Section: Scientific Foundations

From the Schrödinger equation to Boltzmann-like equations

Participant : François Castella.

The Schrödinger equation is the appropriate way to describe transport phenomena at the scale of electrons. However, for real devices, it is important to derive models valid at a larger scale.

In semi-conductors, the Schrödinger equation is the ultimate model that allows to obtain quantitative information about electronic transport in crystals. It reads, in convenient adimensional units,

 $\begin{array}{c}\hfill i{\partial }_{t}\psi \left(t,x\right)=-\frac{1}{2}{\Delta }_{x}\psi +V\left(x\right)\psi ,\end{array}$ (11)

where $V\left(x\right)$ is the potential and $\psi \left(t,x\right)$ is the time- and space-dependent wave function. However, the size of real devices makes it important to derive simplified models that are valid at a larger scale. Typically, one wishes to have kinetic transport equations. As is well-known, this requirement needs one to be able to describe “collisions” between electrons in these devices, a concept that makes sense at the macroscopic level, while it does not at the microscopic (electronic) level. Quantitatively, the question is the following: can one obtain the Boltzmann equation (an equation that describes collisional phenomena) as an asymptotic model for the Schrödinger equation, along the physically relevant micro-macro asymptotics? From the point of view of modelling, one wishes here to understand what are the “good objects”, or, in more technical words, what are the relevant “cross-sections”, that describe the elementary collisional phenomena. Quantitatively, the Boltzmann equation reads, in a simplified, linearized, form :

 $\begin{array}{c}\hfill {\partial }_{t}f\left(t,x,v\right)={\int }_{{𝐑}^{3}}\sigma \left(v,{v}^{\text{'}}\right)\phantom{\rule{0.277778em}{0ex}}\left[f\left(t,x,{v}^{\text{'}}\right)-f\left(t,x,v\right)\right]d{v}^{\text{'}}.\end{array}$ (12)

Here, the unknown is $f\left(x,v,t\right)$, the probability that a particle sits at position $x$, with a velocity $v$, at time $t$. Also, $\sigma \left(v,{v}^{\text{'}}\right)$ is called the cross-section, and it describes the probability that a particle “jumps” from velocity $v$ to velocity ${v}^{\text{'}}$ (or the converse) after a collision process.