2 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 where is the z -dependent, inverse effective mass for the conduc- tion band electrons and with 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 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 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- 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. H$_{eff}$ = p 2 ‖ 2 m $^{∗}$( z ) − ℏ 2 2 d dz [ 1 m $^{∗}$( z ) d dz ] + V$_{c}$ ( z ) + H$_{so}$, (7) 1 m $^{∗}$( z ) = 1 m$_{0}$ − 2 ˜ P 2 3 m 2 0 [ 2 V$_{v}$ ( z ) + 1 V$_{v}$ ( z ) − Δ$_{0}$( z ) ] (8) H$_{so}$ = α ( z ) ℏ ( p$_{y}$σ$_{x}$ − p$_{x}$σ$_{y}$ ) (9) α ( z ) = ℏ 2 3 m 2 0 d dz [ ˜ P 2 V$_{v}$ ( z ) − ˜ P 2 V$_{v}$ ( z ) − Δ$_{0}$( z ) ] (10) F IG. 1: (color online). Schematic of the band-edge profile o f an A/B semiconductor interface located at z = 0. The c onduction, heavy hole, light hole, and split-off bands are l abelled as C, HH, LH, and SO, respectively. A 0 B 0 B g E ( ) v V z ( ) c V z 0 z A B C HH LH SO A g E