Scattering Rate

A formula may be derived mathematically for the rate of scattering when a beam of electrons passes through a material.

The interaction picture

Define the unperturbed Hamiltonian by $H_0$, the time dependent perturbing Hamiltonian by $H_1$ and total Hamiltonian by $H$.

The eigenstates of the unperturbed Hamiltonian are assumed to be
:$H=H_0+H_1\$
:$H_0 , k\rang = E(k), k\rang$

In the interaction picture, the state ket is defined by
:$, k(t)\rang _I= e^ , k(t)\rang_S= \sum_ c_(t) , k'\rang$

By a Schrödinger equation, we see
:$i\hbar \frac , k(t)\rang_I=H_, k(t)\rang_I$
which is a Schrödinger-like equation with the total $H$ replaced by $H_$.

Solving the differential equation, we can find the coefficient of n-state.
:$c_(t) =\delta_ - \frac \int_0^t dt' \;\lang k', H_1(t'), k\rang \, e^$

where, the zeroth-order term and first-order term are
:$c_^=\delta_$
:$c_^=- \frac \int_0^t dt' \;\lang k', H_1(t'), k\rang \, e^$

The transition rate

The probability of finding $, k'\rang$ is found by evaluating $, c_(t), ^2$.

In case of constant perturbation,$c_^$ is calculated by
:$c_^=\frac(1-e^)$

:$, c_(t), ^2= , \lang\ k', H_1, k\rang , ^2\frac \frac$

Using the equation which is
:$\lim_ \frac \frac= \delta(x)$

The transition rate of an electron from the initial state $k$ to final state $k'$ is given by

:$P(k,k')=\frac , \lang\ k', H_1, k\rang , ^2 \delta(E_-E_k)$

where $E_k$ and $E_$ are the energies of the initial and final states including the perturbation state and ensures the $\delta$-function indicate energy conservation.

The scattering rate

The scattering rate w(k) is determined by summing all the possible finite states k' of electron scattering from an initial state k to a final state k', and is defined by

:$w(k)=\sum_P(k,k')=\frac \sum_ , \lang\ k', H_1, k\rang , ^2 \delta(E_-E_k)$

The integral form is
:$w(k)=\frac \frac \int d^3k' , \lang\ k', H_1, k\rang , ^2 \delta(E_-E_k)$

References

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{{DEFAULTSORT:Scattering Rate
Semiconductor technology