Spin Hall effect in four terminal devices

KWANT script below describes four-terminal graphene device, with gold adatoms in the central square, which generates spin Hall current $I_{2}^{S}$ in the transverse leads as a response to injected longitudinal charge current $I_{1}$. The script output is the spin Hall angle defined as $\theta _{\mathrm {SH} }=I^{S}/I$.

• It may be required to install sympy package. For Windows users, download the .whl file from this link; then, Go to the directory with the .whl file, e.g.:

cd c:\Users\YOUR_USERNAME\Downloads

To install all the .whl-files in the current directory, execute

python -c "import pip, glob; pip.main(['install', '--no-deps'] + glob.glob('*.whl'))"

For more information, refer to Kwant installation instructions here.

from math import sqrt
import random
import itertools as it
import tinyarray as ta
import numpy as np
import matplotlib.pyplot as plt
import kwant
from sympy import S, Eq, solve, Symbol

class Honeycomb(kwant.lattice.Polyatomic):
    """Honeycomb lattice with methods for dealing with hexagons"""

    def __init__(self, name=''):
        prim_vecs = [[0.5, sqrt(3)/2], [1, 0]]  # bravais lattice vectors
        # offset the lattice so that it is symmetric around x and y axes
        basis_vecs = [[-0.5, -1/sqrt(12)], [-0.5, 1/sqrt(12)]]
        super(Honeycomb, self).__init__(prim_vecs, basis_vecs, name)
        self.a, self.b = self.sublattices

    def hexagon(self, tag):
        """ Get sites belonging to hexagon with the given tag.
            Returns sites in counter-clockwise order starting
            from the lower-left site.
        """
        tag = ta.array(tag)
        #         a-sites b-sites
        deltas = [(0, 0), (0, 0),
                  (1, 0), (0, 1),
                  (0, 1), (-1, 1)]
        lats = it.cycle(self.sublattices)
        return (lat(*(tag + delta)) for lat, delta in zip(lats, deltas))

    def hexagon_neighbors(self, tag, n=1):
        """ Get n'th nearest neighbor hoppings within the hexagon with
            the given tag.
        """
        hex_sites = list(self.hexagon(tag))
        return ((hex_sites[(i+n)%6], hex_sites[i%6]) for i in range(6))

def random_placement(builder, lattice, density):
    """ Randomly selects hexagon tags where adatoms can be placed.
        This avoids the edge case where adatoms would otherwise
        be placed on incomplete hexagons at the system boundaries.
    """
    assert 0 <= density <= 1
    system_sites = builder.sites()

    def hexagon_in_system(tag):
        return all(site in system_sites for site in lattice.hexagon(tag))

    # get set of tags for sites in system (i.e. tags from only one sublattice)
    system_tags = (s.tag for s in system_sites if s.family == lattice.a)
    # only allow tags which have complete hexagons in the system
    valid_tags = list(filter(hexagon_in_system, system_tags))
    N = int(density * len(valid_tags))
    total_hexagons=len(valid_tags)
    valuef=random.sample(valid_tags, N)
    return valuef

def cross(W, L):
    def shape(pos):
        return ((-W <= pos[0] <= W and -L <= pos[1] <= L) or  # vertical
                (-W <= pos[1] <= W and -L <= pos[0] <= L))    # horizontal
    return shape

## Pauli matrices ##
sigma_0 = ta.array([[1, 0], [0, 1]])  # identity
sigma_x = ta.array([[0, 1], [1, 0]])
sigma_y = ta.array([[0, -1j], [1j, 0]])
sigma_z = ta.array([[1, 0], [0, -1]])

## Hamiltonian value functions ##

def onsite_potential(site, params):
    return params['ep'] * sigma_0

def potential_shift(site, params):
    return params['mu'] * sigma_0

def kinetic(site_i, site_j, params):
    return -params['gamma'] * sigma_0

def rashba(site_i, site_j, params):
    d_ij = site_j.pos - site_i.pos
    rashba = 1j * params['V_R'] * (sigma_x * d_ij[1] - sigma_y * d_ij[0])
    return rashba + kinetic(site_i, site_j, params)

def spin_orbit(site_i, site_j, params):
    so = (1j/3.) * params['V_I'] * sigma_z
    return so

lat = Honeycomb()
pv1, pv2 = lat.prim_vecs
ysym = kwant.TranslationalSymmetry(pv2 - 2*pv1)  # lattice symmetry in -y direction
xsym = kwant.TranslationalSymmetry(-pv2)  # lattice symmetry in -x direction

# adatom lattice, for visualization only
adatom_lat = kwant.lattice.Monatomic(lat.prim_vecs, name='adatom')

def site_size(s):                                          
    return 0.25 if s.family in lat.sublattices else 0.35

def site_color(s): 
    return '#000000' if s.family in lat.sublattices else '#de962b'

def create_lead_h(W, symmetry, axis=(0, 0)):
    lead = kwant.Builder(symmetry)
    lead[lat.wire(axis, W)] = 0. * sigma_0
    lead[lat.neighbors(1)] = kinetic
    return lead

def create_lead_v(L,W, symmetry):
    axis=(0, 0)
    lead = kwant.Builder(symmetry)
    lead[lat.wire(axis, W)] = 0. * sigma_0
    lead[lat.neighbors(1)] = kinetic
    return lead

def create_hall_cross(W, L, adatom_density=0.2, show_adatoms=False):
    ## scattering region ##
    sys = kwant.Builder()
    sys[lat.shape(cross(W, L), (0, 0))] = onsite_potential
    sys[lat.neighbors(1)] = kinetic

    adatoms = random_placement(sys, lat, adatom_density)

    sys[(lat.hexagon(a) for a in adatoms)] = potential_shift
    sys[(lat.hexagon_neighbors(a, 1) for a in adatoms)] = rashba
    sys[(lat.hexagon_neighbors(a, 2) for a in adatoms)] = spin_orbit
    if show_adatoms:
        # no hoppings are added so these won't affect the dynamics; purely cosmetic
        sys[(adatom_lat(*a) for a in adatoms)] = 0
    ## leads ##
    leads = [create_lead_h(W, xsym)]
    leads += [create_lead_v(L,W, ysym)]
    leads += [lead.reversed() for lead in leads]  # right and bottom leads
    for lead in leads:
        sys.attach_lead(lead)
    return sys

def plot_ldos(sys, Es, params):
    fsys = sys.finalized()
    ldos = kwant.ldos(fsys, energy=Es, args=(params,))
    ldos = ldos[::2] + ldos[1::2] # sum spins
    kwant.plotter.map(fsys, ldos, vmax=0.1)

def plot_conductance(sys, energies,lead_i,lead_j, params):
    fsys = sys.finalized()
    data = []

    for energy in energies:
        smatrix = kwant.greens_function(fsys, energy, args=(params,))
        data.append(smatrix.transmission(lead_i,lead_j))

    plt.figure()
    plt.plot(energies, data)
    plt.xlabel("energy (t)")
    plt.ylabel("conductance (e^2/h)")
    plt.show()

def plot_spin_conductance(sys, energies,lead_i,lead_j, params):
    fsys = sys.finalized()
    data = []

    for energy in energies:
        smatrix = kwant.greens_function(fsys, energy, args=(params,))
        ttdagger = smatrix._a_ttdagger_a_inv(lead_i, lead_j)
        sigma_z_matrix = np.kron(np.eye(len(ttdagger)/2),sigma_z)    
        data.append(np.trace(sigma_z_matrix.dot(ttdagger)).real)
    plt.figure()
    plt.plot(energies, data)
    plt.xlabel("energy (t)")
    plt.ylabel("spin conductance (e^2/h)")
    plt.show()

def solvematrix4(G,I0,I2,V1,V3):
    # I=GxV
    I1,I3,V0,V2=S('I1 I3 V0 V2'.split())
    equations = [Eq(I0, G[0][0]*V0+G[0][1]*V1+G[0][2]*V2+G[0][3]*V3), \
                 Eq(I1, G[1][0]*V0+G[1][1]*V1+G[1][2]*V2+G[1][3]*V3), \
                 Eq(I2, G[2][0]*V0+G[2][1]*V1+G[2][2]*V2+G[2][3]*V3), \
                 Eq(I3, G[3][0]*V0+G[3][1]*V1+G[3][2]*V2+G[3][3]*V3)]
    return solve(equations, [I1,I3,V0,V2], set=True, dict=False, rational=None, manual=True, minimal=True, particular=True, implicit=False, quick=True)

def plot_SH_angle(sys, energies, params):
    fsys = sys.finalized()
    data = []

    I0=0
    I2=0
    V1=1.0
    V3=0

    for energy in energies:
        smatrix = kwant.greens_function(fsys, energy, args=(params,))
        G=np.zeros((4,4))
        for i in range(4):
            a=0
            for j in range(4):
                G[i,j]=smatrix.transmission(i, j)
                if i != j: a+=G[i,j]
            G[i,i]=-a
        a=solvematrix4(G,I0,I2,V1,V3)[1]
        result=list(a)

        I1=result[0][0]
        I3=result[0][1]
        V0=result[0][2]
        V2=result[0][3]
        print(I1,I3,V0,V2)

        Gspin=np.zeros((4,4))
        for i in range(4):
            a=0
            for j in range(4):
                ttdagger = smatrix._a_ttdagger_a_inv(i, j)
                sigma_z_matrix = np.kron(np.eye(len(ttdagger)/2),sigma_z)    
                Gspin[i,j]=np.trace(sigma_z_matrix.dot(ttdagger)).real
                if i != j: a+=Gspin[i,j]
            Gspin[i,i]=-a

        IS0=Gspin[1,0]*(V1-V0)+Gspin[2,0]*(V2-V0)+Gspin[3,0]*(V3-V0)
        data.append(IS0/I1)

    plt.figure()
    plt.plot(energies, data)
    plt.xlabel("energy (t)")
    plt.ylabel("spin hall angle")
    plt.show()

if __name__ == '__main__':
    params = dict(gamma=1., ep=0, mu=0.1, V_I=0.007, V_R=0.0165)  # Au adatoms

    W=16
    L=32
    adatom_density=0.15

    sys = create_hall_cross(W, L, adatom_density, show_adatoms=True)
    kwant.plot(sys, site_color=site_color, site_size=site_size)

    sys = create_hall_cross(W, L, adatom_density, show_adatoms=False)
    plot_ldos(sys, 1e-5, params)
    
    Es = np.linspace(-0.1, 0.1, 100)
    plot_spin_conductance(sys, Es, 1, 0, params)   # T01
    plot_conductance(sys, Es, 1, 0, params)   # TS01

    plot_SH_angle(sys, Es, params)