3 $^{0}$( ) c V z $^{0}$( ) v V z ( ) c V z v V ext E (a) (b) 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 and with 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 Equation (14) has the same structure as Eq. (6), 18 which confirms that, indeed, H v so corresponds to the stan- dard SOC generated, purely, by an electric field; in this is the SOC created by the electric field in the valence band. is the mass gradient induced SOC and Thus, one can rewrite the SOC in Eq. (9) as H$_{so}$ = H m $_{so}$+ H v $_{so}$, where 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 Combining Eqs. (8) and (10) one can rewrite the SOC p arameter as 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). F IG. 2: (color online). Schematics of the conduction- and v alence-band potentials in a linearly graded quantum well with infinite barriers, in the absence (a) and presence (b) o f an external electric field E$_{ext}$ . The external electric field c ompensates the electric field in the valence band in such a way that the total electric field in the valence band vanishes (i .e., ∇ V f v = 0). H m,v so = α$_{m,v}$ ℏ δ ( z )( p$_{y}$σ$_{x}$ − p$_{x}$σ$_{y}$ ) ( 1 5) α$_{m}$ = ℏ 2 ( 1 m ∗ B − 1 m ∗ A ) , ( 1 6) α$_{v}$ = ℏ 2 m 2 0 ( ˜ P 2 A E A g − ˜ P 2 B E B g ) , ( 1 7) α ( z ) = ℏ 2 2 d dz [ 1 m $^{∗}$( z ) ] + ℏ 2 m 2 0 d dz [ ˜ P 2 V$_{v}$ ( z ) ] . (11) α ( z ) = ℏ 2 2 d dz [ 1 m $^{∗}$( z ) ] − ℏ $^{2}$( ˜ P/m$_{0}$ ) 2 E 2 g d dz V$_{v}$ ( z ) . (12) H m so = ℏ 2 [ ∇ ( 1 m ∗ ) × p ] · σ (13) H v so = − ℏ ( ˜ P/m$_{0}$ ) 2 E 2 g ( ∇ V$_{v}$ × p ) · σ (14)