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

A. Ambrosetti, F. Pederiva and E. Lipparini

The Rashba 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 2D2D space (plane).

It is a spin-orbit-like interaction, coupling momentum with spin.

It can be tuned in strenght through gate voltage.

The Rashba 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:

)( xyyxSO PPV

yPxPB xy ˆˆ

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

N

iixyyx

i PPmPH

1

2

)(2

kmk

k 2

)(2

2,1

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

HOW DOES DMC WORK?

Take an initial wave function

Make it evolve in imaginary time

Expand over the Hamiltonian eigenstates:

Multiply bywhere = ground state energy:

Let go to infinity

All excited states will be multiplied by the factor

Projection over the ground state is obtained!rojection over the ground state is obtained!

)0(

)0()( )( 0 He

it

i

iiiec )(

0e0

)( 0 ie

DMC algorithm

Suppose that our Hamiltonian contains only a kinetic term 

The solution is given bywhere we used the free particle Green’s function

In terms of walkers, free propagation means generating displacements -> DIFFUSION.

),(2

),(1

22

Rm

RN

ii

')0,'(),'(),( dRRRRGR

DRR

DRRG N 4

)'(exp

)4(1

)'(2

mD

2

2

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

),(])([ 0 Re RV

),(0 Re

)()( oeeee TVVTH

),()(),(2

),(1

22

RRVRm

RN

ii

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 

)0,(R

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.

CoulombRashba VVTH

TVVH eeee RashbaCoulomb

DR

xyDi

ee yx 2)(

)(2

R

DRR

yxPP e

xi

yiIRRGe yxxy 2

)'(

0)(

2

)]([),',(

Checking DMC with SO propagator

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)

CHECK

Slaterji

ij DrusR *))(exp(),(

RESULTS

Ground state energies are shown at constant density for different values of Rashba strenght.

The minimum is shifted when interaction strenght increases NVrs /2

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.

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