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Journal of the Physical Society of Japan Vol. 77, No. 3, March, 2008, 031009 #2008 The Physical Society of Japan

SPECIAL TOPICS Advances in Spintronics

Spin Current in Metals and Superconductors Saburo TAKAHASHI1;2 and Sadamichi M AEKAWA1;2;3 1 Institute for Materials Research, Tohoku University, Sendai 980-8577 CREST, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012 3 Centre for Advanced Study, Norwegian Academy of Science and Letters, Drammensveien 78, N-0271 Oslo, Norway 2

(Received September 4, 2007; accepted November 6, 2007; published March 10, 2008)

The nonlocal spin transport in a nanostructure device consisting of ferromagnetic spin injector (F1) and spin detector (F2) electrodes connected to a normal conductor (N) or a superconductor (SC) is discussed. We explore how the spin transport in such a device depends on the nature of interface (metallic-contact or tunnel junction) and material parameters of electrodes by solving the spin-dependent transport equations, and obtain the conditions for efficient spin injection, spin accumulation, and spin current. In a device with SC, the spin transport is strongly affected by the onset of superconductivity. Nonlocal Spin Hall effect in metals and SCs are investigated. A superconducting qubit with a  junction is discussed. KEYWORDS: spin transport, spin injection, spin accumulation, spin current, spin absorption, spin Hall effect, magnetic nanostructure DOI: 10.1143/JPSJ.77.031009

1.

Introduction

There has been intensive study in the field of spin transport in magnetic nanostructures. This is not only because a variety of new quantum phenomena emerge, but also because the potential applications to spin-based electronic devices are expected.1–4) Recent experimental and theoretical studies have revealed that the spin-polarized carriers injected from a ferromagnet (F) into a nonmagnetic material (N), such as a normal metal, semiconductor, and superconductor (SC), create nonequilibrium spin accumulation and spin current in N. The efficient spin injection, spin accumulation, spin transfer, and spin detection are of crucial importance in utilizing the spin degree of freedom as a new functionality in spin-based electronic devices. In this review article, a basic aspect of spin injection, spin transport, and spin detection in magnetic nanostructures containing normal metals or SCs is discussed. Particular emphasis is placed on the spin accumulation and spin current in a nonlocal spin device of F1/N/F2 structure, where F1 is a spin injector and F2 a spin detector. Using spin-dependent transport equations for the electrochemical potentials (ECPs) of up and down spin electrons and applying them to a structure with arbitrary electrode resistance and junction resistance ranging from a metallic contact to a tunneling regime, we examine the optimal conditions for spin accumulation and spin current. The spin accumulation detected by F2 depends strongly on whether the junction interface is a metallic contact or a tunnel barrier; it is greatly improved when a tunnel barrier is used instead of a metallic contact, and efficient spin injection and detection are achieved when both interfaces are tunnel junctions. By contrast, a large spin-current injection into F2 is realized when N has a metallic contact with F2 which has a short spin diffusion length like a Permalloy (Py), because F2 works as a strong absorber (spin sink) for spin current flowing in N. These findings imply that tunnel junctions are favorable for a large spin accumulation, while metallic contacts are favorable for a large spin-current injection. Interesting and useful

devices are those containing a material with a low density of carriers that carry spins. In a nonlocal spin injection device with SC, the spin accumulation signal is greatly enhanced by opening the superconducting energy gap, because a large spin-splitting of the chemical potentials is necessary to maintain the same amount of spin current at the interface. A large spin accumulation signal is likewise expected in semiconductor devices. The spin-current and charge-current induced spin Hall effects in normal metals and SCs in the present of spin–orbit scattering are investigated in detail. A superconducting qubit containing a  junction is discussed. 2.

Spin Injection, Spin Accumulation, and Spin Current

In 1985, Johnson and Silsbee5) demonstrated that nonequilibrium spin injected from ferromagnets diffuses into Al films over the spin diffusion length of the order of 1 mm (or several hundred mm for pure Al). In 1993, Johnson proposed a spin injection technique6) in a F1/N/F2 structure (F1 is an injector and F2 a detector), in which the output voltage at F2 depends on the relative orientation of the magnetizations (parallel or antiparallel) between F1 and F2. Recently, Jedema et al. made spin injection and detection experiments in a permalloy/copper/permalloy (Py/Cu/Py) structure using a nonlocal geometry for measurement, and observed a clear spin accumulation signal at room temperature.7) Subsequently, they measured a large spin accumulation signal in a cobalt/aluminum/cobalt (Co/I/Al/I/Co) structure with tunnel barriers (I = Al2 O3 ) between Co and Al electrodes.8) Figures 1(a) and 1(b) show a F1/N/F2 structure and an expected spin accumulation signals. The bias current I is sent from F1 and taken out from the left end of N, and the spin accumulation at distance L from F1 is detected by measuring the voltage (V2 ) between F2 and N. Sweeping the magnetic field B from the negative side (or positive side), the nonlocal resistance V=I changes sign from positive to negative, and switches symmetrically around zero, as shown in Fig. 1(b). The positive and negative values of the nonlocal

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J. Phys. Soc. Jpn., Vol. 77, No. 3

(a)

(a) Side view

V2

I

S. T AKAHASHI and S. M AEKAWA

V2

I F1

y

wF

I

dN

wN

x

N

I

dF

F2 N

(b) Electrochemical potential (ECP) in N μ Nσ

L F1

F2

jN↑

μ N↑

x

0

μN

1.0

(b)

P

x=L

x =0

0.5

V/I (mΩ)

jN↓

(c) Detection of spin accumulation by F2 0

F2 (Antiparallel) E

−0.5

AP −50

AP 0

F2 (Parallel)

N E

+V 2P

μ PF2

μ AP F2

50

E

−V2AP

B (mT) Fig. 1. Spin injection and detection in the nonlocal geometry for measurement. (a) Schematic diagram of a F1/N/F2 structure and (b) nonlocal resistance V=I as a function of in-plane magnetic field B, where ‘‘P’’ and ‘‘AP’’ represent the parallel and antiparallel orientations of magnetizations in F1 and F2.

resistance correspond to the parallel and antiparallel magnetizations, the difference of which corresponds to the spin accumulation signal. These behaviors have been observed experimentally in many groups using the nonlocal measurement with normal metals,9–20) graphenes,21,22) and semiconductors.23) 2.1 Spin transport in nonlocal geometry We consider a spin injection and detection device which consists of a nonmagnetic metal N connected to ferromagnets of injector F1 and detector F2 as shown in Fig. 2(a). F1 and F2 are ferromagnetic electrodes with width wF and thickness dF , and are separated by distance L, and N is a normal-metal electrode with width wN and thickness dN . The magnetizations of F1 and F2 are aligned either parallel or antiparallel. In this device, spin-polarized electrons are injected from F1 into N, and the spin accumulation is detected by F2 at distance L from F1 by measuring the voltage V2 between F2 and N. Because of the absence of a voltage source on the right part of the device, there is no charge current in the electrodes that lie on the right side of F1. By contrast, the injected spins are diffused equally in both directions, creating spin accumulation on the right side. Accordingly, the spin and charge degrees of freedom are transported separately in the device. The advantage of the nonlocal measurement is that F2 probes only the spin degrees of freedom. The electrical current density j for spin channel  ( ¼ "; #) in a conductor is driven by the electric field E and the gradient of the carrier density n : j ¼  E  eD rn , where  and D are the electrical conductivity and the diffusion constant with spin  and e ¼ jej. Using rn ¼ N r"F (N is the density of states in the spin subband and "F

Fig. 2. (Color online) (a) Spin injection and detection device (side view). The bias current I is applied from F1 to the left end of N. The spin accumulation at x ¼ L is probed by F2 by measuring the voltage V2 between F2 and N. (b) Spatial variation of the electrochemical potential (ECP) for up and down spin electrons in N. (c) Densities of states for the up and down spin bands in N and F2.

is the Fermi energy) and the Einstein relation  ¼ e2 N D , we have j" ¼ ð" =eÞr" ;

j# ¼ ð# =eÞr# ;

ð1Þ

where  ¼ "F þ e is the ECP and  the electric potential. The continuity equations for charge and spin in the steady state are r  ðj" þ j# Þ ¼ 0;

ð2aÞ n"  n" n#  n# r  ðj"  j# Þ ¼ e þe ; ð2bÞ "# #" where n is the equilibrium carrier density with spin  and  0 is the scattering time of an electron from spin state  to  0 . Making use of these continuity equations and detailed balance N" ="# ¼ N# =#" , one obtains the basic equations for ECP5,24–28) r2 ð" " þ # # Þ ¼ 0; 1 r2 ð"  # Þ ¼ 2 ð"  # Þ;  where  is the spin-diffusion length pffiffiffiffiffiffiffiffiffi  ¼ Dsf ;

ð3aÞ ð3bÞ

with the spin-relaxation time sf and the diffusion constant D:27) 1 1=sf ¼ ð1="# þ 1=#" Þ; ð4Þ 2 1=D ¼ ðN" =D# þ N# =D" Þ=ðN" þ N# Þ: ð5Þ The physical parameters of N are spin-independent, e.g.,

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SPECIAL TOPICS

the conductivity N" ¼ N# ¼ ð1=2ÞN , etc., while those of F are spin-dependent, e.g., F" 6¼ F# (F ¼ F" þ F# ), etc. The spin-diffusion lengths of transition-metal ferromagnets are F  5 nm for Permalloy (Py), F  12 nm for CoFe, and F  50 nm for Co from studies of CPP-GMR (currentperpendicular-plane giant magnetoresistance),29) whereas those of nonmagnetic metals are N  1 mm for Cu,7,12) and N  0:65 mm for Al.8) The fact that F of typical ferromagnets is much shorter than N in nonmagnetic metals, such as Al or Cu, plays a crucial role in spin transport in devices made from those materials as shown below. The interfacial current across the junctions is described by the model used in CPP-GMR developed by Valet and Fert.25) In the presence of spin-dependent interface resistance Ri at junction i (i ¼ 1; 2), the ECP changes discontinuously at the interface when the current passes through the junction. The spin-dependent interfacial current I1 ðI2 Þ from F1 (F2) to N is given by the ECP difference at the interface:25–27)   1  1  I1 ¼  F1  N ; I2 ¼  F2  N ; ð6Þ eR1 eR2 where the distribution of the current is assumed to be uniform over the interface. The total charge and spin currents across the ith interface are Ii ¼ Ii" þ Ii# and Iis ¼ Ii"  Ii# . Equation (6) is applicable not only for a transparent metallic-contact but also for tunnel junctions. In the transparent metallic-contact (Ri ! 0), ECPs are continuous at the interface, so that the spin accumulation on the N side is strongly influenced by the spin splitting of ECPs on the F side. In the tunneling case, the discontinuous change of ECPs at the interface is much larger than the spin splitting, so that the spin splitting on the F side is irrelevant for the spin accumulation on the N side. In real devices, the distribution of the current across the interface depends on the relative magnitude of the interface resistance to the electrode resistance.30) When the interface resistance is much larger than the electrode resistance as in a tunnel junction, the current distribution is uniform in the contact area,31) which validates our assumption of uniform interface current. However, when the interface resistance is comparable to or smaller than the electrode resistance as in a metallic contact junction, the current distribution becomes inhomogeneous; the interface current has a large current density around a corner of the contact.15,32) In this case, the effective contact area through which most of the current flows is smaller than the actual contact area AJ ¼ wN wF of the junctions. When the bias current I flows from F1 to the left side of N (I1 ¼ I) and there is no charge current on the right side (I2 ¼ 0), the solution of eqs. (3a) and (3b) takes the form N ¼  N þ ða1 ejxj=N  a2 ejxLj=N Þ;

ð7Þ

where  N ¼ ½eI=ðN AN Þx (AN ¼ wN dN ) describes the charge transport for x < 0 and  N ¼ 0 (ground level of ECP) for x > 0, and the second term is the shift in ECP of up ( ¼ þ) and down ( ¼ ) spin electrons; the a1 -term in the bracket represents the spin accumulation due to spin injection by F1, while the a2 -term is the spin depletion due to spin extraction by F2. Note that the pure spin current flow in the region of x > 0, i.e., the charge current ( jN ¼ j"N þ j#N ) is absent and only the spin current ( jsN ¼ j"N  j#N ) flows.

S. T AKAHASHI and S. M AEKAWA

In the F1 and F2 electrodes in which the thickness is much larger than the spin diffusion length (dF  F ) as in the case of Py or CoFe, the solutions close to the interfaces may have the forms F1 ¼  F1 þ b1 ez=F ;

F2 ¼  F2  b2 ez=F ; ð8Þ

where  F1 ¼ ½eI=ðF AJ Þz þ eV1 is linear in z and carries the charge current I in F1,  F2 ¼ eV2 is a constant potential with no charge current in F2, and V1 and V2 are the voltage drops across junctions 1 and 2. Using the matching condition that the charge and spin currents are continuous at the interfaces, we determine ai , bi , and Vi as listed in the Appendix. The detected voltages, V2P and V2AP , in the parallel (P) and antiparallel (AP) alignments of magnetizations yield the spin accumulation signal Rs ¼ ðV2P  V2AP Þ=I:

ð9Þ

2.2 Spin accumulation signal The spin accumulation signal Rs is calculated as28) Rs ¼ RN

ð2P1 r1 þ 2pF rF Þð2P2 r2 þ 2pF rF ÞeL=N ; ð10Þ ð1 þ 2r1 þ 2rF Þð1 þ 2r2 þ 2rF Þ  e2L=N

with the normalized resistances: ri ¼

1 Ri ; 1  P2i RN

rF ¼

1 RF ; 1  p2F RN

ð11Þ

where Ri (1=Ri ¼ 1=R"i þ 1=R#i ) is the interface resistance of junction i, RN and RF are the spin accumulation resistances of N and F: RN ¼ ðN N Þ=AN ;

RF ¼ ðF F Þ=AJ ;

ð12Þ

with the resistivity N and the cross-sectional area AN ¼ wN dN of N, the resistivity F of F and the contact area AJ ¼ wN wF of the junctions, and Pi is the interfacial current spin-polarization and pF the spin-polarization of F1 and F2:   Pi ¼ R"i  R#i =ðR"i þ R#i Þ; ð13Þ  "  " # #  ð14Þ pF ¼ F  F =ðF þ F Þ; where F is the electrical resistivity for spin  channel in F. In junctions with metallic contact, the spin polarization, Pi and pF , range around 50 – 70% from GMR experiments29) and point-contact Andreev-reflection experiments,33) whereas in tunnel junctions, Pi ranges around 30 – 55% for alumina tunnel barriers,34–36) and 85% for MgO barriers37,38) from superconducting tunneling spectroscopy experiments. The spin accumulation signal Rs strongly depends on whether the junction is a metallic contact or a tunnel junction, and on the relative magnitude between R1 , R2 , RF , and RN . When RF is much smaller than RN , e.g., RN  RF , as in the Cu and Py electrodes, we have the following limiting cases. When both junctions are tunnel junctions (R1 ; R2  RN  RF ),5,8) Rs ¼ P1 P2 RN expðL=N Þ:

ð15Þ

When one of the junctions is a tunnel junction and the other is a transparent contact, we have28) Rs ¼

2pF P1 RF expðL=N Þ; 1  p2F

for R1  RN  RF  R2 , and

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J. Phys. Soc. Jpn., Vol. 77, No. 3

10

–1

S. T AKAHASHI and S. M AEKAWA

F1/I/N/I/F2

Co/I/Al/I/Co

10

10

ΔR s (m Ω)

ΔR s / R N

10 –2

F1/I/N/F2 or F1/N/I/F2 –3

1

Py/Cu/Py

F1/N/F2 10

–4

0

1

L /λ N

2

0.1

3

0

500

1000

1500

L(nm)

Fig. 3. Spin accumulation signal Rs as a function of distance L between F1 and F2.

ð17Þ

Fig. 4. Spin accumulation signal Rs as a function of distance L between F1 and F2 in a tunnel device and a metallic-contact device. The symbols ( , ) are the experimental data of Co/I/Al/I/Co,8) and ( , ) are those of Py/Cu/Py,45,46) where ( , ) are measured at 4.2 K and ( , ) at room temperature.

for R2  RN  RF  R1 . When both junctions are transparent contacts (RN  RF  R1 ; R2 ),7,26,27,39)   2p2F RF Rs ¼ ð18Þ RF ½sinhðL=N Þ1 : 2 2 ð1  pF Þ RN

700 nm, RN ¼ 2 ,49) and ðRF =RN Þ ¼ 1  102 at 293 K for pF ¼ 0:7. In the tunneling regime, the spin splitting of ECP at position x in N is given by

Rs ¼

2pF P2 RF expðL=N Þ; 1  p2F

Note that Rs in the above limiting cases is independent of Ri . Figure 3 shows the spin accumulation signal Rs for RF =RN ¼ 0:01,7) pF ¼ 0:7, and P1 ¼ P2 ¼ 0:4. We see that Rs increases by one order of magnitude by replacing a metallic-contact with a tunnel barrier, since the resistance mismatch, which is represented by ðRF =RN Þ  1, is removed by replacing a metallic-contact with a tunnel junction. Note that the mismatch originates from a large difference in the spin diffusion lengths between N and F (F  N ). When a nonmagnetic semiconductor is used for N, the resistance mismatch arises from the resistivity mismatch (N  F ).39–41) A question is raised on whether the contacts of the metallic Py/Cu/Py structure7) are transparent (Ri =RF  1) or tunnel-like (Ri =RN  1).42) If one uses the experimental values (Ri AJ  2  1012  cm2 , F  5 nm,29) F  105  cm), one has Ri =RF  0:4, indicating that Py/Cu/Py lies in the transparent regime, so that eq. (18) may be used to analyze the experimental data.7) The value RN ¼ 3  estimated from the experimental values of Cu43) in the Py/Cu/Py structure yields Rs ¼ 1 m at L ¼ N . If the cross-shaped Cu electrode7) is taken into account, we have one-third of the above value,44) which is consistent with the experimental value of Rs ¼ 0:1 m.7) Figure 4 shows the experimental data of Rs as a function of distance L in Co/I/Al/I/Co8) and Py/Cu/Py.45,46) In the tunnel device of Co/I/Al/I/Co (I = Al2 O3 ), fitting of eq. (15) to the data yields N ¼ 650 nm at 4.2 K, N ¼ 350 nm at 293 K, P1;2 ¼ 0:1, and RN ¼ 3 .47) The relation N2 ¼ Dsf with N ¼ 650 nm and D ¼ 1=½2e2 Nð0ÞN   40 cm2 /s leads to sf ¼ 100 ps at 4.2 K, which is of the same order of magnitude as that evaluated from the spin–orbit parameter b ¼ h =ð3sf Al Þ  0:0137) in superconducting tunneling spectroscopy. In the metallic-contact device of Py/ Cu/Py, fitting of eq. (18) to the data yields N ¼ 920 nm, RN ¼ 5 ,48) ðRF =RN Þ ¼ 0:7  102 at 4.2 K, and N ¼

2 N ðxÞ ¼ "N ðxÞ  #N ðxÞ ¼ P1 eRN Iejxj=N :

ð19Þ 8)

For a device with P1  0:1, RN ¼ 3 , and I ¼ 100 mA, the value of N ðxÞ is at most N ð0Þ  15 mV, which is much smaller than the superconducting gap   200 meV of Al films. The effect of superconductivity on spin accumulation is discussed in §3. It is noteworthy that when F1 and F2 are both half-metals (pF ’ 1, rF  1), we have the largest value Rs RN expðL=N Þ;

ð20Þ

without tunnel barriers, which is the advantage of using halfmetals with 100% spin polarization. 2.3 Nonlocal spin injection and manipulation We next investigate how the spin current flows through the structure, particularly the spin current across the N/ F2 interface [Fig. 5(a)], because of an interest in spincurrent induced magnetization switching50–53) in nonlocal devices.54,55) The distribution of spin accumulation and spin current is strongly influenced by the relative magnitude of the interface resistances (Ri ) and the spin accumulation resistances ðRF ; RN Þ. Figure 5(b) shows the spatial variation of spin accumulation N in the N electrode of the F1/I/N/F2 structure. The dashed curve indicates N in the absence of F2. When F2 contacts with N at position of L=N ¼ 0:5, the spin accumulation is strongly suppressed by F2 with small RF , leaving little spin accumulation on the right side of F2. This behavior has been observed in a nonlocal device with three Py electrodes.11,12) We note that the slope of the curve between F1 and F2 (0 < x < L) becomes steeper than that of the dashed curve, indicating that the spin currents INs between F1 and F2 become larger than that in the absence of F2 as seen in Fig. 5(c). In addition, the large discontinuous drop of INs to nearly zero at x ¼ L is caused by spin current absorption by F2, indicating that most of the spin current

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J. Phys. Soc. Jpn., Vol. 77, No. 3

I

(a)

3.

F1

F2 N

I

Is

(b) Spin accumulation

δμN /(eR N I)

0.2

0.1

L/λ N = 0.5 0

(c)

Spin current

0.3

L/λ N = 0.5

s

IN /I

0.2 0.1 0

–0.1 –0.2 –0.5

0

0.5

x /λ N

1

Fig. 5. (a) Nonlocal spin current injection device of F1/I/N/F2, where the first junction is a tunnel junction and the second is a metallic contact. The spin current Is is absorbed by F2. Spatial variation of (b) spin accumulation N and (c) spin current INs flowing in the N electrode for L=N ¼ 0:5 and 1. The discontinuous change of spin current at L=N ¼ 0:5 indicates that the most of spin current flows out through the N/F2 interface. The parameters are the same as those in Fig. 3.

S. T AKAHASHI and S. M AEKAWA

Spin Injection into Superconductors

It is of great interest how the spin accumulation signal Rs is modified by replacing N with a SC in a nonlocal spin injection device. When SC becomes superconductive below the superconducting critical temperature Tc , the superconducting gap  opens at the Fermi energy, and the population of quasiparticles (QPs) above the energy gap  decreases with decreasing temperature, which makes SC a low-carrier system for QP’s spin and charge transport. We will show that the signal Rs in an F1/I/S/I/F2 tunneling device increases due to the increase of RN in the superconducting state.28) In the following, we consider the situation where the spin splitting of ECP is smaller than the gap  [see discussion below eq. (19)], and neglect the suppression of  due to spin accumulation.57–59) While Andreev reflection plays an important role in metallic-contact junctions, it may be disregarded in tunneling junctions. In the superconducting state, the spin diffusion length in the superconducting state is the same as that in the normal state,60–62) which is intuitively understood as follows. Since the dispersion pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficurve of the QP excitation energy is given by Ek ¼ 2k þ 2 with one-electron energy k ,63) the QP’s velocity v~k ¼ ð1=h Þð@Ek =@kÞ ¼ ðj k j=Ek Þvk is slower by the factor j k j=Ek compared with the normal-state velocity vk ( vF , the Fermi velocity). By contrast, the impurity scattering time64) ~ 0 ¼ ðEk =j k jÞ0 , is longer by the inverse of the factor. Consequently, the spin-diffusion length ~ 1=2 ~ ¼ ð1=3Þv~2k ~tr ¼ ðj k j=Ek ÞD and S ¼ ðD P~sf Þ 1, where D 1 1 ~tr ¼  0 ~0 ¼ ðEk =j k jÞtr ,64) becomes S ¼ ðD~ ~sf Þ1=2 ¼ ðDsf Þ1=2 ¼ N ;

flows out to F2 through the N/F2 interface. In the N region on the right side of F2 (x > L), the spin current is very small. This implies that F2 with very low RF like Py and CoFe and with metallic contact with N works as a strong spin absorber (an ideal spin sink). The outgoing spin current I2s across the N/F2 interface is calculated as (see Appendix) jI2s j ¼

2ðP1 r1 þ pF1 rF1 ÞeL=N I ; ð1 þ 2r1 þ 2rF1 Þð1 þ 2r2 þ 2rF2 Þ  e2L=N

ð21Þ

from which a large spin-current injection from N into F2 is realized in a device with a tunnel junction (r1  1) at junction 1 and a metallic contact (r2  1) at junction 2, and with a strong spin absorber F2 (RF2  RN ) such as Py or CoFe, yielding jI2s j P1 I expðL=N Þ:

ð22Þ

The spin current in N between F1 and F2 is INs P1 I expðx=N Þ (0 < x < L), which is two times as large as Is in the absence of F2, while INs 0 in the right side of F2 (0 < x < L). If a nano-scale island of F2 with dimensions much smaller than N but larger than F is placed on the N electrode within the distance N form F1, the F2 island works as a strong absorber (sink) for spin current; the spin-angular momentum is efficiently transferred from F1 to a small F2 by nonlocal spin injection, providing a method for manipulating the orientation of magnetization due to spin transfer torque in nonlocal devices.56)

ð23Þ

owing to the cancellation of the factor j k j=Ek . Therefore the spin splitting ð"  # Þ in SC satisfies the same equation as eq. (3b) in the normal state. By contrast, the spin-relaxation time is strongly influenced by opening of . The spin accumulation in SC is determined by balancing the spin injection rate with the spin-relaxation rate:   X @S Iis þ e ¼ 0; ð24Þ @t sf i where S is the total spins accumulated in SC, and I1s and I2s are the spin injection rates through junctions 1 and 2, respectively. At low temperatures, the main origin of spin relaxation is spin–flip scattering via the spin–orbit interaction Vso at nonmagnetic impurities or grain boundaries. The spin–orbit scattering matrix elements between QP states jki with momentum k and spin  are hk0  0 jVso jki ¼ i so ðuk0 uk  vk0 vk Þ½ 0   ðk  k0 ÞVimp ; where so is the spin–orbit coupling parameter, Vimp is the impurity potential,  is the Pauli spin matrix, and u2k ¼ 1  v2k ¼ ð1=2Þð1 þ k =Ek Þ are the coherent factors.63) Using the golden rule for the spin–flip scattering, one obtains the spin-relaxation rate65,66)   Z Z1 @S Nð0Þ ¼ dr ð fk"  fk# Þ dEk @t sf sf SC  Z 1  2 f0 ðÞNð0Þ S ðrÞ dr; ð25Þ sf SC

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J. Phys. Soc. Jpn., Vol. 77, No. 3

where sf is the spin–flip scattering time in the normal state and fk the distribution function for a QP with momentum k and spin . In eq. (25), we made use of the expansion with respect to the ECP shift S : fk  f0 ðEk Þ  ½@ f0 ðEk Þ=@Ek  S ; where f0 ðEÞ is the Fermi distribution function. On the other hand, the spin accumulation is given by Z X 1 S¼ dr fk"  fk# 2 SC k Z S ðrÞ dr; ð26Þ s ðTÞNð0Þ

S. T AKAHASHI and S. M AEKAWA

decreasing T according to s ðTÞ ’ ð=2kB TÞ1=2 sf at low temperatures. Since the spin diffusion length in the superconducting state is the same as that in the normal state, the ECP shift in SC has the same form as in eq. (7), S ¼ a~1 expðjxj=N Þ  a~2 expðjx  Lj=N Þ, where the coefficients are calculated as follows. In the tunnel device, the tunnel spin currents are I1s ¼ P1 I and I2s 0, so that eqs. (24) and (25) yield the coefficients a~1 ¼ P1 RN eI=½2 f0 ðÞ and a~2 0, leading to the spin splitting of ECP in the superconducting state 2 S ðxÞ ¼ P1

SC

where s ðTÞ is the normalized spin-susceptibility (Yosida function):   Z1 Ek @ f0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

s ðTÞ ¼ 2 ð27Þ dEk ; @Ek Ek2  2  which is s ðTÞ  1  ½7 ð3Þ=42 ð=kB TÞ2 near Tc and

s ðTÞ  ð=2kB TÞ1=2 exp½=kB T well below Tc . The spin-relaxation time s of S in the superconducting state is determined from ð@S=@tÞsf ¼ S=s , and given by s ðTÞ ¼

s ðTÞ sf ; 2 f0 ðÞ

ð28Þ

which is the same result as Yafet66) in the electron-spin resonance (ESR).67) Figure 6 shows the temperature dependence of s ðTÞ=sf . In the normal state above Tc , the spinrelaxation time s ðTÞ coincides with sf , while in the superconducting state below Tc , s ðTÞ becomes longer with 10

(a)

1.0

χs

τs /τsf

0.5

5

1

0

0

2 f0(Δ)

0

0.5

T/Tc

0.5

1

T/Tc 10

ΔR s /Rs (Tc)

(b)

5

1 0.4

0.5

0.6

0.7

0.8

0.9

1

T/Tc Fig. 6. (a) Spin-relaxation time s as a function of temperature T in the superconducting state. The inset shows s and 2 f0 ðÞ vs T, which are used to calculate s . (b) Spin accumulation signal Rs as a function of T in a F/I/S/I/F structure. The values of s and Rs are normalized to those at the superconducting critical temperature Tc .

ð29Þ

indicating that the spin splitting is larger by the factor 1=½2 f0 ðÞ than that in the normal state. The detected voltage V2 by F2 at distance L is given by V2 ¼ P2 S ðLÞ for the P and AP orientations, and therefore the spin accumulation signal Rs detected by F2 at distance L becomes28) Rs ¼ P1 P2

RN expðL=N Þ: 2 f0 ðÞ

ð30Þ

The above result is readily obtained by the replacement N ! ½2 f0 ðÞN in the normal-state result of eq. (15), and originates from the fact that the carrier density decreases in proportion to 2 f0 ðÞ, and SCs become a low carrier system for spin transport. Figure 6(b) shows the temperature dependence of Rs below Tc . The rapid increase of Rs below Tc reflects the strong T-dependence in the resistivity of QPs. When the spin splitting S  ð1=2ÞeP1 RN I=½2 f ðÞ at x ¼ 0 becomes comparable to or larger than , the superconductivity is suppressed by pair breaking due to spin accumulation,57) so that Rs deviate from the curve of Fig. 6. Large enhancement in Rs has observed in recent experiments using nonlocal devices of Co/I/Al/I/Co68) and Py/I/Al/I/Py69) in the superconducting state. 4.

1

RN eI expðjxj=N Þ; 2 f0 ðÞ

Spin Hall Effect

The basic mechanism of the anomalous Hall effect (AHE) is the relativistic interaction between the spin and orbital motion of electrons (spin–orbit interaction) in metals or semiconductors. Conduction electrons in a crystal are scattered by local potentials created by defects or impurities in crystals. The spin–orbit interaction at local potentials causes a spin-asymmetry in the scattering of conduction electrons.70) In ferromagnetic materials, up-spin (majority) electrons are scattered preferentially in one direction and down-spin (minority) electrons in the opposite direction, resulting in an anomalous Hall current that flows in the direction perpendicular to both the applied electric field and the magnetization directions. Another mechanism for AHE is the spin chirality in frustrated ferromagnets.71) Spin injection technique in nanostructure devices makes it possible to observe AHE in nonmagnetic conductors, which is called the spin Hall effect (SHE). If spin-polarized electrons flow in a nonmagnetic electrode (N), these electrons are deflected by spin–orbit scattering to induce spin and charge Hall currents in the transverse direction and accumulate spin and charge at the edges of N.72–77) In a spin injection device, SHE is observable in the following way. When a spin current without accompanying charge current

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J. Phys. Soc. Jpn., Vol. 77, No. 3

(pure spin current) is created in N via nonlocal spin injection, the up and down spin currents, which ďŹ&#x201A;ow in opposite directions, are deďŹ&#x201A;ected in the same direction to induce a charge current in the transverse direction and accumulate charge on the edges of N. Inversely, when an unpolarized charge current ďŹ&#x201A;ows in N by an applied electric ďŹ eld, the up and down spin current, which ďŹ&#x201A;ow in the same direction, are deďŹ&#x201A;ected in the opposite direction to induce a spin current in the transverse direction and accumulates spin near the edges of N. In this way, the spin (charge) degree of freedom are converted to charge (spin) degrees of freedom due to spinâ&#x20AC;&#x201C;orbit scattering by impurities, which is important for spin electronic applications. In addition to these extrinsic SHEs, intrinsic SHEs have been intensively studied in semiconductors and transition metals which do not require impurities or defects.78â&#x20AC;&#x201C;84) In the following, we consider the eďŹ&#x20AC;ect of spinâ&#x20AC;&#x201C;orbit scattering on the spin and charge transport in a nonmagnetic metal (N) such as Cu and Al, and discuss SHE in the presence of spin current (or charge current) in N, taking into account the two mechanisms: side jump (SJ) and skew scattering (SS),70,85â&#x20AC;&#x201C;87) and derive formulas for the extrinsic SHE induced by spin and charge current.75) 4.1 Basic formulation The spinâ&#x20AC;&#x201C;orbit coupling in the presence of nonmagnetic impurities in a metal is derived as follows.88) The impurity potential VĂ°rĂ&#x17E; creates an additional electric ďŹ eld E Âź Ă°1=eĂ&#x17E;rVĂ°rĂ&#x17E;. When an electron passes through the ďŹ eld with velocity p^ =m Âź Ă°h =iĂ&#x17E;r=m, the electron feels the eďŹ&#x20AC;ective magnetic ďŹ eld Beff Âź Ă°1=mcĂ&#x17E;p^  E. This yields the spinâ&#x20AC;&#x201C;orbit coupling Vso Âź B   Beff Âź so   ½rVĂ°rĂ&#x17E;  r=i, where  is the Pauli spin operator and so is the spinâ&#x20AC;&#x201C;orbit coupling parameter. In the free-electron model, so Âź h 2 =4m2 c2 , which is corrected by the Thomas factor of one-half, and in real metals so is enhanced by several orders of magnitude for Bloch electrons.70) The total impurity potential UĂ°rĂ&#x17E; is the sum of the ordinary impurity potential and the spinâ&#x20AC;&#x201C;orbit potential: UĂ°rĂ&#x17E; Âź VĂ°rĂ&#x17E; Ăž Vso Ă°rĂ&#x17E;. The one-electron Hamiltonian H in the presence of the impurity potential UĂ°rĂ&#x17E; is given by X XX 0 HÂź

k ayk ak Ăž Uk0 k ayk0 0 ak : Ă°31Ă&#x17E; k;

k;k0 ; 0

Here, the ďŹ rst term is the kinetic energy of conduction electrons with energies k Âź Ă°h kĂ&#x17E;2 =2m  "F , and the second term describes the scattering of conduction electrons whose 0 scattering amplitude Uk0 k is given by  X iĂ°kk0 Ă&#x17E;ri 0 Uk0 k Âź Vimp 0  Ăž i so  0   Ă°k  k0 Ă&#x17E; e ; Ă°32Ă&#x17E; P

i iĂ°kk0 Ă&#x17E;ri

are the P matrix elements of weak where Vimp i e -function potential VĂ°rĂ&#x17E; Vimp i Ă°r  ri Ă&#x17E; for impurities at position ri . The velocity of electrons is calculated from the velocity operator v^ Âź dr=dt Âź Ă°1=ih Ă&#x17E;½r; H by taking the matrix element between the scattering state jkĂž i in the presence of impurities:89) vk Âź hkĂž j^vjkĂž i Âź

h k Ăž !k ; m

Ă°33Ă&#x17E;

S. T AKAHASHI and S. M AEKAWA

where the ďŹ rst term is the usual velocity and the second term is the anomalous velocity arising from the spinâ&#x20AC;&#x201C;orbit scattering !k Âź

so Ăž hk j  rVjkĂž i; h

Ă°34Ă&#x17E;

with a scattering state jkĂž i in the Born approximation X 0 Vimp eiĂ°kk Ă&#x17E;ri X i jkĂž i Âź jki Ăž ; Ă°35Ă&#x17E; jk0 i



k0 Ăž i k k0 where jki is a one-electron state of conduction electrons. Substituting eq. (35) into eq. (34), the anomalous velocity is calculated up to ďŹ rst order in so :

h k  SJ !k Âź H    ; Ă°36Ă&#x17E; m with the dimensionless coupling parameter of the side jump SJ H Âź

h  so  so Âź ; 2"F tr0 kF limp

Ă°37Ă&#x17E;

where  so Âź kF2 so the dimensionless spinâ&#x20AC;&#x201C;orbit coupling 2 parameter, tr0 Âź 1=½ð2=h Ă&#x17E;nimp NĂ°0Ă&#x17E;Vimp  is the momentum relaxation time due to impurity scattering, and limp is the mean-free path. Introducing the current operator for electrons with spin  X h k  ^J Âź e Ăž !k ayk ak ; Ă°38Ă&#x17E; m k (e Âź jej is the electronic charge), the total charge current Jq Âź J" Ăž J# and the total spin current Js Âź J"  J# are expressed as  ^  J0s ; Jq Âź J0q Ăž SJ Ă°39Ă&#x17E; H z  ^  J0q ; Ă°40Ă&#x17E; Js Âź J0s Ăž SJ H z where J0q and J0s are the charge and spin currents which do not include the side-jump contribution: X h k  J0q Âź e fk" Ăž fk# ; Ă°41aĂ&#x17E; m k X h k  J0s Âź e fk"  fk# ; Ă°41bĂ&#x17E; m k and fk Âź hayk ak i is the distribution function of an electron with energy k and spin . The second terms in eqs. (39) and (40) are the charge and spin Hall currents induced by side jump. In addition to the side jump contribution, there is the skew scattering contribution which originates from the anisotropic scattering due to the spinâ&#x20AC;&#x201C;orbit interaction and modiďŹ es the distribution function. The distribution function fk is calculated based on the Boltzmann transport equation in the next section. The spin density S accumulated in N is given by i X Xh SÂź hayk" ak" i  hayk# ak# i Âź Ă° fk"  fk# Ă&#x17E;: Ă°42Ă&#x17E; k

k

4.2 Scattering probability and Boltzmann equation The Boltzmann transport equation in steady state has a form

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J. Phys. Soc. Jpn., Vol. 77, No. 3

eE  rk fk Âź h



@ fk @t

S. T AKAHASHI and S. M AEKAWA

 Ă°43Ă&#x17E;

1=tr Âź

where vk Âź h k=m, E is the external electric ďŹ eld, and the right-hand term is the collision term due to impurity scattering. The carrier scattering by impurities can be described by the T^ matrix, which is in the second-order Born approximation: " # X Vk0 q Vqk hk0  0 jT^ jki Âź Vk0 k Ăž  0  q k  q Ăž i

1=sf Ă°Ă&#x17E; Âź

vk  r fk Ăž

;

0

where Vk0 k Âź hk0 jVjki. The scattering probability Pk0 k from the state jki to the state jk0  0 i is calculated by 0

Pk0 k Âź

2 nimp jhk0  0 jT^ jkij2 Ă° k  k0 Ă&#x17E;; h

and is related to the change of the distribution function fk Ă°rĂ&#x17E; due to impurity scattering as   X 0 @ fk 0  0 Âź P kk0 fk  0  Pk0 k fk @t scatt k0  0 X  0 Âź Pk0 k Ă°1Ă&#x17E; fk0 0  fk k0  0

Ăž

X

 0 Pk0 k Ă°2Ă&#x17E; fk0  Ăž fk ;

Ă°1Ă&#x17E; P Âź kk0

1 Ă°1 Ăž 2  2so =3Ă&#x17E;; tr0

Ă°48aĂ&#x17E;

Ă°1Ă&#x17E; P"# Âź kk0

 2so Ă°1 Ăž cos2 Ă&#x17E;; 3tr0

Ă°48bĂ&#x17E;

0

k0  0

scatt

Ăž i so Vk0 k Ă°k  k0 Ă&#x17E;   0  ;

X X k0

with the angle  between k and the x axis. Then, the Boltzmann equation (43) with the collision term (47) is74,91) 0 @ fk eE @ fk gĂ°1Ă&#x17E; f 0  fk  Ăž Âź  k  k ; Ă°49Ă&#x17E; h @r @k tr sf Ă°Ă&#x17E; where the ďŹ rst term in the r.h.s. describes the momentum relaxation due to impurity scattering and the second term the spin relaxation due to spinâ&#x20AC;&#x201C;ďŹ&#x201A;ip scattering. Since tr  sf , the momentum relaxation occurs ďŹ rst and is followed by the slow spin relaxation. The ďŹ rst-order solution due to momentum relaxation is obtained as   eE Ă°1Ă&#x17E; 0 gk tr vk  r Ăž  rk fk : Ă°50Ă&#x17E; h

vk 

0 The distribution function fk is a local equilibrium one  with Fermi energy "F Ă°rĂ&#x17E; Âź "F Ăž  "F Ă°rĂ&#x17E; shifted by "F Ă°rĂ&#x17E; from equilibrium for the up and down spin bands, i.e., 0 fk Âź f0 Ă° k   "F Ă&#x17E;, which is expanded as 0 fk f0 Ă° k Ă&#x17E;  

Ă°44Ă&#x17E;

k0  0

where the ďŹ rst term in the brackets is the scattering-in term (k0  0 ! k) while the second term is the scattering-out term (k ! k0  0 ), and  0  Ă°1Ă&#x17E; 0

Pk k

0 Pk0 k Ă°2Ă&#x17E;

2 2 nimp Vimp Âź h  ½  0 Ăž j so Ă°k0  kĂ&#x17E;    0 j2  Ă° k0  k Ă&#x17E;;

Ă°45bĂ&#x17E;

are the ďŹ rst-order symmetric and the second-order asymmetric contributions to the scattering probability. In solving the Boltzmann transport equation, it is convenient to separate fk into three parts90) Ă°2Ă&#x17E; 0 fk Âź fk Ăž gĂ°1Ă&#x17E; k Ăž gk ;

"N Ă°rĂ&#x17E; Âź "F Ăž e Ăž "F Ă°rĂ&#x17E;;

Ă°53aĂ&#x17E;

#N Ă°rĂ&#x17E;

Ă°53bĂ&#x17E;

Âź "F Ăž e  "F Ă°rĂ&#x17E;;

r2 Ă°"N  #N Ă&#x17E; Âź

0 fk

Ă°1Ă&#x17E; RandĂ°iĂ&#x17E;gk and gk dk Âź

Ă°52Ă&#x17E;

and the electric potential  (E Âź r). The spinâ&#x20AC;&#x201C;ďŹ&#x201A;ip scattering by the spinâ&#x20AC;&#x201C;orbit interaction causes a slow relaxation for spin accumulation "N  #N Âź 2 N . Substituting eq. (50) in the Boltzmann equation (49) and summing over k, one obtains the spin diďŹ&#x20AC;usion equation:

Ă°46Ă&#x17E;

where is an undirectional distribution function deďŹ ned by the average of fk with respect to the solid angle k of k: Z dk 0 fk Âź fk ; 4

@ f0 Ă° k Ă&#x17E; vk  rN Ă°rĂ&#x17E;; @ k

with the ECP

2

Ă°2Ă&#x17E; 3 Âź NĂ°0Ă&#x17E; nimp Vimp h  ½ so Ă°k0  kĂ&#x17E;      0 Ă° k0  k Ă&#x17E;;

Ă°51Ă&#x17E;

where f0 Ă° k Ă&#x17E; is the Fermi distribution function. Therefore, eq. (50) becomes gĂ°1Ă&#x17E; k tr

Ă°45aĂ&#x17E;

@ f0 Ă° k Ă&#x17E; "F Ă°rĂ&#x17E;; @ k

with N Âź and

1 Ă°"  #N Ă&#x17E;; N2 N

Ă°54Ă&#x17E;

pďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192; Dsf , D Âź Ă°1=3Ă&#x17E;tr v2F , vF is the Fermi velocity, 1=sf Âź h1=sf Ă°Ă&#x17E;iav Âź 4  2so =Ă°9tr0 Ă&#x17E;:

Ă°55Ă&#x17E;

gĂ°2Ă&#x17E; k

are directional distribution functions, i.e., 0, and are associated with the ďŹ rst-order and the second-order transitions, respectively, which are calculated in the following way. 4.2.1 First-order solution The ďŹ rst term in eq. (44) is written as 0 X 0 gĂ°1Ă&#x17E; f 0  fk k Ă°1Ă&#x17E; 0  0  Ă°1Ă&#x17E; 0 P 0 f  P 0 f  k ; Ă°47Ă&#x17E; Âź  k k kk kk tr sf Ă°Ă&#x17E; k0  0

where tr is the transport relaxation time and sf Ă°Ă&#x17E; is the spinâ&#x20AC;&#x201C;ďŹ&#x201A;ip relaxation time given by

4.2.2 Second-order solution The second-order term in the Boltzmann equation is X 0 Ă°2Ă&#x17E; Ă°1Ă&#x17E;  0  Ă°2Ă&#x17E; Ă°1Ă&#x17E; ½Pk0 k Ă°1Ă&#x17E; Ă°gĂ°2Ă&#x17E; Ă°56Ă&#x17E; k  gk0  0 Ă&#x17E;  Pk0 k Ă°gk Ăž gk0  0 Ă&#x17E; Âź 0: k0  0

Making use of eqs. (45a), (45b), and (52), the solution of the second-order (skew scattering) term becomes SS gĂ°2Ă&#x17E; k Âź H tr

@ f0 Ă° k Ă&#x17E; Ă°   vk Ă&#x17E;  rN Ă°rĂ&#x17E;;

k

Ă°57Ă&#x17E;

where SS H is the dimensionless parameter of skew scattering

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J. Phys. Soc. Jpn., Vol. 77, No. 3

 so NĂ°0Ă&#x17E;Vimp : SS H Âź Ă°2=3Ă&#x17E;

Ă°58Ă&#x17E;

4.3 Spin and charge Hall currents Using the solutions of the Boltzmann equation in the preceding sections, the distribution function becomes @ f0 Ă° k Ă&#x17E; N Ă°rĂ&#x17E; fk f0 Ă° k Ă&#x17E;   @ k @ f0 Ă° k Ă&#x17E;   Ăž tr vk  SS Ă°59Ă&#x17E; H    vk  rN Ă°rĂ&#x17E;; @ k from which the spin and charge currents in eq. (41) z  jq  and J0q Âź jq Ăž are calculated as J0s Âź js Ăž SS H ½^ SS H ½^z  js , where z^ the polarization vector and the second terms are the Hall spin and charge currents induced by the charge and spin currents: jq Âź N E;

js Âź 

N r N ; e

Ă°60Ă&#x17E;

where N Âź 2e2 NĂ°0Ă&#x17E;D is the electrical conductivity and N Âź Ă°1=2Ă&#x17E;Ă°"N  #N Ă&#x17E; is the chemical potential shift. Therefore, the total charge and spin currents in eqs. (39) and (40) are written as  Jq Âź jq Ăž H z^  js ; Ă°61Ă&#x17E;  Ă°62Ă&#x17E; Js Âź js Ăž H z^  jq ; SS where H Âź SJ H Ăž H . Equations (61) and (62) indicate that the spin current js induces the transverse charge current jH z  js , while q Âź H ½^ the charge current jq induces the transverse spin current jH z  jq , as shown in Fig. 7, and are expressed in the s Âź H ½^ matrix forms

Ex Jq;x xx xy ; Ă°63Ă&#x17E; Âź Js;y xy xx ry N =e

Js;x xx xy rx N =e Âź : Ă°64Ă&#x17E; Jq;y Ey xy xx

y

charge current

x z

(a) Spin-current induced spin Hall effect (SHE)

y

spin current

z charge current

jq

Here, xx Âź N is the longitudinal conductivity and xy Âź N H is the Hall conductivity contributed from the sidejump and skew-scattering: SJ SS xy Âź xy Ăž xy ;

Ă°65Ă&#x17E;

where e2 Ă°66Ă&#x17E; so ne ; h e2 2 SS kF limp hNĂ°0Ă&#x17E;Vimp i; Ă°67Ă&#x17E; Âź SS so ne xy H N Âź 3 h with ne Âź Ă°4=3Ă&#x17E;NĂ°0Ă&#x17E;"F the carrier (electron) density. It is SJ noteworthy that the side-jump conductivity xy is independent of impurity concentration. The ratio of the SJ and SS Hall conductivities is SJ Âź SJ xy H N Âź

SJ xy SS xy

Âź

3 1 1 nimp hĂ°NĂ°0Ă&#x17E;Vimp Ă&#x17E;2 i Âź2 : Ă°68Ă&#x17E; 2 kF limp hNĂ°0Ă&#x17E;Vimp i ne hNĂ°0Ă&#x17E;Vimp i

When the impurity potentials have a narrow distribution around a positive (or negative) potential and jhNĂ°0Ă&#x17E;Vimp ij  1, the SS contribution dominates the SJ contribution because of ne  nimp (kF limp  1). On the other hand, when the impurity potentials are distributed symmetrically around zero and jhNĂ°0Ă&#x17E;Vimp ij  1, then the SJ contribution dominates the SS contribution. 2 2 2 The spin Hall resistivity H Âź xy =Ă°xx Ăž xy Ă&#x17E; xy =xx has the linear and quadratic terms in N representing the contributions from side-jump and skew scattering: H Âź aSS N Ăž bSJ 2N ;

Ă°69Ă&#x17E;

where aSS Âź

2  so hNĂ°0Ă&#x17E;Vimp i; 3

bSJ Âź

2 e2  so kF : 3 h

Ă°70Ă&#x17E;

4.4 Spinâ&#x20AC;&#x201C;orbit coupling parameter It is interesting to note that the product N N is expressed in terms of the spinâ&#x20AC;&#x201C;orbit coupling parameter  so as55,76,94) pďŹ&#x192;ďŹ&#x192;ďŹ&#x192; pďŹ&#x192;ďŹ&#x192;ďŹ&#x192; rďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192; 3 RK sf 3 3  RK 1 N N Âź ; Ă°71Ă&#x17E; Âź 2 kF2 tr 4 kF2  so where RK Âź h=e2  25:8 k is the quantum resistance. The relation (71) provides a method of evaluating the spinâ&#x20AC;&#x201C;orbit coupling in nonmagnetic metals. Using the experimental data of N and N for Al, Cu, Ag, and Au in eq. (71), we obtain the values of the spinâ&#x20AC;&#x201C;orbit coupling parameter  so in those metals, which are listed in Table I together with the data of N and N . Although N (N ) has diďŹ&#x20AC;erent values for the same material but diďŹ&#x20AC;erent samples,  so has almost the same value:  so  0:01 for Al,  so  0:03 { 0:04 for Cu, and  so  0:1 for Ag. We note that the values of  so estimated by the spin injection method are 102 â&#x20AC;&#x201C;103 times as large as the free-electron values.

spin current

js

S. T AKAHASHI and S. M AEKAWA

x

(b) Charge-current induced spin Hall effect (SHE) Fig. 7. (a) Spin-current induced spin Hall eďŹ&#x20AC;ect (SHE) in which the spin current js ďŹ&#x201A;owing along the x direction with the polarization parallel to the z axis induces the charge current jH q in the y-direction. (b) Chargecurrent induced SHE in which the charge current jq along the x direction induces the spin current jH s in the y-direction with the polarization parallel to the z-axis.

4.5 Nonlocal spin Hall eďŹ&#x20AC;ect We consider a nonlocal spin-injection Hall device shown in Fig. 8.55,76,94,95) The magnetization of F at x Âź 0 points in the z-direction. The spin injection is made by ďŹ&#x201A;owing the current I from F to the left end of N (the negative direction of x), while the Hall voltage (VH Âź VHĂž  VH ) is measured by the Hall probe in the right side of N (x > 0). In the

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J. Phys. Soc. Jpn., Vol. 77, No. 3

Table I. Spinâ&#x20AC;&#x201C;orbit coupling parameter  so of Cu, Al, and Ag. Here we use the Fermi momenta: kF Âź 1:75  108 cm1 (Al), 1:36  108 cm1 (Cu), 1:20  108 cm1 (Ag), and 1:21  108 cm1 (Au).92Ă&#x17E; N (nm)

N (m cm)

tr =sf

 so

Ref.

Al Al

650 455

5.90 9.53

0:36  104 0:36  103

0.009 0.008

8 96

Al

705

5.88

0:33  103

0.008

96

Cu

1000

1.43

0:70  103

0.040

7

Cu

1500

1.00

0:64  103

0.037

46

Cu

546

3.44

0:41  103

0.030

17

Ag

162

4.00

0:55  102

0.113

93

Ag

195

3.50

0:50  102

0.107

93

I

+

VH + +

+

+

+

+

I

Is N F

â&#x2C6;&#x2019; â&#x2C6;&#x2019;

â&#x2C6;&#x2019;

L m//z

â&#x2C6;&#x2019;

â&#x2C6;&#x2019;

â&#x2C6;&#x2019;

VHâ&#x2C6;&#x2019;

Fig. 8. (Color online) Nonlocal spin-injection Hall device. The magnetization of F is pointed perpendicular to the plane. The nonlocal Hall voltage VH Âź VHĂž  VH is generated in the transverse direction by injection of pure spin current Is .

positive region of x, the pure spin current js ďŹ&#x201A;ows in the xdirection without accompanying the charge current jq . Therefore, from eqs. (61) and (62), Js  Âź js and  Jq Âź H z^  js Ăž N E; Ă°72Ă&#x17E; where the ďŹ rst term is the Hall current induced by the spin current, and the second term is the Ohmic current which builds up in the transverse direction as opposed to the Hall current. In an open circuit condition in the transverse direction, the y component of Jq in eq. (72) vanishes, which leads to the relation Ey Âź H N js between the Hall electric ďŹ eld Ey and the spin current js Âź Ă° js ; 0; 0Ă&#x17E;. Integrating Ey over the width of N, we obtain the Hall voltage at x Âź L VH Âź H wN N js ;

Ă°73Ă&#x17E;

where wN is the width of N. When the F/N junction is a tunnel junction, the spin current at x Âź L is   1 I j s P1 Ă°74Ă&#x17E; expĂ°L=N Ă&#x17E;; 2 AN so that the nonlocal Hall resistance RH Âź VH =I is given by55,76,94,95) RH Âź

1 N L=N P1  H e : 2 dN

When the junction is a metallic contact,94)   1 p F RF N L=N RH Âź e : H 2 1  p2F RN dN

Ă°75Ă&#x17E;

Ă°76Ă&#x17E;

S. T AKAHASHI and S. M AEKAWA

For the parameter values P1  0:3,  so  0:1, N  1 m cm, and dN  10 nm, RH is of the order of 0.1â&#x20AC;&#x201C;1 m, indicating that the spin-current induced SHE is measurable in nonlocal Hall devices. Recently, the spin Hall eďŹ&#x20AC;ect has been observed using nonlocal spin injection techniques in CoFe/Al,96,97) Py/Cu/ Pt,98) and FePt/Au99) lateral Hall devices, and ferromagnetic resonance (FMR) technique in a Py/Pt bilayer ďŹ lm.100) 4.6 Spin Hall eďŹ&#x20AC;ect in superconductors It is interesting to see whether SHE appears in the superconducting state. The spin current carried by QPs in a SC is deďŹ&#x201A;ected by spinâ&#x20AC;&#x201C;orbit impurity scattering to accumulate the QP charge (charge imbalance) in the transverse direction. The QP charge accumulation is compensated by the Cooper pair charge due to overall charge neutrality, thereby creating the electric potential to maintain the ECP of pairs to be constant in space (otherwise the pairs are accelerated). This spin and charge coupling leads to the spin Hall eďŹ&#x20AC;ect in SCs.75) In a nonlocal spin Hall device in which N is replaced by SC in Fig. 8, the nonlocal Hall voltage is very much enhanced by the onset of superconductivity below the superconducting critical temperature Tc . The calculated Hall voltage VH induced by the spin current carried by QPs is given by75)  SJ  1 H

s Ă°TĂ&#x17E; SS N L=N RH Âź P1  e ; Ă°77Ă&#x17E; Ăž 2 2 f0 Ă°Ă&#x17E; ½2 f0 Ă°Ă&#x17E;2 H dN where f0 Ă°Ă&#x17E; is the QP population at the gap energy  and

s Ă°TĂ&#x17E; is given in eq. (27). The spin Hall resistivity RH for QPs increases very rapidly in the superconducting state below Tc , reďŹ&#x201A;ecting the strong T-dependence of spin and charge imbalance of QPs in a SC. 5.

Superconducting  Qubit

In quantum computers, a quantum coherent two-level system is a basic element used as a quantum bit (qubit). The superposition of the two-level states is utilized in the processing of quantum computing. So far various qubits have been proposed, e.g., ion traps, nuclear spins, and photons. Among them, solid state qubits have the advantage of scalability and ďŹ&#x201A;exibility of layout. Recently, qubits based on the Josephson eďŹ&#x20AC;ect have been proposed; one is a charge qubit which uses the charging eďŹ&#x20AC;ect of excess Cooper-pairs in a box,101) and another is a ďŹ&#x201A;ux qubit which uses the superconducting phase in a superconducting loop with three Josephson junctions.102) In the ďŹ&#x201A;ux qubit, a quantum coherent two-level state is realized in the superconducting phase space under the external magnetic ďŹ eld at which the loop ďŹ&#x201A;ux corresponds to half the unit magnetic ďŹ&#x201A;ux. Microwave-induced qubit rotation (Rabi oscillation), the direct coupling between the two qubits, and the entangled states have been demonstrated.103â&#x20AC;&#x201C;107) Recent advance in nanofabrication techniques has brought a variety of Josephson devices using ferromagnets.108,109) In a Josephson junction with a ferromagnet sandwiched between SCs, a novel phenomenon, â&#x20AC;&#x2DC;â&#x20AC;&#x2DC;â&#x20AC;&#x2122;â&#x20AC;&#x2122; state, arises from the interplay of ferromagnetism and superconductivity. In the -junction, the current-phase relation is shifted by  from that of a conventional Josephson junction (0-junction).

031009-10


SPECIAL TOPICS

U Âź EJ0 Ă°cos 1 Ăž cos 2 Ă&#x17E;  EJ cosĂ° Ăž Ă&#x17E; Ăž Ă°  ext Ă&#x17E;2 =Ă°2Ls Ă&#x17E;;

Ă°79Ă&#x17E;

where  is the total ďŹ&#x201A;ux in the loop, ext is the external ďŹ&#x201A;ux, and Ls is the self-inductance of the loop. A single-valuedness of the wave function along the loop gives the relation 1 Ăž 2 Ăž  Âź 2Ă°=0 Ă&#x17E;  2l;

Ă°80Ă&#x17E;

where 0 is the unit ďŹ&#x201A;ux and l is an integer. Using eq. (80) in eq. (79), we obtain U Âź EJ0 Ă°cos 1 Ăž cos 2 Ă&#x17E;    Ă°  ext Ă&#x17E;2 Ăž EJ cos 2  1  2 Ăž : Ă°81Ă&#x17E; 0 2Ls Minimizing U with respect to , we obtain the selfconsistent equation   sin 2Ă°=0 Ă&#x17E;  1  2 Âź 2Ă°  ext Ă&#x17E;=0 ; Ă°82Ă&#x17E; where  Âź EJ =EJ0 and  Âź Ă°2=0 Ă&#x17E;I0 Ls . The numerical solution  Âź Ă°1 ; 2 Ă&#x17E; of eq. (82) is used to calculate U Âź UĂ°1 ; 2 Ă&#x17E; as a function of 1 and 2 . In the following, we assume Ec0 =Ec Âź EJ =EJ0 , and take  Âź 0:8 and  Âź 3:0  103 . The value of  is controllable by changing the junction area, the barrier height, and the thickness of the ferromagnet. The value of  is suitable for the micrometer-size loop and the Josephson junction with the critical current of several hundred nA.

S. T AKAHASHI and S. M AEKAWA

F

(a) S

S I

I

S 2

(b)

2

(c)

1

1

θ 2 /Ď&#x20AC;

θ 2 /Ď&#x20AC;

0

â&#x20AC;&#x201C;1 â&#x20AC;&#x201C;2 â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;1

0

θ1 /Ď&#x20AC;,

1

2 1

2

ÎŚ ext = 0

â&#x20AC;&#x201C;1

â&#x20AC;&#x201C;1

0

θ1 /Ď&#x20AC;,

1

2

3

(e)

0

â&#x20AC;&#x201C;2 (â&#x20AC;&#x201C;1,â&#x20AC;&#x201C;1)

â&#x20AC;&#x201C;2 â&#x20AC;&#x201C;2

2

3

(d)

0

â&#x20AC;&#x201C;1

U/E 0J

Recent experimental observation of the  state110â&#x20AC;&#x201C;113) in Josephson junctions is quite promising for practical application of the  state to superconducting quantum devices. In particular, in a superconducting loop which consists of a -junction, the two level states are formed in the phase space without external magnetic ďŹ elds because a half magnetic ďŹ&#x201A;ux is spontaneously generated by the -junction.114,115) Here we consider a superconducting loop with two conventional Josephson junctions (0-junctions) and one junction as shown in Fig. 9(a). In the three-junction system, a qubit states are formed in the phase space without external magnetic ďŹ elds, and a small external magnetic ďŹ eld is enough to manipulate the state. These features lead to a smaller size of qubit that is resistant to decoherence by external noises. In the loop, the 0-junctions have the Josephson energy U0 Ă°Ă&#x17E; Âź EJ0 cos  and the current-phase relation is I Âź I0 sin , with the Josephson coupling energy EJ0 and the critical current I0 Âź Ă°2e=h Ă&#x17E;EJ0 . On the other hand, the -junction has U Ă°Ă&#x17E; Âź EJ cosĂ° Ăž Ă&#x17E; and I Âź I sin  with the Josephson coupling energy EJ and the critical current I Âź Ă°2e=h Ă&#x17E;EJ . The total energy of the system consists of the electrostatic energy and the potential energy of the junctions. The electrostatic energy K is written as 4E0 KÂź  c 0 E Ăž 2Ec

c  2  @ @2 @2 0  Ă°Ec Ăž Ec0 Ă&#x17E; Ăž  2E ; Ă°78Ă&#x17E; c @12 @22 @1 @2 where 1 and 2 are the phase diďŹ&#x20AC;erences of the ďŹ rst and second 0-junctions,  is the phase diďŹ&#x20AC;erence of the junction, Ec0 Âź e2 =Ă°2C0 Ă&#x17E; and Ec Âź e2 =Ă°2C Ă&#x17E; are the charging energies of the zero and -junctions, and C0 and C are the capacitances of those junctions. The potential energy U is

U/E 0J

J. Phys. Soc. Jpn., Vol. 77, No. 3

1

ÎŚ ext = 0.05ÎŚ0

0

â&#x20AC;&#x201C;1 (0,0)

(θ1 /Ď&#x20AC;,θ2 /Ď&#x20AC;)

(1,1)

â&#x20AC;&#x201C;2 (â&#x20AC;&#x201C;1,â&#x20AC;&#x201C;1)

(0,0)

(θ1 /Ď&#x20AC;,θ2 /Ď&#x20AC;)

(1,1)

Fig. 9. (Color online) (a) Schematic diagram of a superconducting loop with a -junction and two 0-junctions. Contour plot of the total potential U=EJ0 in the 1 â&#x20AC;&#x201C;2 plane in zero external magnetic ďŹ eld (b) and in a small external magnetic ďŹ eld of ext Âź 0:050 (c) for  Âź 0:8 and  Âź 3:0  103 . The phase dependence of U along the dashed lines in (b) and (c) in zero external magnetic ďŹ&#x201A;ux (d) and in external magnetic ďŹ&#x201A;ux ext Âź 0:050 (e), respectively.

Figures 9(b) and 9(d) show the potential energy U in the 1 â&#x20AC;&#x201C;2 plane in zero external magnetic ďŹ&#x201A;ux (ext Âź 0). As seen in Fig. 9, UĂ°1 ; 2 Ă&#x17E; has double minima in the phase space. The degenerate j"i and j#i states at the minima have the circulating supercurrents of magnitude 0:8I0 in the clockwise and anticlockwise directions, respectively. Quantum tunneling between the degenerate states creates the bonding j0i / j"i Ăž j#i and antibonding j1i / j"i  j#i states, which are used as a quantum bit. The j0i and j1i states are the states of vanishing circulating current because of the superposition of j"i and j#i with equal weight, and have the energy gap E which is measured by the microwave resonance. From the numerical calculation, the resonance frequency E=h 4:4 GHz for EJ0 =Ec0 Âź 30. Figures 9(c) and 9(e) show the potential energy U in the 1 â&#x20AC;&#x201C;2 plane in the external magnetic ďŹ&#x201A;ux ext Âź 0:050 . As seen in Fig. 9(e), the degeneracy of the j"i and j#i states is lifted by small external magnetic ďŹ elds. In the bonding j0i state, the j"i component increases and the j#i component decreases, while in the antibonding j1i state, the components change in a reverse way. Therefore, spontaneous circulating currents ďŹ&#x201A;ow in the clockwise and anticlockwise directions at the j0i and j1i states, respectively, producing the spontaneous magnetic ďŹ&#x201A;ux in the loop. One can detect the states of the qubit through the measurement of the spontaneous ďŹ&#x201A;ux by a SQUID placed around the loop.

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SPECIAL TOPICS

A most prominent feature of three junction  qubits is the formation of the coherent two-level states without external magnetic fields; thus a small external magnetic field is enough to manipulate and detect the states, as compared to the external half unit flux 0 =2 required in the flux qubit.102) For example, a small magnetic field of the order of mT is enough to manipulate our qubit with a small dimensions of several 100 nm. This feature enables us to make qubits of smaller size which is advantageous for large-scale integration and small decoherence. 6.

Josephson Current through Strong Ferromagnets

A growing new interest is the Josephson current through strong ferromagnets, such as transition-metal ferromagnets or oxide ferromagnets. Spin-singlet cooper pairs is easily destroyed in strong ferromagnets, and the penetration of singlet pairs into strong ferromagnets is at most of the order of 1 nm. However, recent observation of the Josephson supercurrent through a half-metallic ferromagnet CrO2 116) with several hundreds nanometer thickness strongly suggested that spin-triplet superconductivity is induced in the half-metallic ferromagnet (HM). The mechanism for occurring spin-triplet superconductivity is a conversion from spin singlet to spin triplet pairs due to the spin-mixing or spin– flip scattering at the interface.117,118) The dynamics of magnetization and its effect on the proximity effect in S/F structures will open a new research field. In particular, the effects of spin wave emission and absorption and precession of magnetization on the proximity effect, the Andreev reflection,119) and the Josephson effect120–122) have been studied theoretically. In a Josephson junction with a HM between two conventional SC, the magnon-assisted tunneling electrons across the interface creates a composite state of a spin-triplet pair and magnon in the HM; a triplet-pair carries not only supercurrent but also spin current, while a magnon carries the backflow of spin current, and therefore both spin and charge currents are conserved throughout the junction. We expect that the Josephson current is amplified by coherent excitation of magnons due to precession of magnetization caused by microwaves at ferromagnetic resonance (FMR).121) 7.

Summary

In this article, we discuss a variety of new quantum phenomena caused by spin injection into normal metals and superconductors, and clarify the conditions for efficient spin injection, accumulation, and transport in nanostructured devices. Using the nonlocal spin injection technique, a pure spin current is created in nonmagnetic conductors without accompanying the charge current, so that we have the opportunity to observe the spin-current induced spin Hall effect (SHE) in nonmagnetic conductors (N) via the spin–orbit scattering. The observation of the SHE provides direct verification of the existence of spin current flowing in N. Inversely, the electrical current creates the spin current by SHE, which provides a spin-current source without using ferromagnetic materials. The nonlocal spin injection also makes it possible to realize a nonlocal spin manipulation, in which a small ferromagnet is attached to N and its magnetization direction is switched

S. T AKAHASHI and S. M AEKAWA

by the spin transfer torque due to nonlocal spin-current absorption. Spin injection technique provides a new type of ferromagnet/superconductor devices based on the interplay of superconductivity and spin accumulation created in SC. This technique allows us to control electronically the mutual influence between superconductivity and magnetism in these structures. The  state in a Josephson junction offers a new route for studying the interplay of superconductivity and magnetism. In particular, in a superconducting  qubit with a -junction in a superconducting circuit, a quantum coherent two-level state is realized without an external magnetic field, which allows us to reduce the size of the qubit for small decoherence. Development of new spin-based devices described in this article is a new challenge in the research field of spin electronics and quantum computing. Acknowledgment The authors would like to thank M. Ichimura, H. Imamura, J. Ieda, T. Yamashita, M. Mori, S. Hikino, R. Sugano, J. Martinek, G. E. W. Bauer, T. Kimura, T. Seki, S. Mitani, Y. Otani, E. Saitoh, and K. Takanashi for helpful discussions and their collaboration. This work was supported by the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT), and the Next Generation Supercomputer Project of MEXT. Appendix: Electrochemical Potentials in F1/N/F2 The coefficients in the electrochemical potentials (ECPs) in §2.1 are presented. Using the matching conditions that the charge and spin currents are continuous at the interfaces of F1/N/F2, we determine the coefficients a1 and a2 in eq. (7) in the N electrode: ðP1 r1 þ pF1 rF1 Þð1 þ 2r2 þ 2rF2 ÞRN eI a1 ¼ ; ðA:1Þ ð1 þ 2r1 þ 2rF1 Þð1 þ 2r2 þ 2rF2 Þ  e2L=N ðP1 r1 þ pF1 rF1 ÞeL=N RN eI ; ðA:2Þ a2 ¼ ð1 þ 2r1 þ 2rF1 Þð1 þ 2r2 þ 2rF2 Þ  e2L=N where ri and rFi (i ¼ 1; 2) are the normalized resistances: ri ¼

1 Ri ; 1  P2i RN

rFi ¼

1 RFi : 1  p2Fi RN

ðA:3Þ

The spin accumulation voltage V2 detected by F2 is given by 2ðP1 r1 þ pF1 rF1 ÞðP2 r2 þ pF2 rF2 ÞeL=N ð1 þ 2r1 þ 2rF1 Þð1 þ 2r2 þ 2rF2 Þ  e2L=N  RN I; ðA:4Þ

V2 ¼

where ‘‘þ’’ and ‘‘’’ correspond to the parallel and antiparallel alignments of magnetizations, respectively. The coefficients, b1 and b2 , in ECP of eq. (8) in F1 and F2 are related to a1 and a2 through   b1 ¼ ðF1 =F1 Þ pF1 RF1 eI=2  ðRF1 =RN Þa1 ; ðA:5Þ  ÞðRF2 =RN Þa2 : ðA:6Þ b2 ¼ ðF2 =F2 The interfacial spin currents across junctions 1 and 2 are given by I1s ¼ ð2=eRN Þa1 ;

I2s ¼ ð2=eRN Þa2 ;

where a1 and a2 are given in (A·1) and (A·2).

031009-12

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J. Phys. Soc. Jpn., Vol. 77, No. 3

SPECIAL TOPICS

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S. T AKAHASHI and S. M AEKAWA

113) M. Weides, M. Kemmler, H. Kohlstedt, R. Waser, D. Koelle, R. Kleiner, and E. Goldobin: Phys. Rev. Lett. 97 (2006) 247001. 114) T. Yamashita, K. Tanikawa, S. Takahashi, and S. Maekawa: Phys. Rev. Lett. 95 (2005) 097001. 115) T. Yamashita, S. Takahashi, and S. Maekawa: Appl. Phys. Lett. 88 (2006) 132501. 116) R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G. Miao, G. Xiao, and A. Gupta: Nature 439 (2006) 825. 117) M. Eschrig, J. Kopu, J. C. Cuevas, and G. Scho¨n: Phys. Rev. Lett. 90 (2003) 137003. 118) Y. Asano, Y. Tanaka, and A. A. Golubov: Phys. Rev. Lett. 98 (2007) 107002. 119) E. McCann and V. I. Fal’ko: Europhys. Lett. 56 (2001) 583. 120) J.-X. Zhu, Z. Nussinov, A. Shnirman, and A. V. Balatsky: Phys. Rev. Lett. 92 (2004) 107001. 121) S. Takahashi, S. Hikino, M. Mori, J. Martinek, and S. Maekawa: Phys. Rev. Lett. 99 (2007) 057703. 122) S. Hikino, S. Takahashi, M. Mori, J. Martinek, and S. Maekawa: Physica C 460 – 462 (2007) 1323. Saburo Takahashi was born in Gunma Prefecture, Japan in 1952. He obtained his B. Sc. (1976), M. Sc. (1978), and D. Sc. (1983) degrees from Tohoku University. He was a research associate (1984 – 2006) and an assistant professor since 2007 at Institute for Materials Research, Tohoku University. He has been working on the theoretical condensed matter physics on the interplay of superconductivity and magnetism. Recent works include theory of spin dependent transport, especially, nonlocal spin injection, spin Hall effect, and Josephson effect in hybrid nanostructures. Sadamichi Maekawa was born in Nara Prefecture, Japan in 1946. He obtained his B. Sc. (1969), M. Sc. (1971) degrees from Osaka University, and D. Sc. (1975) degree from Tohoku University. He was a research associate (1971–1982) and an associate professor (1982 –1988) at Institute for Materials Research, Tohoku University, and a professor (1988 –1997) at Faculty of Engineering, Nagoya University. Since 1997 he has been a professor at Institute for Materials Research, Tohoku University. His main research topics include theory of electronic properties in strongly correlated electron systems, in particular, hightemperature superconductors and orbital physics in transition metal oxides and theory of transport in magnetic nanostructures.

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Spin Current in Metals and Superconductors