Boltzmann approach to high-order transport: The nonlinear and nonlocal responses

The phenomenological textbook equations for charge and heat transport are extensively used in a number of fields ranging from semiconductor devices to thermoelectricity. We provide a rigorous derivation of transport equations by solving the Boltzmann equation in the relaxation-time approximation and show that the currents can be rigorously represented by an expansion in terms of the “driving forces”. Besides the linear and nonlinear response to the electric field, the gradient of the chemical potential and temperature, there are also terms that give the response to the higher-order derivatives of the potentials. These nonlocal responses might play an important role for some materials and/or under certain conditions, such as extreme miniaturization. Our solution provides the general solution of the Boltzmann equation in the relaxation-time approximation (or, equivalently, the particular solution for the specific boundary conditions). It differs from the Hilbert expansion, which provides only one of infinitely many solutions which may or may not satisfy the required boundary conditions.

Figure 1

(a) Geometry of the metal-semiconductor junction and the direction of positive current. The interface is at $x=0$. (b) Modification of the bottom of the band structure as in Eq. (51). (c) Position-dependent chemical potential calculated for different currents at $T=300$ K for the band structure with a finite $\delta$, but using the textbook transport equations, i.e., setting ${\sigma }^{\left[{\mathbf{\nabla }}_{}\mathbf{E}\right]}=0$ in the complete set of equations. The results depend on the value of the current running through the device (see the inset, which also provides the color code for the currents). (d) Same as (c), but calculated with the correct transport equations, including the additional term ${\sigma }^{\left[{\mathbf{\nabla }}_{}\mathbf{E}\right]}$. The insets in (c) and (d) provide the current vs voltage characteristic of the semiconductor layer.