Massless Dirac Quasiparticles at the Interface of a Topological Insulator and a d-Wave Superconductor Adam C. Durst Department of Physics and Astronomy, Hofstra University
Introduction Condensed matter physicists (like me) use quantum mechanics to understand the electrical, thermal, optical, magnetic, structural, etc properties of complex materials and devices, for the purpose of advancing fundamental physics as well as developing new technology. My current research focuses on the properties of superconductors (materials that have been fascinating physicists for over a century) as well as topological insulators (fascinating materials that were only discovered in the past decade). The work reported here describes the thermal properties of the interface layer sandwiched between a topological insulator and a d-wave superconductor. Such a system is a playground for studying massless Dirac fermions, excitations that behave more like ultrarelativistic particles (i.e. neutrinos) than ordinary electrons.
TI-dSC Interface State
Topological Insulators Topological insulators behave as insulators in the interior, but due to spin-orbit interactions and time-reversal symmetry, have symmetryprotected conducting states on the surface.
Analytical Results in Large and Small |m| Limits For |m| >> G0
2D Topological Metal with Proximity-Induced d-Wave Superconductivity
≠
Exactly half the ordinary dSC expression, since the TI surface state is not spin-degenerate
Topologically distinct from ordinary insulators in the same sense as this trefoil knot is distinct from a circle.
Demonstration of the sense in which the TI surface topological metal is “half” of an ordinary 2DEG [L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008)]
For |m| << G0
Metals, Insulators, and Superconductors In a metal, like copper, electrons are free to travel far from their host atoms, thereby conducting electrical current in response to an applied voltage. As they travel, they occasionally scatter from imperfections in, and thermal vibrations of, the crystal lattice of atoms, resulting in a nonzero resistance to current flow.
For 3D TIs, like Bi2Se3, these surface states are 2D massless Dirac fermions, with spin locked at a right angle to momentum.
Ordinary dSC expression for a single isotropic node (rather than the usual four): (1+1)/4 = 1/2
Note: Ordinary dSC expression is
Disorder-independent in both limits
Interface of Topological Insulator and d-Wave Superconductor
Thermal Conductivity Calculation
Universal-Limit Thermal Conductivity as a Function of Chemical Potential
Green’s Function 4×4 matrix
TI-dSC Interface: 2D Topological Metal with Proximity-Induced d-Wave Superconductivity
In an insulator, like plastic, electrons are tightly bound to their host atoms and therefore unable to conduct electricity.
Numerical Results Comparison with Limits G0 = 0.01 v/a
Disorder Dependence Breakdown of independent node approx
Disorder
Heat Current
Massless Dirac Fermions Abound
Shrinking annular peak blurs into single isotropic peak
Greater disorder hastens transition
Kubo Formula In a superconductor, like YBCO, electrons work together, forming a collective state of matter that is robust to scattering and can therefore conduct electricity with zero resistance. This collective state also exhibits perfect diamagnetism, which is why magnets can be levitated above superconductors, as shown in the photo.
Disorder-dependent transition between disorder-independent limits
Quasiparticles of d-Wave Superconductor
Topological Insulator Surface State
• Four anisotropic Dirac points • Couples particle to hole • Dirac points tied to chemical potential
• Single isotropic Dirac point (for Bi2Se3 family) • Couples spin-up to spin-down • Can tune chemical potential through Dirac point
Summary s-Wave vs d-Wave Superconductors
Evolution of k-Space Structure of k0 Integrand
Universal-Limit Thermal Conductivity in Ordinary d-Wave Superconductors
Fully gapped quasiparticle excitations
• In both the large and small |m| limits, we obtain simple, disorder-independent, closed-form expressions for k0/T.
|m| >> G0
Ek
Four well-separated, anisotropic peaks
Same image, zoomed in by factor of 20
Anisotropic Dirac Cones k1
k2 Peaks grow more anisotropic and begin to curve around Fermi circle
Quasiparticle Excitation Spectrum Thermal Conductivity in Zero-Temperature Limit
Quasiparticle gap vanishes at four nodal points
Independent node approximation starts breaking down
Independent of disorder P. A. Lee, Phys. Rev. Lett. 71, 1887 (1993) M. J. Graf, S.-K. Yip, J. A. Sauls, and D. Rainer, Phys. Rev. B 53, 15147 (1996) A. C. Durst and P. A. Lee, Phys. Rev. B 62, 1270 (2000)
• The large-|m| expression is exactly half the value expected for an ordinary d-wave superconductor, a demonstration of the sense in which the TI surface topological metal is half of an ordinary 2DEG. • Numerical results for intermediate |m| reveal a two-stage transition between these limits that depends on disorder in a well-defined manner.
|m| = G0 Width and radius of annular peak now nearly equal
|m| << G0 Peaks have curved into each other, forming a single annular peak
• We have calculated the low-temperature thermal conductivity, k0, as a function of chemical potential, m, due to quasiparticle excitations of the proximity-induced superconducting state at the 2D interface of a topological insulator and a d-wave superconductor.
Annular peak blurs into a single isotropic peak at the origin – the isotropic node inherited from the TI surface state
Reference A. C. Durst, “Low-temperature thermal transport at the interface of a topological insulator and a d-wave superconductor,” Physical Review B 91, 094519 (2015)