gas flow of 4 sccm and an argon gas flow of 100 sccm. Onto this film, a 150 nmthick layer of hydrogen silsesquioxane resist was spin-coated. This layer was patterned using electron-beam lithography (100 keV, ∼ 2 mC/cm2 ). The resist was developed in a tetra-methyl ammonium hydroxide solution and a postdevelopment exposure was performed to harden the resulting etch mask [14]. Finally, the pattern was transferred into the NbTiN film by reactive-ion etching 80


6.3 Reduction of the critical current due to current crowding

in a 50 W SF6 /O2 plasma, using interferometric end point monitoring. Both a large-area test structure and the wires showed a critical temperature of 9.5 K. Devices were inspected using electron microscopy after electrical testing. All structures designed to be 1 μm wide, had an actual width of ∼970 nm. The inner curves of nominally zero-radius corners were observed to have an inner radius of ∼60 nm. We ascribe this curvature to imperfect fabrication resolution. The devices were tested in a dipstick probe in liquid helium with a base temperature of 4.2 K. The temperature of the sample stage was controlled by a heater element. We tested the devices by ramping a low-noise home-built current supply with a programmed ramp from 0 to 500 μA and back down to 0 μA with a 100 Hz repetition rate and measuring the resulting voltage over the wire. We determined the critical current by recording the current at which a threshold voltage across the device was passed during the positive portion of the current ramp. All results presented here were measured on a single chip and in a single testing session.

6.3

Reduction of the critical current due to current crowding

Fig. 6.3 shows the resulting histograms from the critical current measurements at 4.2 K. We recorded 1000 successive critical currents for each device. The straight wires (a) and wires with an optimally rounded inner corner (b) show averaged critical currents ranging from 382 μA to 560 μA. All three devices with a sharp inner corner (c) exhibit a lower critical current between 280 μA and 292 μA. Note that two of the sharp-corner histograms are overlapping at 280 μA. The device with a straight wire with a stub (d) has a critical current of 358 μA. The devices with sharp corners and the device with a stub all exhibit a narrower distribution of critical currents over their 1000 measurements, which can be seen from the height of the peak in the histogram. We interpret the small variation in critical currents between the three different wires with a sharp corner (c) as an indication that indeed the corner is the dominant weakest link. For the straight (a) and rounded (b) wires there is no natural point for the switching to nucleate. Therefore, the critical current is sensitive to possible inhomogeneities. For example, on inspection we discovered that occasionally the wires contain a pinhole of ∼60-nm-diameter. These were absent in the devices with the highest critical current. The results for the wires with a sharp corner were independent of the presence of a pinhole. We emphasize that wire-to-wire variations are expected to be more likely to occur for uniform wires than for wires with a clearly defined weakest link. We compared the measured values of the critical current to the expected critical pair-breaking current. Assuming a uniform current distribution, this 81


6. Critical-Current Reduction in Thin Superconducting Wires Due to Current Crowding

number of counts

800 600

a: Straight wire b: Rounded corner c: Sharp corner d: Stub

c c

400 200 0

ba

a

300

400

( A) C

a b

500

600

I

Figure 6.3. Histograms of 1000 measured critical currents each, for 9 distinct devices: (a) straight lines (3 devices) 1-μm wide; (b) corners with optimally designed inner corner curves (2 devices, straight segments were also 1-μm wide); (c) corners as in (b) but in which the inner corner was fabricated to be as sharp as possible (3 devices); and (d) lines with a wider region (stub) extending 1 μm to one side (1 device). The bin width is 0.5 μA.

current density is temperature dependent, and all material-dependent parameters are captured in a scaling parameter described by Romijn et al. [8]. Using the experimentally determined values for Tc = 9.5 K and Rs = 217 Ω, the only unknown in this equation is the diffusion constant D. The value of D is unknown for our film, but if we use the value D = 0.45 cm2 /s that was measured for a comparable NbN film [15], we arrive at a critical current of Ic ≈ 815 μA for our 1 μm wide wire at T = 4.2 K. Given the uncertainty in the diffusion constant D, the observed critical current for our straight wires falls into the right range. Using the results of Ref. [11], for a corner with the observed inner radius of curvature of 60 nm, we expect R ≈ 0.33 but observe a reduction of only 0.51 relative to the largest Ic measured. For the suddenly widening stub device, assuming zero inner radius of curvature of the corners, we expect R ≈ 0.46, but observe instead R ≈ 0.64. This could in part be due to the finite minimal radius of curvature achievable in our fabrication process, but we note that the theory by Clem and Berggren does not take into account the superconducting coherence between the different current filaments. Taking this into account would require solving the Ginzburg-Landau equations or the more elaborate Usadel equations [16]. Nevertheless, the overall pattern of suppression is quali82


6.4 Temperature dependence of the critical current distribution

Figure 6.4. Temperature dependence of (a) the critical current Ic , and (b) the width of the critical current distribution Î&#x201D;Ic for straight wires (black squares), optimally rounded corners (blue circles), sharp corners (red triangles), and a wire with a stub (green diamonds). The dashed curve in (a) is a ďŹ t following Kupriyanov and Lukichev [7].

tatively clear: structures with sharp inner corners exhibit systematically lower values of Ic than straight wires and structures with rounded inner corners. 83


6. Critical-Current Reduction in Thin Superconducting Wires Due to Current Crowding

6.4

Temperature dependence of the critical current distribution

To provide further evidence for the impact of sharp corners on the critical current, in Fig. 6.4 we plot the temperature dependence of the critical current and of the width of the distribution of the critical current between 5.5 K and 8 K. For even higher temperatures, the current-voltage curves no longer exhibit a sharp transition from the superconducting to the normal state. At each temperature, histograms like the ones in Fig. 6.3 were taken with 1000 current ramps for each wire. The critical current Ic was taken to be the median of the critical current distribution. The width of the critical current distribution ΔIc was defined as the difference between the third and first quartile. As shown in Fig. 6.4, the wires containing a sharp corner exhibit a significantly decreased critical current and a decreased width of their distribution of critical current, as seen before in Fig. 6.3. For all wires, the critical current decreases with increasing temperature, as expected. The difference between the unaffected and reduced critical currents decreases as the temperature is increased towards Tc . The dashed curve in Fig. 6.4(a) is a fit following Kupriyanov and Lukichev [7]. This fit gives us a critical temperature Tc = 11.1 K, in contrast to Tc determined from low-bias R(T ) measurements of 9.5 K. We systematically observe such a difference, also in other NbTiN devices. Recently Benfatto et al. discussed the effects of a broadened BerezinskiiKosterlitz-Thouless transition in disordered superconductors, which might explain this observation [17]. Another difference between wires is the width of their critical-current distribution, as shown in Fig. 6.4(b). For straight wires, the critical-current variance is approximately proportional to the critical current over the temperature range studied, ΔIc /Ic (T ) is constant. However, the wires with sharp geometric features exhibit a qualitatively different temperature dependence of ΔIc . First, the critical-current distribution is consistently narrower. Furthermore, the absolute width of their critical-current distribution varies little with temperature, while the critical current decreases with increasing temperature. We interpret this difference also as being due to the sharp features, noting that they provide a natural weak spot at which the wire will initiate its transition to the normal state. In contrast, in the straight and optimally rounded devices, a fluctuation at any point in the wire could cause it to switch, and so the switching effect is not localized. Finally, we return to the stub device (Fig. 6.2(d)). It is counterintuitive that a wider cross-section would lead to a lower critical current. As shown in Fig. 6.3 however, the stub device is at the lower end of the critical currents. Moreover, in Fig. 6.4(a) the temperature dependence is deviating from that of the straight wires, analogous to that of the sharp corners. Similarly Fig. 6.4(b) shows that 84


6.5 Conclusion

the width of the critical current distribution is narrow, again analogous to the sharp corners. These observations taken together are taken as evidence that the stub indeed creates a dominant weakest link.

6.5

Conclusion

In summary, we have shown a clear reduction of the critical current in devices containing a sharp geometric feature. It is shown that the reduction of the critical current can be avoided by the inclusion of an optimally rounded corner. We further note that even in cases when W > Λ, which we have not treated theoretically or experimentally, as long as the condition ξ < W is still satisfied, one would expect these phenomena to be important. The avoidance of critical current reduction may have an immediate impact on the performance of superconducting single-photon detectors and parametric amplifiers based on superconducting nanowires.

85


6. Critical-Current Reduction in Thin Superconducting Wires Due to Current Crowding

References [1] G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, Picosecond superconducting single-photon optical detector, Appl. Phys. Lett. 79, 705 (2001). [2] A. J. Kerman, E. A. Dauler, J. K. W. Yang, K. M. Rosfjord, V. Anant, K. K. Berggren, G. N. Gol’tsman, and B. M. Voronov, Constriction-limited detection efficiency of superconducting nanowire single-photon detectors, Appl. Phys. Lett. 90, 101110 (2007). [3] S. N. Dorenbos, E. Reiger, U. Perinetti, V. Zwiller, T. Zijlstra, and T. M. Klapwijk, Low noise superconducting single photon detectors on silicon, Appl. Phys. Lett. 93, 131101 (2008). [4] M. Rosticher, F. R. Ladan, J. P. Maneval, S. N. Dorenbos, T. Zijlstra, T. M. Klapwijk, V. Zwiller, A. Lupa¸scu, and G. Nogues, A high efficiency superconducting nanowire single electron detector, Appl. Phys. Lett. 97, 183106 (2010). [5] B. H. Eom, P. K. Day, H. G. LeDuc, and J. Zmuidzinas, A wideband, lownoise superconducting amplifier with high dynamic range, Nature Phys. 8, 623 (2012). [6] J. Pearl, Current distribution in superconducting films carrying quantized fluxoids, Appl. Phys. Lett. 5, 65 (1964). [7] M. Y. Kupriyanov and V. F. Lukichev, Temperature dependence of pairbreaking current in superconductors, Sov. J. Low Temp. Phys. 210, 210 (1980). [8] J. Romijn, T. M. Klapwijk, M. J. Renne, and J. E. Mooij, Critical pairbreaking current in superconducting aluminum strips far below Tc , Phys. Rev. B 26, 3648 (1982). [9] A. Anthore, H. Pothier, and D. Esteve, Density of states in a superconductor carrying a supercurrent, Phys. Rev. Lett. 90, 127001 (2003). [10] B. Sac´ep´e, C. Chapelier, T. I. Baturina, V. M. Vinokur, M. R. Baklanov, and M. Sanquer, Disorder-induced inhomogeneities of the superconducting state close to the superconductor-insulator transition, Phys. Rev. Lett. 101, 157006 (2008). [11] J. R. Clem and K. K. Berggren, Geometry-dependent critical currents in superconducting nanocircuits, Phys. Rev. B 84, 174510 (2011). [12] J. K. W. Yang, A. J. Kerman, E. A. Dauler, B. Cord, V. Anant, R. J. Molnar, and K. K. Berggren, Suppressed critical current in superconducting nanowire single-photon detectors with high fill-factors, IEEE Trans. Appl. Supercond. 19, 318 (2009). [13] L. N. Bulaevskii, M. J. Graf, and V. G. Kogan, Vortex-assisted photon counts and their magnetic field dependence in single-photon superconducting detectors, Phys. Rev. B 85, 014505 (2012). 86


6.5 References

[14] J. K. W. Yang, V. Anant, and K. K. Berggren, Enhancing etch resistance of hydrogen slisesquioxane via postdevelop electron curing, J. Vac. Sci. Tech. B 24, 3157 (2006). [15] A. D. Semenov, G. N. Gol’tsman, and A. Korneev, Quantum detection by current carrying superconducting film, Physica C 351, 349 (2001). [16] K. D. Usadel, Generalized diffusion equation for superconducting alloys, Phys. Rev. Lett. 25, 507 (1970). [17] L. Benfatto, C. Castellani, and T. Giamarchi, Broadening of the Berezinskii-Kosterlitz-Thouless superconducting transition by inhomogeneity and finite-size effects, Phys. Rev. B 80, 214506 (2009).

87


6. Critical-Current Reduction in Thin Superconducting Wires Due to Current Crowding

88


Chapter 7 Possible Indications of Electronic Inhomogeneities in Superconducting Nanowire Detectors

The voltage-carrying state of superconducting NbTiN nanowires, used for single-photon detectors, is analyzed. Upon lowering the current, the wire returns to the superconducting state in a steplike pattern, which diďŹ&#x20AC;ers from sample to sample. Elimination of geometrical inhomogeneities, such as sharp corners, does not remove these steplike features. They appear to be intrinsic to the material. Since the material is strongly disordered, electronic inhomogeneities are considered as a possible cause. A thermal model, taking into account random variations of the electronic properties along the wire, is used as an interpretative framework.

This chapter is based on H.L. Hortensius, E.F.C. Driessen, and T.M. Klapwijk, IEEE transactions on Applied Superconductivity 23, 2200705 (2013)

89


7. Possible Indications of Electronic Inhomogeneities in Superconducting Nanowire...

7.1

Introduction

Superconducting nanowire single-photon detectors (SNSPD’s) are promising devices for the detection of single photons because of their fast response time, broadband sensitivity, and low dark-count rate [1]. A variety of materials are currently being used, with the common denominator that their resistivity in the normal state is unusually large, consistent with an electronic mean free path in the order of the interatomic distance. These materials are chosen for their high critical temperature and fast electron-phonon scattering. In view of the increased interest, the need for reproducible device-fabrication has become an issue. It has been found that often variations of detection efficiency occur from device to device [2] and as a function of position on a single device [3], which are not easily understood. Part of this lack of control may signal the need for improved materials control on an atomic level. However, since SNSPD’s are made from highly resistive superconductors, an intrinsic cause might be present as well. Due to the short elastic-scattering length, localization is competing with superconductivity, and a tendency to an insulating state is accompanied by a tendency to become superconducting. It is predicted that intrinsic electronic inhomogeneities are formed irrespective of the specific atomic inhomogeneity [4]. Recent experimental work on TiN has clearly demonstrated the occurrence of such electronic inhomogeneities by measuring the local values of the superconducting energy-gap by scanning tunneling microscopy [5]. Stimulated by these observations, we have recently studied the microwave electrodynamics of such films [6], and analyzed the data in the context of a recent theory proposed by Feigel’man and Skvortsov [7]. Unfortunately, for SNSPD’s a limited number of parameters is available to characterize them: the critical temperature, the critical current, the resistivity and the resistive transition. It is, however, reasonable to assume that variations of the superconducting gap, as observed by Sac´ep´e et al. play a role in the observed variation of detection efficiency from device to device and from position to position on the device. In this manuscript, we focus on one of the few available extra sources of information; the return of the device to the superconducting state, the so-called retrapping current (Fig. 7.1), which is known to vary from sample to sample and shows a clear steplike structure.

7.2

Experiments

The measurements described in this paper, were performed on the same NbTiN wires described in [8], where the reduction of the critical current due to current crowding was studied. The NbTiN films have a thickness d of ∼8 nm, a critical temperature Tc of 9.5 K, and a resistivity ρ of 170 μΩcm. The wires have 90


7.2 Experiments

Figure 7.1. Examples of the steplike decrease in voltage observed for decreasing current-bias for various wire geometries. The inset shows the temperature dependence of the resistance of one of the 20 μm-long, 1 μm-wide, and 8 nm-thick wires, including the gradually widening contact pads [8].

a width w of 1 μm. We have chosen this width, though it is wider than typical SNSPD devices, since wider wires are less sensitive to fabrication defects and show stronger current-crowding effects. Similar results were obtained for samples as narrow as 50 nm. The length L is 20 μm. Four different wire geometries were studied: straight wires, corners with optimally designed inner curves, corners with sharp inner curves, and a straight line that included a stub extending 1 μm to one side. For each of the devices the current-voltage characteristics were measured in a 4.2 K liquid helium dipstick. The samples were mounted on a ceramic PCB that was in direct contact with a copper sample stage with thermally anchored wires, to ensure a temperature equal to the sample-stage. The temperature of the sample stage itself was controlled by a heater element and measured by a temperature sensor embedded in the sample stage. The measurements reported here are all current-biased in a two-point measurement configuration. The wires were driven into the normal state by applying a current above the critical current of the wire. Then we monitored for each sample for decreasing current the return from the normal state into the superconducting state. 91


7. Possible Indications of Electronic Inhomogeneities in Superconducting Nanowire...

7.3

Steplike transitions

Fig. 7.1 shows the retrapping characteristics for four geometries. This set of measurements was taken at 4.2 K. In all samples a stepwise transition from the normal state to the superconducting state is observed. The intermediate states between the normal and superconducting state of the wire are stable for long periods of time (measured up to ∼10 minutes) if the bias current is kept stable at this point. The specific pattern of a wire is reproducible over multiple measurements. We observe similar stepwise retrapping characteristics in 100 nm wide NbTiN and TiN nanowires. We find that the presence and stability of these steps does not depend on the geometrical shape of the wire. All geometries show stepwise retrapping patterns. This shows that, for these geometries, the presence of a geometrical ‘constriction’ does not dominate the retrapping characteristics. This is clearly different from the superconducting to normal-state transition, where the critical current is significantly suppressed in geometries with sharp features [8]. The available evidence suggests that these disordered superconducting wires have a tendency to become superconducting by forming one or more superconducting domains, which expand in a steplike discontinuous way. We have recently performed a detailed study of a model system consisting of a superconducting wire between two normal reservoirs. The analysis was executed taking the full non-equilibrium processes into account [9]. In the strongly disordered wires of NbTiN (and TiN) the origin of the steps needs to be determined prior to a more detailed analysis. It has been theoretically suggested that for a superconductor with homogeneous disorder the superconducting state tends to develop mesoscopic fluctuations in the value of the energy gap and a flattening of the peak in the density of states [4]. Scanning tunneling microscopy measurements on thin films of TiN, with a resistivity even higher than our films: ρ > 1000μΩcm, show such a variation of the gap energy over the film [5]. The maximum variation of the gap energy is about 20% of the total gap energy, and the typical size of the regions is about 50 nm, much larger than the grain size of the films. Variations in the geometry of the nanowire, such as thickness variations due to steps in the underlying substrate, might lead to similar behavior as modeled here [10]. We use SEM-inspection to guaranty the homogeneity of the width. We observe similar behavior in ALD-deposited TiN nanowires on an amorphous silicon oxide substrate. The persistent presence of these steplike features for wires that appear ‘perfect’, leads us to conjecture that electronic inhomogeneities play a role. 92


7.4 Thermal model with randomly fluctuating parameters

(a)

(b) PDC Pdi

,e (x)

Pdi

,p (x)

electrons Te Pe−ph

dx Tb

w

,e (x

+ dx)

Pdi

,p (x

+ dx)

Pph−e phonons Tp

Pesc Tb

L

Pdi

substrate Tb

Figure 7.2. Schematic representation of the model. (a) To model inhomogeneities, one can let the critical temperature, heat transfer coefficients, or resistivity vary as a function of position in the wire. (b) In each slice of the wire, the electron and phonon system are assumed to have a well-defined temperature. Energy enters the electron system due to Ohmic dissipation in the normal regions. Heat diffusion along the wire occurs both in the electron and in the phonon system. Heat is exchanged between the electrons and the phonons, and between the phonons and the substrate.

7.4

Thermal model with randomly fluctuating parameters

We assume that the transition occurs by the formation of expanding superconducting domains, with the observed resistance given by the backscattering resistance in the remaining normal part of the wire. In principle, for a driven system consisting of superconducting and normal domains one would have to include current conversion processes at the interfaces. Since we are interested in the possible influence of random electronic inhomogeneities, we use the much more tractable thermal model of the superconducting wire introduced by Skocpol, Beasley, and Tinkham [11]. In Fig. 7.2 we illustrate our model. The bulk contacts are assumed to be in the superconducting state and both the electron- and phonon-temperature are taken to be at the bath temperature of 4.2 K at the contacts (at x = 0 and x = L in Fig. 7.2(a)). The heat flow model in each segment of length dx is shown in Fig. 7.2(b). We assume that in each slice of the wire the electrons and phonons each have a thermal distribution, with temperatures Te and Tp respectively. The wire is biased with a constant current I, and has a normal state resistance Rn . We also assume the temperature is constant over the width of the wire. The electron system looses heat by diffusion along the wire and by electron-phonon interaction. The phonon system looses energy by phonondiffusion along the wire and escape of phonons from the film to the substrate. 93


7. Possible Indications of Electronic Inhomogeneities in Superconducting Nanowire...

We write a static heat balance equation for both the electron and phonon baths: PDC + Pdiff,e (x) − Pdiff,e (x + dx) +Pph−e − Pe−ph = 0

(7.1)

−Pesc + Pdiff,ph (x) − Pdiff,ph (x + dx)+ Pe−ph − Pph−e = 0

(7.2)

We describe the heat transport by electron and phonon diffusion with Fourier’s law: dTe wd, dx dTp wd, = −λph dx

Pdiff,e = −λe Pdiff,ph

(7.3) (7.4)

with wd the wire cross section, and λe and λph the electron and phonon heat transfer coefficients:  2 π 2 kB σTe , (7.5) λe = 3 e (7.6) λph = cph Dph , with σ the normal state conductivity, cph the phonon heat capacity, and Dph the phonon diffusion coefficient. Assuming two-dimensional phonons, the phonon heat capacity is given by  2 T , (7.7) cph ≈ 43.3Nmodes kB ΘD with Nmodes the number of phonon modes per unit volume, and ΘD the Debye temperature. The phonon diffusion coefficient is given by Dph =

1 ud, 2

(7.8)

with u the speed of sound in the material, and d the film thickness. The electron-phonon coupling is parameterized by a time constant τe−ph . The energy flows are given by ce Pe−ph = Te · w · d · dx, (7.9) τe−ph cph Tp · w · d · dx, (7.10) Pph−e = τph−e 2

with ce = π3 ( keB )2 Dσe Te the electronic heat capacity, estimated with the Fermi free electron gas model, with De the electronic diffusion constant. Phonon 94


7.4 Thermal model with randomly fluctuating parameters

escape to the substrate, which is assumed to be at the bath temperature, is also parameterized by an escape time τesc : Pesc =

cph (Tp − Tb ) w · d · dx. τesc

(7.11)

In this straightforward model, the crucial part is the electron temperature and electronic heat flow. Assuming that electronic inhomogeneities play a role, it may lead to random variations in Tc . However, we find that the resistive transition R(T ) curves of the wires are very sharp (inset Fig. 7.1). An alternative source of fluctuations might be contained in the electronic heat conductivity, which would then also be present in the resistivity. In principle, the superconducting state is insensitive to elastic scattering. Variations in elastic scattering can easily be tolerated, while maintaining a uniform Tc [12]. However, in these strongly disordered films the superconducting properties do depend on disorder and a variation of Tc with position is possible [6]. For simplicity, we have chosen to model the intrinsic electronic inhomogeneities by letting the critical temperature vary randomly along the length of the wire. At points at which the electron temperature exceeds the local critical temperature, the wire is considered to be in the normal state and Ohmic dissipation is present. At points at which the electron temperature is lower than the critical temperature, the wire is considered to be in the superconducting state and no dissipation is present:  I 2 Rn dx L , if Te (x) > Tc (x) PDC = (7.12) 0, if Te (x) ≤ Tc (x) Several parameters play a role but many are known by design (the geometric parameters w, d, and L), or can be measured independently (De , Rn , σ). The following parameters were estimated from literature values for thin NbN films and applied to our NbTiN films: • τe−ph . We use an empirical relation τe−ph ≈ 500Te−1.6 ps, that was reported for NbN thin films [13]. We use the value at Tc and ignore the slight temperature dependence. • τesc . The value reported for 3.5 nm NbN on sapphire is τesc = 38 ps [14]. • Nmodes . This value is taken equal to the atomic density, for NbN this is Nmodes = 4.8 · 1028 m−3 [15]. 3D • ΘD . The two-dimensional Debye temperature is Θ2D D ≈ 0.91ΘD [16]. 3D The value of the latter is ΘD = 250 K [15].

• u = 2.3 km/s for NbN [16]. 95


7. Possible Indications of Electronic Inhomogeneities in Superconducting Nanowire...

Te (K)

(a) 15

10

5

Te (K)

(b) 15

10

5 0

5

10

x ( m)

15

20

Figure 7.3. Electron temperature profiles along a wire for decreasing current-bias from 122 (red) to 61 μA (blue). The black line represents the randomly varying critical temperature along the wire. The stepwise transition from normal state to superconducting state can be seen from the temperature profiles in which only a part of the wire has an electrontemperature above the critical temperature.

7.5

Simulated characteristics

In Fig. 7.3, we show two sets of electron temperature profiles for a 20 μm long, 1 μm wide, 8 nm thick, NbTiN wire, simulated using the model described in section 7.4. The black line represents the critical temperature, which varies randomly around the experimentally determined value Tc,avg = 9.5 K. The used parameters are ΔTc = 0.5 K, where Tc (x) = Tc,avg + χ · ΔTc , with χ a random number between − 12 and 12 . The length of the Tc regions of the random pattern is 500 nm, a value arbitrarily chosen but suggested by the experiments by Sacepe et al. [5]. In the two panels of Fig. 7.3 a different random pattern with the same parameters is chosen, clearly leading to a different approach 96


7.6 Conclusions

to the superconducting state. The bias current is decreased from 122 μA, at which the entire wire is normal. As the current is lowered, the electron temperature in the wire decreases smoothly, until a stepwise jump occurs in which the superconducting regions at the boundary increase to a new stable position. This continues until finally the entire wire is in the superconducting state. The phonon-temperature follows the same profile as the electron-temperature. In the normal region it lies between 5.9 and 6.3 K for the currents shown. Because of the low electronic diffusion constant for our films, D ≈ 1 cm2 /s, and strong electron-phonon coupling, the energy flow is dominated by energy transfer to the substrate via the phonon system. In other words, the size of the normal regions is always much larger than the thermal healing length in the wire. Therefore, under current-bias, the electron- and phonon-temperatures are constant over the normal region and their value is independent of the size of the normal region. Fig. 7.4 shows the current-voltage characteristics which results from the electron-temperature profiles in Fig. 7.3. Elements of the wire where the electron temperature is above the critical temperature are taken to be in the normal state, with a resistance, whereas elements of the wire where the electron temperature is below the critical temperature are taken to be in the superconducting state. Jumps in the transition from the normal state to the superconducting state can be seen around 95 μA. The precise value of the retrapping current differs from the experimentally observed value of ∼ 70 μA. However, we are not aiming for a quantitatively accurate prediction of the retrapping current, since a number of thermal parameters are not well-known. We are aiming for possible contributions to the observed steplike features. The exact shape depends on the particular random inhomogeneity pattern of the sample. Steps in the current-voltage characteristics are also reported in [17], where they are ascribed to phase slip centers. These current-voltage characteristics are however qualitatively different from the ones we observe. In [17], the stepwise transition occurs mainly in the increasing current branch, in which we observe an immediate transition from the superconducting- to the normal-state.

7.6

Conclusions

We observe a persistent stepwise transition from the normal to the superconducting state under decreasing current-bias in NbTiN wires, for all wire geometries. It is argued that the stepwise pattern may be due to disorder fluctuations or electronic inhomogeneities intrinsic to disordered superconductors, modeled by local variations in the critical temperature.

97


7. Possible Indications of Electronic Inhomogeneities in Superconducting Nanowire...

V (mV)

1000

Fig. 3(a) Fig. 3(b)

500

0 85

90

95

I ( A)

100

105

Figure 7.4. Current-voltage characteristics deduced from the simulated electron temperature profiles in Fig. 7.3. Steps in the transition from the normal to the superconducting state are seen for decreasing currentbias.

References [1] C. M. Natarajan, M. G. Tanner, and R. H. Hadfield, Superconducting nanowire single-photon detectors: physics and applications, Supercond. Sci. Technol. 25, 063001 (2012). [2] A. J. Kerman, E. A. Dauler, J. K. W. Yang, K. M. Rosfjord, V. Anant, K. K. Berggren, G. N. Gol’tsman, and B. M. Voronov, Constriction-limited detection efficiency of superconducting nanowire single-photon detectors, Appl. Phys. Lett. 90, 101110 (2007). [3] M. Rosticher, F. R. Ladan, J. P. Maneval, S. N. Dorenbos, T. Zijlstra, T. M. Klapwijk, V. Zwiller, A. Lupa¸scu, and G. Nogues, A high efficiency superconducting nanowire single electron detector, Appl. Phys. Lett. 97, 183106 (2010). [4] A. Ghosal, M. Randeria, and N. Trivedi, Role of spatial amplitude fluctuations in highly disordered s-wave superconductors, Phys. Rev. Lett. 81, 3940 (1998). [5] B. Sac´ep´e, C. Chapelier, T. I. Baturina, V. M. Vinokur, M. R. Baklanov, and M. Sanquer, Disorder-induced inhomogeneities of the superconducting state close to the superconductor-insulator transition, Phys. Rev. Lett. 101, 157006 (2008). [6] E. F. C. Driessen, P. C. J. J. Coumou, R. R. Tromp, P. J. de Visser, and T. M. Klapwijk, Strongly disordered TiN and NbTiN s-wave superconduc98


7.6 References

[7]

[8]

[9]

[10]

[11] [12] [13]

[14]

[15]

[16] [17]

tors probed by microwave electrodynamics, Phys. Rev. Lett. 109, 107003 (2012). M. V. Feigel’man and M. A. Skvortsov, Universal broadening of the Bardeen-Cooper-Schrieffer coherence peak of disordered superconducting films, Phys. Rev. Lett. 109, 147002 (2012). H. L. Hortensius, E. F. C. Driessen, T. M. Klapwijk, K. K. Berggren, and J. R. Clem, Critical-current reduction in thin superconducting wires due to current crowding, Appl. Phys. Lett. 100, 182602 (2012). N. Vercruyssen, T. G. A. Verhagen, M. G. Flokstra, J. P. Pekola, and T. M. Klapwijk, Evanescent states and nonequilibrium in driven superconducting nanowires, Phys. Rev. B 85, 224503 (2012). Y. Noat, T. Cren, C. Brun, F. Debontridder, V. Cherkez, K. Ilin, M. Siegel, A. Semenov, H. W. H¨ ubers, and D. Roditchev, Break-up of long-range coherence due to phase fluctuations in ultrathin superconducting NbN films, Phys. Rev. B 88, 014503 (2013). W. J. Skocpol, M. R. Beasly, and M. Tinkham, Self-heating hotspots in superconducting thin-film microbridges, J. Appl. Phys. 45, 4054 (1974). P. W. Anderson, Theory of dirty superconductors, J. Phys. Chem. Solids 11, 26 (1959). Y. P. Gousev, G. N. Gol’tsman, A. D. Semenov, E. M. Gershenzon, R. S. Nebosis, M. A. Heusinger, and K. F. Renk, Broadband ultrafast superconducting NbN detector for electromagnetic radiation, J. Appl. Phys. 75, 3695 (1994). K. S. Il’in, M. Lindgren, M. Currie, A. D. Semenov, G. N. Gol’tsman, R. Sobolewski, S. I. Cherednichenko, and E. M. Gershenzon, Picosecond hot-electron energy relaxation in NbN superconducting photodetectors, Appl. Phys. Lett. 76, 2752 (2000). A. D. Semenov, R. S. Nebosis, Y. P. Gousev, M. A. Heusinger, and K. F. Renk, Analysis of the nonequilibrium photoresponse of superconducting films to pulsed radiation by use of a two-temperature model, Phys. Rev. B 52, 581 (1995). R. Barends, Master’s thesis, TU Delft, 2004. A. K. Elmurodov, F. M. Peeters, D. Y. Vodolazov, S. Michotte, S. Adam, F. de Menten de Horne, L. Piraux, D. Lucot, and D. Mailly, Phase-slip phenomena in NbN superconducting nanowires with leads, Phys. Rev. B 78, 214519 (2008).

99


7. Possible Indications of Electronic Inhomogeneities in Superconducting Nanowire...

100


Chapter 8 EďŹ&#x20AC;ect of spatial variations of the superconducting properties on common observables. The insensitivity of superconductivity to local disorder, known as Andersonâ&#x20AC;&#x2122;s theorem, is deeply rooted in the understanding of superconductivity. As a consequence the materials control needed for commonly used superconductors is embarrassingly simple compared to the one needed for semiconductors. The recently found, unavoidable, spatial variation in strongly disordered superconductors has changed this view. In this chapter we address three commonly used experiments to characterize a superconducting ďŹ lm in order to assess where and to what extent these spatial variations become part of the observations.

101


8. Effect of spatial variations of the superconducting properties on common observables.

8.1

Introduction

For many years, the general consensus in conventional superconductivity has been that the superconducting state is robust against underlying microscopic disorder of the material, as discussed originally by P.W. Anderson [1]. Polycrystallinity and variations in normal state resistivity have hardly any effect on the important superconducting properties such as the critical temperature or the energy gap. An exception is formed by the characteristic lengths such as the penetration depth and the coherence length, which both depend on the elastic mean free path. In addition for Type-II superconductors, used in a high magnetic field, the resistivity emerges due to vortex movement. Maintaining zero resistance means avoiding vortex movement, which is done by pinning of the vortices. Defects in the film are engineered in order to provide the needed pinning sites [2]. In the case of very strong disorder however, interesting new observations are made. There is increasing evidence that spatial variations of superconducting gaps appear (Sac´ep´e et al. [3]), not as a consequence of poor materials processing, but due to the competition between localization, increased electronelectron interaction and superconductivity. In the past decade it has been shown that effects on the macroscopic superconducting state can be strong, both theoretically [4, 5] and experimentally [3, 6, 7]. In Chapter 7 we modeled the influence that a position-dependent critical temperature has on the retrapping current-voltage characteristics of a onedimensional wire. The focus in this chapter is on the commonly observed parameters like the resistive transition, the critical current and the voltagecarrying state of a superconducting film in cases of very strong disorder. All three would normally not depend on elastic scattering, but we will address cases where they do, suggested by the experiments reported in the previous chapter.

8.2

Atomic-layer-deposited-TiN nanowires

We study Titanium-Nitride films deposited using Atomic Layer Deposition (ALD). This technique has been introduced for the use of thin TiN as diffusion barriers in the semiconductor industry. It is ideally suited for the fabrication of smooth very thin films (details of the deposition can be found in Ref. [8]). Excellent control over the thickness of the film can be achieved, since the growth occurs in well-controlled self-limiting cycles. By controlling the thickness of the film the resistance per square changes. This causes a reduction in diffusivity due to strong localization and an enhancement of the electron-electron interaction, both of which compete with the attractive superconducting interaction. For TiN, one can conveniently tune the system through 102


8.2 Atomic-layer-deposited-TiN nanowires

Figure 8.1. False-color scanning electron microscope image of sample D20. The gold-colored region indicates the gold contact wiring and the blue the TiN wire. The overlap of the gold on the TiN can be seen. The TiN-region gradually narrows into a 100 nm wide, 20 μm long wire.

the so-called superconducting-insulator transition. Although initially the focus was on macroscopic properties like the resistivity as a function of temperature and magnetic field, in recent years the focus has shifted to the local properties. The tuning of the normal state resistivity appears accompanied by an increase in the density and amplitude of electronic inhomogeneities. It is important to underline that the inhomogeneities appear within the electron system, the atomic arrangement can be fully homogeneous. In this chapter we study three films, with thicknesses of 11, 22, and 45 nm. These correspond to films B, C, and D in the work by Driessen et al. and Coumou et al. [7, 8], where the electrodynamics of these films were reported, studied by using microwave resonators. Figure 8.1 shows the geometry used for the wires. All wires are 100 nm wide and have a length of 10, 20, or 100 μm. The thickness d of the films ranges from 11 to 45 nm and the width of all wires is 100 nm. The wire width gradually increases at the contacts to avoid any current crowding effects, as discussed in Chapter 6. Two gold contacts are available on either side of the wire, to enable four-point probe measurements. The dimensions of the wires studied in this chapter are given in table 8.1. The resistances of the wires, measured at 4.2 K using a four-point probe lock-in technique, are also given in Table 8.1. The relative contribution to the 103


8. Effect of spatial variations of the superconducting properties on common observables.

Table 8.1. Dimensions of the TiN nanowires, all wires are 100 nm wide. R4.2K denotes the resistance measured for the samples, including the gradually widening contact regions, at 4.2 Kelvin. Rwire denotes the resistance of the 100 nm wide part of the wire with length L, corrected for the resistance of the widening contact regions.

Sample

L (μm)

d (nm)

R4.2K (kΩ)

Rwire (kΩ)

ρ (μΩcm)

B10 B20 C20 D20 D100

10 20 20 20 100

11 11 22 45 45

43.2 83.1 30.0 9.0 47.8

34.6 73.9 26.7 8.0 46.6

381 407 294 180 210

measured resistance of the widening region of the contact pads is calculated using SONNET. We find that the normal state resistance of the widening region corresponds to an additional straight wire segment of length 2.25 μm and width 100 nm. In the column Rwire , the resistances of only the straight wire parts are given, corrected for the resistance of the widening regions. We use that resistance to calculate the resistivity of the films and find similar values as reported for these films in Ref. [7]. The superconducting coherence length ξ0 for these films is ∼5 nm [9]. We use the temperature dependence of the coherence length for a dirty superconductor close to the critical temperature as given by Tinkham [2]: ξ(T ) = √ 0.855 √ ξ0 l . Clearly, when we approach the critical temperature, the wires 1−T /TC

cross over from a three-dimensional limit to lower dimensionality, taking the relation of the coherence length to the geometry as a measure for dimensionality.

8.3

The resistive transition

Ever since the discovery of superconductivity by Kamerlingh Onnes, the first and most basic measurement on a superconducting sample is the temperature dependence of the resistance. It is used to infer the critical temperature of the material, which is influenced by for instance the enhanced Coulomb interaction [10] and changes in the density of states [7] in highly disordered superconductors. Moreover, the temperature dependence of the resistance also traces how the conductivity gets gradually enhanced upon approaching the critical temperature from the high temperature side. Temporal superconducting fluctuations are assumed consisting roughly of emerging and decaying ’blobs of 104


8.3 The resistive transition

superconductivity’. They suppress the resistance of the superconductor up to temperatures well above the critical temperature of the material [11–13]. In addition coming from lower temperatures the resistance gradually emerges. In one dimension these processes are described by thermally activated phase slips at the low end of the R(T ) curves [2]. In two-dimensional samples, the phase-slip process requires a different treatment. It is understood that a true superconducting state with long-range order does not exist. Instead the groundstate consist of paired topological excitations, vortex-anti-vortex pairs. With increasing temperature some of these pairs unbind and free vortices become available, which when moving under the influence of a transport current contribute to the resistance. With increasing resistivity the temperature at which these vortices unbind, the TBKT , is pushed to lower temperatures [14, 15]. Because of this richness, the determination of the critical temperature of a highly disordered superconductor is a complicated problem [16, 17]. In the analysis to date it has usually been assumed that the underlying superconducting properties such as the TC and the energy gap Δ are uniform throughout the film. So far, spatial variations of superconductivity have only been taken into account in the context of a broadening of the Berezinski-Kosterlitz-Thouless transition [17]. If spatial electronic inhomogeneities occur in these homogeneously disordered materials, we might expect the critical temperature of the film itself to become a function of position. We therefore start our experimental exploration of the consequences of spatial variations of superconducting parameters with a look at the superconducting transition as a function of temperature. Measurements are done using a He-3 sorption cooler mounted in a liquid helium cryostat, with the sample space surrounded by a superconducting Pb/Sn magnetic shield. We measure the zero-bias resistance of the wires with a four-point lock-in technique using a small (1 nA) oscillating current-bias. The temperature of the sample stage is swept during these measurements. We measure both for increasing and decreasing temperature to ensure proper thermalization of the sample, seen by the lack of hysteresis in the resistance versus temperature graphs. Fig. 8.2 shows the temperature dependence of the low-bias resistance of the three 20 μm-long wires with increasing disorder from D to B. Clearly, the critical temperature of the wire is strongly dependent on the amount of disorder, as reported previously in Ref. [7]. This is a sign that the nature of the disorder influencing the superconducting parameters, is homogeneous instead of granular [18]. We follow the definition of the critical temperature as the temperature at which the wire resistance of the wire is 90 % of its normal state resistance, from Ref. [7]. The critical temperatures of the nanowires determined in this way match with those obtained for the films previously, as shown in Table 8.2. 105


8. Effect of spatial variations of the superconducting properties on common observables.

75

B

R (kΩ)

50 C

25 D

0

2.0

2.5

3.0

T (K) Figure 8.2. The temperature dependence of the resistance of the three 20 μm-wires. The vertical lines denote the width of the superconducting transition, determined as the temperature difference between points at 60 % and 10 % of the normal resistance. The most disordered wire (B) has the widest superconducting transition.

Table 8.2. The critical temperature determined from the resistive transition (TC,90% ), determined from a fit to Aslamazov-Larkin theory of superconducting fluctuations (TC,AL ), and determined from the temperature dependence of the critical current (TC,IC ). ΔTC denotes the width of the resistive transition. Increasing disorder in the film leads to: a lower critical temperature, a wider resistive transition, and a larger spread between the critical temperatures determined using the different methods.

106

Sample

TC,90% (K)

ΔTC (K)

TC,AL (K)

TC,IC (K)

B10 B20 C20 D20 D100

2.2 2.2 2.7 3.1 3.1

0.07 0.09 0.06 0.03 0.03

2.0 2.0 2.5 3.05 3.05

1.85 1.85 2.4 2.95 2.95


8.3 The resistive transition

4 C 16.4 kΩ

3

-1 -1

(Rsq - RN ) (kΩ)

B 20.2 kΩ

-1

2

D 13.5 kΩ

1

0

2.0

2.5

3.0

3.5

T (K) Figure 8.3. Reduced resistance as a function of temperature for the three 20 μm-wires. Temporal superconducting fluctuations would give a linear dependence on temperature close to TC,AL with slope RAL /TC,AL [11]. We find a factor 3 to 5 stronger conductivity enhancement than expected from Aslamazov-Larkin.

Temporal superconducting fluctuations above the critical temperature of a superconductor, can lead to an excess conductivity in the normal state. These fluctuations are known to persist up to temperatures well above the critical temperature. This effect is described for a two-dimensional film (d < ξ < w, L) by the theory of Aslamazov and Larkin [11]. The superconducting fluctuations lead to an additional term in the conductivity of the material: σtot (T ) = σN + σAL (T ) T e2 · σAL (T ) = , 16d T − TC,AL

(8.1) (8.2)

with σtot (T ) the measured conductivity, σN the normal state conductivity, and σAL (T ) the superconducting fluctuation contribution. If we define RAL = 16 e2 ≈ 65 kΩ, and look at the resistances per square to take out geometry dependencies, we arrive at the following equation for the ’reduced resistance’: (

1 1 −1 T − TC,AL · RAL − ) = Rsq RN,sq T

(8.3)

Fig. 8.3 shows the reduced resistance of the 20 μm wires versus temperature. If the excess conductance of the wires above the critical temperature is caused by temporal superconducting fluctuations, as described by Aslamazov and Larkin [11], close to the critical temperature this data should follow 107


8. Effect of spatial variations of the superconducting properties on common observables.

8

4

w = 400 nm

2.5

2.0

1.5 0

100

200 300 w (nm)

400

-1

-1 -1

(Rsq - RN ) (kΩ)

6

AL prefactor

3.0

w = 50 nm

2

0 10.0

10.5

11.0

11.5

12.0

T (K)

Figure 8.4. Reduced resistance as a function of temperature for NbTiN-wires of varying width. We find a factor 1.7 to 2.8 stronger conductivity enhancement than expected from Aslamazov-Larkin.

a linear dependence on temperature starting at TC,AL and rising with a slope given by RAL /TC,AL [19]. Additional terms for temporal fluctuations can be considered, such as proposed by Maki [12] and Thompson [13]. These terms however rely on the remaining coherence between quasiparticles after a superconducting fluctuation and are expected to be negligible in dirty superconductors [20]. Nevertheless, for thin TiN films a full fitting procedure including the Aslamazov-Larkin, Maki-Thompson, and other terms, seems to describe the resistive transition well [16, 21]. As seen in Fig. 8.3, we do indeed find a linear dependence of the reduced resistance on temperature close to TC . The critical temperature determined using this method (TC,AL ) is significantly lower than that determined using the 90%-criterium. The slope ranges from 20.2 kΩ/K for film B to 13.5 kΩ/K for film D. We note that the excess conductivity above the critical temperature is a factor of 3 to 5 larger than that expected from the Aslamazov-Larkin theory for temporal superconducting fluctuations. As a comparison, we look at the resistive transition for NbTiN nanowires of different widths. These wires were produced from a ∼8 nm thin NbTiN film, that was deposited on a thermal silicon oxide layer using DC magnetron sputtering from a Nb0.7 Ti0.3 target. Apart from the substrate the deposition is the same as in Chapter 6. The nanowires were patterned using the same process. Note that no care was taken in these samples to avoid current-crowding at the contact area of the nanowires. The experiment was however performed 108


8.3 The resistive transition

at low bias current, and the resulting TC and Rsq do not depend on wire width, from which we conclude that for this measurement, current crowding does not play an important role. All wires were 10 μm long, and their width was varied from 50 nm to 400 nm. The resistive transition of these wires was measured using a lock-in technique. Fig. 8.4 shows the reduced resistance of the NbTiN wires from a width of 50 nm to 400 nm. All NbTiN wires have a resistive transition at the same temperature (TC,AL = 10.5 K) and a similar square resistance (Rsq ≈ 220 Ω), indicating good film uniformity. There is however a clear dependence of the reduced resistance on the width of the wire. The inset shows the prefactor needed for the Aslamazov-Larkin theory as a function of wire width. For the widest wires, the excess conductivity is about a factor 1.7 stronger than predicted, whereas for the narrowest wires, the excess conductivity is more than a factor of 2.5 stronger than predicted. Whereas the increase of excess conductivity for decreasing wire width could be explained as a transition going from two-dimensional wires to one-dimensional wires, the fact remains that even for the widest wires the reduced resistance is a factor of 1.7 smaller than expected from Aslamazov-Larkin theory. These observations lead to the conclusion that both size effects and effects of the strong disorder play an important role in the fluctuation conductivity around the critical temperature. Fig. 8.5 shows the critical temperatures determined using three different methods for the three films as a function of disorder. The IC -data refers to the critical temperature determined from the temperature dependence of the critical current, as will be discussed in Section 8.4. Independently of the choice of definition of the critical temperature, it is clear that the critical temperature of the TiN-films decreases as the sheet resistance increases. The spread in critical temperatures determined from the 90%-criterium in the temperature dependence of the resistance, the fit to superconducting fluctuation enhanced conductivity above the critical temperature, and that determined from the temperature dependence of the critical current, also becomes larger as the disorder is increased. We believe this to be a sign that different measurement techniques probe the temporal fluctuations and/or spatial variations of superconductivity in strongly-disordered materials in a different way. We now turn our attention to the width of the superconducting transition as indicated by the vertical lines in Fig. 8.2. The width of the superconducting transition is taken as the difference in temperature between the points where the resistance of the wire is 10 % and 60 % of the resistance at 4.2 K. These values are chosen to probe the steepest part of the superconducting transition, which will be the most sensitive to possible spatial variations of the superconducting temperature. At lower temperatures, the resistance is dominated by thermallyassisted phase-slips, while at higher temperatures temporal fluctuations of the superconductivity are expected to dominate. These values are given as ΔTC in 109


8. Effect of spatial variations of the superconducting properties on common observables.

90% Aslamazov-Larkin IC

TC (K)

3.0

2.5

2.0 0

100

200

300

400

Rsq (Ω) Figure 8.5. The critical temperature as a function of film disorder, determined using the 90 % of the normal state resistance criterium, the fit of superconducting fluctuations in the normal state using AslamazovLarkin [11], and the critical temperature determined from the temperature dependence of the critical current using Kupriyanov-Lukichev [22]. The spread between these values increases as the resistivity of the film increases.

0.05

ΔTC / TC

0.04

0.03

0.02

0.01

0.00

0

100

200

300

400

500

Rsq (Ω) Figure 8.6. Relative width of the superconducting transition as a function of film disorder. The transition is broader for the more disordered wires.

110


8.4 The critical current

table 8.2 and plotted relative to the critical temperature in Fig. 8.6. We clearly observe a wider superconducting transition for the more disordered nanowires. The presence of a spatially varying critical temperature due to disorder-induced electronic inhomogeneities, which would be stronger in the wires with more disorder and higher resistivity, agrees with this observation.

8.4

The critical current

The critical current of a superconducting nanowire is a critical parameter for applications such as single photon detectors or parametric amplifiers. In these applications, a higher critical current leads directly to enhanced device operation [23, 24]. Moreover, as shown in Chapter 6, the statistics of the critical current measurements as a function of temperature, can give an indication of the presence of a weakest spot in a superconducting wire. In contrast to the temperature dependence of the resistance and the retrapping characteristics, the critical current of a superconducting wire is generally sensitive only to it’s weakest point. We measure the critical current in the same way as in Chapter 6. We ramp the current through the wire from zero to slightly above the base critical current of a wire in 200 ms and record the current at which the voltage over the wire jumps to the normal state. 1000 critical currents are measured at each temperature. Due to the high resistivity of the materials, we used an idle period of 400 ms after each current ramp to let the wire return completely to the bath temperature. Close to the critical temperature, the temperature dependence of the critical current is given by the Ginzburg-Landau depairing current: jc = jc (0) · (1 − T /TC )3/2 [2]. To compute the critical current over the full temperature range, one has to solve the Eilenberger equations. We follow the approach of Kupriyanov and Lukichev [22] to calculate the temperature dependence of the critical current, following Romijn et al. [25]. The temperature dependence of the critical current of a wire is described fully by a critical temperature TC,IC , and a scaling factor IC,0 : IC,0

√  8π 2 2π (kB Tc )3 w , = 21ζ(3)e 3D Rsq

(8.4)

with ζ Riemann’s zeta function, e the electron charge, kB Boltzmann’s constant, D the diffusion constant, and Rsq the normal-state square resistance of the wire. Figure 8.7 shows the measured critical current densities as a function of temperature. Films B and D have data for two different wires with different lengths, the longer wire giving the lower critical current. The critical temperature TC,IC deduced from the temperature dependence of the critical current 111


8. Effect of spatial variations of the superconducting properties on common observables.

D: 20 μm D: 100 μm C: 20 μm B: 10 μm B: 20 μm

6

6

4.0x10

j

2/3

(A/m )

2 2/3

6.0x10

6

2.0x10

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

T (K) Figure 8.7. Critical current densities, with fits of KupriyanovLukichev over the whole temperature range [22, 25]. Films B and D have data for two different wires with different lengths, the longer wire giving the lower critical current.

however, does not change between wires from the same film. The measured data can be well described by the Kupriyanov-Lukichev temperature dependence. The critical temperatures determined from the fits in Fig. 8.7, were discussed in Section 8.3. They are consistently smaller than the critical temperature determined from the temperature dependence of the zero-bias resistance, via either the 90%-criterium or the Aslamazov-Larkin theory. This is consistent with a weakest spot in a wire with spatially varying strength of superconductivity. The values of the diffusion constant D determined from the temperature dependence of the critical current are given in Table 8.3. The diffusion constant has also been determined from the temperature dependence of the magnetic critical field [9]. The diffusion constant from the magnetic field measurements for film B was found to be 0.36 cm2 /s. The diffusion constant was not determined for films C and D, but a slightly thicker (55 nm) film from the same fabrication cycle was determined to have a diffusion constant of 0.94 cm2 /s. The diffusion constants determined from the critical currents seem to be slightly higher. This might be due to the fact that the critical current probes the weakest spot of the wire, while the magnetic field measurements of wide strips probe averaged values. Figure 8.8 shows the width of the critical current distribution as a function of temperature for all five wires. As in Chapter 6, the width of the critical 112


8.5 The voltage-carrying state

Table 8.3. The base-temperature critical currents of the samples and the diffusion constant D determined from the Kupriyanov-Lukichev fits in Fig. 8.7.

Sample

IC,350mK (μA)

D (cm2 /s)

B10 B20 C20 D20 D100

5.3 4.5 18.3 76.1 58.4

0.50 0.62 0.64 0.77 0.99

current distribution is defined as the difference between the third and the first quartile of the collection of 1000 measured critical currents at each point. Even though the value of the critical current differs for two wires from the same film with different length, the value of the width of the critical current distribution appears to be the same. We find that the width of the critical current distribution increases with increasing temperature, until it drops off close to the critical temperature. In the context of Chapter 6, this was interpreted as a weak spot in the wire at which the switch from the superconducting to the normal state preferentially occurs. Both the apparent presence of a weakest link for the critical current in the geometrically unconstrained nanowires, as well as the consistently lower critical temperature determined from the temperature dependence of the critical current than that determined from the temperature dependence of the resistance, are consistent with the hypothesis of spatially varying superconducting parameters. Once the wire switches to the normal state in a single point, thermal runaway will cause the entire wire to switch. The weakest section of the random distribution will form the weakest link. The local critical current at that spot will determine the critical current, whereas the temperature dependence of the resistance probes a spatially averaged result.

8.5

The voltage-carrying state

In Chapter 7, we found that the details of the current-voltage characteristics of a strongly-disordered superconducting wire gives possible indications of intrinsic spatial variations of the superconducting parameters. We now study these characteristics for the TiN-wires with varying disorder, to gain a further understanding into the role that disorder plays. We also add voltage-biased measurements of the resistive state of the wire and provide a simple extension 113


8. Effect of spatial variations of the superconducting properties on common observables.

0.15

ΔIC (μA)

D C B

0.10

0.05

0.00

0

1

2

3

T (K) Figure 8.8. Width of the critical current distribution at each temperature. The open symbols denote the 20 μm wires, the closed symbols denote the wires with varying length. The width increases with temperature, which in the current crowding data on NbTiN-wires in Chapter 6 was interpreted as an indication of a (geometrical) weakest link.

of the thermal model of Chapter 7 to the voltage-biased case. Current-voltage characteristic were measured at the base temperature of the cryostat of 310 mK. Current-biased measurements were done using a fourpoint measurement setup, where the voltage is probed at the two widening regions at the ends of the wire. For voltage-biased measurements, we use a two-probe technique and we are therefore also sensitive to the resistance of the cryostat wiring. When we start increasing the bias-current from zero, we stay in the superconducting state with zero voltage over the wire until the wire jumps to the normal state at the critical current. We discussed measurements of this critical current and the involved statistics in Section 8.4. Once the normal state is reached, Ohmic dissipation is present in the wire. This raises the electron temperature and causes hysteresis in the current-voltage characteristics. The wire stays in the normal state as the bias-current is lowered due to the self-heating of the wire, as modeled originally by Skocpol, Beasley, and Tinkham [26], and adapted for a spatially varying critical temperature by us in Chapter 7. At the retrapping current, the wire will no longer sustain a normal region and return to the superconducting state. The return from the self-heated normal state under lowering bias is shown 114


8.5 The voltage-carrying state

B

5

V (mV)

4

3

2

1

0 50

55

60

65

70

75

230

240

250

I (nA) C

V (mV)

6

4

2

0 200

210

220

I (nA) D

V (mV)

6

4

2

0 750

800

850

900

I (nA)

Figure 8.9. Stepwise transition from the normal to the superconducting state on the return branch of the current-voltage characteristics. The data for the three 20 Îźm TiN-wires is shown under current bias (black) and voltage bias (red). There is a unique pattern for a particular wire, which is reproducible also between current- and voltage-biased measurements and after heating above the critical temperature.

115


8. Effect of spatial variations of the superconducting properties on common observables.

in Fig. 8.9. Under current-bias (black dots), we see a similar stepwise transition on the return branch of the current from the normal to the superconducting state as in Chapter 7. We interpret this as a possible consequence of intrinsic electronic inhomogeneities in the highly disordered superconducting nanowires. All three wires show similar retrapping characteristics under current-bias, apart from the value of the retrapping current. The thicker, less resistive, wire switches back earlier to the superconducting state. This is expected in the self-heating model we employ, because of the lower current density and lower resistivity and therefore lower Ohmic dissipation. The red dots in Fig. 8.9 show the current-voltage characteristics measured under voltage-bias. In this case, we use a two-point measurement technique and the resistance of the cryostat wiring is corrected for by setting the voltage on the superconducting branch to zero. For the data on wire D, a small offset current of 6 nA was corrected for in the voltage-bias data. Both the currentand voltage-biased data have been measured multiple times for each wire and are completely reproducible, also after thermal cycling to temperatures above the critical temperature. There is an excellent agreement between the current-bias (black) and voltagebias (red) current-voltage characteristics, as shown in Fig. 8.9. The currentbiased retrapping characteristics follow the voltage-biased data when lowering the current. Under voltage-bias, we are able to probe the transition more completely. We see a large series of oscillations of the measured current through the wire around the retrapping current. In our model, when a part of the wire becomes superconducting under decreasing voltage-bias, the total resistance of the wire goes down and the current through the wire will increase. This leads to increased dissipation in the remaining normal part of the wire, until the next segment switches. Voltage-bias data should therefore give us a more complete picture of the disordered superconducting nanowires.

8.6

Qualitative model of inhomogeneity in a voltage-biased self-heating hotspot

To understand the relationship between inhomogeneous superconductivity and the erratic pattern of the return current in the voltage-biased current-voltage characteristic we develop the following picture. Our starting point is that a wire in the normal state under bias, clamped between two large pads, develops an electronic temperature profile (see previous Chapter). Since for TiN, like for NbTiN in Chapter 7, the diffusivity is small and the electron-phonon interaction time is relatively short the electron temperature is constant throughout the wire, dominated by the heat transfer from the electron system to the phonon-system. In the experiments by Sac´ep´e et al. [3], it was shown that 116


8.6 Qualitative model of inhomogeneity in a voltage-biased self-heating hotspot

Figure 8.10. Scanning electron microscope image of the 50 nm wide, 10 μm long, ∼8 nm thick NbTiN wire, current-voltage characteristics of which are shown in fig. 8.11.

for high resistance per square the superconducting film develops large patches with a higher or a lower energy gap, as revealed by local STM spectroscopy. These patches are in the 2-dimensional films of a scale of tens of nanometers, much larger than the coherence length and the size of the crystallites in the polycrystalline film. Unfortunately, similar STM measurements for a narrow wire are not available. However, we assume that the structure, which is observed in the currentvoltage characteristics, is related to these type of domains. Each domain can in the driven situation be in the normal state, leading to a finite voltage for a given current. Taking the normal state resistivity as a guideline, we estimate for the observed voltages, that the domains must have a size in the order of 1 μm. These domains are larger or of the same order as the thermal healing length in the homogeneous model. A similar domain size can be inferred from the current-voltage characteristics of NbTiN nanowires. Fig. 8.10 shows a scanning electron microscope image of a 50 nm wide, 10 μm long, ∼8 nm thick NbTiN wire. The fabrication process was the same as described in Chapter 6. Fig. 8.11 shows the return-branch of the current-voltage characteristics for this wire for temperatures from 5 K to 10 K. The steplike pattern in the transition from the normal to the superconducting state clearly persists over this range of temperatures. The inset shows the measured differential resistance at 5 K as a function of bias-current. From the change in resistance between the steps, we find for this 50 nm wide NbTiN nanowire a domain size of ∼2μm. Therefore, we assume that the wire can be considered to consist of a series 117


8. Effect of spatial variations of the superconducting properties on common observables.

T = 5K T = 10K

0.3

V (V)

0.2

dV/dI (kΩ)

60

0.1

30

0 0

5

10

I (μA)

0.0 0

2

4

6

8

10

I (μA) Figure 8.11. Return-branch of the current-voltage characteristics for the NbTiN nanowire shown in fig. 8.10 at a range of temperatures from 5 K to 10 K. The size of the major steps corresponds to a length scale for the regions of ∼2 μm. The inset shows the differential resistance at 5 K.

arrangement of pieces of superconductor, hereafter called ’segments’, with different Tc ’s. The Tc ’s are random over a certain bandwidth. Upon decreasing the voltage, starting from the normal state the dissipated power decreases, leading to a decrease in temperature of the wires, with some segments approaching the point where superconductivity will set in. However, the segments close to the contact pads, are on one side connected to the bath temperature, assuming the pads to be in equilibrium. Therefore, it is to be expected that one of these end-segments will be the first to become superconducting. This is analogous to what has been found in a more rigorous model analysis by Vercruyssen et al. [27] for a NSN-system. In this case two blobs of superconductivity appear at both ends of the S-wires, upon lowering of the bias voltage. In order to generate a current-voltage characteristic in the spirit of such a qualitative picture we proceed as shown in Fig. 8.12. The temperature of each segment is dominated by a local equilibrium between dissipation and heat transfer to the phonon-bath. Upon lowering the voltage the current carried by the segment decreases and for a certain value of the current the dissipation is low enough to allow the segment to become superconducting. Therefore we assign to each segment a randomly distributed minimum current. Since the resistance decreases, due to the addition of a segment which turns to the R = 0 118


8.6 Qualitative model of inhomogeneity in a voltage-biased self-heating hotspot

Tc,1 Tc,2

...

IR(Tc,1) IR(Tc,2)

Tc,N IR(Tc,N)

...

Figure 8.12. To model the voltage-biased current-voltage characteristics, we divide the wire in N sections. Each section has a randomized critical temperature, which corresponds to a local random retrapping current. We use the measurements of the resistive transition as input for the model. When the retrapping current of a section is reached, we let it switch from the normal to superconducting state. When a section switches, the total resistance of the wire decreases and the current through the wire increases.

state, the current at the given voltage will increase and the temperature will adjust itself accordingly. Now we have two new end-segments, one connected to a large pad, the other one to a piece of superconductor, which is part of the narrow wire. The latter has a temperature between the temperature of the pads and the other segments, below the critical temperature. Through this process, the segments will flip to the superconducting state on by one, implying reduced resistance and regularly occurring increases in current for decreasing voltage values. To estimate the minimum current under which a normal region is stable in our superconducting wires, we use the analytical results of the hotspot model of Skocpol, Beasley, and Tinkham [26]. This model is identical to the one used in Chapter 7, except for two facts. It assumes a uniform Tc and it assumes that the electron temperature is identical to the phonon-temperature of the superconducting metal. The heat transfer is limited by the Kapitza-resistance for phonons between the metal and the substrate and parameterized by the heat transfer coefficient α. Within this model they find a minimum current which generates sufficient heat to keep the temperature of the sample above the critical temperature [26]: 119


8. Effect of spatial variations of the superconducting properties on common observables.

I1 =

 αw2 d(TC − Tbath )/ρ

(8.5)

with w the width of the wire, d the thickness, TC the critical temperature, Tbath the bath-temperature (in fact the substrate-temperature), ρ the resistivity, and α the heat transfer coefficient per unit area. For  a wire of length L  η, where η is the thermal healing length (given by Kd/α) [26], with contacts at the bath temperature, the cooling at the contacts leads to a higher minimal current:  Im =

1+

KS I1 KN

(8.6)

with KS and KN the thermal conductivity in the superconducting and normal regions respectively. For the purposes of our model, we ignore the difference between these two thermal conductivities, since all processes occur close to the Tc ’s, where the differences are small. This leads to a minimum current of: Im =

2I1

(8.7)

In the SBT-model an important assumption is that the electron- and phonon temperature in the metal are the same. In many subsequent experiments this assumption was relaxed, sometimes making the electron-phonon processes in the metal the limiting process, leading to an electron temperature higher than the phonon-temperature and the latter locked to the substrate-temperature. We use a model, which allows for both the phonon-temperature and the electron temperature being elevated compared to the substrate temperature, as well as heat conduction provided by both electron-diffusion and phonon-diffusion. In practice, given the parameters for our samples, we find that the electronic processes dominate. In our TiN samples, the bottleneck in the heat transfer to the substrate is then between the electrons and the phonons of the metal. The parameter α in Equation 8.5 can be replaced by [28]: ce d (8.8) α= τe,ph with ce the electronic heat capacity and τe,ph the electron-phonon scattering time.With these assumptions the thermal healing length is given by [28]:  (8.9) η = Dτe,ph The electronic heat capacity is estimated from the Fermi free electron gas model [29], for T = TC : ce = 120

LTC ρD

(8.10)


8.6 Qualitative model of inhomogeneity in a voltage-biased self-heating hotspot

Table 8.4.

Sample

ρ (μΩcm)

TC,AL (K)

ΔTC (K)

D (cm2 /s)

B20 C20 D20

407 294 180

2.0 2.5 3.05

0.09 0.06 0.03

0.62 0.64 0.77

2

with L = π3 ( keb )2 the Lorenz number, TC the critical temperature, ρ the resistivity, and D the diffusion constant. The electron-phonon scattering time is estimated using the theory for electronphonon scattering in dirty metals of A. Schmid [30]: τe,ph ≈

2 3 ωD (kB T )3 B

(8.11)

with ωD the Debye frequency and B a parameter describing the effect of the short mean free path in dirty metals. The Debye temperature for TiN ΘD ≈ 580 K [31]. For our films the disorder parameter kF l ≈ 5, with kF the Fermi wave vector and l the elastic mean free path [7]. At the critical temperature, following the work of Schmid, this leads to a B ≈ 10. Table 8.5 shows the electron phonon scattering time, thermal healing length, minimum current for sustaining a stable normal region, determined using Eqs. 8.11, 8.9, and 8.5 and 8.7 respectively. The input parameters used, as previously measured in sections 8.2, 8.3, 8.4, are repeated in Table 8.4. The last column of Table 8.5 gives the bandwidth over which the minimum stable current varies (ΔIm ), given the width of the resistive transition (ΔTC ) determined in Section 8.3. The relatively large difference in minimum stable current between the different wires, originates from the difference in critical temperature, resistivity, and thickness of the wires. The minimum stable currents given in Table 8.5 are calculated for segments at the ends of the normal region of the wire, with a contact at√the bath temperature. From Eq. 8.7 it follows that this current is a factor of 2 larger than the minimum stable current in the center of the wire. The bandwidth over which the minimum stable current varies (ΔIm ) is always significantly smaller than the difference between the minimum current with cooling at the contacts Im and the minimum current with heat flow only to the phonons I1 . Therefore, we expect the normal region to shrink from the contacts under decreasing bias, even though our wires may contain electronic inhomogeneities. Fig. 8.13 shows the simulated voltage-biased current-voltage characteristics for decreasing voltage-bias, going from a biased wire in the normal state to a wire completely in the superconducting state. The curve is simulated by 121


8. Effect of spatial variations of the superconducting properties on common observables.

Table 8.5.

Sample

τe,ph (ns)

η (μm)

Im (nA)

ΔIm (nA)

B20 C20 D20

20 14 7.5

1.1 0.9 0.8

85 411 2087

10 26 52

20

V (mV)

15

10

5

0

0

100

200

300

I (nA) Figure 8.13. Simulated voltage-biased current-voltage characteristics for wire B20. An increase of the current at a given bias voltage can be seen each time an end-segment of the wire switches from the normal to the superconducting state and the total resistance of the wire is decreased.

considering a one-dimensional wire of 20 segments, as schematically shown in fig. 8.12. Each segment has a randomly generated minimum stable current within a bandwidth ΔIm around Im as given in Table 8.5. When the minimum stable current of one of the two end-segments is reached, the resistance of the wire is reduced accordingly and the current increases. The decrease of bias voltage is continued for the remaining normal region of N − 1 segments, until the minimum stable current of one of the new end-segments is reached. This process continues until all the segments have switched to the zero-resistance state. There is a qualitative agreement between the simulated voltage-biased current122


8.7 Conclusions

voltage characteristics in Fig. 8.13 and the measured data of Fig. 8.9. The random increases in current under decreasing voltage-bias are reproduced both in simulation and experiment. In contrast to the simulation, experimentally the slopes of the steps do not extrapolate to the origin. This indicates that in reality the size of the normal region is not completely constant even before the next large segment switches. The quantitative disagreement between the calculated and measured minimum stable current is not surprising given the uncertainty in the used model parameters.

8.7

Conclusions

The deeply rooted assumption in superconductivity that superconducting parameters are insensitive to material disorder [1], underlies many commonly accepted predictions for experimental observables. In strongly-disordered superconductors however, the superconducting parameters are known to be sensitive to the resistivity, which itself is a local parameter. We ďŹ nd multiple experimental indications of position dependent electronic properties in onedimensional wires of titanium nitride. The steplike retrapping characteristics in both current- and voltage-bias, are consistent with a thermal balance in the return-branch of the current-voltage characteristics, where the electronic properties of the wire vary as a function of position. The more disordered ďŹ lms show a wider transition region in the resistance as a function of temperature, consistent with a wider distribution of critical temperatures through the wire. Finally, the width of the critical current distribution increases with increasing temperature, which is consistent with a weak-spot in the wire. Invalidity of Andersonâ&#x20AC;&#x2122;s theorem in strongly-disordered superconductors necessitates a critical re-evaluation of most commonly used experiments.

123


8. Effect of spatial variations of the superconducting properties on common observables.

References [1] P. W. Anderson, Theory of dirty superconductors, J. Phys. Chem. Solids 11, 26 (1959). [2] M. Tinkham, Introduction to superconductivity, McGraw-Hill, New York, (1975). [3] B. Sac´ep´e, C. Chapelier, T. I. Baturina, V. M. Vinokur, M. R. Baklanov, and M. Sanquer, Disorder-induced inhomogeneities of the superconducting state close to the superconductor-insulator transition, Phys. Rev. Lett. 101, 157006 (2008). [4] A. Ghosal, M. Randeria, and N. Trivedi, Role of spatial amplitude fluctuations in highly disordered s-wave superconductors, Phys. Rev. Lett. 81, 3940 (1998). [5] M. V. Feigel’man and M. A. Skvortsov, Universal broadening of the Bardeen-Cooper-Schrieffer coherence peak of disordered superconducting films, Phys. Rev. Lett. 109, 147002 (2012). [6] Y. Noat, T. Cren, C. Brun, F. Debontridder, V. Cherkez, K. Ilin, M. Siegel, A. Semenov, H. W. H¨ ubers, and D. Roditchev, Break-up of long-range coherence due to phase fluctuations in ultrathin superconducting NbN films, Phys. Rev. B 88, 014503 (2013). [7] E. F. C. Driessen, P. C. J. J. Coumou, R. R. Tromp, P. J. de Visser, and T. M. Klapwijk, Strongly disordered TiN and NbTiN s-wave superconductors probed by microwave electrodynamics, Phys. Rev. Lett. 109, 107003 (2012). [8] P. C. J. J. Coumou, M. R. Zuiddam, E. F. C. Driessen, P. J. de Visser, J. J. A. Baselmans, and T. M. Klapwijk, Microwave properties of superconducting atomic-layer deposited TiN films, IEEE Trans. Appl. Supercond. 23, 7500404 (2013). [9] R. R. Tromp, Competition between localization and superconductivity in TiN films, Master’s thesis, TU Delft, 2011. [10] A. M. Finkelstein, Superconductings transition-temperature in amorphous films, JETP Lett. 45, 46 (1987). [11] L. G. Aslamazov and A. I. Larkin, The influence of fluctuation pairing of electrons on the conductivity of normal metal, Physics Letters A 26, 238 (1968). [12] K. Maki, The critical fluctuation of the order parameter in type-II superconductors, Progress of Theoretical Physics 39, 897 (1968). [13] R. S. Thompson, Microwave, flux flow, and fluctuation resistance of dirty type-II superconductors, Phys. Rev. B 1, 327 (1970). [14] M. R. Beasley, J. E. Mooij, and T. P. Orlando, Possibility of vortexantivortex pair dissociation in two-dimensional superconductors, Phys. Rev. Lett. 42, 1165 (1979). 124


8.7 References

[15] B. I. Halperin and D. R. Nelson, Resistive transition in superconducting films, J. Low Temp. Phys. 36, 599 (1979). [16] T. I. Baturina, S. V. Postolova, A. Y. Mironov, A. Glatz, M. R. Baklanov, and V. M. Vinokur, Superconducting phase transitions in ultrathin TiN films, Europhysics Letters 97, 17012 (2012). [17] L. Benfatto, C. Castellani, and T. Giamarchi, Broadening of the Berezinskii-Kosterlitz-Thouless superconducting transition by inhomogeneity and finite-size effects, Phys. Rev. B 80, 214506 (2009). [18] A. Frydman, The superconductor insulator transition in systems of ultrasmall grains, Physica C: Superconductivity 391, 189 (2003). [19] A. T. Fiory, A. F. Hebard, and W. I. Glaberson, Superconducting phase transitions in indium/indium-oxide thin-film composites, Phys. Rev. B 28, 5075 (1983). [20] J. W. P. Hsu and A. Kapitulnik, Superconducting transition, fluctuation, and vortex motion in a two-dimensional single-crystal Nb film, Phys. Rev. B 45, 4819 (1992). [21] A. Glatz, A. A. Varlamov, and V. M. Vinokur, Fluctuation spectroscopy of disordered two-dimensional superconductors, Phys. Rev. B 84, 104510 (2011). [22] M. Y. Kupriyanov and V. F. Lukichev, Temperature dependence of pairbreaking current in superconductors, Sov. J. Low Temp. Phys. 210, 210 (1980). [23] S. N. Dorenbos, P. Forn-D´ıaz, T. Fuse, A. H. Verbruggen, T. Zijlstra, T. M. Klapwijk, and V. Zwiller, Low gap superconducting single photon detectors for infrared sensitivity, Appl. Phys. Lett. 98, 251102 (2011). [24] B. H. Eom, P. K. Day, H. G. LeDuc, and J. Zmuidzinas, A wideband, lownoise superconducting amplifier with high dynamic range, Nature Phys. 8, 623 (2012). [25] J. Romijn, T. M. Klapwijk, M. J. Renne, and J. E. Mooij, Critical pairbreaking current in superconducting aluminum strips far below Tc , Phys. Rev. B 26, 3648 (1982). [26] W. J. Skocpol, M. R. Beasly, and M. Tinkham, Self-heating hotspots in superconducting thin-film microbridges, J. Appl. Phys. 45, 4054 (1974). [27] N. Vercruyssen, T. G. A. Verhagen, M. G. Flokstra, J. P. Pekola, and T. M. Klapwijk, Evanescent states and nonequilibrium in driven superconducting nanowires, Phys. Rev. B 85, 224503 (2012). [28] M. Stuivinga, T. M. Klapwijk, J. E. Mooij, and A. Bezuijen, Self-heating of phase-slip centers, J. of Low Temp. Phys. 53, 673 (1983). [29] N. W. Ashcroft and M. D. Mermin, Solid state physics, Thomson Learning, 1976. 125


8. EďŹ&#x20AC;ect of spatial variations of the superconducting properties on common observables.

[30] A. Schmid, Electron-phonon scattering in dirty metals, in Localization, Interaction, and Transport Phenomena, volume 61 of Springer Series in Solid-State Sciences, pages 212â&#x20AC;&#x201C;220, Springer Berlin Heidelberg, 1985. [31] D. Chen, J. Chen, Y. Zhao, B. Yu, C. Wang, and D. Shi, Theoretical study of the elastic properties of titanium nitride, Acta Metallurgica Sinica (English Letters) 22, 146 (2009).

126


Summary

The electrodynamic response of mesoscopic objects to radiation allows us both to study the mesoscopic processes occurring in these samples, and to use the devices as sensitive detector elements. We studied two distinct mesoscopic systems. First, suspended carbon nanotubes, which show aspects of collective modes of the electron system. And second, superconducting nanowires made of strongly disordered material, which approaches the superconductor-insulator transition and starts to show inhomogeneous behavior of its superconducting properties. Carbon nanotubes provide a unique system for the study of the consequences of electron confinement in reduced dimensionality, due to their intrinsic low-dimensional nature and excellent charge transport properties. The absence of screening of the long-range Coulomb interaction between electrons, leads to a collective nature of the electron system, described by the Tomonaga-Luttinger liquid theory. In this theory, the elementary excitations of the system are no longer described by single fermionic particles, but instead by bosonic excitations. Charge and spin excitations exist separately and the charge waves are predicted to move at a significantly increased velocity. Superconducting nanowires can be excellent single-photon detectors. They display a high detection efficiency, fast reset times, low timing jitter, and low dark count rate. The device characteristics are for a large part determined by the weakest link in the superconducting nanowire. Therefore, homogeneity of the system is crucial for optimal application of these devices. The superconducting materials typically chosen are niobium nitride and niobium titanium nitride, because of their high critical temperature and fast energy relaxation rates. However, thin films of such materials are currently being studied in the context of the superconducting-insulator transition, induced by their strongly 127


disordered nature. On the superconducting side of this transition, inhomogeneity of the superconducting properties is observed experimentally. In this thesis, we discuss experiments on the charge transport characteristics of suspended carbon nanotubes and strongly disordered superconducting nanowires. The irradiation of antenna-coupled suspended single carbon nanotubes with 108 GHz radiation leads to a selective heating of the electrons in the nanotube with respect to those in the leads. This influences the charge transport characteristics in two ways: an increase of the conductance and the development of an offset voltage. The temperature dependent suppression of the low-bias conductance is indicative of Tomonaga-Luttinger liquid behavior. The DC offset voltage is interpreted as a thermovoltage due to the strong temperature gradient of the electron system at the interface between the nanotube and the electrodes. We studied the frequency dependent response of the suspended carbon nanotubes to radiation from 1.5 to 3 terahertz at the free-electron laser for infrared experiments in Nieuwegein. The response is analogous to the response to 108 GHz radiation. There is a strong frequency dependence of the response even after correction for the frequency dependence of the incident power. For both a 2 μm and a 4 μm long suspended carbon nanotube, we observe a large variation in the response in the frequency range from 1.5 to 2 terahertz and a relatively flat spectrum from 2 to 3 terahertz. Superconducting nanowire single-photon detectors are typically fabricated in a meandering pattern to cover an increased detection area. Sharp corners in superconducting circuits, for instance at the bends of the meandering pattern, are predicted to lead to a decreased critical current due to current-crowding. We experimentally demonstrate that sharp corners in superconducting niobium titanium nitride wires cause a clear reduction of the critical current. We also demonstrate that this effect is avoidable by the use of optimally rounded corners. The avoidance of critical current reduction due to geometry has an immediate impact on the performance of superconducting single-photon detectors and other applications or experiments in which the homogeneity of the critical current is important. We studied the return from the voltage-carrying state to the superconducting state of niobium titanium nitride nanowires. In the voltage carrying state, the wire remains resistive due to self-heating to a temperature above the critical temperature. Upon lowering the current, the wire returns to the superconducting state in a characteristic steplike pattern. The pattern is reproducible over multiple measurements, but differs between identical wires. The occurrence of the pattern does not depend on the presence of sharp geometric features. Intrinsic electronic inhomogeneities due to the strongly disordered nature of the material are considered. We describe a thermal model with a random variation 128


of the critical temperature along the wire and ďŹ nd that it produces similar current-voltage characteristics as those observed experimentally. To further elucidate the consequences of spatial variations of the superconducting parameters in strongly disordered superconductors, we performed a set of typical charge transport experiments on titanium nitride wires of varying thickness. In temperature-dependent measurements of the resistance, we ďŹ nd that the more disordered wires show a wider transition region between the normal and superconducting state. The statistics of the critical current measurements indicates the possible presence of a weak-spot in the wires. We observe a characteristic steplike retrapping pattern in the wires both under current- and voltage-bias. Hendrik Lude Hortensius Delft, November 2013

129


130


Samenvatting

De elektrodynamische respons van mesoscopische objecten op straling maakt het aan de ene kant mogelijk om de mesoscopische processen te bestuderen die in deze objecten plaats vinden. Aan de andere kant kunnen de objecten dienen als gevoelige detector elementen. We hebben twee mesoscopische systemen bestudeerd. Ten eerste vrijhangende koolstofnanobuisjes, die aspecten van collectief gedrag van het elektronensysteem laten zien. Ten tweede supergeleidende nanodraden van sterk wanordelijk materiaal, wat in de buurt van de supergeleider-isolator overgang komt en inhomogeen gedrag van de supergeleidende eigenschappen laat zien. Koolstofnanobuisjes zijn een uniek systeem om de gevolgen van de inperking van elektronen in een gereduceerde dimensionaliteit te bestuderen. Dit vanwege hun intrinsieke ´e´en-dimensionale karakter en uitstekende elektrische geleiding. De afwezigheid van afscherming van de Coulomb interactie tussen elektronen leidt tot een collectief gedrag van het elektronen systeem, wat beschreven wordt door de Tomonaga-Luttinger liquid theorie. In deze theorie worden de elementaire excitaties van het systeem niet langer beschreven door enkele fermionische deeltjes, maar door bosonische excitaties. Lading en spin excitaties bestaan afzonderlijk en de ladingsgolven worden verwacht zich voort te planten met een grotere snelheid. Supergeleidende nanodraden kunnen uitstekende enkel-foton detectoren zijn. Ze bezitten een hoge detectie effici¨entie, snelle reset tijden, lage tijdsonzekerheid en een lage valse-detectie waarschijnlijkheid. De detectie eigenschappen worden voor een groot deel bepaald door de zwakste schakel in de supergeleidende nanodraad. Daarom is homogeniteit van het systeem cruciaal voor optimaal gedrag van deze detectoren. De supergeleidende materialen die typisch gebruikt worden zijn niobium nitride en niobium titaan nitride, vanwege hun 131


hoge kritische temperatuur en snelle energie relaxatie. Dunne films van deze materialen worden echter bestudeerd in de context van de supergeleider-isolator overgang, welke ge¨ınduceerd wordt door hun sterk wanordelijke aard. Aan de supergeleidende kant van deze overgang is inhomogeniteit van de supergeleidende eigenschappen experimenteel waargenomen. In deze thesis bespreken we experimenten aan de elektrische transport eigenschappen van vrijhangende koolstofnanobuisjes en sterk wanordelijke supergeleidende nanodraden. Het elektronensysteem in vrijhangende koolstofnanobuisjes warmt selectief op ten opzichte van de omgeving onder de bestraling met 108 GHz licht. Dit be¨ınvloedt de elektronische transport eigenschappen op twee manieren: een hogere geleiding en het ontstaan van een offset spanning. De temperatuurafhankelijke suppressie van de geleiding bij lage spanning is een karakteristiek gevolg van Tomonaga-Luttinger liquid gedrag. De offset spanning wordt ge¨ınterpreteerd als een thermospanning die ontstaat door de sterke temperatuurgradi¨ent bij de contacten tussen het nanobuisje en de elektrodes. We hebben de frequentie afhankelijkheid van de respons van vrijhangende koolstofnanobuisjes op straling van 1.5 tot 3 THz bestudeerd met behulp van de vrije elektronen laser in Nieuwegein. De respons is vergelijkbaar met de respons op 108 GHz straling. Er is een sterke frequentie afhankelijkheid van de respons, ook na correctie voor de frequentie afhankelijkheid van de inkomende straling. We observeren een grote variatie in de respons in het frequentiegebied van 1.5 tot 2 terahertz en een relatief vlak spectrum tussen 2 en 3 terahertz, voor zowel een 2 μm als een 4 μm lang koolstofnanobuisje. Supergeleidende nanodraad enkel-foton detectoren worden doorgaans in een meanderend patroon gefabriceerd om een zo groot mogelijk detectie gebied te bedekken. Scherpe hoeken in supergeleidende circuits, bijvoorbeeld in de bochten van het meanderend patroon, worden verwacht te leiden tot een verminderde kritische stroom. We demonstreren experimenteel dat scherpe hoeken in supergeleidende niobium titaan nitride draden leiden tot een reductie van de kritische stroom. We demonstreren ook dat dit effect vermijdbaar is door het gebruik van optimaal afgeronde bochten. Het vermijden van de reductie van de kritische stroom door de geometrie leidt tot een directe verbetering van supergeleidende enkel-foton detectoren. Ook in andere toepassingen of experimenten waar de homogeniteit van de kritische stroom een rol speelt, dient hiermee rekening gehouden te worden. We hebben de overgang bestudeerd van de spanningsdragende toestand naar de supergeleidende toestand in niobium titaan nitride nanodraden. In de spanningsdragende toestand blijft de draad resistief door zelfverhitting tot een temperatuur boven de kritische temperatuur. Wanneer de stroom door de draad verlaagd wordt, gaat de draad over naar de supergeleidende toestand in een karakteristiek trapvormig patroon. Dit patroon is reproduceerbaar over 132


verschillende metingen, maar verschilt tussen identieke draden. Het optreden van dit patroon hangt niet af van de aanwezigheid van scherpe geometrische elementen. We overwegen intrinsieke elektronische inhomogeniteiten die optreden door de sterk wanordelijke eigenschappen van het materiaal. We beschrijven een thermisch model met een willekeurige variatie van de kritische temperatuur over de draad. Uit deze simulaties vinden we vergelijkbare elektronische transport karakteristieken als degene die we experimenteel hebben geobserveerd. Om de invloed van willekeurige ruimtelijke variaties van de supergeleidende eigenschappen in sterk wanordelijke supergeleiders verder te verhelderen, hebben we een set typische transport metingen verricht op titaan nitride draden van verschillende diktes. Temperatuur afhankelijke metingen van de weerstand laten zien dat de meer wanordelijke draden een bredere overgang tussen de normale en de supergeleidende toestand vertonen. De statistische eigenschappen van de kritische stroom metingen wijzen op de mogelijke aanwezigheid van een zwakke schakel in de draden. We observeren een karakteristiek patroon in de overgang van de normale naar de supergeleidende toestand in de draden, zowel onder stroom- als onder spannings-bias. Hendrik Lude Hortensius Delft, November 2013

133


134


Curriculum Vitae

Hendrik Lude Hortensius 17–02–1985

Born in Zeist

1998–2003

Grammar school, Koningin Wilhelmina College, Culemborg

2003–2006

B.Sc. Physics, Leiden University Bachelor Thesis: ”The conduction of a single water molecule” Advisor: Prof.dr. J. M. van Ruitenbeek

2006–2008

M.Sc. Nanoscience, Delft University of Technology Master Thesis: ”Introduced two-level systems in NbTiN highquality thin film superconducting resonators” Advisor: Prof.dr.ir. T. M. Klapwijk

2008 2008–2013

Internship, Thales Research & Technology, Palaiseau France Ph.D. Research, Delft University of Technology Thesis: ”Electrodynamic response of mesoscopic objects: carbon nanotubes and superconducting nanowires” Promotor: Prof.dr.ir. T. M. Klapwijk

135


136


List of publications

Journal articles ”Possible indication of electronic inhomogeneities in superconducting nanowire detectors”, IEEE transactions on Applied Superconductivity, 23 (2200705) 2013 H.L. Hortensius, E.F.C. Driessen, and T.M. Klapwijk

”Critical-current reduction in thin superconducting wires due to current crowding”, Applied Physics Letters, 100 (182602) 2012 H.L. Hortensius, E.F.C. Driessen, T.M. Klapwijk, K.K. Berggren, and J.R. Clem

”Microwave-induced nonequilibrium temperature in a suspended carbon nanotube”, Applied Physics Letters, 100 (223112) 2012 H.L. Hortensius, A. Ozturk, P. Zeng, E.F.C. Driessen, and T.M. Klapwijk

”Noise in NbTiN, Al and Ta superconducting resonators on silicon and sapphire substrates”, IEEE transactions on Applied Superconductivity, 19 (936) 2009 R. Barends, H.L. Hortensius, T. Zijlstra, J.J.A. Baselmans, S.J.C. Yates, J.R. Gao, and T.M. Klapwijk

137


”Contribution of dielectrics to frequency and noise of NbTiN superconducting resonators”, Applied Physics Letters, 92 (223502) 2008 R. Barends, H.L. Hortensius, T. Zijlstra, J.J.A. Baselmans, S.J.C. Yates, J.R. Gao and T.M. Klapwijk

Selected Conferences & Workshops Applied Superconductivity Conference (oral) 2012 Casimir Spring School (oral)

Portland, United States

Arnemuiden, the Netherlands 2012

International Conference on Low Temperature Physics (poster) China 2011 International Conference on Molecular Electronics (poster) Switzerland 2010 European School on Nanosciences and Technologies (poster) France 2009 VIP’07 (oral)

138

Beijing, Emmetten, Grenoble,

Amsterdam, the Netherlands 2007


Acknowledgements

My PhD has been a process that I could not have completed without the guidance, support, and sometimes distraction of many people. I have benefited a lot from the knowledge and expertise of a large number of people in the daily live as an experimental physicist. I am very happy to have the opportunity here to thank some of you. First of all, of course, my promotor Teun. Thank you for your inspiration, advice, and guidance. I feel that during this process you have helped me to grow, evolve, and find the person that I am. I have gotten to know you as not only an expertly knowledgeable and intelligent supervisor, but also someone who cares greatly about the people under his guidance on a personal level. I would like to remember Niels here, without his expertise, patience, and enthusiasm I could never have performed my experiments. Moreover he was a binding element in the group on so many occasions. I would also like to remember John Clem, who inspired the current crowding experiments. I only had the pleasure to meet him once, but he made a lasting impression. I am grateful to Eduard and Karl, who I am very happy to have on my defense committee, although it will be quite a switch from working side by side to being in opposition. Eduard, you have been my mentor not just in the cleanroom, but also in most other aspects of the PhD. Karl, thank you for your willingness to discuss and try out new ideas. One of the most rewarding aspects of the PhD for me was the guidance of students in their graduation projects. Teun, thank you for your creativity and enthusiasm, I am sure they will bring you far. Peng, thank you for all your hard work in the cleanroom. Now I will be following you into the world of trading. And finally Robbert-Jan, it is great to see you having started your own PhD now, best of luck! 139


I have been lucky enough to share my office with some wonderful people. First of all, Alibey. This thesis was originally planned to be the work of us together. And although I am now defending it, I still feel it is. Thank you for the great times we had together. After that there was Morris, who always brought a good atmosphere to the room (or was sleeping on his keyboard). Then there was Cristina, thank you for the running, the table tennis, and the talks. And finally David, may Atletico win all their matches (except of course against Ajax). In general my favorite part of the PhD was being a part of the wonderful CosmoNano-group. Maria, Dorine, Monique, and Irma, thank you for all your hard work. I cannot imagine what disasters would have happened without you. Rami, thank you for getting me started on this path and teaching me to ”See the data, feel the data, and be the data”. Pieter, thank you for all the table tennis games and being so polite to wait with your defense until I am done. Reinier, thank you and your sister for providing what will surely be the most massive cakes I will ever see. Pieter-Jan, I fondly remember our Portland trip and particularly the jazz bar you found. Jochem and Akira, thank you for always being willing to discuss anything, whether it is physics, photography or promoting the group. Gao, thanks for providing advice when needed and discussing any sports. David, thank you for always being willing to have a chat and introducing so many new expressions. Alessandro, thank you for the discussions on gaming and technology. Merlijn, thanks for all the help in the job hunting. Yuan, thank you for always being ready to discuss the football. Thanks also to Mihai, Chris, Tarun, and Zhu. Marc-Peter, Jaime, and Kate, I always enjoyed your visits. Bastian, Tom, Rutger, Maryna, Werner, Nuri, and all former students and visitors, thank you for keeping the group young. And finally Nathan, you were there when I came and you are here when I leave. I could never imagine what this group would have been like without you. Thank you all for all the great memories and I am sure there are many more still to come. Thanks also to all the people in Delft that I met outside of the CosmoNanogroup. In particular to Ben, Gary, Ferry, and Edgar for helping me get started with carbon nanotubes and electromigration. Aurele, thanks for the near-field measurements. Also thanks to Jack, Ruud, Raymond, Aad and Mascha, for all the electronical, mechanical, and technical help. And to all the cleanroom staff for maintaining such an excellent facility. I am grateful to my friends outside of physics for keeping me (more or less) sane and linked to the real world. Dinesh and Joel, Marije and Tim, Siebe, Tess, Marieke (next stop New York or Melbourne?), Martijn, Tim, and all the other people from Velocitas for providing a place to get my mind away from research. Thanks to Roos and Cris for the wonderful introduction to diving. Noha and Safaa, thank you for being friends from afar. Pinar, Marta, and 140


Valerie, thank you for all the nice drinks, dinners, concerts, and trips. Thanks to my grandparents, aunts and uncles, nieces and nephews for being a wonderfully supportive family. Thanks to my paranymphs Sander and Ries for being there throughout my PhD and the many FIFA nights. And thanks to Alette and Iris for putting up with all of those. Finally my sister and soonto-be brother-in-law Lian and Steve, thank you for your support. And last but not least, my parents. Thank you for everything. You have always provided a safe place and supported me to grow.

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Casimir series 2014-4 (Hortensius)