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# Spin-orbit coupling induced by a mass gradient 

A. Matos-Abiague 

Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany 

(Dated: October 29, 2018) 

The existence of a spin-orbit coupling (SOC) induced by the gradient of the effective mass in low-dimensional heterostructures is revealed. In structurally asymmetric quasi-two-dimensional semiconductor heterostructures the presence of a mass gradient across the interfaces results in a SOC which competes with the SOC created by the electric field in the valence band. However, in graded quantum wells subjected to an external electric field, the mass-gradient induced SOC can be finite even when the electric field in the valence band vanishes. 

Semiconductor spintronics is an emerging field based on the controlled manipulation of the carrier spins for data processing and device operations. 1,2 Most proposals for spintronic devices rely on the ability of manipulating electron spins by using the spin-orbit coupling (SOC), which is the most fundamental spin-dependent interac- tion in nonmagnetic semiconductors. 2 In semiconductor heterostructures the SOC results from the lack of inver- sion symmetry. The bulk inversion asymmetry (BIA) of zinc-blende semiconductors leads to the so-called Dres- selhaus SOC, 3 while the structure inversion asymmetry (SIA) of the heterostructure itself results in the Bychkov- Rashba (BR) SOC. 4 

In this paper I focus on the investigation of the BR- type SOC and show that, in addition to the SOC gen- erated by the electric field in the valence band, 5–7 there is a mass-gradient contribution to the SIA-induced SOC. In general, the two contributions compete. However, in some specific cases the mass-gradient induced SOC dom- inates. 

The emergence of SOC due to the existence of a mass gradient can be better understood by establishing an analogy between the nonrelativistic limit of Dirac’s the- ory and the effective-mass Hamiltonian for conduction electrons in semiconductor heterostructures. 

For simplicity and without loss of generality, I consider the case of a time-independent system in the presence of an electrostatic potential V . Starting with the time- independent Dirac equation the upper ψ$_{u}$ and lower ψ$_{l}$ components of the four-component spinor Ψ = ( ψ$_{u}$, ψ$_{l}$ ) T are found to be coupled through the equations 8 

where ϵ is the particle energy (measured from the rest energy m$_{0}$c $^{2}$), σ is the vector of Pauli matrices and p , m$_{0}$ , and c , refer to the momentum operator, the bare electron mass, and the velocity of light, respectively. It follows from Eq. (2) that ψ$_{l}$ is smaller than ψ$_{u}$ by a factor ∼ v/c . Thus, in the nonrelativistic limit ( v ≪ c ) the main contribution to the four-component spinor comes 

PACS numbers: 71.70.Ej,73.61.Ey,73.20.Qt,73.21.Fg 

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from ψ$_{u}$ . Using Eq. (2) one can eliminate ψ$_{l}$ from Eq. (2) and obtain, after some algebra, an equation that involves only ψ$_{u}$ , 

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where I have introduced the position-dependent potential mass 

Note that no approximation has been made in deriv- ing Eq. (3). The two-component spinor ψ$_{u}$ is, how- ever, not normalized. The standard procedure to over- come this problem in the nonrelativistic limit is to in- troduce a new normalized two-component spinor ˜ ψ = (1 + p $^{2}$/ 8 m$_{0}$c $^{2}$) ψ$_{u}$ . 8 In doing this, however, some approx- imations has to be made and the Hamiltonian for the normalized spinor ˜ ψ acquires additional terms. 8 However, these additional terms are irrelevant for our discussion here and will be omitted in our analysis. 

The Hamiltonian in Eq. (3) is Hermitian and resem- bles the effective-mass Hamiltonian describing the mo- tion of electrons in a solid with position-dependent ef- fective mass. Interestingly, the spin-orbit coupling seems to originate from the gradient of the potential mass µ . In the Dirac theory the energy gap, E$_{0}$ = 2 m$_{0}$c $^{2}$, which separates the energy spectra of the free particles and an- tiparticles is a position-independent constant. Therefore, the only position dependence in µ which can lead to a fi- nite SOC has to come from the electrostatic potential V . Thus, in the Dirac theory, the SOC emerges purely from the electric field E = ( − 1 /e ) ∇ V (here e denotes the elec- tron charge). In the nonrelativistic approximation the dominant energy is the vacuum gap E$_{0}$ and the inverse potential mass can be approximated as [see Eq. (4)] 

As a result the SOC reduces to the well-known form 8 

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