# Transmission through semiconductor heterojunction

1. Mar 31, 2015

### John Loven

I have calculated the electron transmission function T(E) over a potential step of height V0 using T-matrices. I model a semiconductor heterojunction, which requires different effective electron masses on either side of the step.

The wave functions on either side of the step are planar waves:

Y1(x)=Aexp(ik1x)+Bexp(-ik1x),
Y2(x)=Cexp(ik2x)+Dexp(-ik2x),

where A=1, B=r, C=t, D=0 and

k1 = sqrt(2m1E/hbar),
k2 = sqrt(2m2(E-V0)/hbar).

We have 2 boundary conditions at the step at x = x0:

Y1(x0) = Y2(x0),
1/m1*d/dxY1(x0) = 1/m2*d/dxY2(x0).

Note that the second boundary condition is not the standard one, since we have to account for the different masses, in order to have current conservation.

I calculate the T-matrix M and get the transmitted wave amplitude

t = (M(1,1)*M(2,2)-M(1,2)*M(2,1)) / M(2,2).

I then calculate the transmission flux

T(E) = (k2m1)/(k1m2)*abs(t)^2.

When plotting T(E) it only approaches unity for increasing E, if the masses are equal. If they are not equal T(E) approaches a value less than unity.

I'm just wondering if this result is correct? Is there always some reflection at a heterojunction potential step with different effective electron masses, even for very large energies?

Thanks,

John

Last edited: Mar 31, 2015
2. Apr 2, 2015

### DrDu

Sounds plausible to me as k's are directly proportional to $m^{1/2}$.
Same as for light reflection when refractive indices n1 and n2 are different.