4 TABLE I: Interface spin-orbit coupling parameters (in eV ˚ A $^{2}$) for A/B abrupt interfaces composed of arsenides. α$_{m}$ and α$_{v}$ correspond to the contributions due to the mass gradient and valence-band electric field, respectively, while α$_{int}$ is the total interface SOC strength. A B α$_{m}$ α$_{v}$ α$_{int}$ AlAs InAs 138.93 -169.02 -30.09 GaAs AlAs -35.88 39.07 3.19 InAs GaAs -103.05 129.95 26.9 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., 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. 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 I am grateful to J. Fabian for fruitful discussions. This work was supported by the Deutsche Forschungsgemein- schaft via SFB 689. x ( z ) = x$_{max}$z/d. (18) Acknowledgments 1 I. ˇ Zuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76 , 323 (2004). 2 J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. ˇ Zuti´ c, Acta. Phys. Slovaca 57 , 565 (2007). 3 G. Dresselhaus, Phys. Rev. 100 , 580 (1955). 4 Y. A. Bychkov and E. I. Rashba, J. Phys. C 17 , 6039 (1984). 5 R. Lassnig, Phys. Rev. B 31 , 8076 (1985). 6 R. Winkler, Spin-orbit coupling effects in two-dimensional electron and hole systems (Springer, Berlin, 2003). 7 R. Winkler, Physica E 22 , 450 (2004). 8 J. J. Sakurai, Advanced Quantum Mechanics (Addison- Wesley, Reading, Massachusetts, 1967). 9 E. O. Kane, Lecture Notes in Physics 133 , 13 (1980). 10 P. L¨ owdin, J. Chem. Phys. 19 , 1396 (1951). 11 E. A. de Andrada e Silva, G. C. La Rocca, and F. Bassani, Phys. Rev. B 55 , 16293 (1997). 12 I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, J. Appl. Phys. 89 , 5815 (2001). 13 A. D¨ arr, J. P. Kotthaus, and T. Ando, in Proceedings of the 14 J. J. Sakurai, Modern Quantum Mechanics (Addison- Wesley, Reading, Massachusetts, 1994). 15 F. J. Ohkawa and Y. Uemura, J. Phys. Soc. Jpn. 37 , 1325 (1974). 16 F. Malcher, G. Lommer, and U. R¨ ossler, Superlatt. Mi- crostruct. 2 , 267 (1986). 17 W. Zawadzki and P. Pfeffer, Phys. Rev. B 64 , 235313 (2001). 18 Here Kane’s momentum matrix element ˜ P plays the role of m$_{0}$c in the Dirac theory. This correspondence has been previously recognized in Ref. 7. 19 B. Jusserand, D. Richards, G. Allan, C. Priester, and B. Etienne, Phys. Rev. B 51 , 4707 (1995). 20 J. B. Miller, D. M. Zumb¨ uhl, C. M. Marcus, Y. B. Lyanda- Geller, D. Goldhaber-Gordon, K. Campman, and A. C. Gossard, Phys. Rev. Lett. 90 , 076807 (2003). 13th International Conference of Semiconductors , edited by F. G. Fumi (North-Holland, Amsterdam, 1976), p. 774.