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Transport Properties

Boltzmann transport equation

The Boltzmann transport equation for electrons under a small electric field, F\mathbf{F}, can be written

eFkfT(k,T)=(δfT(k,T)δt). \frac{e \mathbf{F}}{\hbar} \nabla_\mathbf{k} f_\mathrm{T} (\mathbf{k}, T) = \left ( \frac{\delta f_\mathrm{T} (\mathbf{k}, T)}{\delta t} \right ).

Under low field conditions, the total distribution function, fTf_\mathrm{T}, is

fT(k)=f(k)+xg(k), f_\mathrm{T}(\mathbf{k}) = f(\mathbf{k}) + xg(\mathbf{k}),

where xx is the cosine of the angle between F\mathbf{F} and k\mathbf{k}, gg is the perturbation part of the distribution. and ff is the distribution at equilibrium (Fermi–Dirac distribution), given as

f=1e(EEF)/kBT+1 f = \frac{1}{e^{(E-E_\mathrm{F})/k_\mathrm{B}T} + 1}

Elastic versus inelastic scattering

The perturbation is dependent on the scattering rate ss. The differential scattering rate from state k\mathbf{k} to k\mathbf{k}^\prime is given as

s(k,k)=sinel(k,k)+sel(k,k) s(\mathbf{k}, \mathbf{k}^\prime ) = s_\mathrm{inel} (\mathbf{k}, \mathbf{k}^\prime ) + s_\mathrm{el} (\mathbf{k}, \mathbf{k}^\prime )

where the subscripts el\mathrm{el} and inel\mathrm{inel} refer to elastic and inelastic scattering processes, respectively.

Elastic scattering is completely randomizing and has no effect on ff, i.e.,

s(k,k)s(k,k), s(\mathbf{k}, \mathbf{k}^\prime) \equiv s(\mathbf{k}^\prime, \mathbf{k}),

whereas ineleastic scattering is not randomizing and does affect ff, i.e.,

s(k,k)s(k,k). s(\mathbf{k}, \mathbf{k}^\prime) \neq s(\mathbf{k}^\prime, \mathbf{k}).

The non-randomizing behaviour follows from the different rates for emission and absorption processes in inelastic scattering. Accordingly, we define the scattering out rate from k-point, k\mathbf{k}, as

sout(k)=[s(k,k)(1f)+s(k,k)f]dk, s_\mathrm{out}(\mathbf{k}) = \int \left [ s( \mathbf{k}, \mathbf{k}^\prime ) (1 - f^\prime) + s(\mathbf{k}^\prime, \mathbf{k}) f^\prime \right ] \mathrm{d}\mathbf{k}^\prime,

and scattering in rate to k-point, k\mathbf{k}, as

sin(k)=Xg[s(k,k)(1f)+s(k,k)f]dk, s_\mathrm{in}(\mathbf{k}) = \int X g^\prime \left [ s(\mathbf{k}^\prime, \mathbf{k}) (1 - f) + s(\mathbf{k}, \mathbf{k}^\prime) f \right ] \mathrm{d}\mathbf{k}^\prime,

where f=f(k)f=f(\mathbf{k}), f=f(k)f^\prime = f(\mathbf{k}^\prime), g=g(k)g^\prime = g(\mathbf{k}^\prime) and XX is the angle between k\mathbf{k} and k\mathbf{k}^\prime.

Iterative Boltzmann transport

As detailed in Rode1, the perturbation to the total distribution function can be written,

g(k)=sin(g,k)eFkfT(k,T)sout(k)+sel(k) g (\mathbf{k}) = \frac{s_\mathrm{in}(g, \mathbf{k})- \frac{e \mathbf{F}}{\hbar} \nabla_\mathbf{k} f_\mathrm{T} (\mathbf{k}, T)} {s_\mathrm{out}(\mathbf{k}) + s_\mathrm{el}(\mathbf{k})}

As gg depends upon itself, an iterative procedure is needed to solve the perturbation to the distribution function. Fortunately, the functional form of gg ensures it converges exponentially, in most cases requiring fewer than 5 iterations to achieve accurate results.

In the absence of inelastic scattering, sin(g)=0s_\mathrm{in}(g) = 0 and g1g_1, is the exact solution. In addition, the g1g_1 solution is often termed the relaxation time approximation (RTA). Currently, AMSET only support the RTA but the full iterative solution will be added in a future release.

Calculating transport properties

Transport properties are calculated using the transport density of states (DOS), defined as:

σ(E,T)=nvnkvnkτnk,Tδ(EEnk)dk8π3, \sigma(E, T) = \int \sum_n \mathbf{v}_{n\mathbf{k}} \otimes \mathbf{v}_{n\mathbf{k}} \tau_{n\mathbf{k},T} \delta (E - E_{n\mathbf{k}}) \frac{\mathrm{d}\mathbf{k}}{8 \pi^3},

where vnk\mathbf{v}_{n\mathbf{k}} is the group velocity of k-point k\mathbf{k} and band nn, τnk\tau_{n\mathbf{k}} is the lifetime defined as 1/sn(k)1/s_n(\mathbf{k}) and EE is energy.

The transport DOS is used to calculate the moments of the generalized transport coefficients

L(α)(EF,T)=q2σ(E,T)(EEF)α(δfT(E,EF,T)δE)dE, \mathcal{L}^{(\alpha)}(E_\mathrm{F}, T) = q^2 \int \sigma(E, T)(E - E_\mathrm{F})^\alpha \left ( - \frac{ \delta f_\mathrm{T}(E, E_\mathrm{F}, T)}{\delta E} \right ) \mathrm{d}E,

where EFE_\mathrm{F} is the Fermi level at doping concentration, cc.

Finally, the electrical conductivity, σ\sigma, Seebeck coefficient, SS, and electronic contribution to the thermal conductivity, κe\kappa_\mathrm{e}, are calculated according to

σ=L(0),S=1qTL(1)L(0),κe=1q2T[(L(1))2L(0)L(2)]. \begin{aligned} \sigma = {}& \mathcal{L}^{(0)},\\ S = {}& \frac{1}{qT} \frac{\mathcal{L}^{(1)}}{\mathcal{L}^{(0)}},\\ \kappa_\mathrm{e} = {}& \frac{1}{q^2T} \left [ \frac{(\mathcal{L}^{(1)})^2}{\mathcal{L}^{(0)}} - \mathcal{L}^{(2)} \right ] . \end{aligned}

  1. Rode, D. L. Low-field electron transport. Semiconductors and semimetals 10, (Elsevier, 1975).