Band offset and strain-induced valence band splitting in semiconductor compounds

Question

My aim is to draw a plot of $GaAs_{1-x}Sb_{x}$/InP (GaAsSb on InP substrate) band offset as a function of As/Sb composition.

The first part is easy if I'm doing this correctly: I found VBO (valence band offset) parameters for GaAs and GaSb, then the bowing parameter for $GaAs_{1-x}Sb_{x}$ VBO and finally the bowing parameter for $GaAs_{1-x}Sb_{x}$ band gap. This is what I got:

However, when I put GaAs or GaSb on InP, strains appear and the valence band is split into heavy hole and light hole subbands. In some articles I found the way to calculate the splitting:

$\Delta E_{HH}=\partial E_H^V + \partial E_S$ $\Delta E_{LH}=\partial E_H^V - \partial E_S$

$\partial E_S=b(1-2\frac{C_{12}}{C_{11}})\epsilon_{||}$ $\partial E_H^V=2 a^V (1-\frac{C_{12}}{C_{11}})\epsilon_{||}$

a, b and Cs are material properties and can be found in this article: http://jap.aip.org/resource/1/japiau/v89/i11/p5815_s1 (Vurgraftman, Meyer, Ram-Mohan)

However, I'm not sure what the $\epsilon_{||}$ is. Is it the direction of growth? Whatever it is, I can't see where I should consider the substrate, as I assume the strain to be substrate-dependent. Is it hidden in the $\epsilon$?

Answer 1

I think I found a solution in a book "Physics of Optoelectronic devices" by S.L Chuang. It's written there that:

$\epsilon=\frac{a_{substrate}-a_{layer}}{a_{substrate}}$

It makes sense - the higher the difference between lattice constants, the higher the VB splitting. It's of order of several meV, which is also a reasonable value. I attach a plot in case someone has a similar problem in the future. Red lines represent an unstrained GaAsSb, LH level is plotted in green, HH and CB splitting in blue. It's interesting to see that the band gap is lower when the layer is stretched and higher when it's "squeezed".