Hong Kong Forum of Condensed Matter Physics:

Past, Present, and FutureDecember 20, 2006

Adam Durst

Department of Physics and AstronomyStony Brook University

Dirac Quasiparticles in Condensed Matter Physics

Hong Kong Forum of Condensed Matter Physics:

Past, Present, and FutureDecember 20, 2006

Adam Durst

Department of Physics and AstronomyStony Brook University

Dirac Quasiparticles in Condensed Matter Physics

(mostly d-wave superconductors)

Outline

I. Background

II. d-Wave Superconductivity

III. Universal Limit Thermal Conductivity (w/ aside on Graphene)

IV. Quasiparticle Transport Amidst Coexisting Charge Order

V. Quasiparticle Scattering from Vortices

VI. Summary

Dirac Fermions

Relativistic Fermions (electrons)

Massless Relativistic Fermions (neutrinos)

•We need only non-relativistic quantum mechanics and electromagnetism

•But in many important cases, the low energy effective theory is described by Dirac Hamiltonian and Dirac energy spectrum

•Examples include:

• Quasiparticles in Cuprate (d-wave) Superconductors

• Electrons in Graphene

• etc…

•Low energy excitations are two-dimentional massless Dirac fermions

What does this have to do with Condensed Matter Physics?

High-Tc Cuprate Superconductors

T

x

AF dSC

s-Wave Superconductor

Fully gapped quasiparticle excitations

d-Wave Superconductor

Quasiparticle gap vanishes at four nodal points

Quasiparticles behave more like massless relativistic particles than normal electrons

Two Characteristic Velocities

Quasiparticle Excitation Spectrum

Anisotropic Dirac Cone

d-Wave Superconductivity

Disorder-Induced Quasiparticles

N()

Disorder-InducedQuasiparticles

Density of States

L. P. Gorkov and P. A. Kalugin, JETP Lett. 41, 253 (1985)

Universal Limit Transport Coefficients

Disorder generates quasiparticles

Disorder scatters quasiparticlesDisorder-independent conductivities

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)

Disorder-dependent Disorder-independent

Low Temperature Thermal Conductivity Measurements

YBCO:

BSCCO:

Taillefer and co-workers, Phys. Rev. B 62, 3554 (2000)

Graphene

Single-Layer Graphite

Universal Conductivity?

Bare Bubble:

Novosolov et al, Nature, 438, 197 (2005)

max

(h/

4e2 )

1

0

(cm2/Vs)

0 8,000

2

4,000

15 devices

Missing Factor of !!!

Can vertex corrections explain this?

Shouldn’t crossed (localization) diagrams be important here?

Low Temperature Quasiparticle Transport in a d-Wave Superconductor with Coexisting Charge Density Wave Order

(with S. Sachdev (Harvard) and P. Schiff (Stony Brook))

STM from Davis Group, Nature 430, 1001 (2004)

Checkerboard Charge Order in Underdoped

Cuprates

x

T

underdoped

Hamiltonian for dSC + CDW

Current Project: Doubles unit cell

Future:

CDW-Induced Nodal Transition

K. Park and S. Sachdev, Phys. Rev. B 64, 184510 (2001)

Nodes survive but approach reduced Brillouin zone boundary

Nodes collide with their “ghosts” from 2nd reduced Brillouin zone

Nodes are gone and energy spectrum is gapped

Thermal Conductivity Calculation

Green’s Function

Disorder

Heat Current

Kubo Formula

4×4 matrix

Analytical Results in the Clean Limit

Beyond Simplifying Approximations

Realistic Disorder

Vertex Corrections

4×4 matrix

•Self-energy calculated in presence of dSC+CDW

•32 real components in all (at least two seem to be important)

•Not clear that these can be neglected in presence of charge order

Work in Progress with Graduate Student, Philip Schiff

Scattering of Dirac Quasiparticles from Vortices

H

x

2R

Two Length Scales

Scattering from Superflow

+

Aharonov-Bohm Scattering (Berry phase effect)

(with A. Vishwanath (UC Berkeley), P. A. Lee (MIT), and M. Kulkarni (Stony Brook))

Model and Approximations

•Account for neighboring vortices by cutting off superflow at r = R

•Neglect Berry phase acquired upon circling a vortex

- Quasiparticles acquire phase factor of (-1) upon circling a vortex

- Only affects trajectories within thermal deBroglie wavelength of core

•Neglect velocity anisotropy vf = v2

R

R r

Ps

Single Vortex Scattering

Momentum Space

Coordinate Space

Cross Section Calculation

•Start with Bogoliubov-deGennes (BdG) equation

•Extract Berry phase effect from Hamiltonian via gauge choice

•Shift origin to node center

•Separate in polar coordinates to obtain coupled radial equations

•Build incident plane wave and outgoing radial wave

•Solve inside vortex (r < R) and outside vortex (r > R) to all orders in linearized hamiltonian and first order in curvature terms

•Match solutions at vortex edge (r = R) to obtain differential cross section

Small by k/pF

Contributions to Differential Cross Section from Each of the Nodes

Calculated Thermal Conductivity

YBa2Cu3O6.9

9

Experiment (Ong and co-workers (2001)) Calculated

What about the Berry Phase?

Should be important for high field (low temperature) regime where deBroglie wavelength is comparable to distance between vortices

Over-estimated in single vortex approximation

Branch Cut

Better to consider double vortex problem

Elliptical Coordinate

s

Work in Progress with Graduate Student, Manas Kulkarni

Summary

•The low energy excitations of the superconducting phase of the cuprate superconductors are interesting beasts – Dirac Quasiparticles

•Cuprates provide a physical system in which the behavior of these objects can be observed

•In turn, the study of Dirac quasiparticles provides many insights into the nature of the cuprates (as well as many other condensed matter systems)