Dresselhaus linear and cubic terms

This is VERY late.. but neither of these are the cubic term (there is no explicit $p^3$). These are both the result of the $<p_z^2>$ term; i.e. the linear term. And the confinement direction is assumed to be the z-direction.

For a zincblende structure, in its full glory, the Dresselhaus Hamiltonian is:

$H_D = frac{gamma}{hbar}Big(p_x(p_y^2-p_z^2)sigma_x + p_y(p_z^2 - p_x^2)sigma_y + p_z(p_x^2 - p_y^2)sigma_zBig)$.

Assuming $p_z^2 rightarrow <p_z^2>$ and $p_z rightarrow <p_z>=0$, due to confinement in a well:

$H_D = frac{gamma}{hbar}Big(p_x(p_y^2-<p_z^2>)sigma_x + p_y(<p_z^2> - p_x^2)sigma_y + <p_z>(p_x^2 - p_y^2)sigma_zBig)$

$H_D = frac{gamma}{hbar}Big(p_x(p_y^2-<p_z^2>)sigma_x + p_y(<p_z^2> - p_x^2)sigma_y Big)$.

Finally, $frac{gamma}{hbar}<p_z^2> = beta$,

$H_D = underbrace{frac{beta}{hbar}Big(p_y sigma_y - p_x sigma_x Big)}_text{linear} + underbrace{frac{gamma}{hbar}Big(p_xp_y^2sigma_x - p_yp_x^2sigma_yBig)}_text{cubic}$.

This reproduces your second equation's form. Your first has the incorporation of the Rashba effect and has a specific, "privileged orientation," as my advisor has said in one of her papers, such that the linear Dresselhaus looks much like the Rashba effect (this better shows how $alpha = beta$ generates pure spin current, as well as some other points of interest).

My understanding of when the cubic or linear terms are more appropriate has to do with the width of the confinement. Essentially, when the $beta$ definition is used, you have that the expectation of the wave-vector squared is $<k_z^2> approx big( pi/W big)^2$, where $W$ is the confinement width. The cubic term, and the bulk parameter $gamma$ are not $k_z$ dependent and will then dominate when $W$ is large. Hence, why the linear term is usually taken for 2D systems, but is less important in 3D bulk systems.

Here are a couple links that may be useful:

• Understanding the point at which cubic term takes over, Marinescu (2017).
• Learning about the orientation used in the first form as presented in the question, Pan (2019)