Mouyang Cheng

Mouyang Cheng

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

Transport Regimes: Boltzmann, Ballistic, and Quantum

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:

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

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.

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