D. From Boltzmann Distribution to Drift-Diffusion Current Equations

We consider a steady state situation and, for simplicity, a one-dimenstional geometry [208]. With the use of a relaxation time approximation as in (3.2) the BOLTZMANN equation becomes

$$\frac{e F}{m^*} \frac{\partial f(v,x)}{\partial v} + v \frac{\partial f(v,x)}{\partial x} = \frac{f_{eq}(v,x)-f(v,x)}{\tau}$$ (D.1)

Here the relation $m^* \vec{v} = \vec{p} = \hbar \vec{k}$ was used, which is valid for a parabolic energy band. Note that the charge $e$ has to be taken with the proper sign of the particle (positive for holes and negative for electrons). A general definition of current density is given by

$$J(x)=e \int v f(v,x) dv$$ (D.2)

where the integral on the right hand side represents the first moment of the distribution function. This definition of current can be related to (D.1). After multiplying both sides by $v$ and integrating over v one gets

$$\frac{e F}{m^*} \int v \frac{\partial f(v,x)}{\partial v} dv + \i... ...brace{\int v f_{eq}(v,x) dv}_0 - \int v f(v,x) dv\right] = -\frac{J(x)}{e \tau}$$ (D.3)

since the function $v f_{eq}(v,x)$ is odd in $v$, and its integral is therefore zero. Thus, one has from (D.3)

$$J(x) = -e \frac{e \tau}{m^*} F \int v \frac{\partial f}{\partial v} dv - e \tau \frac{d}{dx} \int v^2 f(v,x) dv$$ (D.4)

Integrating by parts yields

$$\int v \frac{\partial f}{\partial v} dv = \underbrace{\left[v f(v,x)\right]_{-\infty}^{\infty}}_0 - \int f(v,x) dv = -n(x)$$ (D.5)

and

$$\int v^2 f(v,x) dv = n(x) \langle v^2 \rangle$$ (D.6)

can be written, where $\langle v^2 \rangle$ is the average of the square of the velocity defined as

$$\langle v^2 \rangle = \frac{1}{n} \int v^2 f(v,x) dv$$ (D.7)

Because of the equipartition theorem, for a purely one-dimensional treatment, the $-\frac{3}{2}$ exponent in (3.3) may be replaced with $-\frac{1}{2}$, while the appropriate thermal kinetic energy becomes $\frac{k_B T}{2}$ instead of $\frac{3 k_B T}{2}$.
The drift-diffusion equations are derived introducing the mobility $\mu = \frac{e \tau}{m^*}$ and replacing $\langle v^2 \rangle$ with its average equilibrium value $\frac{k_B T}{m^*}$, therefore neglecting thermal effects. The diffusion coefficient $D = \frac{\mu k_B T_0}{e}$ (Einstein's relation) is also introduced, and the resulting drift-diffusion current is

$$J_n = q n(x) \mu_n F(x) + q D_n \frac{dn}{dx}$$

$$J_p = q p(x) \mu_p F(x) - q D_p \frac{dp}{dx}$$ (D.8)

where q is used to indicate the absolute value of the electronic charge. Although no direct assumptions on the non-equilibrium distribution function $f(v,x)$ was made in the derivation of (D.8), the choice of equilibrium (thermal) velocity means that the drift-diffusion equations are only valid for very small perturbations of the equilibrium state (low fields). The validity of the drift-diffusion equations is empirically extended by introducing field-dependent mobility $\mu(F)$ and diffusion coefficient $D(F)$, obtained from experimental measurements.