Boltzmann Transport

Boltzmann transport is the semiclassical kinetic theory of how quasiparticle occupations move through phase space and are redistributed by collisions. Its central object is not a single drift velocity but a band- and momentum-resolved distribution

$$f_n(\mathbf r,\mathbf k,t).$$

For fermionic Bloch quasiparticles, $0\le f_n\le1$. The Boltzmann equation balances streaming under the band dynamics against a collision integral:

$$\frac{\partial f_n}{\partial t} + \dot{\mathbf r}\cdot \boldsymbol\nabla_{\mathbf r}f_n + \dot{\mathbf k}\cdot \boldsymbol\nabla_{\mathbf k}f_n = \mathcal I_n[f].$$

Unlike a single drift velocity, the distribution resolves how velocity and scattering vary among states. Collisions scatter particles both out of and into each state, while electric fields, thermal gradients and boundaries drive different distortions. The approach requires resolved quasiparticles and a regime in which their coherent evolution can be reduced to occupations and transition rates.

Drude Theory gives the one-time benchmark. Transport Coefficients explains broader transport limits, and Kubo Formula supplies the correlation-function formulation.

Required background. Band Theory Overview supplies the Bloch-band energies, state labels, occupations, and velocities used in $f_n(\mathbf k)$ without rederivation.

Semiclassical Dynamics of Bloch Electrons gives the packet trajectories used in the streaming term. The kinetic equation combines them with occupations and collisions to predict a current.

Helpful background. Drude Theory supplies the one-time benchmark; Fermi Surface and Density of States support metallic phase-space and state counting; Phonons supplies lattice-scattering inputs; and Transport Coefficients Preview supplies general limit, constitutive, and Onsager taxonomy.

Distribution Functions and State Counting

Let $n$ label a band and let $\mathbf k$ lie in the first Brillouin zone. Spin, valley, and other discrete labels may be included in $n$ or supplied as explicit degeneracy factors, but never both.

For a $d$-dimensional bulk system, the carrier density is

$$n_{\mathrm c} = \sum_n \int_{\mathrm{BZ}} \frac{d^dk}{(2\pi)^d} f_n(\mathbf k).$$

This normalization gives density per $d$-dimensional volume. A two-dimensional sheet density and a three-dimensional volume density are not interchangeable.

At local equilibrium,

$$f_n^0(\mathbf r,\mathbf k) = \frac{1}{ \exp\left[ \dfrac{ \varepsilon_n(\mathbf k)-\mu(\mathbf r) }{ k_{\mathrm B}T(\mathbf r) } \right] +1 }.$$

The derivative

$$- \frac{\partial f^0} {\partial\varepsilon} = \frac{1}{ 4k_{\mathrm B}T } \operatorname{sech}^2 \left[ \frac{ \varepsilon-\mu }{ 2k_{\mathrm B}T } \right]$$

is positive, integrates to one over an unrestricted energy axis, and selects a window of width of order $k_{\mathrm B}T$ around $\mu$.

Charge and heat currents

For carriers with signed charge $q$, the uniform transport currents are

$$\mathbf j = q \sum_n \int_{\mathrm{BZ}} \frac{d^dk}{(2\pi)^d} \mathbf v_n(\mathbf k) f_n(\mathbf k),$$ $$\mathbf j_Q = \sum_n \int_{\mathrm{BZ}} \frac{d^dk}{(2\pi)^d} \left[ \varepsilon_n(\mathbf k)-\mu \right] \mathbf v_n(\mathbf k) f_n(\mathbf k).$$

The heat current transports energy relative to the electrochemical work per particle. In transverse thermal response, equilibrium magnetization currents must also be separated from transport currents; that refinement is outside the zero-field longitudinal derivation below.

Semiclassical Streaming

For an isolated nondegenerate band without Berry-curvature corrections,

$$\dot{\mathbf r} = \mathbf v_n(\mathbf k) = \frac{1}{\hbar} \boldsymbol\nabla_{\mathbf k} \varepsilon_n(\mathbf k),$$

and

$$\hbar\dot{\mathbf k} = q \left[ \mathbf E + \dot{\mathbf r} \times \mathbf B \right].$$

The wave packet moves with band velocity, while the Lorentz force changes crystal momentum. These equations assume that the external fields vary slowly across the packet and do not induce appreciable interband transitions.

Substitution into the kinetic equation gives

$$\frac{\partial f_n}{\partial t} + \mathbf v_n\cdot \boldsymbol\nabla_{\mathbf r}f_n + \frac{q}{\hbar} \left( \mathbf E + \mathbf v_n\times\mathbf B \right) \cdot \boldsymbol\nabla_{\mathbf k}f_n = \mathcal I_n[f].$$

Berry curvature can add an anomalous velocity and modify the phase-space measure. Strong fields can produce Landau quantization or magnetic breakdown. Those effects are not corrections to be inserted selectively: the equations of motion, density of states, orbital moment, and current must be updated consistently.

Collision Integrals

The collision integral is where microscopic scattering enters. It is a gain-minus-loss functional, not merely a linewidth.

One-particle transitions

For transitions between resolved states $a$ and $b$,

$$\mathcal I_a[f] = \sum_b \big[ W_{b\to a} f_b \left( 1-f_a \right) - W_{a\to b} f_a \left( 1-f_b \right) \big].$$

The factors $1-f$ enforce Pauli blocking. Static disorder, phonon absorption and emission, boundary scattering, and interband processes produce different transition kernels $W$.

Equilibrium requires stationarity, $\mathcal I[f^0]=0$. A sufficient stronger condition, assumed for the reciprocal collision models below, is pairwise detailed balance:

$$W_{b\to a} f_b^0 \left( 1-f_a^0 \right) = W_{a\to b} f_a^0 \left( 1-f_b^0 \right).$$

For a thermal phonon bath in this reciprocal treatment, the cancellation pairs absorption and emission with the correct Bose factors. Omitting one member of these inverse processes spoils that equilibrium cancellation.

Pairwise equality is not a universal condition on rates at a fixed magnetic state. Equilibrium can instead follow from cancellation of the summed gain and loss terms. The asymmetric rates responsible for skew scattering are an important example; Nagaosa and collaborators explain the distinction between reciprocal first-Born rates and higher-order chiral scattering. Do not impose pairwise detailed balance on such a kernel simply to force equilibrium.

Two-particle collisions

A schematic fermionic two-to-two collision term for state $1$ is

$$\mathcal I_1[f] = \sum_{234} W_{12\to34} \big[ f_3f_4 \left( 1-f_1 \right) \left( 1-f_2 \right) - f_1f_2 \left( 1-f_3 \right) \left( 1-f_4 \right) \big].$$

This displayed form assumes the reciprocal-kernel approximation $W_{12\to34}=W_{34\to12}$; otherwise the gain and loss terms carry their respective directional rates. Here $W_{12\to34}$ includes energy conservation, crystal-momentum selection, matrix elements, and any symmetry factor required by the state convention.

Normal electron–electron collisions conserve total crystal momentum within a reciprocal cell. Umklapp processes satisfy momentum conservation only up to a reciprocal vector and transfer momentum to the lattice. Whether a process relaxes electrical current depends on the overlap of current with all conserved modes, not simply on the number of collisions.

Conservation tests

For number-conserving scattering,

$$\sum_a \mathcal I_a[f] = 0.$$

For an isolated energy-conserving collision operator,

$$\sum_a \varepsilon_a \mathcal I_a[f] = 0.$$

Momentum-weighted sums vanish only for processes that conserve the chosen momentum. Impurities, boundaries, phonons treated as a bath, and Umklapp can exchange momentum with degrees of freedom outside the electronic distribution.

These identities are unit tests for an analytic or numerical collision operator. Small violations can create a false finite resistivity, incorrect steady temperature, or spurious particle source.

Linearizing Near Equilibrium

Write

$$f_a = f_a^0 - \frac{\partial f_a^0} {\partial\varepsilon_a} \phi_a.$$

Since $\partial f^0/\partial\varepsilon<0$, a positive $\phi_a$ increases occupation. The nonequilibrium information is carried by the state-dependent function $\phi_a$.

The linearized collision integral can be organized as

$$\mathcal I_a^{(1)} = \frac{\partial f_a^0} {\partial\varepsilon_a} \sum_b \mathcal C_{ab}\phi_b.$$

The kinetic problem becomes

$$\sum_b \mathcal C_{ab}\phi_b = \mathcal X_a,$$

where $\mathcal X_a$ is the streaming drive. Thus transport is a linear-operator inversion problem.

For the detailed-balance models just specified, the properly weighted linearized collision operator is self-adjoint and positive semidefinite. Its null space contains perturbations associated with quantities conserved by the retained collisions. More general stationary kernels may also have an antisymmetric part; entropy production constrains the dissipative part rather than making the full operator self-adjoint. A steady inversion must respect its conserved modes and compatibility conditions; physical momentum-relaxing processes or boundary conditions are needed when a protected mode prevents a driven steady state.

Relaxation-time approximation

The simplest replacement is

$$\mathcal I_a^{(1)} \approx - \frac{\delta f_a}{\tau_a},$$

or

$$\mathcal C_{ab} \approx \frac{\delta_{ab}}{\tau_a}.$$

This discards scattering into state $a$ and assigns each state an independent decay time. It can be accurate for a symmetry channel dominated by one eigenvalue of the collision operator, but it is not the definition of Boltzmann transport.

For a generic disturbance,

$$\sum_a \mathcal I_a^{(1)} \ne 0$$

under this approximation. Number conservation may hold accidentally for an odd electric-field distortion, while failing for density or thermal perturbations. For a common $\tau$, a conserving model relaxes toward local equilibrium with chemical potential, temperature and drift chosen to match the conserved moments. With state-dependent $\tau_a$, the condition instead contains the rates:

$$\sum_a Q_a\frac{f_a-f_{\mathrm{le},a}}{\tau_a}=0$$

for every quantity $Q_a$ conserved by those collisions. Matching only the unweighted moments need not satisfy this condition.

Vector mean free path

For reciprocal elastic rates $W_{a\to b}=W_{b\to a}$, with magnetic streaming absent, the electric-field problem can be written in terms of a vector mean free path $\boldsymbol\Lambda_a$:

$$\mathbf v_a = \sum_b W_{a\to b} \left( \boldsymbol\Lambda_a - \boldsymbol\Lambda_b \right).$$

The term containing $\boldsymbol\Lambda_b$ is the scattering-in or vertex contribution. The conductivity is

$$\sigma_{ij} = q^2 \sum_a \left( - \frac{\partial f_a^0} {\partial\varepsilon_a} \right) v_{a,i} \Lambda_{a,j}.$$

The diagonal relaxation-time approximation takes $\boldsymbol\Lambda_a=\tau_a\mathbf v_a$. Solving the coupled equation retains angular and interband redistribution.

Electric-Field Response

Consider a uniform, steady system at zero magnetic field. Define the effective electrochemical field

$$\boldsymbol{\mathcal E} := \mathbf E - \frac{1}{q} \boldsymbol\nabla\mu.$$

Here $\mu$ is the local chemical potential measured relative to the stated band energies. For an electrostatic potential $\varphi$, the electrochemical potential is $\zeta=\mu+q\varphi$, so $q\boldsymbol{\mathcal E}=-\boldsymbol\nabla\zeta$. A static equilibrium gradient has constant $\zeta$, not necessarily constant $\mu$.

In the relaxation-time approximation,

$$\phi_{n\mathbf k} = q\tau_{n\mathbf k} \mathbf v_{n\mathbf k} \cdot \boldsymbol{\mathcal E},$$

and therefore

$$\delta f_{n\mathbf k} = q\tau_{n\mathbf k} \left( - \frac{\partial f^0} {\partial\varepsilon} \right) \mathbf v_{n\mathbf k} \cdot \boldsymbol{\mathcal E}.$$

For any differentiable band with a momentum-independent $\tau$, this first-order distortion is a small shift of the equilibrium distribution: $\delta f=-\delta\mathbf k\cdot\boldsymbol\nabla_{\mathbf k}f^0$, with

$$\delta\mathbf k = \frac{q\tau}{\hbar} \boldsymbol{\mathcal E}.$$

Parabolicity is not needed for this linear-order translation. It is needed for the further reduction to one constant mass. Different band lifetimes give different shifts, while momentum-dependent lifetimes or a coupled scattering-in solution generally deform the distribution. At finite drive, the full stationary distribution is not generally a rigidly translated Fermi function.

Top: the first-order displacement $\delta\mathbf k=q\tau\mathbf E/\hbar$ at uniform chemical potential, illustrated with a circular Fermi sea. The shift is exaggerated for visibility. Bottom: $w_T(\varepsilon)=k_BT_0(-\partial f^0/\partial\varepsilon)$ versus $x=(\varepsilon-\mu)/(k_BT_0)$ at $T=T_0$ and $2T_0$. Both curves are positive and have unit integral over the full $x$ axis; increasing temperature broadens and lowers this window.

Conductivity tensor

The resulting conductivity is

$$\sigma_{ij} = q^2 \sum_n \int_{\mathrm{BZ}} \frac{d^dk}{(2\pi)^d} \tau_{n\mathbf k} v_{n,i}(\mathbf k) v_{n,j}(\mathbf k) \times \left( - \frac{\partial f^0} {\partial\varepsilon} \right).$$

All resolved bands contribute. The integrand weights velocity correlation, lifetime, and thermal accessibility, not density of states alone.

For an isotropic parabolic band and constant $\tau$, integration by parts gives

$$\sum_n \int \frac{d^dk}{(2\pi)^d} v_i v_j \left( - \frac{\partial f^0} {\partial\varepsilon} \right) = \frac{n_{\mathrm c}}{m^*} \delta_{ij}.$$

Thus

$$\sigma_{ij} = \frac{ n_{\mathrm c}q^2\tau }{ m^* } \delta_{ij},$$

which recovers Drude Theory as a special reduction.

Transport lifetime

For elastic scattering on an isotropic Fermi surface with a rotationally invariant angular kernel,

$$\frac{1}{\tau_{\mathrm{tr}}} = \int d\Omega_{\mathbf k'} W_{\mathbf k\to\mathbf k'} \left( 1-\cos\theta \right).$$

The factor $1-\cos\theta$ comes from solving for current relaxation, not from the total escape rate. The quantum or single-particle rate

$$\frac{1}{\tau_q} = \int d\Omega_{\mathbf k'} W_{\mathbf k\to\mathbf k'}$$

counts forward scattering fully. Consequently $\tau_{\mathrm{tr}}$ can greatly exceed $\tau_q$.

Thermal Gradients and Thermoelectric Response

For a weak temperature gradient, the local-equilibrium streaming term adds an energy-dependent drive. In the relaxation-time approximation,

$$\begin{aligned} \phi_{n\mathbf k} = \tau_{n\mathbf k} \mathbf v_{n\mathbf k} \cdot \left[ q\boldsymbol{\mathcal E} - \frac{ \varepsilon_{n\mathbf k}-\mu }{T} \boldsymbol\nabla T \right]. \end{aligned}$$

Particles above and below $\mu$ are driven oppositely by $\boldsymbol\nabla T$. That odd energy weighting is why thermopower is sensitive to particle–hole asymmetry.

Transport distribution

Define the transport distribution tensor

$$\Xi_{ij}(\epsilon) := \sum_n \int_{\mathrm{BZ}} \frac{d^dk}{(2\pi)^d} \tau_{n\mathbf k} v_{n,i}(\mathbf k) v_{n,j}(\mathbf k) \times \delta \left( \epsilon-\varepsilon_{n\mathbf k} \right).$$

It combines the density of available states with squared velocity and lifetime. A large density-of-states peak can contribute little to conductivity if the corresponding states are flat or short lived.

Define energy moments

$$\mathcal L_{ij}^{(r)} = \int d\epsilon\, \left( \epsilon-\mu \right)^r \left( - \frac{\partial f^0} {\partial\epsilon} \right) \Xi_{ij}(\epsilon).$$

At zero magnetic field,

$$\mathbf j = q^2 \boldsymbol{\mathcal L}^{(0)} \boldsymbol{\mathcal E} - \frac{q}{T} \boldsymbol{\mathcal L}^{(1)} \boldsymbol\nabla T,$$ $$\mathbf j_Q = q \boldsymbol{\mathcal L}^{(1)} \boldsymbol{\mathcal E} - \frac{1}{T} \boldsymbol{\mathcal L}^{(2)} \boldsymbol\nabla T.$$

The electrical conductivity is

$$\boldsymbol\sigma = q^2 \boldsymbol{\mathcal L}^{(0)}.$$

For a scalar or a common principal axis, the open-circuit Seebeck coefficient is defined by

$$S := \left. \frac{\mathcal E} {\nabla T} \right\rvert_{\mathbf j=0} = \frac{1}{qT} \frac{ \mathcal L^{(1)} }{ \mathcal L^{(0)} }.$$

This agrees with the common voltage convention $S=-\Delta V/\Delta T$ because $\mathcal E=-\boldsymbol\nabla V$ when the chemical contribution is fixed.

The electronic thermal conductivity measured at zero electric current is

$$\boldsymbol\kappa_e = \frac{1}{T} \left[ \boldsymbol{\mathcal L}^{(2)} - \boldsymbol{\mathcal L}^{(1)} \left( \boldsymbol{\mathcal L}^{(0)} \right)^{-1} \boldsymbol{\mathcal L}^{(1)} \right].$$

Using $\boldsymbol{\mathcal L}^{(2)}/T$ alone corresponds to zero electrochemical field, not open circuit. The subtraction accounts for the field that builds up to cancel charge current.

Mott relation

For a degenerate metal with a smooth scalar $\Xi(\epsilon)$,

$$\mathcal L^{(0)} = \Xi(\mu) + O(T^2),$$ $$\mathcal L^{(1)} = \frac{\pi^2}{3} \left( k_{\mathrm B}T \right)^2 \Xi'(\mu) + O(T^4).$$

Therefore

$$S \approx \frac{ \pi^2k_{\mathrm B}^2T }{ 3q } \left. \frac{d\ln\Xi} {d\epsilon} \right\rvert_{\mu}.$$

For electrons, $q=-e$, so a transport distribution increasing with energy gives negative thermopower. The sign is not universally determined by the word electron: multiband compensation, energy-dependent scattering, and nearby singularities can reverse it.

Wiedemann–Franz limit

The same expansion gives

$$\mathcal L^{(2)} = \frac{\pi^2}{3} \left( k_{\mathrm B}T \right)^2 \Xi(\mu) + O(T^4).$$

Hence

$$\frac{\kappa_e}{\sigma T} \longrightarrow L_0, \qquad L_0 = \frac{\pi^2}{3} \left( \frac{k_{\mathrm B}}{e} \right)^2.$$

This limit requires degenerate carriers and sufficiently elastic, smooth transport near $\mu$. Phonon heat conduction, bipolar diffusion, inelastic scattering, hydrodynamic flow, and non-Fermi-liquid dynamics can invalidate a naive comparison.

Metals

At low temperature,

$$- \frac{\partial f^0} {\partial\varepsilon} \longrightarrow \delta \left( \varepsilon-\mu \right).$$

The conductivity becomes a Fermi-surface integral:

$$\begin{aligned} \sigma_{ij} = q^2 \sum_n \int_{\mathrm{FS}_n} \frac{dS}{ (2\pi)^d\hbar } \frac{ \tau_{n\mathbf k} v_{n,i}v_{n,j} }{ |\mathbf v_n| }. \end{aligned}$$

Transport in a metal is therefore organized by the geometry, velocity, and scattering around the Fermi Surface, rather than by all occupied electrons equally.

Typical scattering regimes

• Elastic disorder: produces a residual resistivity to leading order, but anisotropic or resonant defects can make $\tau(\mathbf k,\epsilon)$ strongly structured.
• Acoustic phonons: in a simple three-dimensional metal, the Bloch–Grüneisen result crosses from $\rho_{\mathrm{ph}}\propto T^5$ at low temperature to approximately linear $T$ above the characteristic phonon scale. Dimensionality, screening, Fermi-surface geometry, and phonon spectrum change these powers.
• Electron–electron scattering: a Fermi-liquid quasiparticle rate often scales as $T^2$, but electrical resistivity requires current relaxation through Umklapp, multiband drag, compensation, disorder-assisted processes, or another momentum sink.
• Boundaries: matter when the mean free path approaches a sample dimension; scattering specularity and geometry then enter.

Phonons explains the lattice modes and electron–phonon kinematics. Lifetime and Spectral Weight distinguishes the transport time from spectral, coherence, and energy-relaxation times.

Matthiessen’s rule,

$$\tau_{\mathrm{tr}}^{-1} \approx \sum_a \tau_a^{-1},$$

is an approximation to the spectrum of a combined collision operator. It is not guaranteed when channels are anisotropic, energy dependent, mutually dressing, or constrained by shared conservation laws.

Semiconductors

In a nondegenerate conduction band,

$$f^0(\epsilon) \approx \exp \left[ - \frac{ \epsilon-\mu }{ k_{\mathrm B}T } \right].$$

Carrier density, mobility, and thermopower can then vary exponentially with chemical potential and sensitively with scattering.

For one isotropic three-dimensional parabolic conduction band,

$$\varepsilon = E_c + \frac{\hbar^2k^2}{2m^*},$$

and a power-law lifetime

$$\tau(\epsilon) \propto \left( \epsilon-E_c \right)^r,$$

the transport distribution scales as

$$\Xi(\epsilon) \propto \left( \epsilon-E_c \right)^{r+3/2}.$$

If this power law extends to the band edge, require $r>-5/2$ so the transport moments converge. A low-energy cutoff or crossover in the lifetime instead changes the moments. In the convergent model, the nondegenerate thermopower is

$$S_n = \frac{1}{qT} \left[ E_c-\mu + \left( r+\frac{5}{2} \right) k_{\mathrm B}T \right].$$

For electrons $q=-e$, this expression is negative. The constant $5/2$ is specific to a three-dimensional parabolic band and the stated definition of $r$.

Mobility is model dependent

The drift mobility

$$\mu_{\mathrm d} = \frac{\sigma} {n_{\mathrm c}|q|}$$

is an average over the occupied transport distribution. It need not equal $|q|\tau/m^*$ evaluated at a single energy. Valley degeneracy, anisotropic masses, nonparabolicity, ionized-impurity scattering, polar optical phonons, and intervalley transitions can all change the average.

Electrons and holes together

For independent electron and hole channels,

$$\sigma = \sigma_n+\sigma_p,$$

and

$$S = \frac{ \sigma_nS_n+\sigma_pS_p }{ \sigma_n+\sigma_p }.$$

Opposite thermopower signs can cancel even when both carrier populations are large. Bipolar diffusion also adds heat conduction, so an intrinsic semiconductor can violate a one-carrier Wiedemann–Franz estimate without exotic physics.

Magnetic Field and Chambers Formula

At nonzero magnetic field, the Lorentz term moves a quasiparticle along a constant-energy orbit:

$$\hbar\dot{\mathbf k} = q \mathbf v_n(\mathbf k) \times \mathbf B.$$

For one band with constant $\tau$, the formal steady solution leads to Chambers’ formula:

$$\sigma_{ij}(\mathbf B) = q^2 \int_{\mathrm{BZ}} \frac{d^dk}{(2\pi)^d} \left( - \frac{\partial f^0} {\partial\varepsilon} \right) v_i\!\left[ \mathbf k(0) \right] \times \int_{-\infty}^{0} dt\, e^{t/\tau} v_j\!\left[ \mathbf k(t) \right].$$

The current at the present state remembers the velocity history over approximately one relaxation time. At $\mathbf B=0$, $\mathbf k(t)$ is constant and the inner integral becomes $\tau v_j$.

This expression exposes why magnetotransport depends on whole Fermi-surface orbits, open trajectories, and anisotropic lifetimes. The elementary isotropic Hall tensor is derived in Drude Theory; a full Hall analysis must also track tensor inversion, multiband response, Berry curvature, and magnetization-current conventions.

Conservation, Entropy, and Reciprocity

For real distribution distortions, define the weighted inner product

$$\langle a,b\rangle = \sum_n \int_{\mathrm{BZ}} \frac{d^dk}{(2\pi)^d} \left( - \frac{\partial f^0} {\partial\varepsilon} \right) a_{n\mathbf k} b_{n\mathbf k}.$$

For a detailed-balance collision operator,

$$\langle\phi,\mathcal C\phi\rangle \ge 0.$$

Up to convention-dependent normalization, this quadratic form is $T$ times the collision-generated entropy-production rate. Equality identifies a collision invariant.

This structure supports variational solutions: trial functions give controlled approximations to the current-carrying deformation when the null modes are handled correctly. It also makes a negative longitudinal conductivity from a passive, equilibrium, linearized calculation a diagnostic of a sign, discretization, or collision-kernel error.

Microscopic reversibility relates a system to the state with all time-reversal-odd parameters reversed. If there is no additional magnetic order or other fixed time-reversal-odd background, the electrical Onsager–Casimir relation is

$$\sigma_{ij}(\mathbf B) = \sigma_{ji}(-\mathbf B).$$

When magnetization or another time-reversal-odd background is present, reverse it together with $\mathbf B$ in the partner state. Thermoelectric cross coefficients obey corresponding relations after current definitions and time-reversal parities are matched. A numerical solver should recover these symmetries within convergence error.

Kinetic theory and its local limit

Resolved quasiparticles, slowly varying fields on the wave-packet scale and a justified collision kernel are conditions on the kinetic description. A local conductivity requires more: the distribution must relax over a distance short compared with the scale of its spatial drive. Thus $\ell_{\mathrm{tr}}\ll L_{\mathrm{drive}}$ is a useful condition for a local constitutive relation, not a universal prerequisite for solving the Boltzmann equation. Boundary distributions and spatial streaming can describe ballistic or nonlocal transport when quantum interference is negligible. Battiato and collaborators discuss this distinction explicitly.

Similarly, fast momentum-conserving collisions do not invalidate kinetic theory. They produce a hydrodynamic limit once the remaining slow modes and boundary conditions are treated properly. A diagonal relaxation approximation that damps every mode would lose that physics. Hydrodynamics develops the resulting long-wavelength equations.

Linear response requires the field-induced distortion to remain small; a useful local estimate is $|q\mathcal E|\ell_{\mathrm{tr}}\ll E_{\mathrm{window}}$, with the relevant thermal or band scale specified. At very low temperature the derivative of a sharp Fermi surface is understood as a limiting distribution rather than a pointwise-small change at every energy. Strong interband coherence, localization or a broad incoherent spectrum can require density-matrix kinetics or a different quantum description. For a smooth magnetic response, Landau quantization must be negligible in the observable of interest. Comparing $\hbar|\omega_c|$ with thermal and physical spectral broadening is one check; instrumental smoothing alone does not restore the classical dynamics.

Solving the collision equation

A constant relaxation time isolates the band contribution to transport but leaves its absolute rate unspecified. State-dependent escape times add scattering information while still omitting the coupled scattering-in term. An iterative or variational solution treats that redistribution and the conserved null modes explicitly. Poncé and collaborators review this progression in first-principles carrier calculations.

For a discrete collision matrix, particle and energy conservation, equilibrium stationarity and the allowed reciprocity relations provide independent checks. A small linear-solver residual alone cannot establish any of them. Convergence must also resolve the electron and phonon momentum grids and the energy-conservation kernels. Coarse quadrature can make a positive continuum operator nonphysical or shift the dominant scattering channels.

Electrical conductivity and thermopower need separate convergence checks. A common constant time cancels from thermopower because it is a ratio of transport moments. Agreement of that ratio does not verify the absolute scattering rate or justify the constant-time approximation. Transport measurements explains the geometry and contacts needed to compare the calculated coefficient with experiment.

Exercises

1. Recover the Drude conductivity

For an isotropic parabolic band with constant $\tau$, use integration by parts in $\mathbf k$ to prove

$$\sigma_{ij} = \frac{n_{\mathrm c}q^2\tau}{m^*} \delta_{ij}.$$

Solution

Since

$$\frac{\partial f^0} {\partial k_i} = \hbar v_i \frac{\partial f^0} {\partial\varepsilon},$$

we have

$$\begin{aligned} \int \frac{d^dk}{(2\pi)^d} v_iv_j \left( - \frac{\partial f^0} {\partial\varepsilon} \right) &= - \frac{1}{\hbar} \int \frac{d^dk}{(2\pi)^d} v_j \frac{\partial f^0} {\partial k_i} \\ &= \frac{1}{\hbar} \int \frac{d^dk}{(2\pi)^d} f^0 \frac{\partial v_j} {\partial k_i}. \end{aligned}$$

For $\varepsilon=\hbar^2k^2/(2m^*)$,

$$\frac{\partial v_j} {\partial k_i} = \frac{\hbar}{m^*} \delta_{ij}.$$

The remaining integral is $n_{\mathrm c}$, so inserting the result gives the Drude form. The integration uses a continuum parabolic band with a vanishing boundary term at large momentum. A periodic band also allows integration by parts across a Brillouin zone, but its curvature is generally momentum dependent and cannot be replaced by one constant mass.

2. Number conservation in the RTA

Let

$$\delta f_a = C \left( - \frac{\partial f_a^0} {\partial\varepsilon_a} \right)$$

represent a small uniform chemical-potential shift. Does $\mathcal I_a=-\delta f_a/\tau$ conserve particle number for constant $\tau$?

Solution

The integrated collision term is

$$\sum_a \mathcal I_a = - \frac{C}{\tau} \sum_a \left( - \frac{\partial f_a^0} {\partial\varepsilon_a} \right).$$

The sum is positive for a compressible system, so the result is nonzero. The diagonal RTA incorrectly relaxes a uniform density shift into or out of existence.

A conserving model must relax toward a local equilibrium $f_{\mathrm{le}}$ with a shifted chemical potential chosen so that

$$\sum_a \left( f_a-f_{\mathrm{le},a} \right) = 0.$$

3. Angular scattering

In two dimensions, take an elastic angular kernel

$$W(\theta) = W_0 \left( 1+a\cos\theta \right), \qquad |a|\le1.$$

Find $\tau_q^{-1}$ and $\tau_{\mathrm{tr}}^{-1}$ up to the common state-density factor.

Solution

The quantum rate is

$$\begin{aligned} \tau_q^{-1} &= \int_0^{2\pi} d\theta\, W(\theta) \\ &= 2\pi W_0. \end{aligned}$$

The transport rate is

$$\begin{aligned} \tau_{\mathrm{tr}}^{-1} &= \int_0^{2\pi} d\theta\, W(\theta) \left( 1-\cos\theta \right) \\ &= \pi W_0 \left( 2-a \right). \end{aligned}$$

Thus

$$\frac{ \tau_{\mathrm{tr}}^{-1} }{ \tau_q^{-1} } = 1-\frac{a}{2}.$$

Forward-favoring scattering has $a>0$ and relaxes current less efficiently than it broadens a single-particle state.

4. Independent collision operators need not give additive resistivities

After removing conserved modes, consider two real orthonormal distortion modes. In units of a common collision rate, two independent processes have

$$C_1=\begin{pmatrix}1&0\\0&9\end{pmatrix},\qquad C_2=\begin{pmatrix}9&0\\0&1\end{pmatrix}.$$

For the unit current vector $v=(1,1)^T/\sqrt2$, define the dimensionless conductivity $\sigma=v^TC^{-1}v$ and resistivity $\rho=1/\sigma$. Compare the resistivity for $C=C_1+C_2$ with $\rho_1+\rho_2$. Why can independent scattering rates fail to give Matthiessen’s rule for resistivity?

Solution

Each isolated process gives $\sigma_i=(1+1/9)/2=5/9$ and $\rho_i=9/5$. Their scalar resistivities sum to $18/5$. Together, however, $C_1+C_2=10I$ gives $\sigma=1/10$ and $\rho=10$.

The collision operators add, but solving for the driven distribution requires an inverse. Each separate process permits the current to concentrate in its more weakly scattered mode; the second process scatters that mode strongly. The optimally conducting distortion changes when both are present. Additivity of independent collision operators is therefore weaker than additivity of their fitted scalar resistivities.

5. Nondegenerate semiconductor thermopower

For a three-dimensional parabolic conduction band with $\tau\propto(\epsilon-E_c)^r$ down to the band edge and $r>-5/2$, show that

$$\frac{ \mathcal L^{(1)} }{ \mathcal L^{(0)} } = E_c-\mu + \left( r+\frac{5}{2} \right) k_{\mathrm B}T.$$

Solution

The transport distribution is

$$\Xi(\epsilon) \propto x^{r+3/2}, \qquad x=\epsilon-E_c.$$

In the nondegenerate limit, the common weight is proportional to $e^{-x/(k_{\mathrm B}T)}$. The weighted mean of $x$ is

$$\langle x\rangle = \frac{ \int_0^\infty dx\, x^{r+5/2} e^{-x/(k_{\mathrm B}T)} }{ \int_0^\infty dx\, x^{r+3/2} e^{-x/(k_{\mathrm B}T)} } = \left( r+\frac{5}{2} \right) k_{\mathrm B}T,$$

using $\Gamma(z+1)=z\Gamma(z)$. Since $\epsilon-\mu=E_c-\mu+x$, the result follows. Dividing by $qT$ gives the Seebeck coefficient.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976.
• M. Battiato, V. Zlatić, and K. Held, “Boltzmann Approach to High-Order Transport: The Nonlinear and Nonlocal Responses,” Physical Review B 95, 235137 (2017), doi:10.1103/PhysRevB.95.235137.
• L. Boltzmann, “Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen,” Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Classe 66, 275–370 (1872).
• R. G. Chambers, “The Kinetic Formulation of Conduction Problems,” Proceedings of the Physical Society. Section A 65, 458–459 (1952), doi:10.1088/0370-1298/65/6/114.
• M. Cutler and N. F. Mott, “Observation of Anderson Localization in an Electron Gas,” Physical Review 181, 1336–1340 (1969), doi:10.1103/PhysRev.181.1336.
• M. Jonson and G. D. Mahan, “Mott’s Formula for the Thermopower and the Wiedemann–Franz Law,” Physical Review B 21, 4223–4229 (1980), doi:10.1103/PhysRevB.21.4223.
• N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, “Anomalous Hall effect,” Reviews of Modern Physics 82, 1539–1592 (2010), doi:10.1103/RevModPhys.82.1539.
• L. Onsager, “Reciprocal Relations in Irreversible Processes. I,” Physical Review 37, 405–426 (1931), doi:10.1103/PhysRev.37.405.
• L. Onsager, “Reciprocal Relations in Irreversible Processes. II,” Physical Review 38, 2265–2279 (1931), doi:10.1103/PhysRev.38.2265.
• S. Poncé, W. Li, S. Reichardt, and F. Giustino, “First-Principles Calculations of Charge Carrier Mobility and Conductivity in Bulk Semiconductors and Two-Dimensional Materials,” Reports on Progress in Physics 83, 036501 (2020), doi:10.1088/1361-6633/ab6a43.
• J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids, Oxford University Press, 1960; Oxford Classic Texts reissue, 2001, doi:10.1093/acprof:oso/9780198507796.001.0001.