Dirac Quasiparticles in
Condensed Matter Physics
Adam Durst
Department of Physics and Astronomy
Stony Brook University
Hong Kong Forum of
Condensed Matter Physics:
Past, Present, and Future
December 20, 2006
Dirac Quasiparticles in
Condensed Matter Physics
(mostly d-wave superconductors)
Adam Durst
Department of Physics and Astronomy
Stony Brook University
Hong Kong Forum of
Condensed Matter Physics:
Past, Present, and Future
December 20, 2006
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)
What does this have to do with Condensed Matter Physics?
•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
High-Tc Cuprate Superconductors
T
AF
dSC
x
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
d-Wave Superconductivity
Quasiparticle Excitation Spectrum
Two Characteristic Velocities
Anisotropic
Dirac Cone
Disorder-Induced Quasiparticles
Density of States
N(w)
Disorder-Induced
Quasiparticles
w
L. P. Gorkov and P. A. Kalugin, JETP Lett. 41, 253 (1985)
Universal Limit Transport Coefficients
Disorder generates quasiparticles
Disorder scatters quasiparticles
Disorder-dependent
Disorder-independent conductivities
Disorder-independent
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)
Low Temperature Thermal Conductivity Measurements
YBCO:
BSCCO:
Taillefer and co-workers, Phys. Rev. B 62, 3554 (2000)
Graphene
Single-Layer Graphite
Universal Conductivity?
max (h/4e2)
2
15 devices
Bare Bubble:
1
Missing Factor of p!!!
0
0
4,000
8,000
 (cm2/Vs)
Novosolov et al, Nature, 438, 197 (2005)
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))
Checkerboard Charge
Order in Underdoped
Cuprates
T
x
underdoped
STM from Davis Group, Nature 430, 1001 (2004)
Hamiltonian for dSC + CDW
Current Project:
Future:
Doubles unit cell
CDW-Induced Nodal Transition
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
K. Park and S. Sachdev, Phys.
Rev. B 64, 184510 (2001)
Thermal Conductivity Calculation
Green’s Function
4×4 matrix
Disorder
Heat Current
Kubo Formula
Analytical Results in the Clean Limit
Beyond Simplifying Approximations
Realistic Disorder
4×4 matrix
•Self-energy calculated in presence of dSC+CDW
•32 real components in all (at least two seem to be important)
Vertex Corrections
•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
(with A. Vishwanath (UC Berkeley), P. A. Lee
(MIT), and M. Kulkarni (Stony Brook))
Scattering from Superflow
2R
+
Aharonov-Bohm Scattering
(Berry phase effect)
Two Length Scales
H
x
Model and Approximations
•Account for neighboring vortices by cutting off superflow at r = R
Ps
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
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
Small by k/pF
•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
Contributions to Differential Cross Section from Each of the Nodes
Calculated Thermal Conductivity
Experiment (Ong and co-workers (2001))
YBa2Cu3O6.
99
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
Coordinates
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)