Condensed Matter Physics 2016 Lecture 13/12: Charge and heat transport.
1. 2. 3. 4.
Theoretical tool: Boltzmann equation (review). Electrical and thermal conductivity in metals. Ballistic transport and conductance quantization. Integer quantum Hall effect.
References: Ashcroft & Mermin, 16, 26 Taylor & Heinonen, 8.1, 8.2, 8.4, 8.5, 9.1, 10.1, 10.2 Cohen and Louie*, 12.2 – 12.4, 16.2
* Fundamentals of Condensed Matter Physics (Cambridge University Press, 2016)
1. Theoretical tool: Boltzmann equation (review) Given a piece of metal, consider the local electron distribution f(k,r,t)
from external fields
Semiclassical phase space picture!
from scattering processes in the sample (impurity and phonon scattering)
1. Theoretical tool: Boltzmann equation (review) Given a piece of metal, consider the local electron distribution f(k,r,t)
from external fields
Semiclassical phase space picture!
from scattering processes in the sample (impurity and phonon scattering)
Given a piece of metal, consider the local electron distribution f(k,r,t)
from external fields
Semiclassical phase space picture!
Homogeneous sample in equilibrium: (Fermi-Dirac distribution)
from scattering processes in the sample (impurity and phonon scattering)
Given a piece of metal, consider the local electron distribution f(k,r,t)
from external fields
from scattering processes in the sample (impurity and phonon scattering)
Goal of transport studies: Determine f(k, r, t; T, , H) for a perturbed system. Use f to extract response functions! magnetic temperature
electric field
field
To properly include band structure effects, we should put a band index n on the distribution function f ––> fn. Ditto for scattering probabilities (two band indices to allow for interband scattering), energies, velocities, etc. ”For ease of notation”, these indices have been suppressed in the following.
Given a piece of metal, consider the local electron distribution f(k,r,t)
:
Given a piece of metal, consider the local electron distribution f(k,r,t)
:
Now, let’s turn to the collision term…
probability for scattering from k’ to k
time-reversal invariance:
Now, let’s turn to the collision term…
probability for scattering from k’ to k
time-reversal invariance:
Adding the drift and collision terms:
Boltzmann equation
Many experiments and applications are concerned with the steady-state response.
Drift processes must balance collision processes!
Now consider spatially slowly varying perturbations (electric field, temperature,…) Expand f in powers of and ∆T:
Many experiments and applications are concerned with the steady-state response.
Drift processes must balance collision processes!
Many experiments and applications are concerned with the steady-state response.
Drift processes must balance collision processes!
(no time-dependence in f0)
Many experiments and applications are concerned with the steady-state response.
Drift processes must balance collision processes!
(no time-dependence in f0)
Assume spatial variation in f only due to variation in temperature: Also use:
linea
rized
Bolt zma
nn e
quat io
n
2. Electrical and thermal conductivity in metals A. Elastic scattering against impurities (dilute impurity approximation) Scattering probability (Fermi’s golden rule):
impurity density
Assume no temperature gradient, hence no spatial variation in f:
since
by energy conservation
A. Elastic scattering against impurities (dilute impurity approximation)
Insert this expression into the linearized Boltzmann equation, and remove the term with a temperature gradient (no temperature variation by assumption!)
A. Elastic scattering against impurities (dilute impurity approximation)
write
Solve for How does enter the conductivity?
! Pull out the physics!
A. Elastic scattering against impurities (dilute impurity approximation) Current density
cross-sectional area
conductivity tensor with components i,j = x,y,z
A. Elastic scattering against impurities (dilute impurity approximation)
conductivity tensor with components i,j = x,y,z
Specialize to the case with no magnetic field diagonal conductivity tensor. Spherical Fermi surface: v and parallel to k. At low temperatures, ∂f0/∂E = – . It follows that
u Dr
d
r o f e
! a l mu
A. Elastic scattering against impurities (dilute impurity approximation)
Dr
u
for e d
la! u m
mean free path
= relaxation time = probability/unit time of an electron having a collision in which it loses any momentum gained from the applied electric field
What about effects from adding a magnetic field? Fascinating (and difficult!) subject! The change of the resistance due to an applied magnetic field: magnetoresistance.
Magnetoresistance is the tendency of a material to change the value of its electrical resistance in an externally-applied magnetic field. There are a variety of effects that can be called magnetoresistance: some occur in bulk non-magnetic metals and semiconductors, such as geometrical magnetoresistance, Shubnikov de Haas oscillations, or the common positive magnetoresistance in metals.[1] Other effects occur in magnetic metals, such as negative magnetoresistance in ferromagnets[2] or anisotropic magnetoresistance (AMR). Finally, in multicomponent or multilayer systems (e.g. magnetic tunnel junctions, giant magnetoresistance (GMR), Tunnel magnetoresistance (TMR), and Extraordinary magnetoresistance (EMR) can be observed.
Wikipedia
The typical increase in metallic resistance due to a magnetic field (positive magnetoresistance in metals) can be calculated directly from the linearized Boltzmann equation. (But rather challenging…)
B. Inelastic electron-phonon scattering We must put in a new collision term in the Boltzmann equation, coding for the inelastic e-p scattering!
Scattering probability (Fermi’s golden rule):
Fröhlich e-p interaction Hamiltonian
B. Inelastic electron-phonon scattering Scattering probability (Fermi’s golden rule):
B. Inelastic electron-phonon scattering To obtain the collision term: Sum (or integrate) over all k’, subtracting/adding outgoing/incoming scattering events (just as we did for electron-impurity scattering!):
Very cumbersom analysis of the resulting Boltzmann equation!
Instead, at this point, let’s do some qualitative observations at the blackboard… (…. the full next lecture on mesoscopic transport will also be on the blackboard!)
