Rashba effect

The Rashba effect, named for Emmanuel Rashba, is a momentum-dependent spin-orbit interaction associated with broken spatial inversion symmetry. In an isotropic two-dimensional model, choose $\lambda >0$ and write the Hamiltonian $H={\hslash }^2{k}^{2}I/(2m)+\lambda (\hat{\mathbf{z}}\times \mathbf{k})\cdot \mathbf{\sigma }$, where $I$ is the identity matrix and $\mathbf{\sigma }$ is the vector of Pauli matrices. The eigenvalues of the spin term give $E_{\pm }(k)={\hslash }^2{k}^2/(2m)\pm \lambda k$. Define $k_R=m\lambda /{\hslash }^2$ and $E_R={\hslash }^2{k}_R^2/(2m)$. The lower branch is $E_{-}(k)={\hslash }^2(k-k_R{)}^2/(2m)-E_R$ and has a ring of minima at $k=k_R$. A negative $\lambda$ reverses the spin directions; ordering the two energies still uses $|\lambda |$. The alternative parameter $q_R=2k_R$ must not be mistaken for the minimum radius.