content_image/442662.jpg content_image/442641.jpg content_image/442661.jpg content_image/442663.jpg content_image/442664.jpg content_image/442658.jpg content_image/442659.jpg content_image/442660.jpg
t erpreted as resulting from the competition between the h ere proposed mass-gradient SOC and the SOC induced p urely by E$_{v}$ . Interestingly, the mass gradient contri- b ution (and therefore the BR SOC) can be finite even when the electric field in the valence band vanishes (i.e, E$_{v}$ = 0). 

Combining Eqs. (8) and (10) one can rewrite the SOC p arameter as 

When modelling semiconductor heterostructures, the m omentum matrix element, ˜ P , is commonly considered t o be position independent. 6 Within this approximation a nd taking into account that for most semiconductors V$_{v}$ is of the same order of the band gap (i.e., | V$_{v}$ − E$_{g}$ | ≪ E$_{g}$ , o ne can approximate Eq. (11) as 

<img src='content_image/442662.jpg'>

Thus, one can rewrite the SOC in Eq. (9) as H$_{so}$ = H m $_{so}$+ H v $_{so}$, where 

is the mass gradient induced SOC and 

is the SOC created by the electric field in the valence band. 18 

Equation (14) has the same structure as Eq. (6), which confirms that, indeed, H v so corresponds to the stan- dard SOC generated, purely, by an electric field; in this 

<img src='content_image/442641.jpg'>

<img src='content_image/442661.jpg'>

<img src='content_image/442663.jpg'>

<img src='content_image/442664.jpg'>

case by the valence-band electric field E$_{v}$ = ( − 1 /e ) ∇ V$_{v}$ . Consequently, H v so vanishes when E$_{v}$ = 0. On the co n- trary, it is clear from Eqs. (8) and (13) that even in the case of vanishing E$_{v}$ , the mass gradient induced SO C, H m $_{so}$, remains, in general, finite. 

For an estimation of the strengths of the mass gradient and valence-band electric field induced SOCs, I consider the case of an A/B abrupt interface between two III-V semiconductors [see Fig. 1]. In such a case the band pa- rameters, and therefore m ∗ and V$_{v}$ , are step-like functio ns of z and the mass-gradient and valence-band electric field induced SOCs in Eq. (11) reduce to 

with 

and 

respectively. Here the values of the parameters in regions A and B are indicated with the respective labels. Note also, that for a better accuracy, the step-like position dependence of the momentum matrix element has also been considered in Eq. (17). 

The calculated values of α$_{m}$ , α$_{v}$ , and the total interfa ce SOC strength, α$_{int}$ = α$_{m}$ + α$_{v}$ , for different interfaces a re listed in Table I. For all the considered interfaces, α$_{m}$ a nd α$_{v}$ are of the same order but with opposite signs. Thus, the competition between the mass-gradient and valence- band electric field induced SOC contributions results in the decrease of the total interface SOC. 

In systems in which α$_{m}$ and α$_{v}$ are of the same ord er (as the ones considered above) the mass-gradient contri- bution to the SOC is masked by the SOC induced by the valence-band electric field. Therefore the experiment al measurement of α$_{m}$ alone may be difficult in such sy s- tems. To overcome this problem, I propose to measure α$_{m}$ in a graded, semiconductor quantum well subjected to an external electric field. 

For illustration, I consider a Ga$_{1}$$_{−}$$_{x}$Al$_{x}$As-based quan- tum well with high potential barriers, so that interface effects play a little role and can be neglected. In a lin- early graded quantum well (i.e, with Al concentration varying linearly with the position) the energy gap to- gether with the band parameters become position de- pendent. For small grading, the band parameters in- terpolate linearly with the Al concentration. Conse- quently, both the potential profile of the conduction ( V 0 c ) and valence ( V 0 v ) band edges change linearly with z [see Fig. 2(a)]. In the presence of a constant external electric field, E$_{ext}$ , oriented along the growth direction the band edge profiles are modified as V$_{c}$ = V 0 c ( z ) − e E$_{ext}$ z and V$_{v}$ = V 0 v ( z ) − e E$_{ext}$ z . Since V 0 v ( z ) is a linear function 

<img src='content_image/442658.jpg'>

<img src='content_image/442659.jpg'>

<img src='content_image/442660.jpg'>

