Physics, asked by anoushi3776, 6 hours ago

The dependence between density and the Fermi energy to 2Dfree electronic gas

Answers

Answered by ankitabareth200787
2

Answer:

I am assuming that you are referring to the free electron model in 2d.

Consider N electrons confined to a rectangular box with sides lx,ly .The Schrodinger equation for a free 2d system is

−ℏ22m(∂2∂x2+∂2∂y2)ψ=Eψ

The solutions are

ψ(x,y)=2lxly−−−√sin(nxπlxx)sin(nyπlyy)

The allowed energy levels are

E=ℏ22m(n2xπ2l2x+n2yπ2l2y)

Let

k⃗ =(nxπlx,nyπly)

Then

k2=n2xπ2l2x+n2yπ2l2y

So

E=ℏ2k22m

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