Mouyang Cheng

Mouyang Cheng

AI for materials | Computational physics | Mingda Li Group, MIT

Boltzmann Transport and Why It Breaks Down

This note is a compact map for deciding which transport model is appropriate. I use four scales: mean free path $\ell_{\mathrm{mfp}}$, phase-coherence length $L_\phi$, Fermi wavelength $\lambda_F$ (equivalently $k_F=2\pi/\lambda_F$), and device/bulk length $L$.

1. Boltzmann Transport in One Page

The semiclassical Boltzmann transport equation (BTE) evolves the distribution $f(\mathbf r,\mathbf k,t)$:

$$ \frac{\partial f}{\partial t} +\mathbf v_{\mathbf k}\cdot\nabla_{\mathbf r}f +\frac{q\mathbf E}{\hbar}\cdot\nabla_{\mathbf k}f = \left.\frac{\partial f}{\partial t}\right|_{\mathrm{coll}}. $$

With the relaxation-time approximation (RTA),

$$ \left.\frac{\partial f}{\partial t}\right|_{\mathrm{coll}} =-\frac{f-f_0}{\tau}, \qquad \ell_{\mathrm{mfp}}=v_F\tau. $$

In linear response, this gives Drude-like conductivity:

$$ \sigma=\frac{nq^2\tau}{m^\ast}, \qquad \mu=\frac{q\tau}{m^\ast}. $$

For thermoelectrics, we keep the full energy dependence and explicitly linearize around equilibrium:

$$ f_{n\mathbf k}=f_0(\varepsilon_{n\mathbf k})+\delta f_{n\mathbf k}, \qquad |\delta f_{n\mathbf k}|\ll f_0. $$

For steady, weakly non-uniform transport (linear response), keep only first-order terms:

$$ \frac{\partial f}{\partial t}=0,\qquad \mathbf v\cdot\nabla_{\mathbf r}\delta f\approx 0,\qquad \left.\frac{\partial f}{\partial t}\right|_{\mathrm{coll}} =-\frac{\delta f}{\tau_{n\mathbf k}}. $$

So BTE becomes

$$ \mathbf v_{n\mathbf k}\cdot\nabla_{\mathbf r} f_0 +\frac{(-e)\mathbf E}{\hbar}\cdot\nabla_{\mathbf k} f_0 =-\frac{\delta f_{n\mathbf k}}{\tau_{n\mathbf k}}. $$

Use chain rules

$$ \nabla_{\mathbf k} f_0 =\frac{\partial f_0}{\partial \varepsilon}\nabla_{\mathbf k}\varepsilon_{n\mathbf k} =\hbar \mathbf v_{n\mathbf k}\frac{\partial f_0}{\partial \varepsilon}, $$

and (taking $\nabla\mu=0$ in the common BoltzTraP-style setup)

$$ \nabla_{\mathbf r} f_0 =\frac{\partial f_0}{\partial T}\nabla T =\frac{\varepsilon_{n\mathbf k}-\mu}{T} \left(-\frac{\partial f_0}{\partial \varepsilon}\right)\nabla T. $$

Substituting these into BTE gives the perturbation

$$ \delta f_{n\mathbf k} = -\tau_{n\mathbf k}\, \mathbf v_{n\mathbf k}\!\cdot\! \left[ -e\mathbf E +\frac{\varepsilon_{n\mathbf k}-\mu}{T}\nabla T \right] \left(-\frac{\partial f_0}{\partial \varepsilon}\right), $$

Why this directly gives transport properties: electric and heat currents are moments of the nonequilibrium part $\delta f$:

$$ J_\alpha = \frac{-e}{N_k\Omega}\sum_{n\mathbf k} v_{n\mathbf k,\alpha}\,\delta f_{n\mathbf k}, \qquad J_{Q,\alpha} = \frac{1}{N_k\Omega}\sum_{n\mathbf k}(\varepsilon_{n\mathbf k}-\mu)\,v_{n\mathbf k,\alpha}\,\delta f_{n\mathbf k}. $$

Insert $\delta f_{n\mathbf k}$, collect terms proportional to $E_\beta$ and $-\nabla_\beta T$, and define the common moments

$$ \mathcal L_{\alpha\beta}^{(m)} = \frac{1}{N_k\Omega} \sum_{n\mathbf k} \tau_{n\mathbf k}\, v_{n\mathbf k,\alpha}v_{n\mathbf k,\beta} (\varepsilon_{n\mathbf k}-\mu)^m \left(-\frac{\partial f_0}{\partial \varepsilon}\right). $$

Then

$$ J_\alpha = e^2\mathcal L_{\alpha\beta}^{(0)}E_\beta +\frac{e}{T}\mathcal L_{\alpha\beta}^{(1)}(-\nabla_\beta T), $$
$$ J_{Q,\alpha} = e\mathcal L_{\alpha\beta}^{(1)}E_\beta +\frac{1}{T}\mathcal L_{\alpha\beta}^{(2)}(-\nabla_\beta T). $$

So the “$2\times2$ matrix” is just the compact Onsager form of these two coupled linear equations (charge and heat currents vs electric and thermal driving forces):

Using Onsager form for charge/heat currents,

$$ \begin{pmatrix}\mathbf J\\ \mathbf J_Q\end{pmatrix} = \begin{pmatrix} e^2\mathcal L^{(0)} & \dfrac{e}{T}\mathcal L^{(1)} \\ e\mathcal L^{(1)} & \dfrac{1}{T}\mathcal L^{(2)} \end{pmatrix} \begin{pmatrix}\mathbf E\\ -\nabla T\end{pmatrix}, $$

the electronic thermoelectric tensors are obtained by taking $\sigma$ from $\mathbf J$ vs $\mathbf E$ at $\nabla T=0$, $S$ from the open-circuit condition $\mathbf J=0$, and $\kappa_e$ from $\mathbf J_Q$ vs $-\nabla T$ at $\mathbf J=0$:

$$ \sigma_{\alpha\beta}=e^2\mathcal L_{\alpha\beta}^{(0)}, \qquad S_{\alpha\beta} =-\frac{1}{eT} \left(\mathcal L^{(0)-1}\mathcal L^{(1)}\right)_{\alpha\beta}, $$
$$ \kappa_{e,\alpha\beta} =\frac{1}{T} \left[ \mathcal L^{(2)} -\mathcal L^{(1)}\mathcal L^{(0)-1}\mathcal L^{(1)} \right]_{\alpha\beta}, \qquad \mathrm{PF}_{\alpha}=S_{\alpha}^2\sigma_{\alpha}. $$

Finally, the figure of merit is

$$ zT_{\alpha} = \frac{S_{\alpha}^2\sigma_{\alpha}T} {\kappa_{e,\alpha}+\kappa_{L,\alpha}}. $$

Use BTE when carriers can be treated as wave packets with well-defined momentum between scattering events.

2. Practical Boltzmann Implementations

For electronic materials from first principles, the two common BTE pipelines are:

  • BoltzTraP/BoltzTraP2: interpolate DFT bands and compute transport in constant-$\tau$ or model-$\tau$ settings.
  • EPW (electron-phonon Wannier): compute $e$-ph matrix elements and scattering rates, then solve BTE with ab initio lifetimes.

Both are still semiclassical BTE frameworks; they differ mainly in how accurately $\tau$ and scattering are modeled.

For an ab initio thermoelectric workflow (RTA):

  1. Electronic structure from DFT: compute $\varepsilon_{n\mathbf k}$ on dense meshes (often via Wannier/Fourier interpolation).
  2. Band velocities: evaluate $\mathbf v_{n\mathbf k}=\hbar^{-1}\nabla_{\mathbf k}\varepsilon_{n\mathbf k}$.
  3. Lifetimes $\tau_{n\mathbf k}$:
    • constant-$\tau$ / fitted-$\tau$ (BoltzTraP-style), or
    • explicit $e$-ph scattering from supercell finite-displacement data (e.g., VASP + phono3py style) or Wannier interpolation (EPW-style), e.g. from Fermi’s golden rule.
  4. Transport moments: build $\mathcal L^{(0,1,2)}(\mu,T)$ and obtain $\sigma$, $S$, $\kappa_e$, and PF.
  5. Lattice thermal conductivity: compute $\kappa_L$ from phonon BTE using 2nd/3rd-order IFCs (typically finite-displacement supercells).
  6. Figure of merit: combine to get $zT(\mu,T)$ and map optimal carrier concentration/temperature windows.

This is the standard ab initio route used in recent RTA thermoelectric studies (including the methodology of arXiv:2511.15249), with differences mainly in how $\tau_{n\mathbf k}$ and $\kappa_L$ are treated.

3. Quantum Transport (Landauer + NEGF)

For coherent two-terminal transport, current is

$$ I(V)=\frac{2e}{h}\int dE\;T(E,V)\bigl[f_L(E)-f_R(E)\bigr]. $$

In NEGF, transmission is

$$ T(E)=\mathrm{Tr}\!\left[\Gamma_L G^r \Gamma_R G^a\right], $$

with

$$ G^r(E)=\bigl[E S-H_C-\Sigma_L^r-\Sigma_R^r\bigr]^{-1}, \qquad \Gamma_\alpha=i\left(\Sigma_\alpha^r-\Sigma_\alpha^a\right). $$

$H_C$ and $S$ are the device-region Hamiltonian and overlap (non-orthogonal basis in many DFT codes), and $\Sigma_{L/R}$ encode open-boundary coupling to semi-infinite electrodes.

4. How SIESTA/TranSIESTA-Style Quantum Transport Works

SIESTA provides localized-orbital DFT Hamiltonians; TranSIESTA performs NEGF transport on top.

Workflow. DFT+NEGF transport calculation
  1. Run electrode DFT for left/right bulk leads to obtain converged lead Hamiltonians.
  2. Build a device region (left lead layers + scatterer + right lead layers).
  3. Compute lead self-energies $\Sigma_{L/R}(E)$ from electrode surface Green's functions.
  4. Solve the open-device Green's function $G^r(E)$ self-consistently with charge density.
  5. Evaluate $T(E)$, then integrate to get $I(V)$ and differential conductance.

Important interpretation:

  • DFT gives the effective single-particle Hamiltonian.
  • NEGF imposes open boundaries and nonequilibrium occupations.
  • Landauer formula converts transmission into measurable current.

This framework naturally captures tunneling, resonances, contact effects, and atomistic chemistry that semiclassical BTE cannot represent.

5. Three Breakdown Tests for Boltzmann

Let $L$ be device length, $\ell_{\mathrm{mfp}}$ mean free path, and $L_\phi$ coherence length.

  1. Ioffe-Regel / quasiparticle test:
    $\lambda_F\ll \ell_{\mathrm{mfp}}$ (equivalently $k_F\ell_{\mathrm{mfp}}\gg1$) supports Fermi-liquid quasiparticles and BTE.
    If $k_F\ell_{\mathrm{mfp}}\sim1$, Boltzmann breaks down (strong disorder/localization) and quantum methods are required.

  2. Diffusive test:
    $L\gg \ell_{\mathrm{mfp}}$ gives diffusive transport (bulk BTE/Drude regime).
    If this fails ($L\lesssim\ell_{\mathrm{mfp}}$), transport is ballistic: bulk diffusive BTE is no longer right, boundary conditions dominate, and for nanoscale electrons Landauer/NEGF is typically the clean choice.

  3. Coherence test:
    $L\gg L_\phi$ gives semiclassical incoherent transport.
    If this fails ($L_\phi\gtrsim L$), phase coherence survives across the device and quantum transport is needed.

In short, Boltzmann is safest when

$$ \lambda_F\ll \ell_{\mathrm{mfp}}\ll L, \qquad L_\phi\ll L, \qquad k_F\ell_{\mathrm{mfp}}\gg1. $$

6. Scattering: Elastic vs Inelastic

Different scattering mechanisms contribute differently to $\ell_{\mathrm{mfp}}$ and $\tau_\phi$:

Elastic scattering (impurities, defects, boundaries):

  • Randomizes momentum $\Rightarrow$ contributes to $1/\ell_{\mathrm{mfp}}$.
  • Preserves phase coherence (no energy loss).
  • Does not reduce $\tau_\phi$ or $L_\phi$.

Inelastic scattering (electron-phonon, electron-electron, magnons, etc.):

  • Randomizes momentum $\Rightarrow$ contributes to $1/\ell_{\mathrm{mfp}}$.
  • Breaks phase coherence (energy/information loss) $\Rightarrow$ defines dephasing time $\tau_\phi$ and thus $L_\phi$.

In BTE frameworks like BoltzTraP or EPW, electron-phonon scattering provides the dominant inelastic contribution to lifetimes and thus to $\tau_\phi$.

7. How to Estimate $L_\phi$

Use

$$ L_\phi=\sqrt{D\tau_\phi}, $$

with $D$ the diffusion constant and $\tau_\phi$ the dephasing time from inelastic processes.

Getting the diffusion constant $D$:

From Einstein relation,

$$ D=\frac{\sigma}{e\cdot N(E_F)}=\frac{v_F^2\tau}{d}=\frac{v_F\ell_{\mathrm{mfp}}}{d}, $$

where $\sigma$ is conductivity, $e$ is elementary charge, $N(E_F)$ is density of states at Fermi level, $d$ is dimensionality, and $\ell_{\mathrm{mfp}}=v_F\tau$. Experimentally, extract from resistivity and Hall effect, or compute from BTE/ab initio (BoltzTraP, EPW).

Getting the dephasing time $\tau_\phi$ via Matthiessen’s rule:

Inelastic scattering rates add inversely:

$$ \frac{1}{\tau_\phi}=\sum_i\frac{1}{\tau_i^{\phi}}, $$

where $\tau_i^{\phi}$ are individual dephasing times ($e$-$e$, $e$-ph, magnon-scattering, etc.). Each temperature-dependent:

  • Electron-electron ($e$-$e$): $1/\tau_{ee}^{\phi}\propto T$ (in 2D) or $\propto T^2$ (in 3D).
  • Electron-phonon ($e$-$ph$): typically weaker; $\propto T$ at high $T$, stronger at low $T$ in some materials.

Then compute $L_\phi=\sqrt{D\tau_\phi}$ combining the Einstein $D$.

Typical experimental extraction routes:

  1. Weak localization / anti-localization magnetoconductance fits (HLN-type).
  2. Universal conductance fluctuation correlation field.
  3. Aharonov-Bohm oscillation visibility vs temperature/size.
  4. Microscopic dephasing rates ($e$-$e$, $e$-ph) combined with $L_\phi=\sqrt{D\tau_\phi}$.

References