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of z one can find a target electric field for which V$_{v}$ be- comes position independent and the total electric field in the valence bands vanishes [see Fig. 2(b)]. Therefore, for such a target external field, α$_{v}$ ≈ 0, while α$_{m}$ remains fi- nite. Under this condition, the SOC is determined solely by the mass-gradient contribution. 

I now estimate the values of α$_{m}$ and the target external field E$_{target}$ for a quantum well with Al concentration varying linearly from x$_{min}$ = 0 to x$_{max}$ = 0 . 1, i.e., 

Here d = 200 ˚ A is the well width. The composition de- pendence of the band parameters is evaluated accord- ing to the interpolation scheme developed in Ref. 12. I then expand Eq. (8) up to the linear order in x [which for the small concentrations considered here ( x ≤ 0 . 1) suffices] and obtain the position dependence of the in- verse effective mass by using (18). For the target ex- ternal field E$_{target}$ ≈ 30 kV/cm the valence-band elec- tric field vanishes (i.e., α$_{v}$ = 0), while the strength of the mass-gradient induced SOC is found to be α$_{m}$ ≈ − 8 . 74 meV ˚ A. Apart from the sign, this value is of the same order but still larger than the SIA SOC parameters experimentally measured in GaAs-AlGaAs asymmetric 

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quantum wells. 19,20 

Beyond the case of semiconductor heterostructures, the mass-gradient induced SOC is expected to be relevant in metal/semiconductor interfaces across which the val- ues of the effective mass have large and abrupt changes. 

In summary, I have revealed the existence of a mass- gradient contribution to the SOC in systems with struc- ture inversion asymmetry. I have shown that for some semiconductor heterostructures, the mass-gradient con- tribution is of the same order as the SOC generated by the electric field in the valence band. However, in the particular case of a linearly graded semiconductor quan- tum well subjected to a conveniently designed external field, the electric field in the valence band vanishes and the remaining SOC is purely induced by the mass gradi- ent. 

I am grateful to J. Fabian for fruitful discussions. This work was supported by the Deutsche Forschungsgemein- schaft via SFB 689. 

13th International Conference of Semiconductors , edited by F. G. Fumi (North-Holland, Amsterdam, 1976), p. 774. 

## Acknowledgments 

