# Supplementary Materials for
Giant Rashba electrical control of magnetism in band models

Wen Li and Stewart E. Barnes  Physics Department, University of Miami, Coral
Gables, FL 33124, USA.

The Kronig-Penney model. This is unique in that permits an analytic study of
the transition from the free electron to the tight bonding model in one, two
or higher dimensions. The separable generalisation to higher dimensions is due
to NearingNearing . A periodic delta-function corresponds to
$V(x)=Va\sum_{n}\delta(x-an)=V\sum_{n}e^{iG_{n}x}$ where the reciprocal
lattice vectors are $G_{n}=\frac{2\pi}{a}n$, $a$ is the lattice spacing and
$V$ the strength of the potential in energy units. In three dimensions, the
separable potential $V(x)+V(y)+V(z)$, defines boxes of side $a$, that reflect
atoms separated by interstitial walls. See Fig. [1] of the main text.

For simplicity, and familiarity, first the results for this model are re-
derived in one dimension and generalised to the relevant two dimensions as is
appropriate. The Bloch theorem dictates the wave function is of the form
$\psi_{q}(x)=e^{iq_{x}x}u_{q}(x)$ where $u_{q}(x)$ is a periodic function,
i.e. $u_{q_{x}}(x)=u_{q_{x}}(x\pm na)$. Without spin-orbit coupling the
Kronig-Penney model $u_{q_{x}}(x)$ can be taken as real. In the case when the
Rashba coupling is included it is still the case that
$u_{q_{x}}(x)=\sum_{n}u_{q_{x}}(G_{n})e^{iG_{n}x}$ but where
$u_{q_{x}}(G_{n})\neq u_{q_{x}}(-G_{n})$. With periodic boundary conditions on
an interval $L$, the normalisation $\frac{1}{L}\int dx|\psi_{q_{x}}(x)|^{2}=1$
implies $\sum_{n}|u_{q}(G_{n})|^{2}=1$. For the Hamiltonian
$H=\frac{1}{2m}p^{2}+V(x)$ the Schrödinger equation is
$H\psi_{q_{x}}(x)=E_{q_{x}}\psi_{q}(x)$ or

$\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+Va\sum_{n}\delta(x-na)\right]\psi_{q_{x}}(x)=E_{q_{x}}\psi_{q}(x)$
(S1)

Corresponding to Bloch’s theorem, for any $q_{x}$, the wave-function can
written as

$\psi_{q_{x}}(x)=e^{iq_{x}x}u_{q}(x)=e^{iq_{x}x}\sum_{n}u_{G_{n}}(q_{x}+G_{n})e^{iG_{n}x}=\sum_{n}u_{G_{n}}(q_{x}+G_{n})e^{i(q_{x}+G_{n})x}$
(S2)

and it follows that
$\psi_{q_{x}}(na)=e^{iq_{x}na}u_{k}(na)=e^{iq_{x}na}u_{q}(0)=e^{iq_{x}na}\psi_{q}(0)$.
Also $\psi_{q_{x}}(0)=\sum_{n}u_{G_{n}}(q_{x}+G_{n})$. Schrödinger’s equation
is then

$\sum_{n}\left[\frac{\hbar^{2}(q_{x}+G_{n_{x}})^{2}}{2m}-E_{q_{x}}\right]u_{G_{n}}(q_{x}+G_{n_{x}})e^{i(q+G_{n_{x}})x}=-V\psi_{q_{x}}(0)\sum_{n}e^{i(q_{x}+G_{n})x}$
(S3)

using $\sum_{n}\delta(x-an)=\sum_{n}e^{iG_{n}x}$ on the right hand side. The
amplitude $u_{G_{n}}$ is extracted using
$\frac{1}{L}\int^{L}_{0}e^{i(G^{\prime}-G)}dx=\delta_{G^{\prime}G}$. Obtained
is

$\left[\frac{\hbar^{2}(q_{x}+G)^{2}}{2m}-E_{q_{x}}\right]u_{G}=V\sum_{n}u_{G_{n}+G}=V\sum_{n}u_{G_{n}}$
(S4)

that reduces to

$u_{G_{n}}={\cal G}_{n}(E_{q_{x}})V\sum_{n}u_{G_{n}}$ (S5)

where

${\cal
G}_{n}(E_{q_{x}})=\frac{1}{E_{q_{x}}-\frac{\hbar^{2}(q_{x}+G_{n})^{2}}{2m}}$
(S6)

is the free electron Green’s function for momentum $\hbar(k_{x}+G_{n})$.
Summing over $n$ then gives

$\frac{1}{V}=\sum_{n}{\cal G}_{n}(E_{q_{x}})$ (S7)

and a key result reproduced as Eq. [1] in the main text. The equal weight of
the ${\cal G}_{n}$ reflects the $\delta$-function nature of the periodic
potential. The sum on $n$ is performed using $\cot x=\sum_{n}\frac{1}{n\pi+x}$
to give the familiar result,

$\cos k_{x}a=\cos q_{x}a+\frac{\pi^{2}V}{2E_{0}}\frac{\sin q_{x}a}{q_{x}a}$
(S8)

and, for a given $q_{x}$, is solved for $k_{x}$. Since the potential is zero
except where the delta-functions act, it can be insisted that the sought for
energy
$E_{q_{x}}(k_{x})=\frac{{p_{x}}^{2}}{2m}=\frac{\hbar^{2}}{2m}{k_{x}}^{2}$,
where $p_{x}=\hbar k_{x}$, i.e., the $k_{x}$ in Eq. S8 corresponds to the full
momentum $p_{x}$ and not the crystal momentum $\hbar q_{x}$. Here
$E_{0}=\frac{\pi^{2}\hbar^{2}}{2ma^{2}}$ is the ground state energy of an
isolated atom in one dimension.

In the tight binding regime the bands develop from “atomic” levels
($V\to\infty$) corresponding to

$\sin q_{x}a=0,\ \ \ \ \ \cos q_{x}a=(-1)^{n_{x}},\ \ \ \
q_{x}=\frac{\pi}{a}n_{x};\ \ \ \ n_{x}=1,2,\ldots$ (S9)

Expanding about $q_{x}a=n_{x}\pi$, following some algebra, it is found

$E_{q_{x}}=E_{n_{x}}-(-1)^{n}E_{n}\frac{4E_{0}}{\pi^{2}V}\cos q_{x}a$ (S10)

and, as described in the text, the bands are centred about the atomic levels
$E_{n_{x}}={n_{x}}^{2}E_{0}$. These results are illustrated in Fig. [1a] of
the main text.

In two dimensions and a total potential $V(x,y,z)=V(x)+V(y)$, with the same
definition of $V(x)$, the wave function
$\Psi_{\vec{k}}(x,y,z)=\psi_{k_{x}}(x)\psi_{k_{y}}(y)$ is separable. There are
two separation constants $E_{k_{x}}$ and $E_{k_{y}}$ and the full Schrödinger
equation

$\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V(x,y,z)\right]\Psi_{\vec{q}}(x,y,z)=E_{\vec{q}}\Psi_{\vec{q}}(x,y,z)$
(S11)

reduces to Eq. S1 for $\psi_{q_{x}}(x)$ with equivalent equations for
$\psi_{q_{y}}(y)$. The constants of separation add to give the energy

$E_{\vec{q}}=E_{q_{x}}+E_{q_{y}}$ (S12)

the simple sum of the energies for the two degrees of freedom.

Since the model is separable, the tight binding regime in two dimensions
corresponds to “atomic” levels $E_{0}({s_{x}}^{2}+{s_{y}}^{2})$ characterised
by quantum numbers $s_{x}$ and $s_{y}$. Adding the one dimensional energies:

$E_{q_{x}q_{y}}=E_{0}({s_{x}}^{2}+{s_{y}}^{2})+\frac{4E_{0}}{\pi^{2}V}[(-1)^{n_{x}}E_{s_{x}}\cos
q_{x}a+y(-1)^{n_{y}}E_{s_{y}}\cos q_{y}a]$ (S13)

The resulting two dimensional band structure is illustrated in Fig. [1b]. The
similar three dimensional band structure develops from the “atomic” levels
$E_{0}({s_{x}}^{2}+{s_{y}}^{2}+{s_{z}}^{2})$ as discussed in the text.

The magnetic Kronig-Penny model  Since the thin film is magnetic, added to the
Hamiltonian is the Stoner exchange interaction
$-J_{0}\vec{S}\cdot\vec{\sigma}$ where $\vec{S}$ is the magnitude of the order
parameter, i.e., the effective spin that is determined self-consistently. This
might be imagined to be a simple version of the exchange correlation energy in
the DFT. The density matrix corresponds to a magnetisation $\vec{M}$ pointing
in the direction of $\vec{S}$. Assuming $\vec{M}=M\hat{\bf z}$, the basic Eq.
S7 is unchanged except the diagonal Green’s function

${\cal
G}_{n}^{-1}(E_{q_{x}})=[(E_{q_{x}}-\frac{\hbar^{2}(q_{x}+G_{n})^{2}}{2m})I+J_{0}S\sigma_{z}].$
(S14)

is now a two-by-two matrix. The Mathematica solutions of this are described in
the text in the connection with Fig. [4a] . The solutions are shifted up or
down in energy by $J_{0}S$ as the band corresponds to minority/majority
electrons.

The periodic Rashba model.  This illustrates the elementary diagonalisation of
the spin problem by a rotation matrix. In two dimensions, the spin-orbit
interaction $\frac{\alpha_{R}}{\hbar}(\sigma_{x}p_{y}-\sigma_{y}p_{x})$ is
added to the Hamiltonian. Unfortunately, because of the Pauli matrices
$\sigma_{x}$ and $\sigma_{y}$ in the Rashba term, this is no longer exactly
separable in two dimensions. In a single dimension the chains are taken to be
the $x$-direction implying a spin-orbit coupling
$\frac{\alpha_{R}}{\hbar}\sigma_{y}p_{x}$. The problem still reduces to Eq. S7
but where now the free electron propagator:

${\cal
G}_{n_{x}}^{-1}(E_{q_{x}})=[(E_{q_{x}}-\frac{\hbar^{2}(q_{x}+G_{n_{x}})^{2}}{2m})I+\alpha_{R}(q_{x}+G_{n})\sigma_{y}],$
(S15)

again a two-by-two matrix but now with off-diagonal matrix elements. This can
be made spin diagonal by a rotation of $\frac{\pi}{2}$ about the $x$-axis. The
rotation matrix $R_{x}(\frac{\pi}{2})$ is such that
$\sigma_{y}=(\frac{\pi}{2})\sigma_{z}R_{x}(\frac{\pi}{2})$. It follows
$[(E_{q_{x}}-\frac{\hbar^{2}(q_{x}+G_{n_{x}})^{2}}{2m})I+\alpha_{R}(q_{x}+G)R_{x}^{{\dagger}}(\frac{\pi}{2})\sigma_{z}R_{x}(\frac{\pi}{2})]=R_{x}^{{\dagger}}(\frac{\pi}{2})[(E_{q_{x}}-\frac{\hbar^{2}(q_{x}+G_{n_{x}})^{2}}{2m})I+\alpha_{R}(q_{x}+G_{n_{x}})\sigma_{z}]R_{x}(\frac{\pi}{2})$.
Since $R_{x}(\frac{\pi}{2})$ is independent of $n_{x}$, these rotation
matrices can be factored out of the sum on $n_{x}$, i.e.,

$\frac{1}{V}=R_{x}(\frac{\pi}{2})\left(\sum_{n_{x}}{\cal
G}_{n_{x}}(E_{q_{x}})\right)R_{x}^{{\dagger}}(\frac{\pi}{2})\to\frac{1}{V}=\sum_{n_{x}}{\cal
G}_{n_{x}}(E_{q_{x}})$ (S16)

multiplying on the left by $R_{x}^{{\dagger}}(\frac{\pi}{2})$ and the right by
$R_{x}(\frac{\pi}{2})$. (The same diagonal result is obtained trivially be
using a representation of the Pauli matrices in which $\sigma_{y}$ is directly
diagonal.) The spin dependent momentum shift
$\alpha_{R}(k+G_{n_{x}})\sigma_{z}$ is absorbed into the kinetic energy, i.e.,
$[(E_{q_{x}}-\frac{\hbar^{2}(q_{x}+G_{n_{x}})^{2}}{2m})I+\alpha_{R}(q_{x}+G_{n_{x}})\sigma_{z}]=[(E_{q_{x}}-\frac{\hbar^{2}(q_{x}+G_{n_{x}})^{2}}{2m})I\pm\alpha_{R}(q_{x}+G_{n_{x}})]=[(E_{q_{x}}-\frac{\hbar^{2}(q_{x}+G_{n_{x}}\mp\Delta
k)^{2}}{2m}-E_{R}]$ upon completing the square. The band structure is shifted
to the left or right by $\Delta k=\frac{m}{\hbar^{2}}{\alpha_{R}}$ and down in
energy by

$E_{R}=\frac{\hbar^{2}{\Delta
k}^{2}}{2m}=\frac{1}{2}m\frac{{\alpha_{R}}^{2}}{\hbar^{2}}$ (S17)

This is the same characteristic energy identified for the free electron
caseBarnes et al. (2014), i.e., when $V=0$. However the identification of an
effective $``E_{R}"$ for the different bands is changed as discussed in the
text in the context of Fig. [4b]. The determination of $\alpha_{R}$ from the
experimental $``E_{R}"$ cannot be justified.

The two dimensional periodic magnetic Rashba model. Now in two dimensions,
both the spin-orbit interaction
$\frac{\alpha_{R}}{\hbar}(\sigma_{x}p_{y}-\sigma_{y}p_{x})$ and exchange term
$-J_{0}\vec{S}\cdot\vec{\sigma}$ are added to the Hamiltonian. Without the
periodic potential, this is the model considered by Barnes et alBarnes et al.
(2014). With the periodic potential the Hamiltonian is therefore

$\hat{H}=\frac{p^{2}}{2m}-J_{0}\vec{S}\cdot\vec{\sigma}+\frac{\alpha_{R}}{\hbar}(\sigma_{x}p_{y}-\sigma_{y}p_{x})+V[\sum_{{G_{n}}_{x}}e^{-i{G_{n}}_{x}x}+\sum_{{G_{n}}_{y}}e^{-i{G_{n}}_{y}y}.]$
(S18)

The model still has both free electron and tight binding limits. Again,
because of the Pauli matrices $\sigma_{x}$ and $\sigma_{y}$ in the Rashba
term, this is no longer exactly separable in two dimensions. How this can be
approximately separated in two or higher dimensions will be taken up again
below.

The one dimensional periodic magnetic Rashba model - in plane  In a single
dimension, first imagine, the equivalent to the two dimensional “in-plane”
case, see Fig. [4c]. The chain is taken to be the $x$-direction implying a
spin-orbit coupling $\frac{\alpha_{R}}{\hbar}\sigma_{y}p_{x}$. In-plane here
means that $\vec{S}=S\hat{\bf y}$, i.e., the exchange $\vec{B}_{E}$ and Rashba
field $\vec{B}_{R}$ are parallel and both involve only $\sigma_{y}$. Again Eq.
S7 applies but now with the matrix

${\cal
G}_{n}^{-1}(E_{q_{x}})=[(E_{q_{x}}-\frac{\hbar^{2}(q_{x}+G_{n_{x}})^{2}}{2m})I+\left(J_{0}S+\alpha_{R}(q_{x}+G_{n_{x}})\right)\sigma_{y}].$
(S19)

As with the non-magnetic case, this can be made spin diagonal by a rotation of
$\frac{\pi}{2}$ about the $x$-axis. Now in Eq. S7 the diagonal Green’s
function is

${\cal
G}_{n}^{-1}(E_{q_{x}})=[(E_{q_{x}}-\frac{\hbar^{2}(q_{x}+G_{n_{x}})^{2}}{2m})I+\left(J_{0}S+\alpha_{R}(q_{x}+G_{n_{x}})\right)\sigma_{z}].$
(S20)

Comparing Eq. S15 with Eq. S20, little changes as compared to the non-magnetic
Rashba problem except the minority/majority shifts band shifts of $\pm J_{0}S$
in addition to those of the simpler problem. Again the solutions exhibited in
the text are obtained with Mathematica.

It is a key observation that the shift in energy due to the Rashba effect is
$E_{R}$ for all bands and is second order in the electric field $E$. This is
perhaps surprising since the Rashba field itself
$B_{R}=\alpha_{R}(q_{x}+G_{n_{x}})$ is linear in $E$. The momentum shifts are
indeed linear, however the process of completing the square reduces the energy
shift to second order.

The one dimensional periodic magnetic Rashba model - out of plane  Turning to
the more interesting one dimensional version of an “out-of-plane”
magnetisation, i.e., when the average order parameter $\vec{M}=M\hat{\bf z}$
is parallel to the direction of the electric field $\vec{E}=E\hat{\bf z}$. The
combination $-J_{0}S\sigma_{z}+\frac{\alpha_{R}}{\hbar}\sigma_{y}p_{x}$ is
diagonalised by a rotation $R_{x}(\theta_{G})$ such that
$R_{x}(\theta_{G}){J}S\sigma_{z}R^{{\dagger}}_{x}(\theta_{G})=J_{0}S\sigma_{z}+\alpha_{R}(q_{x}+G_{n_{x}})\sigma_{y}$
where there is a new effective exchange

$SJ=\sqrt{J_{0}^{2}S^{2}+\alpha_{R}^{2}(k+G)^{2}}\approx
J_{0}S+\frac{\alpha_{R}^{2}(q_{x}+G_{n_{x}})^{2}}{2J_{0}S}$ (S21)

and $\theta_{G_{n_{x}}}=\arctan(\frac{\alpha(q_{x}+G_{n_{x}})}{J_{0}S})$. The
Rashba field $B_{R}=\alpha_{R}(q_{x}+G_{n_{x}})$ is unchanged in magnitude,
but is now perpendicular to the exchange field $B_{E}=J_{0}S$, and inevitably,
via vector addition, leads to a second order, i.e., $E^{2}$, correction
assuming $B_{E}$ is the larger field. As explained in the previous section,
the in-plane magnetic anisotropy energy is also second order in $E$. This is
the reason for the counter intuitive result that a Rashba field perpendicular
the magnetisation can lead to a greater magnetic anisotropy energy that does
the in-plane field of equal magnitude. Schrödinger’s equation reduces to

$\frac{a}{V}=\sum_{n_{x}}R_{x}^{{\dagger}}(\theta_{G_{n_{x}}})[(\frac{1}{2}(1+\sigma_{z}){\cal
G}^{+}_{G_{n_{x}}}+\frac{1}{2}(1-\sigma_{z}){\cal
G}^{-}_{G_{n_{x}}})R_{x}(\theta_{G_{n_{x}}})$ (S22)

where now the free electron Green’s functions ${\cal
G}^{\pm}_{n_{x}}=[E_{q_{x}}-\frac{\hbar^{2}(q_{x}+G_{n_{x}})^{2}}{2m}\pm{JS}]^{-1}$
includes the exchange splitting with the new $JS$ that includes the second
order Rashba correction $\frac{\alpha_{R}^{2}(q_{x}+G_{n_{x}})^{2}}{2J_{0}S}$.
While the Green’s function has been reduced to diagonal form, this has been at
the cost of introducing the non-trivial rotation matrices
$R_{x}(\theta_{G_{n}})$ that can no longer be factored out of the sum and
eliminated.

The appearance of $\sigma_{z}$ in combination with the rotations
$R_{x}(\theta_{G_{n_{x}}})$ implies a non-trivial Berry phase. The quantity
$\sigma_{z}\equiv
e^{i\frac{\pi}{2}(1-\sigma_{z})}=e^{i\frac{\pi}{2}}e^{-i\frac{\pi}{\hbar}s_{z}}$,
and apart from the phase factor $e^{i\frac{\pi}{2}}$, corresponds to a
rotation by $\pi$ about the $\hat{\bf z}$-axis. It follows:

$R_{x}^{{\dagger}}\frac{1}{2}(1\pm\sigma_{z})R_{x}=\frac{1}{2}(1\pm
R_{x}^{{\dagger}}e^{i\frac{\pi}{2}}e^{i\pi s_{z}/\hbar}R_{x})=\frac{1}{2}(1\pm
e^{i\frac{\pi}{2}}e^{-i\frac{\pi}{\hbar}(\cos\theta_{G_{n_{x}}}s_{z}+\sin\theta_{G_{n_{x}}}s_{y})}).$
(S23)

As with the Stern-GerlachRosen and Zener (1932) experiment, an electron
propagates along two similar paths that interfere with each other. The
electrically controlled spin Berry phase corresponds to a “magnetic flux”
enclosed by these paths and for small fields is second order in the electric
field $E$. However, the off-diagonal part
$-\frac{\pi}{\hbar}\sin\theta_{G_{n_{x}}}s_{y}$ implies linear in $E$ spin-
mixing that leads directly to electrically controlled band-structure gaps.
These predicted effects provide a novel experimental possibility in which
spin-Berry phase effects can be explored by electrical gating of thin film
magnets.

In a concise form, the equivalent of Schrödinger’s equation is therefore

$\frac{1}{V}=\frac{1}{2}\sum_{\pm}\sum_{n_{x}}(1\pm
e^{-i\frac{\pi}{2}(\cos\theta_{G_{n_{x}}}+\sigma_{z})\sigma_{z}-\sin\theta_{G_{n_{x}}}\sigma_{y})}){\cal
G}^{\pm}_{G_{n_{x}}}$ (S24)

as reproduced as Eq. [2] in the text. In the phase factor
$e^{-i\frac{\pi}{\hbar}(\cos\theta_{G_{n_{x}}}s_{z}+\sin\theta_{G_{n_{x}}}s_{y})}$,
the quantity
$\sigma^{\prime}_{z}=(\cos\theta_{G_{n_{x}}}\sigma_{z}+\sin\theta_{G_{n_{x}}}\sigma_{y})$
is such that $(\sigma^{\prime}_{z})^{2}=1$ and it follows

$e^{i\frac{\pi}{2}}e^{-i\frac{\pi}{2}\sigma^{\prime}_{z}}=\sigma^{\prime}_{z}=\cos\theta_{G_{n_{x}}}\sigma_{z}+\sin\theta_{G_{n_{x}}}\sigma_{y}$
(S25)

and hence, following some algebra, Eq. S24 reduces to

$\frac{1}{V}={\cal S}+{\cal A}\sigma_{z}+{\cal B}\sigma_{y}$ (S26)

where ${\cal S}\equiv\frac{1}{2}\sum_{n_{x}}({\cal G}^{+}_{G_{n_{x}}}+{\cal
G}^{-}_{G_{n_{x}}})$, ${\cal A}\equiv\frac{1}{2}\sum_{n_{x}}({\cal
G}^{+}_{G_{n_{x}}}-{\cal G}^{-}_{G_{n_{x}}})\cos{\theta_{G_{n_{x}}}}$ and
${\cal B}\equiv\frac{1}{2}\sum_{n_{x}}({\cal G}^{+}_{G_{n_{x}}}-{\cal
G}^{-}_{G_{n_{x}}})\sin{\theta_{G_{n_{x}}}}$. Following yet another spin
rotation by the angle $\Theta=\arctan{\frac{\cal B}{\cal A}}$ the
majority/minority spin-split bands are given by the solutions of
$\frac{1}{V}-{\cal S}+\sqrt{{\cal A}^{2}+{\cal B}^{2}}\sigma_{z}=0$ as
$\sigma_{z}=\pm 1$. Finally, performing a square to eliminate $\sigma_{z}$,
results in

$\left(\frac{1}{V}-{\cal S}\right)^{2}-\left({\cal A}^{2}+{\cal
B}^{2}\right)=0$ (S27)

an implicit equation for $E_{q_{x}}$, including spin mixing effects, that is
fully equivalent to the original Schrödinger equations for both spin
directions. Both of the majority and minority bands are obtained by
Mathematica solutions of this single equation.

Spin Berry physics.  From Ref. [11], with the present definitions of the
rotation matrices, the spin Berry connection is

$\vec{A}_{s}=-\frac{\hbar}{2e}(\sin\theta_{G_{n_{x}}}\vec{\nabla}\phi\sigma_{y}+\vec{\nabla}\theta\sigma_{x})$
(S28)

The somewhat generalised spin Berry phase $\phi_{s}$ is then defined by

$e^{i\phi_{s}}=:e^{i\frac{e}{\hbar}\int_{c}\vec{A}_{s}\cdot d\vec{r}}:$ (S29)

where the $:\ldots:$ implies spatial ordering. The spatial ordering is
important since the $\sigma_{x}$ and $\sigma_{y}$ that occur in the definition
of $\vec{A}_{s}$ do not commute. In general $\phi_{s}$ is an operator written
as a linear combination of Pauli matrices. There is a magnetic monopole at the
centre of the spin sphere, compensated by a Dirac string. The contour over
which the integration in Eq. S29 is preformed is dictated by the rotations

$R_{x}^{{\dagger}}(\theta_{G_{n_{x}}})e^{i\frac{\pi}{\hbar}(1-s_{z})}R_{x}(\theta_{G_{n_{x}}}).$
(S30)

For an initial up spin, this path is illustrated in the text Fig. [1g]. A
beginning up spin is first rotated down by the angle $\theta_{G_{n_{x}}}$ then
about the $\hat{\bf z}$-axis by $\pi$ and finally back to its original
alignment along the $\hat{\bf z}$-direction. Because of the factor of
$e^{i\frac{\pi}{2}}$, the solid angle $\Omega$ is defined relative the the
positive $\hat{\bf z}$-axis for up-spin and the negative axis for down-spin.
That part of $:e^{i\frac{e}{\hbar}\int\vec{A}\cdot d\vec{r}}:$ that arises
from the commutators of $\sigma_{x}$ and $\sigma_{y}$ is given by a
generalised Stoke’s theorem. The proof is straightforward[11] and will not be
given here since it leads to the well known result that the diagonal Berry
phase is $\pm\frac{1}{2}\Omega$ where $\Omega$ is the solid angle subtended by
the path on the Bloch sphere relative to the positive or negative $\hat{\bf
z}$-axis as the spin is up or down. A familiar resultShankar (2014) is the the
solid angle for a full circle about the positive $\hat{\bf z}$-direction is
$\Omega=2\pi(1-\cos\theta)$. For half a circle the (diagnal) Berry phase is
therefore

$\frac{\pi}{2}(1-\cos\theta_{G_{n_{x}}})\sigma_{z}.$ (S31)

If the commutators are ignored, the spatial ordering is not important and the
remaining part of the phase is given directly by the integral

$\frac{1}{2}\int_{c}(\sin\theta_{G_{n_{x}}}d\phi\sigma_{y}+d\theta\sigma_{x})=\frac{\pi}{2}\sin\theta_{G_{n_{x}}}\sigma_{y}.$
(S32)

since the $\int_{c}d\theta=0$ by symmetry. This is intended as merely a sketch
of how the general definition[11], Eq. S28, that defines $\vec{A}_{s}$, leads
to the same result as the algebraic manipulations that lead to Eq. S24, i.e.,
the Berry phase factor

$e^{i\phi_{s}}=:e^{i\frac{e}{\hbar}\int\vec{A}\cdot
d\vec{r}}:=e^{-i\frac{\pi}{2}(\cos\theta_{G_{n_{x}}}+\sigma_{z})\sigma_{z}+\sin\theta_{G_{n_{x}}}\sigma_{y})}$
(S33)

for the appropriate spin up/down contour. When compared with the standard
approach to the Berry phase, this algebraic definition[11] of the real space
connection $\vec{A}_{s}(\vec{r})$, leads in a natural manner, to the off-
diagonal part of the Berry phase factor $e^{i\phi_{s}}$ with the on-diagonal
corresponding to a curvature defined by a Stoke’s theorem. It is however the
off-diagonal part that leads to the electrical control of spin gaps as
described in the text. This prediction is clearly accessible to
experimentation.

Out of plane - tight binding - analytic results.  Analytic results for the
tight binding regime are obtained by using the approximation $SJ\approx
J_{0}S+\alpha_{R}^{2}(q_{x}+G)^{2}/J_{0}S$ valid given the Rashba field
$\alpha_{R}\hbar(q_{x}+G_{n_{x}})$ is small when compared with the exchange
field $J_{0}S$. It follows
$E^{\pm}_{n}(k)\approx\frac{\hbar^{2}(q_{x}+G_{n_{x}})^{2}}{2m}\mp(J_{0}S+{\alpha_{R}}^{2}\frac{\hbar^{2}(q_{x}+G_{n_{x}})^{2}}{2J_{0}S})$.
The Rashba correction
$\mp{\alpha_{R}}^{2}\frac{\hbar^{2}(q_{x}+G_{n_{x}})^{2}}{2J_{0}S}$ is then
absorbed into a spin dependent effective mass $\mu$ defined by
$\frac{1}{\mu^{\pm}}=\frac{1}{m}\mp\frac{{\alpha_{R}}^{2}}{J_{0}S}$. The
centre of tight binding band with index $n_{x}$ is then
$E^{\pm}_{n}=n^{2}\frac{\pi^{2}\hbar^{2}}{2\mu
a^{2}}=E_{n}\mp\frac{1}{4}{G_{n_{x}}}^{2}\frac{\hbar^{2}{\alpha_{R}}^{2}}{2J_{0}S}$
using $\frac{1}{2}G_{n}=n\frac{\pi}{a}$. The resulting energies are

$E^{\pm}_{q_{x}}=E_{n_{x}}-\frac{4E_{0}E_{n}}{V}\cos
q_{x}a\mp\left(J_{0}S+{\alpha_{R}}^{2}\frac{\hbar^{4}}{8J_{0}S}{G_{n_{x}}}^{2}\right)$
(S34)

for the majority/minority bands. In the text, this result is discussed in the
context of the Fig. [4d]. The role of the reciprocal lattice vectors in
determining the Rashba field $B_{R}$ is put in evidence.

Electrically controlled spin gaps - spin mixing effects  If neither
$\alpha_{R}$ or $V$ have extreme values, then it is possible to develop an
approximate theory for the interaction between bands by reducing the sum over
Green’s functions in Eq. S7 to only two terms. The secular equation reduces to

$\frac{1}{V}=\sum_{n_{x}}G_{n_{x}}(E_{k_{x}})=G_{a}(E_{q_{x}})+G_{b}(E_{q_{x}}).$
(S35)

To be specific consider the first such gap near $k_{x}=\frac{\pi}{a}$. The
Green’s function for the lowest band
$G_{0}=[E_{k}-\epsilon_{c}+\frac{\hbar^{2}}{2m}\frac{2\pi}{a}(q-k_{c})]^{-1}$
where $\epsilon_{c}\approx\frac{\hbar^{2}}{2m}(\frac{\pi}{a})^{2}$ is the
energy where the two free electron bands cross. The other function
$G_{2\pi/a}=[E_{k}-\epsilon_{c}-\frac{\hbar^{2}}{2m}\frac{2\pi}{a}(q-k_{c})]^{-1}$
corresponds to the free electron band translated from beyond $-\frac{\pi}{a}$
to near $k_{x}=\frac{\pi}{a}$. Since the free electron bands cross at
$\epsilon_{c}$ the Green’s functions differ only in the direction of
dispersion, i.e., in the sign of the term
$\Delta=\frac{\hbar^{2}}{2m}\frac{2\pi}{a}(q-k_{c})$. The Rashba and exchange
fields are contained in the definition of
$k_{c}=\frac{2m}{\hbar^{2}}\frac{a}{2\pi}JS\approx\frac{2m}{\hbar^{2}}\frac{a}{2\pi}(J_{0}S+{\alpha_{R}}^{2}\frac{\hbar^{2}{{k_{F}}^{2}}}{2J_{0}S})$.
The Rashba effect is clearly $O({\alpha_{R}}^{2})$ in the absence of the spin
mixing rotations
$\theta_{0}\approx-\theta_{2\pi/a}\approx\frac{\alpha_{R}k_{c}}{J_{0}S}\equiv\theta$.
Substituting the definitions of $\cal S$, $\cal A$ and $\cal B$ into
$\frac{1}{V}={\cal S}+{\cal A}\sigma_{z}+{\cal B}\sigma_{y}$, following some
involved algebra, obtained for the energies is

$E_{k}=\epsilon_{c}+V+\sigma_{z}\frac{V}{2}\sqrt{\left(\frac{\alpha_{R}\pi}{J_{0}Sa}\right)^{2}+\left(\frac{\hbar^{2}}{m}\frac{\pi}{V}\right)^{2}(q-k_{c})^{2}}$
(S36)

The two branches near $\epsilon_{c}+V$ disperse up or down as $\sigma_{z}=\pm
1$. The key result is that the gap evident in Fig. [4e] is approximately

$\left|V\frac{1}{J_{0}S}(\alpha_{R}k_{c})\right|$ (S37)

and is linear in the Rashba parameter $\alpha_{R}$ and hence the electric
field $E$.

Electrically controlled spin gaps - measuring the spin Berry phase.  The more
general result for the gap is

$\left|V\theta_{G_{n_{x}}}\right|$ (S38)

where the angle $\theta_{G_{n_{x}}}$ reflects a unitary rotation matrix
$R_{x}(\theta_{G_{n_{x}}})=e^{-i\theta_{G_{n_{x}}}s_{x}/\hbar}$ and where
$n_{x}$ is the band index of the lower of the two bands. This rotation can be
traced back to the spin dependent geometrical phase
$e^{-i\theta_{G_{n_{x}}}s_{x}/\hbar}$ that appears in the wave-function

$\psi_{q_{x}}(x)=\sum_{n}e^{-i\theta_{G_{n_{x}}}s_{x}/\hbar}|u(q_{x}+G_{n_{x}})|e^{i(q_{x}+G_{n_{x}})x}$
(S39)

This causes a spin dependent Aharonov–Bohm (or Stern-Gerlach) type
interference between the different amplitudes $|u(q_{x}+G_{n_{x}})|$ as the
electron is reflected by the periodic potential. Because the momentum
$\hbar(q_{x}+G_{n_{x}})$ changes sign, so does $\theta_{G_{n_{x}}}$ while as
usual for a hard potential there is a $\pi$ change in the phase $\phi$. The
angles $\theta_{G_{n_{x}}}$ can be controlled by the electric field $E$
through $\alpha_{R}$ and hence dictated by experiment. Such measurements
permit the determination of the Berry phase factor
$e^{-i\theta_{G_{n_{x}}}s_{x}/\hbar}$ and again the existence of these spin-
gaps provides a novel test of spin Berry phase physics.

Approximate separation of variables in two plus dimensions.  At first sight,
the two dimensional Rashba term
$\frac{\alpha^{y}_{R}}{\hbar}\sigma_{x}p_{y}-\frac{\alpha^{x}_{R}}{\hbar}\sigma_{y}p_{x}$
is separable since, e.g., $\frac{\alpha^{y}_{R}}{\hbar}\sigma_{x}p_{y}$ only
operates on functions of $y$. This is not the case since the spin wave
function, upon which $\sigma_{x}$ and $\sigma_{y}$ act, cannot be factored.
This not withstanding, the problem is indeed separable if the rotation angles
$\theta_{G_{xn}}(q_{x})$ defined earlier by
$R_{x}(\theta_{G_{xn}}(q_{x})){J}S\sigma_{z}R^{{\dagger}}_{x}(\theta_{{}_{G_{n_{x}}}}(q_{x}))=J_{0}S\sigma_{z}+\alpha^{x}_{R}(q_{x}+G_{n_{x}})\sigma_{y}$
are small. In this case it is possible to write

$\displaystyle
R_{x}(\theta_{G_{n_{x}}}(k_{x})){J}S\sigma_{z}R^{{\dagger}}_{x}(\theta_{{}_{G_{n_{x}}}}(k_{x}))+R_{y}(\theta_{G_{n_{y}}}(k_{y})){J}S\sigma_{z}R^{{\dagger}}_{y}(\theta_{G_{n_{y}}}(k_{y}))-{J_{0}}S\sigma_{z}$
$\displaystyle=J_{0}S\sigma_{z}+\alpha^{x}_{R}(q_{x}+G_{n_{x}})\sigma_{y}-\alpha^{y}_{R}(k_{y}+G_{n_{y}})\sigma_{x}.$
(S40)

A separable solution is of the form:
$E_{q_{x},q_{y}}\psi_{x}(x)\psi_{y}(y)|\rangle=H\psi_{x}(x)\psi_{y}(y)|\rangle$,
where $|\rangle$ is the spin ket. This reduces to

$\Big{[}\frac{1}{\psi_{x}(x)}\hat{H}_{x}\psi_{x}(x)+\frac{1}{\psi_{y}(y)}\hat{H}_{y}\psi_{y}(y)\Big{]}|\rangle=(E_{q_{x},q_{y}}+SJ_{0}\sigma_{z})|\rangle$
(S41)

where, $\hat{H}_{x}$ was given earlier and
$\hat{H}_{y}=(p^{2}/2m)-R_{y}(\theta_{G_{n_{y}}}(q_{y})){J}S\sigma_{z}R^{{\dagger}}_{y}(\theta_{{}_{G_{n_{y}}}}(q_{y}))+V(y)$.
The right hand side of this equation is an, $x$ and $y$ independent, two-by-
two matrix acting on the Pauli spinor $|\rangle$ and the same is true for both
terms on the left hand side. It must be that, e.g.,

$\frac{1}{\psi_{x}(x)}[\frac{1}{2m}{p_{x}}^{2}-R_{x}(\theta_{G_{xn}}(q_{x})){J}S\sigma_{z}R^{{\dagger}}_{x}(\theta_{{}_{G_{xn}}}(q_{x}))+V(x)]\psi_{x}(x)=M_{x}(q_{x})$
(S42)

where the matrix $M_{x}(q_{x})=E^{x}(q_{x}){\bf
I}-J^{x}(q_{x})S\sigma_{z}-\alpha^{x}_{R}e^{x}(q_{x})\sigma_{y}$ and where
$E^{x}(q_{x})$, $J^{x}(q_{x})$ and $e(q_{x})$ are three separation constants.
That $M_{x}(q_{x})$ involves only $\sigma_{y}$ and not $\sigma_{x}$ reflects
the Rashba term $\frac{\alpha^{x}_{R}}{\hbar}\sigma_{y}p_{x}$. Given the
$M_{x}(q_{x})$ and $M_{y}(q_{y})$ are determined, there then remains the spin
Schrödinger’s equation:

$[(M_{x}(q_{x})+M_{y}(q_{y}))-(E_{q_{x},q_{y}}+SJ_{0}\sigma_{z})]|\rangle=0$
(S43)

that is trivially solved by a rotation in spin space. The resulting spin ket
determines $q_{x}$, $q_{y}$ dependent density matrix but leaves the Berry
phase angles, e.g., $\theta_{G_{n_{x}}}$ hidden within $M_{x}(q_{x})$. Since
there are many solutions for a given crystal momentum $\vec{q}$ there are
equally many different separation matrices $M_{x}(q_{x})$.

The above for $\psi_{x}(x)$ can be re-arranged to read

$\frac{1}{\psi_{x}(x)}[\frac{1}{2m}{p_{x}}^{2}-E^{x}(q_{x})+-S(J_{0}-\Delta
E)\sigma_{z}+\alpha^{x}_{R}\sigma_{y}(\frac{p_{x}}{\hbar}-e)]\psi_{x}(x)=V(x)$
(S44)

where
$E^{x}(q_{x})=\frac{1}{2}(E_{\uparrow}^{x}(q_{x})+E_{\downarrow}^{x}(q_{x}))$
and $\Delta
E=\frac{1}{2}[(E_{\uparrow}^{x}(q_{x})-E_{\downarrow}^{x}(q_{x})]$. This is of
the original, one dimensional, form but with the extra parameters
$E^{x}(q_{x})$ and $\Delta E$ that modify the exchange and Rashba terms.
Following the same steps as before, the problem for $M_{x}(q_{x})$ reduces to

$(\frac{1}{V_{x}}+{\cal S}_{x}+{\cal A}_{x}\sigma_{z}+{\cal
B}_{x}\sigma_{y})=0$ (S45)

but with the same definitions of ${\cal S}_{x}$, ${\cal A}_{x}$ and ${\cal
B}_{x}$ in terms of

$\theta_{G_{n_{x}}}=\arctan(\frac{\alpha^{x}(q_{x}+G_{n_{x}})-e(q_{x})}{J_{0}S}).$
(S46)

The Green’s functions are again

${\cal
G}^{\pm}_{G_{n_{x}}}=\frac{1}{E^{x}_{q_{x}}-\frac{\hbar^{2}(q_{x}+G_{xn})^{2}}{2m}\pm{J}S}.$
(S47)

but with the effective exchange including Rashba corrections and the parameter
$\Delta E$:

$SJ=\sqrt{J_{0}^{2}S^{2}+{\alpha^{x}_{R}}^{2}\hbar^{2}(q_{x}+G_{xn})^{2}}-\Delta
E$ (S48)

Different at this point is that the coefficients of $\sigma_{z}$, $\sigma_{y}$
are individually required to be zero, i.e., it is insisted ${\cal A}_{x}=0$
and ${\cal B}_{x}=0$, leaving $\frac{a}{V_{x}}+{\cal S}_{x}=0$. These three
equations yield multiple solutions for the three separation constants
$E^{y}(q_{y})$, $J^{y}(q_{y})$ and $e(q_{z})$ that determine $M_{y}(q_{y})$.
With similar equations for $\psi_{y}(y)$, the problem reduces to the spin
Schrödinger’s equation given above. In principle, these sets of equations
reduce the problem to quadrature but in practice, given the many solutions for
$E^{y}(q_{y})$ they overwhelm Mathematica. They can be solved in simple limits
as described in the text. Extension of these results by limiting the number of
terms in the sums on $n_{x}$ and $n_{y}$ is possible but beyond the work
presented here. The method can evidently extended to higher dimensions

Spin-orbit coupling  The full spin-orbit coupling can be written with a number
of ways:

$V_{\rm
so}=-\frac{e\hbar}{4m^{2}c^{2}}\vec{E}\cdot(\vec{\sigma}\times\vec{p})=-\frac{\lambda_{c}}{8\pi
mc}\vec{E}\cdot(\vec{\sigma}\times\vec{p})=-\mu_{B}\frac{1}{2c^{2}}\vec{E}\cdot(\vec{\sigma}\times\vec{v})=\frac{\alpha_{R}}{\hbar}\vec{\sigma}\cdot(\vec{p}\times\hat{\bf
z})$

where the first version illustrates $V_{\rm so}$ arises from an expansion in
the mass gap $\frac{1}{mc^{2}}$, the second relates this expansion to the
Compton wave-length $\lambda_{c}=\alpha a_{0}\approx 2$pm,
$\alpha\approx\frac{1}{137}$ the fine-structure constant and the expansion
parameter of QED and where $a_{0}\approx 0.5$Å is the Bohr radius. The next
version re-enforces the idea that the Rashba field $B_{R}=\frac{v}{2c^{2}}E$
is just the magnetic field seen in the electron’s frame due to an electric
field in the laboratory frame and where the factor of $\frac{1}{2}$ accounts
for the Thomas rotation. The last version puts in evidence that the $V_{\rm
so}$ is of the form of a Rashba spin-orbit coupling. This is the entire such
coupling if $\vec{E}=E\hat{\bf z}$ is restricted to the applied electric field
and/or the internal field that occurs in the absence of an inversion symmetry.
The apparently subtle point being made here is that this last version involves
the full momentum $\vec{p}=\hbar\vec{k}$ while the usual Rashba term as
derived by Rashba and co-workersBychkov and Rashba (1984) or described in the
exhaustive work found in Winkler’s textWinkler (2011), is rather the crystal
momentum $\hbar\vec{q}$ that occurs in the triple product
$\vec{\sigma}\cdot(\vec{p}\times\hat{\bf z})$, where the wave-vector $\vec{q}$
is defined relative to the appropriate symmetry point, usually the
$\Gamma$-point for semi-conductors such as GaAs or GaIn.

It is however customaryWinkler (2011) to insist $\vec{E}$ is large only in the
core region where the wave-functions are atomic like and it followsMessiah
(2014) $V_{\rm so}$ reduces to the familiar $v_{\rm
so}=\xi(r)\vec{\sigma}\cdot\vec{\ell}\,$ where
$\xi(r)=\frac{e\hbar}{4m^{2}c^{2}}\frac{1}{r}\frac{dV_{1}(r)}{dr}$ and where
$V_{1}(r)$ is the Coulomb potential of the atomic coreWinkler (2011). It is
invariably $v_{\rm so}$ that is accounted for in the current flavours of the
DFT and in the usual theory of the Rashba effect as described, e.g., in Ref.
Winkler (2011). Using a number of different methods it is well
establishedKoseki et al. (2019), for isolated atoms, $v_{\rm so}$ accounts
well for the spin-orbit splitting.

While the wave-length $\lambda_{c}$ is very small when compare with the Bohr
radius $a_{0}$ or the atomic radius of heavy elements, e.g., 135pm for Au, it
is not so small when compared with the radius of the e.g., Au inner 1s
orbital. This goes as $r_{1}=\frac{a_{0}}{Z}<1$pm, where $a_{0}\approx 53$pm
is the Bohr radius and $Z=79$, for Au, the atomic number. If instead $a_{0}$
is replaced by the actual atomic radius of 135pm, it remains the case that
$\frac{135}{Z}\sim 2$pm and fully comparable with $\lambda_{c}$. The speed of
such a core $s$-electron $v_{1}\sim(Z\alpha)c$ where
$\alpha\approx\frac{1}{137}$ is the fine structure constant, and so for Au,
$v_{1}\sim 0.6$c. The electric field seen by this orbital $E_{1}\approx
k_{e}\frac{eZ^{3}}{{a_{0}}^{2}}$, where $k_{e}=8.99\times 10^{9}$Nm2/C2 is
enormous as indicated in the text and below. The magnitude of $E_{\rm
so}\sim\frac{1}{8\pi}\alpha(Z\alpha)k_{e}\frac{eZ^{3}}{{a_{0}}}\sim\frac{1}{8\pi}(Z\alpha)^{2}Z^{2}k_{e}\frac{e}{a_{0}}$,
where $k_{e}\frac{e}{a_{0}}\sim 10$eV, and is much greater than the binding
energy of hydrogen. Landau and LifshitzLandau1991  argue the outer, 6s for Au,
orbitals have a weight $\frac{1}{Z^{2}}$ in the core region. Even then the
question not why is the spin-orbit coupling so big in the solid state, e.g.,
for Au, but rather why is it so small.

In more detail, Landau and LifshitzLandau1991  observe the WKB radial wave-
function in the inner region goes as $|\psi|\sim\frac{1}{r\sqrt{p}}.$ The
logic is that $|\psi|=\frac{u}{r}$ where $u(r)$ is non-singular. The
$1/\sqrt{p}$ is the usual per-factor in the WKB wave-function. This is
$|\psi|$, so the $e^{ipr/\hbar}$ does not appear. While this WKB approximation
has well known problems at the origin the above version should couple smoothly
with the exact solution for $s$-waves near the origin and should be good
enough for the estimation being made here. In the WKB approximation, for
energy $E$, the value of $p$ is given by $\frac{p^{2}}{2m}-|U(r)|=E$ and near
the core $E$ is negligible compared with the potential $|U(r)|$. Without
worrying about the constants of proportionality: $p\sim\sqrt{|U(r)|}$ and for
the lowest energy orbitals
$U(r)\approx\frac{Ze^{2}}{r}\sim\frac{Z^{2}e^{2}}{a_{0}}$
$\sqrt{p}\sim|V(r)|^{1/4}\sim\frac{Z^{1/2}}{{a_{0}}^{1/4}}.$ It follows:

$|\psi|\sim\frac{1}{r\sqrt{p}}\sim\frac{Z}{a_{0}\sqrt{p}}\sim\frac{Z}{Z^{1/2}}\sim
Z^{1/2}$

The factor of $a_{0}$ is dropped since $|\psi|$ must be dimensionless. Then
for an $s$-wave the wave-function $|\psi|$ should be a constant for
$r<a_{0}/Z$ and so the probability of finding the wave-function in the lowest
$s$-orbital is proportional to the volume
$v_{c}=\frac{4\pi}{3}r^{3}\sim\frac{1}{Z^{3}}$ (not a shell) and the
probability $\sim|\psi|^{2}v_{c}\sim Z\frac{1}{Z^{3}}\to\frac{1}{Z^{2}}$ again
not keeping track of the dimension full $a_{0}$. The estimate for the
magnitude $E_{so}$ of the spin-orbit coupling is then based upon
$E_{so}=\frac{e\hbar^{2}}{4m^{2}c^{2}}Ze^{2}\langle\frac{1}{r^{3}}\rangle\to\frac{1}{4}\
Z^{2}(\alpha Z)^{2}\frac{e^{2}}{{a_{0}}}\to\frac{1}{4}\
Z^{2}\frac{e^{2}}{{a_{0}}}$ dividing by the factor of $Z^{2}$. This Landau and
LifshitzLandau1991  estimate is based upon
$\langle\frac{1}{r^{3}}\rangle\sim\frac{Z^{3}}{{a_{0}}^{3}}.$ A better
estimate for other than $s$-states is based upon the resultradialint
$\langle\frac{1}{r^{3}}\rangle=\frac{2}{n^{3}\ell(\ell+1)(2\ell+1){a_{0}}^{3}}.$
With the $Z^{2}$ correction

$E_{so}=\frac{1}{2}\frac{1}{n(\ell+\frac{1}{2})(\ell+1)}(\alpha Z)^{2}|E_{n}|$

where $E_{n}=-\frac{e^{2}}{2{a_{0}}}\frac{1}{n^{2}}$ are the hydrogen energy
levels. This is the estimate reproduced in the text. That for a spherical
potential $V(r)$, the spin-orbit coupling requires a non-zero $\ell$ and hence
the mixing is with core electrons with a finite $n>1$, i.e., a minimum $n=2$
and $\ell=1$. This is clearly important and leads to the moderate values of
$E_{so}$ observed in solids.

Turning more specifically to the estimation of the Rashba spin-orbit coupling.
Again with Au in mind, in the core region, using the above methods $\langle
E\rangle\approx 1.2\times 10^{16}$V/m while for $\ell=1$ the (tangential)
velocity $v\sim 0.15c$. In the context of electrical gating, it is hard to
envisage an electric field much greater than $10^{11}$V/m and, due to the
weighting factor of $1/Z^{2}$, the Rashba magnetic field $B_{R}$ due to the
core is very much negligible, while for the outer electrons the $\ell=1$
tangential velocity is very much too small.

Consider instead the missing radial velocity. The first estimate is the
momentum associated with the Bohr radius. Here $\hbar
k=\hbar/a_{0}=(1.05\times 10^{-34})/(0.5\times 10^{-10})\approx 2.0\times
10^{-24}$ and so the velocity is $v=\hbar k/m=2.0\times 10^{-24}/(9.1\times
10^{-31})=2\times 10^{6}$m/s and a typical Fermi velocity, reflecting a
crystal momentum. However for a n$s$-orbital there are $n-1$ radial nodes
implying $2n-1$ nodes across the atomic diameter. If, as for Au, $n=6$ then
$v\approx 1.2\times 10^{7}$m/s or approx $\frac{c}{25}$.

The effect of the crystal potential is very important. The Au-Au bond length
is $a=2.84$Å and, given a simple cubic structure assumed in the text, it is
this and not the actual lattice parameter that should be used for $a$. It
should really be less than this since the interstitial region has been shrunk
to zero in the derivation of the delta function model. This notwithstanding
for $a=2.84$Å it follows, with $G_{n}=\frac{2\pi}{a}n$, the momentum $\hbar
G_{n}\approx\frac{2\pi}{a}n=2.2n\times 10^{10}$ giving $v=\hbar k/m\approx
n\frac{c}{118}$. So for the Kronig-Penny model with $n_{x}=11$ appropriate to
the outer Au electrons $v\approx\frac{c}{10}$ and already comparable to the
$\ell=1$ tangential velocity in the core. If interstitial region is shrunk to
zero, the minimum $a\sim 1$Å corresponding to two Bohr radial touching each
other, i.e., $a\approx 1$Å and gives an upper limit $v\approx\frac{c}{3.5}$.
Thus while it is impossible to get an experimentally significant Rashba
coupling from the core region, it is possible to have a useful contribution
from the outer electron radial velocity of heavy elements due to the large
principal quantum number $n$. This is the theme taken up in the text.

Zitterbewegung oscillations. For a single free electron described by Dirac’s
equation, the Compton wave-length $\lambda_{c}\approx 2$pm and is about five
orders of magnitude smaller the experimentally determined value in some
solids, e.g, for InAsIwasaki et al. (2017) $\lambda_{c}\sim 100$nm.
Zitterbewegung (ZB) oscillations are a basic measurement that should set the
scale for the key parameter $\lambda_{c}$. It is often suggestedManchon et al.
(2015) that the large value of spin-orbit effect, in general, and specifically
the value of the frequency associated with the ZB oscillations, reflect the
fact that the mass gap $mc^{2}\approx 0.5$MeV is enormously larger than the
spin-orbit band gap $\lesssim 1$eV in the above and many other semi-
conductors. While this argument may be correct in this context it does not
explain the similarly large magnitude of the Rashba coupling.

The central potential spin-orbit coupling as a perturbation  It is important
to understand the origin of the Rashba spin-orbit coupling that arises from
the more usual reduced spin-orbit coupling $v_{\rm
so}=\xi(r)\vec{\sigma}\cdot\vec{\ell}$. In the further reduction of this to
traditional Rashba and/or Dresselhaus form, $\vec{p}$ in the original $V_{\rm
so}$ gets replaced by the crystal momentum $\vec{q}$. The current exercise has
shown that for a simple periodic model this reduction is not valid. It is a
nicety of the model examined in the text that $\vec{E}$ is the “applied”
electric field, i.e., the angular momentum $\vec{E}\times\vec{p}$ has no
intrinsic contribution because the potential within each cell, or effective
atom, has no gradient. A small radial potential $V(r)$ is now added, as a
perturbation, in order to examine the role of $v_{\rm so}$. A typical
perturbative contributionWang et al. (1996) to the magnetic anisotropy, due to
$v_{\rm so}$, has the splitting between, e.g., the t2g to eg levels as the
energy denominator whereas, for the model in the text, this is the exchange
$J_{0}S$. The quantum numbers that couple are different and hence gaps that
might appear correspond to different band crossings. This is of some
considerable experimental importance.

Explicitly
$\xi(r)\vec{\sigma}\cdot(\vec{r}\times\vec{p})=\xi(r)\sigma_{z}(xp_{y}-yp_{x})+(\xi
z)(\sigma_{y}p_{x}-\sigma_{x}p_{y})+\xi(\sigma_{x}y-\sigma_{y}x)p_{z}$. In
order to make calculations more tractable, take $\xi$ to be a constant. It has
been implicitly assumed the wave-function for the $z$-direction is node-less
and that there is symmetric wave-function $\psi_{0}(z)$ corresponding to the
lowest energy. The quantity $p_{z}\psi_{0}(z)$ has a single node and is
orthogonal to this state. The energy to the first excited state is large
compared to other excitation energies and there are therefore no relevant
matrix elements of $\xi(\sigma_{x}y-\sigma_{y}x)p_{z}$ in perturbation theory.
If the expectation value of $z$ is finite, i.e., if inversion symmetry is
broken, the term $(\xi\langle z\rangle)(\sigma_{y}p_{x}-\sigma_{x}p_{y})$
screens the direct Rashba effect. The “ionic” polarisation $-e\langle
z\rangle$ is due to an internal effective field $E_{i}$ that is already
accounted for in the direct Rashba term. If the dielectric constant $\epsilon$
is large, the effect of the polarisation is to reduce $E_{i}\to
E_{i}/\epsilon$. A large pseudo Rashba effect, reflecting $(\xi\langle
z\rangle)(\sigma_{y}p_{x}-\sigma_{x}p_{y})$ and a large polarisation, will
actually cancel the direct Rashba and/or Dresselhaus effects. It is therefore
highly problematic if a DFT code accounts for
$\xi\vec{\sigma}\cdot(\vec{r}\times\vec{p})$ but not the direct Rashba effect
treated here.

It remains to account for the part
$\xi\sigma_{z}\ell_{z}=\xi\sigma_{z}(xp_{y}-yp_{x})$ that involves the matrix
elements of $\ell_{z}=xp_{y}-yp_{x}$. Consider the matrix elements of
$xp_{y}$. The action of $p_{y}=-i\hbar\partial_{y}$ is to add an extra node to
$\psi_{k_{y}}(y)$. As a consequence there are only off-diagonal matrix
elements for the $\psi_{k_{y}}(y)$ part of the factored wave-function. The
periodic
$x=\frac{a}{2\pi}i\sum_{n_{x}=1}^{\infty}\frac{1}{s_{x}}(-1)^{s}\left(e^{iG_{n_{x}}x}-e^{iG_{-n_{x}}x}\right)$
when written in terms of the reciprocal lattice vectors $G_{n_{x}}$. It is
again the case that the factor of $e^{iG_{n_{x}}x}$ in the expansion imply
there are also only off-diagonal matrix elements for the $\psi_{k_{x}}(x)$
part of the wave-function. Perhaps more direct is the observation that a
factor of $x$ turns an odd to an even function or the inverse and also adds a
node. The matrix elements of $\xi\sigma_{z}\ell_{z}$ connect to states that
are doubly off-diagonal. For the tight binding bands it is implies that both
$n_{x}$ and $n_{y}$ differ by at least unity. While possible, because of the
approximation that $\xi$ is a constant, it is not particularly useful to
continue with the evaluation of the matrix elements. It is however important
to understand the magnitude of the magnetic anisotropy. The matrix elements
of, e.g., $xp_{y}\sim\hbar a\frac{2\pi}{a}s_{y}$ and those of
$(xp_{y}-yp_{x})$ squared $\sim 4\hbar^{2}\pi^{2}({s_{x}}^{2}+{s_{y}}^{2})$.
Unlike the Rashba coupling here these matrix elements are between states that
differ by their orbital quantum numbers $s_{x}$ and $s_{y}$ for the tight-
binding regime of the model. These states have energy
$E_{0}({s_{x}}^{2}+{s_{y}}^{2})$ for which $s_{x}$ and $s_{y}$ change by at
least unity. The anisotropy field $B_{A}$ is therefore of order
$4\xi^{2}\hbar^{2}\pi^{2}\sqrt{{s_{x}}^{2}+{s_{y}}^{2}}$. Then since $\xi\sim
E/a$, where $E\sim e/a^{2}$ is the magnitude of the radial electrical field at
the average position of the electrons, it follows $B_{A}\sim
4\pi^{2}\sqrt{{s_{x}}^{2}+{s_{y}}^{2}}\frac{\alpha^{2}}{a^{2}}E^{2}$. In the
present approach, the Rashba field $B_{R}$ rather involves matrix elements
between states that have $\sigma_{z}=\pm 1$ and an energy difference $J_{0}S$
independent of $s_{x}$ and $s_{y}$. This leads to a $B_{R}\sim
4\pi^{2}({s_{x}}^{2}+{s_{y}}^{2})\frac{\alpha^{2}}{a^{2}}E^{2}$ with a
stronger dependence on $s_{x}$ and $s_{y}$. It is again emphasised that
degenerate states with all $s_{x}=s_{y}=s_{z}=s$, and $s$ odd, correspond to
higher “s-bands” with $s=2n-1$ where $n$ is the principal quantum number and
that $({s_{x}}^{2}+{s_{y}}^{2})\to({s_{x}}^{2}+{s_{y}}^{2}+{s_{z}}^{2})$ for
three dimensional atoms imbedded in two dimensions. For heavy elements such as
Au, this increases the Rashba coupling by two orders of magnitude as compared
with free electrons and an order of magnitude relative to the
$\xi(r)\vec{\sigma}\cdot\vec{\ell}$ spin orbit terms, see the discussion in
the text.

The Hohenberg-Kohn theorem.  As BerryBerry (1984) has pointed out, in the
context of adiabatic evolution of a spin dependent Hamiltonian, the spin
dependent phase of a wave-function serves an essential purpose. However since
the Hamiltonian is time dependent this observation does not have direct
repercussions on the Hohenberg-Kohn theorem that insists the ground state
energy is a functional of only the density, i.e., depends only on the modulus
of the wave-function. It is however well knownYe et al. (1999); Taguchi et al.
(2001); Shindou and Nagaosa (2001), in the context of the spin-Hall effect,
that spin texture, i.e., the existence of non-collinear magnetisation,
introduces a spin “flux” $\Phi_{s}$ with non-trivial consequences even for
Hamiltonians that are time independent. More relevant here is that the energy
of electrons on a ring with a certain configuration of static magnetic fields,
i.e., the Stern problemShankar (2014) has a time independent Hamiltonian.
There is a non-degenerate ground state energy that depends on the spin Berry
phase $\phi_{s}=\frac{\hbar}{e}\Phi_{s}$ through the boundary
conditionsShankar (2014). For the free electron Rashba problem there is a
similar configuration of static magnetic fields and this result applies.
However it is easy to argue the boundary conditions do not effect the bulk
density of states and hence the energy in the thermodynamics limit.

In the presence of a periodic potential it has been shown the Kronig-Penney
problem reduces to finding the roots of

$\frac{1}{V}=\frac{1}{2}\sum_{\pm}\sum_{n_{x}}(1\pm
e^{-i\frac{\pi}{2}(\cos\theta_{G_{n_{x}}}+\sigma_{z})\sigma_{z}-\sin\theta_{G_{n_{x}}}\sigma_{y})}){\cal
G}^{\pm}_{G_{n_{x}}}$

where the ${\cal G}^{\pm}_{G_{n_{x}}}$ are real Green’s functions that reflect
the dynamic phase. It has also been seen that the Berry phase factor
$e^{-i\frac{\pi}{2}(\cos\theta_{G_{n_{x}}}+\sigma_{z})\sigma_{z}-\sin\theta_{G_{n_{x}}}\sigma_{y})}$
is a reflection of a real space spin off-diagonal Berry connection
$\vec{A}_{s}(\vec{r})$. Now the spin Berry phase appears not only in the
boundary conditions but also in the reflection amplitudes. It is possible to
define an effective magnetic field, i.e., the Berry curvature, that is
proportional to the commutators $[{A_{s}}_{i},{A_{s}}_{j}]$ of the components
${A_{s}}_{i}$. This is distinct from the Rashba field $\vec{B}_{R}$, but is
the flux of this field that modulates the spin interference effects. Since the
angles $\theta_{G_{n_{x}}}$ are self-consistently determined
$\vec{A}_{s}(\vec{r})$ cannot be considered as an external potential that
might be accommodated within an extended the Hohenberg-Kohn theorem Hohenberg
and Kohn (1964); Vignale and Rasolt (1988).

## References

  * (1) J. Nearing, _Unpublished_.
  * Barnes et al. (2014) S. E. Barnes, J. Ieda, and S. Maekawa, Scientific Reports 4, 4105 (2014), ISSN 2045-2322, URL https://doi.org/10.1038/srep04105.
  * Rosen and Zener (1932) N. Rosen and C. Zener, Phys. Rev. 40, 502 (1932), URL https://link.aps.org/doi/10.1103/PhysRev.40.502.
  * Shankar (2014) R. Shankar, _Principles of Quantum Mechanics_ (Springer, 2014).
  * Bychkov and Rashba (1984) Y. A. Bychkov and É. I. Rashba, Soviet Journal of Experimental and Theoretical Physics Letters 39, 78 (1984).
  * Winkler (2011) R. Winkler, _Spin-orbit coupling effects in two-dimensional electron and Hole Systems_ (Springer, 2011).
  * Messiah (2014) A. Messiah, _Quantum Mechanics_ (Dover Publications, 2014).
  * Koseki et al. (2019) S. Koseki, N. Matsunaga, T. Asada, M. W. Schmidt, and M. S. Gordon, The Journal of Physical Chemistry A 123, 2325 (2019), pMID: 30817150, eprint https://doi.org/10.1021/acs.jpca.8b09218, URL https://doi.org/10.1021/acs.jpca.8b09218.
  * (9) L D Landau, E.M. Lifshitz _Quantum mechanics: non-relativistic theory_ (Butterworth Heinemann, 1991)
  * (10) The integral $\langle\frac{1}{r^{3}}\rangle=\frac{Z^{3}}{n^{3}\ell(\ell+\frac{1}{2})(\ell+1)}\frac{1}{{a_{0}}^{3}}$, so if $\ell=1$ then $n=2$ and there is an additional factor of $\frac{1}{n^{3}\ell(\ell+\frac{1}{2})(\ell+1)}=\frac{1}{24}$. for the estimation of the electric field $\langle\frac{1}{r^{2}}\rangle=\frac{2Z^{2}}{n^{3}(2\ell+1){a_{0}}^{2}}$ is used
  * Iwasaki et al. (2017) Y. Iwasaki, Y. Hashimoto, T. Nakamura, and S. Katsumoto, Scientific Reports 7, 7909 (2017), ISSN 2045-2322.
  * Manchon et al. (2015) A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, Nature Materials 14, 871 (2015), ISSN 1476-4660, URL https://doi.org/10.1038/nmat4360.
  * Wang et al. (1996) X. Wang, R. Wu, D.-S. Wang, and A. J. Freeman, Phys. Rev. B 54, 61 (1996), URL https://link.aps.org/doi/10.1103/PhysRevB.54.61.
  * Berry (1984) M. V. Berry, Proceedings of the Royal Society of London Series A 392, 45 (1984).
  * Ye et al. (1999) J. Ye, Y. B. Kim, A. J. Millis, B. I. Shraiman, P. Majumdar, and Z. Tešanović, Phys. Rev. Lett. 83, 3737 (1999), URL https://link.aps.org/doi/10.1103/PhysRevLett.83.3737.
  * Taguchi et al. (2001) Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa, and Y. Tokura, Science 291, 2573–2576 (2001).
  * Shindou and Nagaosa (2001) R. Shindou and N. Nagaosa, Phys. Rev. Lett. 87, 116801 (2001), URL https://link.aps.org/doi/10.1103/PhysRevLett.87.116801.
  * Hohenberg and Kohn (1964) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964), URL https://link.aps.org/doi/10.1103/PhysRev.136.B864.
  * Vignale and Rasolt (1988) G. Vignale and M. Rasolt, Phys. Rev. B 37, 10685 (1988), URL https://link.aps.org/doi/10.1103/PhysRevB.37.10685.