# Rashba effect vs Dresselhaus effect (spin orbit coupling)

I am studying Transition Metal Dichalcogenides (TMDs) currently, and am having a hard with these two concepts.

I have read that the rashba effect only happens at interfaces. What about in a monolayer where inversion symmetry is broken (2H polymorph of a TMD)?

I understand that both are inversion breaking mechanisms - but rashba relates to structure inversion asymmetry, whereas Dresselhaus relates to bulk inversion asymmetry. I am not really sure what this means. Could someone explain this?

My understanding, and please correct me if I am wrong, is that at the monolayer limit (or odd number of layers), because of the out-of plane inversion asymmetry and in-plane mirror symmetry the crystal electric field, E, flows in-plane. The electrons also move in-plane with momentum K. Lorrentz transforming this, we see that the effective magnetic field is out of plane and the electrons spin will point out of plane (either up or down depending on k). This lifts the spin degeneracy. Is it right to say this is an effective Zeeman splitting with no external magnetic field?

I know rashba splitting lifts the degeneracy by shifting along the momentum axis, whereas this effective Zeeman splitting (is this dresselhaus?!?) would lift it along the energy axis.

I am pretty dang confused.

Think about an interface between two different things. For example, surface is an interface between vacuum and bulk.

Then, because of the discontinuity at the interface, dipoles are often made and induce an electric field towards bulk direction that results in band bending. This field breaks inversion symmetry at the interface and lifts spin degeneracy. This is so-called Rashba effect.

The Dresselhaus spin-splitting arises from the lack of inversion symmetry in the material itself. That is, the crystal lacks this symmetry operation (a reflection symmetry) and therefore so does the reciprocal lattice. In the conduction band of such a material (such as GaAs or InAs), this means that the energy for a wavefunction with wavevector $\mathbf{k}$ is not, in general, equal to that with $-\mathbf{k}$. Specifically, the energy shift is given by

$$\epsilon = \pm \gamma_{D}\left[k^{2}\left(k_x^2k_y^2 + k_y^2k_z^2 + k_z^2k_x^2\right) - 9k_x^2k_y^2k_z^2\right]^{1/2}$$

(where $\gamma_{D}$ is the Dresselhaus coefficient), which lifts the spin degeneracy in all but the $[100]$ and $[111]$ directions. This is given by a Hamiltonian of the form

$$H_{SO} = \gamma_{D}\boldsymbol\sigma\cdot\boldsymbol\kappa$$

where $\boldsymbol\sigma$ is the Pauli pseudo-vector and the components of $\boldsymbol\kappa$ are given by cyclic permutation of

$$\kappa_{x} = k_{x}\left(k_{y}^{2} - k_{z}^{2}\right).$$

(Here $x,y$ and $z$ label the crystallographic axes). Note that this Hamiltonian is the same form as the interaction of a magnetic field with Lamor frequency $\boldsymbol\Omega$

$$H_{SO} = \frac{\hbar}{2}\boldsymbol\sigma\cdot\boldsymbol\Omega.$$

Therefore the spin-orbit-interaction can indeed be viewed as analogous to a Zeeman splitting in an effective magnetic field, except that in the present case, there is no time reversal symmetry.

Inversion symmetry may also be broken by an electric field $\mathbf{E}$, giving rise to the Bychkov-Rashba SOI. This electric field may arise due to the spatial variation of the band edge in asymmetric nanostructures and hence the effect is refered to as structural inversion asymmetry (SIA) SOI. The contribution to the Hamiltonian is given by

$$H_{BR} = \alpha_{0}\boldsymbol\sigma\cdot\left(\mathbf{k}\times\mathbf{E}\right)$$

where $\alpha_{0} = e\eta(2-\eta)P^{2}/(3m_{0}^{2}\epsilon_{g}^{2})$, $e$ is the elementary charge, $m_{0}$ is the free electron mass, $\epsilon_g$ is the band gap and $P$ is the interband momentum matrix element.