Rashba Spin-Orbit Coupling: Spin-Split Bands from Broken Inversion Symmetry

An electron moving through an electric field sees, in its rest frame, a magnetic field \(\propto \mathbf{E}\times\mathbf{v}\). At a surface or an asymmetric quantum well the field points out of the plane, and Bychkov and Rashba (1984) wrote the coupling it produces for a 2D electron gas as

\[H_R = \lambda\,(\boldsymbol\sigma\times\mathbf{k})\cdot\hat{\mathbf{z}}\, .\]

It needs no magnet and keeps time-reversal symmetry, but splits the spin degeneracy of every band away from \(\mathbf{k} = 0\) and locks the spin perpendicular to the momentum. On a square lattice with nearest-neighbour hopping \(t\), set_rashba() gives the bands

\[E_\pm(\mathbf{k}) = 2t(\cos k_x + \cos k_y) \pm \sqrt{b_z^2 + 4\lambda^2(\sin^2 k_x + \sin^2 k_y)}\, ,\]

with \(b_z\) a Zeeman field (set_zeeman()) that breaks time reversal and opens a gap at \(\mathbf{k} = 0\).

import numpy as np
import matplotlib.pyplot as plt

import tbkit.lattices as lattices
from tbkit.orbital import OrbitalSystem
from tbkit.plot import Plot
from tbkit.system import System


L, t, lam = 12, -1., 0.3
SX = np.array([[0, 1], [1, 0]])
SY = np.array([[0, -1j], [1j, 0]])


def rashba_torus(bz=0.):
    lat = lattices.square()
    lat.get_lattice(n1=L, n2=L)
    sys = OrbitalSystem(lat, spin=True, periodic=True)
    sys.set_hopping([{'n': 1, 't': t}])
    sys.set_onsite({'a': 0.})
    sys.set_rashba(lam)
    if bz:
        sys.set_zeeman((0., 0., bz))
    sys.get_ham()
    return sys


def bands(kx, ky, bz=0.):
    eps = 2 * t * (np.cos(kx) + np.cos(ky))
    split = np.sqrt(bz**2 + 4 * lam**2 * (np.sin(kx)**2 + np.sin(ky)**2))
    return eps - split, eps + split


def bloch_block(sys, k):
    r'''The 2x2 spin block of H at momentum k, projected from the real-space torus.'''
    rows = np.array([[sys.row(i, 's', s) for s in (0, 1)] for i in range(sys.lat.sites)])
    wave = np.exp(1j * (k[0] * sys.lat.coor['x'] + k[1] * sys.lat.coor['y'])) / np.sqrt(sys.lat.sites)
    u = np.zeros((sys.ham.shape[0], 2), complex)
    u[rows[:, 0], 0] = wave
    u[rows[:, 1], 1] = wave
    return u.conj().T @ (sys.ham @ u)

The lattice

The square lattice, one site per cell with a spin-1/2 orbital, and its nearest-neighbour bonds. On each bond the Rashba term adds a spin flip \(i\lambda(\boldsymbol\sigma\times\hat{\mathbf{d}})_z\) to the hopping \(t\); the spin is not drawn. The torus used below is \(12\times12\) sites; the bonds that close it are not drawn.

patch = lattices.square()
patch.get_lattice(n1=L, n2=L)
vis = System(patch)
vis.set_hopping([{'n': 1, 't': t}])
fig_lat = Plot(vis).lattice(plt_hop=True, ms=10, figsize=(5, 5))

The spectrum of the real-space model is the Rashba dispersion

On a 12 x 12 torus the momenta are \(2\pi n/12\); the \(2 \times 144\) eigenvalues of the real-space Hamiltonian are exactly \(E_\pm\) at those momenta.

k = 2 * np.pi * np.arange(L) / L
KX, KY = np.meshgrid(k, k)
for bz in (0., 0.4):
    en = np.linalg.eigvalsh(rashba_torus(bz).ham.toarray())
    assert np.allclose(en, np.sort(np.concatenate([b.ravel() for b in bands(KX, KY, bz)])), atol=1e-10)
print('real-space spectrum = E_+-(k) on the torus, with and without Zeeman: OK')

real-space spectrum = E_+-(k) on the torus, with and without Zeeman: OK

Kramers pairs, and the Zeeman field that splits them

Rashba coupling is odd in \(\mathbf{k}\) and in spin, so time reversal survives: \((\mathbf{k}, \uparrow)\) and \((-\mathbf{k}, \downarrow)\) stay degenerate and every level of the torus is (at least) doubly degenerate. A Zeeman field breaks time reversal and lifts it, opening a gap \(2b_z\) at \(\mathbf{k} = 0\).

en = np.linalg.eigvalsh(rashba_torus().ham.toarray())
assert np.allclose(en[0::2], en[1::2], atol=1e-10)
en_z = np.linalg.eigvalsh(rashba_torus(0.4).ham.toarray())
assert not np.allclose(en_z[0::2], en_z[1::2], atol=1e-6)
gamma = np.linalg.eigvalsh(bloch_block(rashba_torus(0.4), (0., 0.)))
assert np.isclose(gamma[1] - gamma[0], 2 * 0.4)
print('Kramers degeneracy of every level without Zeeman, lifted by b_z (gap 2 b_z at k = 0): OK')

Kramers degeneracy of every level without Zeeman, lifted by b_z (gap 2 b_z at k = 0): OK

Spin-momentum locking

Projecting the torus Hamiltonian on plane waves gives its \(2 \times 2\) Bloch block at each momentum. The spin of both eigenstates lies in the plane, perpendicular to \((\sin k_x, \sin k_y)\) (to \(\mathbf{k}\) near \(\Gamma\)), and opposite for the two bands.

sys = rashba_torus()
spins = {}
for kx, ky in zip(KX.ravel(), KY.ravel()):
    if np.hypot(np.sin(kx), np.sin(ky)) < 1e-9:
        continue  # time-reversal-invariant momenta: no splitting, no locking
    e, v = np.linalg.eigh(bloch_block(sys, (kx, ky)))
    assert np.allclose(e, bands(kx, ky))
    s = np.array([[np.vdot(v[:, n], m @ v[:, n]).real for m in (SX, SY)] for n in range(2)])
    assert np.allclose(np.linalg.norm(s, axis=1), 1.)  # fully in-plane
    assert np.allclose(s @ [np.sin(kx), np.sin(ky)], 0., atol=1e-10)  # perpendicular to k
    assert np.allclose(s[0], -s[1])
    spins[(kx, ky)] = s[0]
print('spin in-plane, perpendicular to sin k, opposite on the two bands at {} momenta: OK'.format(len(spins)))

spin in-plane, perpendicular to sin k, opposite on the two bands at 140 momenta: OK

The two spin-split bands

kc = np.linspace(-np.pi, np.pi, 401)
fig, axes = plt.subplots(1, 2, figsize=(10, 4.3))
for bz, ls in ((0., '-'), (0.4, '--')):
    lo, hi = bands(kc, 0., bz)
    axes[0].plot(kc, lo, 'C0' + ls, kc, hi, 'C3' + ls)
fold = k - 2 * np.pi * (k > np.pi)  # torus momenta in [-pi, pi)
axes[0].plot(fold, bands(k, 0.)[0], 'C0o', fold, bands(k, 0.)[1], 'C3o', ms=4)
axes[0].set_xlim(-np.pi / 2, np.pi / 2)
axes[0].set_ylim(-4.6, -1.2)
axes[0].set_xlabel('$k_x$ ($k_y = 0$)')
axes[0].set_ylabel('$E/|t|$')
axes[0].set_title('Rashba (solid), with Zeeman $b_z = 0.4$ (dashed)')
q = np.array(list(spins))
s = np.array(list(spins.values()))
q = q - 2 * np.pi * (q > np.pi)
near = np.hypot(*q.T) < 1.6
axes[1].quiver(q[near, 0], q[near, 1], s[near, 0], s[near, 1], color='C0', pivot='mid')
axes[1].set_aspect('equal')
axes[1].set_xlabel('$k_x$')
axes[1].set_ylabel('$k_y$')
axes[1].set_title('spin of the lower band (torus momenta)')
fig.tight_layout()