Quantum Monte Carlo study of two dimensional electron gas in presence of Rashba interaction

1. Quantum Monte Carlo study of two dimensional electron gas in presence of Rashba interaction A. Ambrosetti, F. Pederiva and E. Lipparini

2. TheRashba Interaction • Rashba interaction has been proved to exist in semiconductor heterostructures, where electrons are subject to a quantum well confinement and therefore move in a 2Dspace (plane). • It is a spin-orbit-like interaction, coupling momentum with spin. • It can be tuned in strenght through gate voltage.

3. TheRashba Interaction • Due to the well asimmetry, electrons are subject to an electric field perpendicular to their plane of motion. • This causes electrons to sense an in-plane effective magnetic field because of relativistic effects • The electron spin couples to the magnetic field giving rise to the Rashba interaction:

4. Switching off Coulomb • In absence of Coulomb interaction the problem is exactly solvable • What we get is two different eigenstates for each wavevector k, consisting of different k-dependent spin states with two different energies • This generates two energy bands, giving “quasi up – quasi down” spin polarization

5. Switching Coulomb on • When Coulomb interaction is introduced the solution to this problem is not known analytically • We need to use a numerical approach. Diffusion Monte Carlo (DMC) is our method of choice: • Widely used for electrons • Very accurate • We know how to treat SO interactions

6. HOW DOES DMC WORK? • Take an initial wave function • Make it evolve in imaginary time • Expand over the Hamiltonian eigenstates: • Multiply by where = ground state energy: • Let go to infinity All excited states will be multiplied by the factor Projection over the ground state is obtained!

7. DMC algorithm Suppose that our Hamiltonian contains only a kinetic term • The solution is given by where we used the free particle Green’s function • In terms of walkers, free propagation means generating displacements -> DIFFUSION.

8. DMC algorithm • Now suppose we have a kinetic term plus a central potential • From Trotter’s formula   • Take into account then the effect of the interaction term over the “renormalized” wavefunction • This can be seen as a weight, i.e. the probability for the walker in R to survive after a time

9. Implementation of DMC A possibile implementation of the projection algorithm is: • Generate initial walkers distribution according to • Diffuse walkers due to free propagation • Kill or multiply walkers due to weight • Repeat steps until convergence is achieved

10. Spin-Orbit propagator • For small time steps • The idea is using Coulomb potential as a weight, and applying the Rashba term right after the free propagator • We can thus rewrite this as • Which means we will need to sample displacements with the free propagator, and then rotate spins according to the just sampled.

11. Checking DMC with SO propagator CHECK • In absence of Coulomb potential the problem is analytically solvable. • The exact ground state solution is a slater determinant of plane waves. • Modify it multiplying by a jastrow factor: • DMC must be able to project over ground state (red dot)

12. RESULTS • Ground state energies are shown at constant density for different values of Rashba strenght. • The minimum is shifted when interaction strenght increases

13. RESULTS Hartree-Fock energy is known analytically, like the energy in absence of Coulomb interaction. In such cases solutions are made of plane waves Slater determinants.

14. CONCLUSIONS • We developed a functioning algorithm based on previous work in nuclear physics, using spin-orbit propagation • We have made some tests on method and trial wavefunction • We are calculating the equation of state for the 2D electron gas in presence of Rashba interaction • We expect to use this method for further research on other systems in presence of spin and momentum dependent interactions