Answer
Answer
Let for electron has wave function
ψ∝eℏikx
The effective momentum of electron is given by
2m∗ℏ2k2=−qϕ
k=iℏ22m∗qϕ
So wavefunction become
ψ∝eiℏ22m∗qϕx
ψ(x=0)ψ(x=d)∝e−ℏ22m∗qϕ.0e−ℏ22m∗qϕ.d∝e−ℏ22m∗qϕd
Let JI
be the incident current and JT
be the transmitted current. Then
Transmission probability
T∝J0JT∝(ψ(x=0)ψ(x=d))2∝e−2−ℏ22m∗qϕd
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