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In the case of semiconductors the analogue to the po- tential mass µ is the effective mass m $^{∗}$, while the equiva- lent to the vacuum gap E$_{0}$ is the energy gap E$_{g}$ separat- ing the energy spectrum of electrons in the conduction band from the hole spectrum in the valence band. In contrast to the vacuum, where E$_{0}$ is a constant, in semi- conductor heterostructures the energy gap E$_{g}$ becomes position dependent. Therefore, the position dependence of the effective mass may originate from both the elec- trostatic potential V and the band gap E$_{g}$ . As a result, in addition to the conventional SOC produced purely by the electric field a finite SOC contribution induced by the position dependence of the effective mass emerges. 

To investigate in more details the mass gradient in- duced SOC, I consider a semiconductor heterostructure grown in the z direction. In such a case the mass gradient induced SOC can be related to the well-known Bychkov- Rashba (BR) SOC observed in quasi two-dimensional systems with structure inversion asymmetry (SIA). 4 

The effective Hamiltonian describing the motion of the conduction band electrons in the heterostructure can be obtained by using the envelope function approximation. I consider the (8 × 8) Kane Hamiltonian 2,6,9 which ac- counts for the Γ$_{6}$$_{c}$, Γ$_{8}$$_{v}$, and Γ$_{7}$$_{v}$ bands [see bands C, HH and LL, and SO bands, respectively, in Fig. 1]. The conduction and valence band states can be decoupled by using the folding-down (L¨ owdin) technique. 2,6,10 Neglect- ing the non-parabolicity effects, the effective Hamiltonian for the conduction electrons is found to be 2,5,11 

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where 

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is the z -dependent, inverse effective mass for the conduc- tion band electrons and 

with 

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is the SIA-induced SOC. In the equations above p$_{x}$ and p$_{y}$ are the components of the in-plane momentum p$_{‖}$ and ˜ P = 〈 S | p$_{x}$ | P 〉 represents the non-vanishing momen- tum matrix elements involving the s-like band edge Bloch state ( | S 〉 ) of the conduction band and the p-like hole states ( | P 〉 = | X 〉 , | Y 〉 , | Z 〉 ). The correction to the effec- tive mass due to the interaction with remote bands can be included by using perturbation theory. 6,12 As shown in Fig. 1, Δ$_{0}$( z ) refers to the spin-orbit splitting energy, while V$_{c}$ ( z ) and V$_{v}$ ( z ) are the potential profiles of the conduction and valence band edges, respectively. 

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For the case of a quantum well (QW) grown along the z -direction one can obtain an effective Hamiltonian de- s cribing the in-plane motion of a tow-dimensional elec- t ron gas by averaging Eq. (7) with the spin-independent z -component, f ( z ), of the wave function. This re- s ults in the so-called BR SOC with α$_{BR}$ = 〈 α ( z ) 〉$_{c}$ = ∫ α ( z ) | f ( z ) | $^{2}$dz as the BR SOC strength. At first glance it seems that the BR spin splitting s hould be proportional to the electric field which brakes 

t he spatial inversion symmetry. However, the fact that f or quasi 2D systems with position-independent effective mass the electric field along the direction of confinement must average to zero 13 (this follows from Ehrenfest’s the- orem which states that the average force on a bound state vanishes $^{14}$) generated intensive discussions about the na- ture of the electric field causing the BR SOC. 5,13,15,16 It was latter found that for a quantum well growth along the z direction, with z -dependent effective mass, the average electric field E$_{c}$ ∼ 〈 p$_{z}$ [( ∂$_{z}$m − $^{1}$) p$_{z}$ ] 〉$_{c}$ may not vanish. 17 

However, the estimated value for this field was found to be too small 16,17 as to explain the experimentally ob- served spin-splitting due to the SIA SOC. Actually, the SOC induced by E$_{c}$ represents a high-order correction which is not even present in the effective Hamiltonian obtained with the standard (8 × 8) Kane approximation. 

In an attempt to clarify the origin of the BR SOC, Lassnig 5 showed that the BR spin splitting in the con- duction band is related to the electric field in the valence band [ E$_{v}$ = ( − 1 /e ) ∇ V$_{v}$ ] whose average (over the con- duction states) does not necessarily vanish (note that in this case Ehrenfest’s theorem does not apply $^{6,7}$). Below I show that in addition to the SOC induced purely by E$_{v}$ a contribution originating from the existence of a mass gra- dient appears. I remark, however, that the mass-gradient contribution discussed here is much larger than the one generated by the mass-gradient induced electric field E$_{c}$ (see discussion in the previous paragraph) and appears readily in the effective Hamiltonian resulting from the (8 × 8) Kane model. In fact, the BR SOC can be rein- 

