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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,

i t ψ(t,x)=-1 2Δ x ψ+V(x)ψ,(11)

where V(x) is the potential and ψ(t,x) 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 :

t f(t,x,v)= 𝐑 3 σ(v,v ' )[f(t,x,v ' )-f(t,x,v)]dv ' .(12)

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