# Rashba spin-orbit coupling and quantum-interference effect for a pair of
spin-correlated electrons in their tunneling and reflection under a step
potential

Po-Hsin Shih1, Godfrey Gumbs1, Danhong Huang2111E-mail contact:
<EMAIL_ADDRESS>Andrii Iurov3 and Yonatan Abranyos1 Department
of Physics, Hunter College, City University of New York, 695 Park Avenue, New
York, NY 10065 USA
2US Air Force Research Laboratory, Space Vehicles Directorate (AFRL$/$RVSU),
Kirtland Air Force Base, New Mexico 87117, USA
3Department of Physics and Computer Science, Medgar Evers College of City
University of New York, Brooklyn, NY 11225, USA

###### Abstract

We present both theory and numerical-computation results for the transmission
and reflection probability currents of a charged particle across a potential
step in the presence of a Rashba spin-orbit interaction. By varying kinetic
energy and angle of incident electrons or barrier height, different features
associated with tunneling and reflection of electrons are revealed by inter-
spin-channel electron tunnelings and reflections. These unique properties are
further accompanied by spin-state quantum interference of either a reflected
or transmitted pairs of spin-correlated electrons with the same kinetic energy
but in different spin-orbital states. Such distinctive features are expected
to give rise to a lot of applications in both spintronics and quantum-
computation devices.

##  I Introduction

It is well known that the spin-orbit interaction (SOI) is essentially a
relativistic effect, i.e., a moving electron under an external electric field
experiences a magnetic field in its rest frame. book1  In a semiconductor, the
SOI enables an electron undergoing a spin procession while it passes through
the material, which constitutes a basis of various proposed semiconductor-
based spintronic devices. add1 ; add2 ; add3  In nanostructures, quantum
confinement can further modify the symmetry of SOI. add4  From physics
perspective, the relativistic motion of an electron can be described by a
Dirac equation. By combining these effects, one expects both an electric
dipole moment as well as Thomas procession thomas  due to the rotational
kinetic energy under an electric field. new6 ; new7  Mathematically, however,
the SOI Hamiltonian can be derived from the Dirac equation after taking the
non-relativistic limit up to terms of $\sim(v/c)^{2}$, where $v$ is the
velocity of an electron. Formally, this limit can be reached by either
expanding the Dirac equation in powers of $\sim(v/c)$ or making use of the
asymptotically exact Foldy-Wouthousen transformation. new8

In recent years, there have been a lot of interests in the research field
dealing with the effect of SOI or the Rashba effect, on the electron transport
and optical properties of low-dimensional semiconductor electronic systems.
new1 ; new2 ; new3 ; new4 ; new19 ; new20 ; new21 ; czhang  Such an SOI
results from the asymmetry in a confining potential for an electron or a hole
gas at the interface of a hetero-structure. Particularly, some of these
studies aim at identifying potential spintronic-device applications, e.g., a
spin-based transistor, in which the spin-current can be manipulated purely by
electrical means (i.e., spin-Hall effect) new5 ; add5 ; add6 . Meanwhile,
special focus has also been put on its effect on plasmon excitations. new9 ;
new10 ; new11 ; new12 ; new13 ; new14 ; new15 ; new16 ; new17 ; new18

Very recently, by utilizing controlled polarization of incident light and
soft-X-ray angle-resolved photoemission spectroscopy, the giant Rashba effect
and spin polarization in GeTe (a component of chalcogenide glasses) thin films
as functions of light polarization and angle of emission were revealed and
analyzed systematically. add6  For non-ferromagnetic GeTe with a giant Rashba
parameter, one expects that a Zeeman gap will open at the Dirac point under a
magnetic field. vijay ; add7  In general, the Fermi surface of spin-orbit
coupled materials consists of two concentric, spin non-degenerate hyper-
surfaces separated in momentum space. new24  In the specific case of Rashba
SOI materials, these two Rashba bands have opposite chiralities with the angle
between the spin and momentum vectors locked and reversed with respect to each
other.

In this paper, base on our calculated energy eigenstate for Rashba-Zeeman
coupled two-dimensional conduction electrons in the presence of a spin-split
gap, we obtain probability currents explicitly for both reflection and
transmission of electrons under a step-like potential barrier. By matching
boundary conditions at the interface of this step barrier, we acquire both
spin-dependent reflection and transmission coefficients as functions of
incident kinetic energy and angle of incidence, as well as of barrier height
and steepness, for various spin directions, Zeeman gaps and Rashba parameter
values. Our current study demonstrates the existence of paired Rashba-Zeeman-
coupled inter-spin-channel tunneling of electrons with different diffraction
angles for possible spin-state quantum interference in the system. Such a
unique and attractive property can be utilized for designing non-magnetic
spintronics and quantum-computation devices.

The rest of paper is organized as follows. In Sec. II, we establish a theory
for calculating exactly energy eigenstates of two-dimensional Rashba-Zeeman-
coupled electrons with a spin-split gap. Moreover, the explicit expressions
for probability currents, as well as for transmission and reflection
coefficients, are derived in Sec. III, corresponding to in-plane Rashba-
Zeeman-coupled electron tunneling into a step-potential barrier. Numerical
results, and their analysis and discussions are given in Sec. IV for tunneling
and reflection coefficients as functions of incident kinetic energy, angle of
incidence, barrier height, and barrier steepness at the interface. Finally, a
brief summary is given in Sec. V.

##  II Energy Eigen-States with a Gap

Let us first consider a two-dimensional electron-gas system in $x$-$y$ plane
with both Rashba spin-orbit and Zeeman couplings 10  for orbital motions of
electrons. For this model system, the Hamiltonian operator $\hat{{\cal H}}$
can be written as

$\hat{{\cal
H}}=\left[\begin{array}[]{cc}\alpha\,(\hat{p}_{x}^{2}+\hat{p}_{y}^{2})+V_{B}(x)-\Delta_{Z}&\alpha_{R}\left(\hat{p}_{y}-i\hat{p}_{x}\right)\\\
\\\
\alpha_{R}\left(\hat{p}_{y}+i\hat{p}_{x}\right)&\alpha\,(\hat{p}_{x}^{2}+\hat{p}_{y}^{2})+V_{B}(x)+\Delta_{Z}\end{array}\right]\
,$ (1)

where $\hbar\alpha_{R}$ stands for the so-called Rashba parameter,
$2\Delta_{Z}$ is the Zeeman gap produced by a localized in-plane magnetic
field, $\alpha=1/2m^{*}$ with an electron effective mass $m^{*}$ (positive for
$n$-doping but negative for $p$-doping),
$\hat{p}_{x}=-i\hbar\,\partial/\partial x$ and
$\hat{p}_{y}=-i\hbar\,\partial/\partial y$ are two momentum operators in the
$x$ and $y$ directions, and $V_{B}(x)=V_{0}\,\Theta(x)$ represents a step
potential with a barrier height $V_{0}>0$ along the $x$ direction.
Consequently, the translation symmetry (or the electron transverse wave number
$k_{y}$) along the $y$ direction retains for this system.

We assume that the trial spinor-type wave function $\Psi(\mbox{\boldmath$r$})$
for our model system takes the form

$\Psi(\mbox{\boldmath$r$})=\frac{\texttt{e}^{ik_{y}y}}{\sqrt{{\cal
L}_{y}}}\,\left[\begin{array}[]{c}\psi_{A}(x)\\\ \\\
\psi_{B}(x)\end{array}\right]\ ,$ (2)

where ${\cal L}_{y}$ stands for the sample size in the $y$ direction and the
subscripts $A,\,B$ label two components of a spinor-type wave function for the
up- and down-pseudo-spin states of electrons. From the static Schrödinger
equation $\hat{{\cal
H}}\,\Psi(\mbox{\boldmath$r$})=\overline{E}\,\Psi(\mbox{\boldmath$r$})$, we
arrive at the following pair of homogeneous eigenvalue equations

$\left[\begin{array}[]{cc}-(\hbar^{2}\alpha)\,\partial^{2}/\partial
x^{2}+V_{B}(x)+\hbar^{2}\alpha
k_{y}^{2}-\Delta_{Z}-\overline{E}&\alpha_{R}\hbar\left(k_{y}-\partial/\partial
x\right)\\\ \\\ \alpha_{R}\hbar\left(k_{y}+\partial/\partial
x\right)&-(\hbar^{2}\alpha)\,\partial^{2}/\partial
x^{2}+V_{B}(x)+\hbar^{2}\alpha
k_{y}^{2}+\Delta_{Z}-\overline{E}\end{array}\right]\left[\begin{array}[]{c}\psi_{A}(x)\\\
\\\ \psi_{B}(x)\end{array}\right]=0\ ,$ (3)

where $\overline{E}$ is the eigen-energy of electrons to be determined.
Mathematically, for such a second-order differential equation, we require two
boundary conditions to select out a specific solution for our considered
system.

In the special case of $V_{B}(x)=0$, we find $\hat{{\cal H}}\to\hat{{\cal
H}}_{0}$, where $\hat{{\cal H}}_{0}$ can be simply obtained from Eq. (1) by
the substitution: $\partial/\partial x\to ik_{x}$. As a result, Eq. (3)
directly leads to a pair of spin-split bands, i.e.,

$E_{s}(k)\equiv\overline{E}_{s}(k)-E_{\rm
min}=\alpha\hbar^{2}k^{2}+s\sqrt{\alpha^{2}_{R}\hbar^{2}k^{2}+\Delta_{Z}^{2}}-E_{\rm
min}=E_{R}\bar{k}^{2}+s\sqrt{4E_{R}^{2}\bar{k}^{2}+\Delta_{Z}^{2}}-E_{\rm
min}\ .$ (4)

In Eq. (4), $\overline{E}_{s}(k)$ reaches a minimum $E_{\rm
min}=-E_{R}(1+\Delta_{Z}^{2}/4E^{2}_{R})<0$ at $k=k_{\rm min}\equiv\pm
k_{R}\sqrt{1-\Delta_{Z}^{2}/4E_{R}^{2}}$, where $s=\pm 1$ specifies the split
upper $(+)$ or lower $(-)$ branch respectively, and the zero-energy point is
selected at the minimum of $E_{s}(k)$ or the middle point of Zeeman gap of
$\overline{E}_{s}(k)$, $\bar{k}=k/k_{R}$, $E_{R}=\alpha_{R}^{2}/4\alpha$, and
$k_{R}=\alpha_{R}/2\hbar\alpha$. The inclusion of $E_{\rm min}$ in Eq. (4)
ensures that $E_{s}(k)\geq 0$, and it formally resembles a split “left-
shifted” ($s=+1$, spin-up) or “right-shifted” ($s=-1$, spin-down) energy
parabola as $\Delta_{Z}=0$, where $k=\sqrt{k_{x}^{2}+k_{y}^{2}}$. This
phenomenon leads to the so-called spin-Hall effect new5 ; add5 ; add6  in the
absence of an external magnetic field. The positive (negative) sign of
$\alpha$ value corresponds to an upward or electron-like (a downward or hole-
like) parabola, resulting in opposite signs for group velocities. At $k=0$,
however, these two spin-split bands are mixed with each other if
$\Delta_{Z}\neq 0$, and therefore, the spin-conservation requirement can be
relaxed in this case. Here, whenever $\Delta_{Z}\neq 0$, $s$ can still be
regarded as a branch index with $s=+1$ for the split-upper branch while $s=-1$
for the split-lower branch. Furthermore, two orthonormal wave functions,
associated with $s=\pm 1$ branches in Eq. (4), are found to be

$\Psi_{s,{\bf
k}}^{\rm(i)}(\mbox{\boldmath$r$})=\frac{\texttt{e}^{ik_{x}x+ik_{y}y}}{\sqrt{{\cal
A}}}\left[\begin{array}[]{c}-is\gamma_{s}(k)\,\texttt{e}^{i\theta_{\bf k}}\\\
\\\ 1\end{array}\right]\,\frac{1}{\sqrt{1+\gamma_{s}^{2}(k)}}\ ,$ (5)

where $\gamma_{s}(k)=\alpha_{R}\hbar
k/(\sqrt{\alpha_{R}^{2}\hbar^{2}k^{2}+\Delta_{Z}^{2}}+s\Delta_{Z})$ and it
reduces to unity for $\Delta_{Z}\to 0$, $\theta_{\bf
k}=\tan^{-1}(k_{y}/k_{x})$ represents a phase angle, and ${\cal A}$ stands for
the surface area of the sample.

If we only focus on the electron states close to $k=0$, by neglecting the
higher-order terms proportional to $\sim k^{2}$ in Eq. (4), we are left with
$E_{\pm}(k)\approx\pm\hbar\alpha_{R}k$ for $\Delta_{Z}=0$, which approximately
resemble the upper ($+$) cone for all allowed electron states and the lower
($-$) cone for excluding disallowed electron states, including both pseudo-
spin states of electrons. However, in this paper, we would always use the full
expressions in Eq. (4) for split energy dispersions in our calculations.

##  III Probability current and transmission coefficient

We now turn attention to an important part of our theory, i.e., the
calculation of probability current as a first step in determining the
transmission or reflection coefficient. In general, the time-dependent
Schrödinger equation for the Hamiltoniam $\hat{{\cal H}}$ in Eq. (1), as well
as the spinor-type wave-function $|\mbox{\boldmath$\Psi$}>=[\psi_{A}\
\psi_{B}]^{T}$, can be written as $i\hbar\,\partial/\partial
t\,|\mbox{\boldmath$\Psi$}>=\hat{{\cal H}}\,|\mbox{\boldmath$\Psi$}>$ from
which we deduce that the probability density
$\rho=<\mbox{\boldmath$\Psi$}|\mbox{\boldmath$\Psi$}>$ satisfies

$i\hbar\frac{\partial\rho}{\partial t}=i\hbar\frac{\partial}{\partial
t}<\mbox{\boldmath$\Psi$}|\mbox{\boldmath$\Psi$}>=<\mbox{\boldmath$\Psi$}|\hat{{\cal
H}}|\mbox{\boldmath$\Psi$}>-\left(\hat{{\cal
H}}|\mbox{\boldmath$\Psi$}>\right)^{\dagger}|\mbox{\boldmath$\Psi$}>\ .$ (6)

Inserting the Hamiltonian in Eq. (1) into Eq. (6), we find, after a lengthy
calculation, that the probability-current density
$\mbox{\boldmath$j$}=\\{j_{x},\,j_{y}\\}$ satisfies the continuity equation
$\nabla\cdot\mbox{\boldmath$j$}+\partial\rho/\partial t=0$, and its two
components, $j_{x}$ and $j_{y}$, are found to be

$\displaystyle j_{x}$ $\displaystyle=2\alpha\hbar\sum_{\nu=A,B}{\rm
Im}\left(\psi_{\nu}^{\ast}\frac{\partial\psi_{\nu}}{\partial
x}\right)+2\alpha_{R}\,{\rm Im}\left(\psi_{A}^{\ast}\psi_{B}\right)\ ,$ (7a)
$\displaystyle j_{y}$ $\displaystyle=2\alpha\hbar\sum_{\nu=A,B}{\rm
Im}\left(\psi_{\nu}^{\ast}\frac{\partial\psi_{\nu}}{\partial
y}\right)+2\alpha_{R}\,{\rm Re}\left(\psi_{A}^{\ast}\psi_{B}\right)\ ,$ (7b)

where the first term on the right-hand-sides relates to the pure orbital
current while the second term results from the spin-orbital-coupling current.
Substituting the wave function in Eq. (5) into Eqs. (7a) and (7b), we obtain
the incident probability current $\mbox{\boldmath$J$}^{(\rm
i)}(s,\mbox{\boldmath$k$})=\\{J_{x}^{\rm(i)}(s,\mbox{\boldmath$k$}),\,J_{y}^{\rm(i)}(s,\mbox{\boldmath$k$})\\}$
for $s=\pm 1$

$\displaystyle J_{x}^{\rm(i)}(s,\mbox{\boldmath$k$})$
$\displaystyle=j_{x}^{\rm(i)}(s,\mbox{\boldmath$k$}){\cal
A}=2\left[s\Gamma_{s}(k)\alpha_{R}\cos\theta_{\bf k}+\alpha\hbar
k_{x}\right]\equiv I_{s}(k)\cos\theta_{\bf k}\ ,$ (8a) $\displaystyle
J_{y}^{\rm(i)}(s,\mbox{\boldmath$k$})$
$\displaystyle=j_{y}^{\rm(i)}(s,\mbox{\boldmath$k$}){\cal
A}=2\left[s\Gamma_{s}(k)\alpha_{R}\sin\theta_{\bf k}+\alpha\hbar
k_{y}\right]\equiv I_{s}(k)\sin\theta_{\bf k}\ ,$ (8b)

where $\Gamma_{s}(k)=\gamma_{s}(k)/[1+\gamma_{s}^{2}(k)]\to 1/2$ for
$\Delta_{Z}=0$, $I_{s}(k)=2[s\Gamma_{s}(k)\alpha_{R}+\alpha\hbar k]$ and
$\\{k_{x},k_{y}\\}=\\{k\cos\theta_{\bf k},\,k\sin\theta_{\bf k}\\}$. In
particular, for an incident particle, we should designate both its branch
index $s$ and kinetic energy $E_{s}(k)$ in Eq. (4) simultaneously. Similarly,
the probability current $\mbox{\boldmath$J$}^{(\rm
t)}(s^{\prime\prime},\mbox{\boldmath$q$};\,|t|^{2})=\\{J_{x}^{\rm(t)}(s^{\prime\prime},\mbox{\boldmath$q$};\,|t|^{2}),\,J_{y}^{\rm(t)}(s^{\prime\prime},\mbox{\boldmath$q$};\,|t|^{2})\\}$
for the transmitted wave with branch index $s^{\prime\prime}=\pm 1$ and the
wave function given by

$\Psi_{s^{\prime\prime},{\bf q}}^{\rm(t)}(\mbox{\boldmath$r$})=t\
\frac{\texttt{e}^{iq_{x}x+iq_{y}y}}{\sqrt{{\cal
A}}}\left[\begin{array}[]{c}-is^{\prime\prime}\gamma_{s^{\prime\prime}}(q)\,\texttt{e}^{i\theta^{\prime\prime}_{\bf
q}}\\\ \\\
1\end{array}\right]\,\frac{1}{\sqrt{1+\gamma_{s^{\prime\prime}}^{2}(q)}}\ ,$
(9)

is calculated as

$\displaystyle J_{x}^{\rm(t)}(s^{\prime\prime},\mbox{\boldmath$q$};\,|t|^{2})$
$\displaystyle=j_{x}^{\rm(t)}(s^{\prime\prime},\mbox{\boldmath$q$};\,|t|^{2}){\cal
A}=2|t|^{2}\left[s^{\prime\prime}\Gamma_{s^{\prime\prime}}(q)\alpha_{R}\cos\theta^{\prime\prime}_{\bf
q}+\alpha^{\prime\prime}\hbar
q_{x}\right]=|t|^{2}I_{s^{\prime\prime}}(q)\cos\theta^{\prime\prime}_{\bf q}\
,$ (10a) $\displaystyle
J_{y}^{\rm(t)}(s^{\prime\prime},\mbox{\boldmath$q$};\,|t|^{2})$
$\displaystyle=j_{y}^{\rm(t)}(s^{\prime\prime},\mbox{\boldmath$q$};\,|t|^{2}){\cal
A}=2|t|^{2}\left[s^{\prime\prime}\Gamma_{s^{\prime\prime}}(q)\alpha_{R}\sin\theta^{\prime\prime}_{\bf
q}+\alpha^{\prime\prime}\hbar
q_{y}\right]=|t|^{2}I_{s^{\prime\prime}}(q)\sin\theta^{\prime\prime}_{\bf q}\
,$ (10b)

where $\theta^{\prime\prime}_{\bf
q}=\tan^{-1}(q_{y}/q_{x})=\tan^{-1}(k_{y}/q_{x})$,
$I_{s^{\prime\prime}}(q)=2[s^{\prime\prime}\Gamma_{s^{\prime\prime}}(q)\alpha_{R}+\alpha^{\prime\prime}\hbar
q]$. In this notation, $\alpha^{\prime\prime}=\pm\alpha$,
$q_{x}=q\cos\theta^{\prime\prime}_{\bf q}$ and
$q_{y}=q\sin\theta^{\prime\prime}_{\bf q}=k_{y}=k\sin\theta_{\bf k}$, where
the spin ($\alpha_{R}$) and the orbital ($\alpha$) currents can have opposite
signs with respect to each other. Especially, for transmission with
$\alpha_{R}=0$, we find $I_{s^{\prime\prime}}(q)\sin\theta^{\prime\prime}_{\bf
q}=(\alpha^{\prime\prime}/\alpha)\,I_{s}(k)\sin\theta_{\bf k}$ for the
conservation of probability current along the $y$ direction under $|t|^{2}=1$,
which is formally similar to the well-known Snell’s law after taking
$\alpha^{\prime\prime}=\alpha$. However, the spin-orbit coupling still enables
breaking down this conservation of probability current along the $y$ direction
even for $s=s^{\prime\prime}$ since $q\neq k$ as $V_{0}\neq 0$. Here, the
branch index $s^{\prime\prime}$ should be assigned in advance for either an
inter-spin-channel ($s\neq s^{\prime\prime}$) or an intra-spin-channel
($s=s^{\prime\prime}$) tunneling of electrons. For given branch indexes
$s,\,s^{\prime\prime}$, incident energy $E_{s}(k)=E_{K}$, incident angle
$\theta_{\bf k}$, barrier height $V_{0}$, and the bandgap $2\Delta_{Z}$, the
longitudinal wave number $q_{x}$ on the barrier side can be found from
Eq.(16b) below. Meanwhile, for the reflected wave with its wave function
written as

$\Psi_{s^{\prime},{\bf k}^{\prime}}^{\rm(r)}(\mbox{\boldmath$r$})=r\
\frac{\texttt{e}^{ik_{y}y-ik^{\prime}_{x}x}}{\sqrt{{\cal
A}}}\left[\begin{array}[]{c}-is^{\prime}\gamma_{s^{\prime}}(k^{\prime})\,\texttt{e}^{i(\pi-\theta_{{\bf
k}^{\prime}})}\\\ \\\
1\end{array}\right]\,\frac{1}{\sqrt{1+\gamma_{s^{\prime}}^{2}(k^{\prime})}}\
,$ (11)

the probability current $\mbox{\boldmath$J$}^{(\rm
r)}(s^{\prime},\mbox{\boldmath$k$}^{\prime};\,|r|^{2})=\\{J_{x}^{\rm(r)}(s^{\prime},\mbox{\boldmath$k$}^{\prime};\,|r|^{2}),\,J_{y}^{\rm(r)}(s^{\prime},\mbox{\boldmath$k$}^{\prime};\,|r|^{2})\\}$
for the branch index $s^{\prime}$ is found to be

$\displaystyle
J_{x}^{\rm(r)}(s^{\prime},\mbox{\boldmath$k$}^{\prime};\,|r|^{2})$
$\displaystyle=j_{x}^{\rm(r)}(s^{\prime},\mbox{\boldmath$k$}^{\prime};\,|r|^{2}){\cal
A}=-2|r|^{2}\left[s^{\prime}\Gamma_{s^{\prime}}(k^{\prime})\alpha_{R}\cos\theta_{{\bf
k}^{\prime}}+\alpha\hbar
k^{\prime}_{x}\right]=-|r|^{2}I_{s^{\prime}}(k^{\prime})\cos\theta_{{\bf
k}^{\prime}}\ ,$ (12a) $\displaystyle
J_{y}^{\rm(r)}(s^{\prime},\mbox{\boldmath$k$}^{\prime};\,|r|^{2})$
$\displaystyle=j_{y}^{\rm(r)}(s^{\prime},\mbox{\boldmath$k$}^{\prime};\,|r|^{2}){\cal
A}=+2|r|^{2}\left[s^{\prime}\Gamma_{s^{\prime}}(k^{\prime})\alpha_{R}\sin\theta_{{\bf
k}^{\prime}}+\alpha\hbar
k_{y}\right]=+|r|^{2}I_{s^{\prime}}(k^{\prime})\sin\theta_{{\bf k}^{\prime}}\
,$ (12b)

where
$I_{s^{\prime}}(k^{\prime})=2[s^{\prime}\Gamma_{s^{\prime}}(k^{\prime})\alpha_{R}+\alpha\hbar
k^{\prime}]$, $k^{\prime}_{x}=k^{\prime}\cos\theta_{{\bf k}^{\prime}}$,
$k_{y}=k^{\prime}\sin\theta_{{\bf k}^{\prime}}=k\sin\theta_{\bf k}$, and the
probability current along the $y$ direction is not always conserved, if
$s^{\prime}\neq s$ or $k^{\prime}\neq k$, even for reflection under
$|r|^{2}=1$.

Physically, we can further introduce a set of boundary conditions for our
model system. Since the connection between two components $\psi_{A}(x)$ and
$\psi_{B}(x)$ of a spinor-type wave function in Eq. (2) have already been
built into eigenvalue equation presented in Eq. (3), we need only consider one
component, e.g., $\psi_{A}(x)$ (in the sense that the obtained $|t|^{2}$ and
$|r|^{2}$ remain the same).

First, for given branch indexes $s,\,s^{\prime\prime}$, by using continuities
for both wave functions and their derivatives on both sides of the step
boundary at $x=0$, we acquire the following relations for unknown $r$ and $t$,
yielding

$\displaystyle-is^{\prime\prime}\texttt{e}^{i\theta^{\prime\prime}_{\bf
q}}\Gamma_{s^{\prime\prime}}(q)\,t$
$\displaystyle=-is\Gamma_{s}(k)\,\texttt{e}^{i\theta_{\bf
k}}+is^{\prime}\Gamma_{s^{\prime}}(k^{\prime})\,\texttt{e}^{-i\theta_{{\bf
k}^{\prime}}}\,r\ ,$ (13a) $\displaystyle
q_{x}s^{\prime\prime}\texttt{e}^{i\theta^{\prime\prime}_{\bf
q}}\Gamma_{s^{\prime\prime}}(q)\,t$
$\displaystyle=k_{x}s\Gamma_{s}(k)\,\texttt{e}^{i\theta_{\bf
k}}+k^{\prime}_{x}s^{\prime}\Gamma_{s^{\prime}}(k^{\prime})\,\texttt{e}^{-i\theta_{{\bf
k}^{\prime}}}\,r\ ,$ (13b)

which leads to the conclusion

$\displaystyle t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})$
$\displaystyle=\left[\frac{s\Gamma_{s}(k)}{s^{\prime\prime}\Gamma_{s^{\prime\prime}}(q)}\right]\,\texttt{e}^{i(\theta_{\bf
k}-\theta^{\prime\prime}_{\bf q})}\left[\frac{k^{\prime}\cos\theta_{{\bf
k}^{\prime}}+k\cos\theta_{\bf k}}{k^{\prime}\cos\theta_{{\bf
k}^{\prime}}+q\cos\theta^{\prime\prime}_{\bf q}}\right]\ ,$ (14a)
$\displaystyle r(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})$
$\displaystyle=-\left[\frac{s\Gamma_{s}(k)}{s^{\prime}\Gamma_{s^{\prime}}(k^{\prime})}\right]\,\texttt{e}^{i(\theta_{\bf
k}+\theta_{{\bf k}}^{\prime})}\left[\frac{k\cos\theta_{\bf
k}-q\cos\theta^{\prime\prime}_{\bf q}}{k^{\prime}\cos\theta_{{\bf
k}^{\prime}}+q\cos\theta^{\prime\prime}_{\bf q}}\right]\ ,$ (14b)

where $r(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})$ depends
on $k^{\prime}_{x},\,q_{x}$ and $s^{\prime},\,s^{\prime\prime}$ if $V_{0}\neq
0$ in addition to its dependence on $s$ and $k_{x}$ corresponding to
$E_{K}=E_{s}(k)$. In Eqs. (14a) and (14b), $\theta^{\prime\prime}_{\bf q}$
depends implicitly on the branch index $s^{\prime\prime}$ for given
$E_{s}(k)=E_{K}$ and $\theta_{\bf k}$. It is important to note that the
condition $E_{s}(k)=E_{K}$ does not determine $k$ uniquely, and therefore, we
should assign $k$ and $s$ simultaneously for incidence of electrons instead of
$E_{K}$ solely. For $s=s^{\prime}=s^{\prime\prime}$ and $V_{0}=0$, we get
$q=k^{\prime}=k$,
$\Gamma_{s^{\prime\prime}}(q)=\Gamma_{s^{\prime}}(k^{\prime})=\Gamma_{s}(k)$
and $\theta_{\bf k}=\theta_{{\bf k}^{\prime}}=\theta^{\prime\prime}_{\bf q}$
since $q_{y}=k_{y}$, which leads to $t=1$ and $r=0$ in Eqs. (14a) and (14b).
On the other hand, even if $V_{0}\to 0$ but $s\neq
s^{\prime},\,s^{\prime\prime}$, we still obtain $r\neq 0$ due to inter-spin-
channel for tunneling and spin-procession for reflection. Here, the spin-
procession process becomes possible for reflection and tunneling of electrons
because of the spin-state mixing by Rashba-Zeeman coupling.

Figure 1: Energy dispersions on the incidence (left) and the step-barrier
(right) sides with a barrier height $V_{0}$, where three different incident
kinetic energies of $E_{K}$ (blue horizontal lines) represent energy ranges
(I), (II) and (III). The blue and red dots refer to selected electronic states
in ranges I, II and III for the incident $(k)$ and barrier $(q)$ sides,
respectively. (A) $V_{0}=0$ with aligned $E_{K}$ states and ranges on the
barrier side, where red-upward (blue-downward) arrows indicate up-spin (down-
spin) states of electrons; (B) $V_{0}>0$ with allowed $E_{K}$ states in
staggered ranges I, II on the barrier side; (C) $V_{0}>0$ with allowed $E_{K}$
states in staggered range II on the barrier side; (D) $V_{0}>0$ with allowed
$E_{K}$ states in staggered range I on the barrier side; (E) $V_{0}>E_{K}$
with no allowed $E_{K}$ state in all ranges on the barrier side.

By using the results in Eqs. (14a) and (14b), the transmission ${\cal
T}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})$
and reflection ${\cal
R}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})$
coefficients are obtained as

$\displaystyle{\cal
T}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})$
$\displaystyle=\left|\frac{J_{x}^{\rm(t)}(s^{\prime\prime},\mbox{\boldmath$q$};\,|t|^{2})}{J_{x}^{\rm(i)}(s,\mbox{\boldmath$k$})}\right|=|t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})|^{2}\left|\frac{I_{s^{\prime\prime}}(q)\cos\theta^{\prime\prime}_{\bf
q}}{I_{s}(k)\cos\theta_{\bf k}}\right|\ ,$ (15a) $\displaystyle{\cal
R}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})$
$\displaystyle=\left|\frac{J_{x}^{\rm(r)}(s^{\prime},\mbox{\boldmath$k$}^{\prime};\,|r|^{2})}{J_{x}^{\rm(i)}(s,\mbox{\boldmath$k$})}\right|=\left|r\right(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})|^{2}\left|\frac{I_{s^{\prime}}(k^{\prime})\cos\theta_{{\bf
k}^{\prime}}}{I_{s}(k)\cos\theta_{\bf k}}\right|\ ,$ (15b)

where ${\cal
T}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})+{\cal
R}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})$
is usually not unity since it only represents the probability current in the
$x$ direction. In Eqs. (15a) and (15b), for fixed branch indexes
$s,\,s^{\prime},\,s^{\prime\prime}$, incident energy $E_{K}=E_{s}(k)$ and
$k_{y}$ (or $\theta_{\bf k}$), $k_{x}$, $q_{x}$, $k^{\prime}_{x}$ for given
$k_{y}$ and $\alpha^{\prime\prime}=\alpha$ are calculated as

$\displaystyle k_{\pm}$
$\displaystyle\equiv\sqrt{k_{x}^{2}+k_{y}^{2}}=\sqrt{2}k_{R}\left\\{1+\frac{E_{K}+E_{\rm
min}}{2E_{R}}\pm\sqrt{1+\frac{E_{K}+E_{\rm
min}}{E_{R}}+\frac{\Delta_{Z}^{2}}{4E_{R}^{2}}}\ \right\\}^{1/2}\ ,$ (16a)
$\displaystyle q_{\pm}$
$\displaystyle\equiv\sqrt{q_{x}^{2}+k_{y}^{2}}=\sqrt{2}k_{R}\left\\{1+\frac{E_{K}-V_{0}+E_{\rm
min}}{2E_{R}}\pm\sqrt{1+\frac{E_{K}-V_{0}+E_{\rm
min}}{E_{R}}+\frac{\Delta_{Z}^{2}}{4E_{R}^{2}}}\ \right\\}^{1/2}\ ,$ (16b)
$\displaystyle k^{\prime}_{\pm}$ $\displaystyle\equiv\sqrt{k_{x}^{\prime
2}+k_{y}^{2}}=\sqrt{2}k_{R}\left\\{1+\frac{E_{K}+E_{\rm
min}}{2E_{R}}\pm\sqrt{1+\frac{E_{K}+E_{\rm
min}}{E_{R}}+\frac{\Delta_{Z}^{2}}{4E_{R}^{2}}}\ \right\\}^{1/2}\ ,$ (16c)

where $k^{\prime}_{\pm}$ can be either same as or different from $k_{\pm}$ if
more than one tunneling or reflection channel exist, $k_{+}$ ($q_{+}$) and
$k_{-}$ ($q_{-}$) represent, separately, the larger and smaller $k$ ($q$)
values, $E_{R}=\alpha_{R}^{2}/4\alpha$ is the rashba energy,
$k_{R}=\alpha_{R}/2\alpha\hbar$ is the Rashba wave number, and $E_{\rm
min}=-E_{R}(1+\Delta_{Z}^{2}/E^{2}_{R})$ for $\overline{E}_{s}(k)$ at $k_{\rm
min}=\pm k_{R}\sqrt{1-\Delta_{Z}^{2}/4E_{R}^{2}}$. The results in Eqs.
(16a)$-$(16c) are illustrated by Fig. 1.

For a given incident energy $E_{K}$, if multiple intersection points of wave
number exist for $E_{s}(k)=E_{K}$ on both sides of incidence and step barrier,
from the conservation of total particle number, we acquire

$\sum\limits_{s,s^{\prime},s^{\prime\prime}}\,\left[{\cal
T}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})+{\cal
R}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})\right]=\sum\limits_{s,s^{\prime},s^{\prime\prime}}\,1\equiv
N_{\rm T}\ ,$ (17)

where $N_{\rm T}$ stands for the total number of inequivalent cases under the
condition of $E_{s}(k)=E_{K}$ for different incidence, reflection and
tunneling channels. Then, by defining average transmission coefficient ${\cal
T}_{\rm
av}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$})$ and
reflection coefficient ${\cal R}_{\rm
av}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$})$
coefficients through

$\displaystyle{\cal T}_{\rm
av}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$})=\frac{1}{N_{\rm
T}}\sum\limits_{s,s^{\prime},s^{\prime\prime}}\,{\cal
T}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})\
,$ (18) $\displaystyle{\cal R}_{\rm
av}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$})=\frac{1}{N_{\rm
T}}\sum\limits_{s,s^{\prime},s^{\prime\prime}}\,{\cal
R}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})\
,\ $ (19)

we arrive from Eq. (17) at the conclusion that

${\cal T}_{\rm
av}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$})+{\cal
R}_{\rm
av}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$})\equiv
1\ .$ (20)

Consequently, we know from Eq. (20) that ${\cal T}_{\rm
av}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$})\leq
1$ and ${\cal R}_{\rm
av}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$})\leq
1$ must be satisfied physically. Moreover, we refer ${\cal
T}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})$
in Eq. (18) and ${\cal
R}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})$
in Eq. (19) as the partial transmission and reflection coefficients,
respectively, which can be larger than one as long as ${\cal
T}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})/N_{\rm
T}$ and ${\cal
R}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})/N_{\rm
T}$ do not exceed unity.

For Fig. 1, as $V_{0}>E_{K}+E_{\rm min}+2E_{R}+\Delta_{Z}^{2}/2E_{R}$,
$q_{\pm}$ becomes an imaginary number, implying no tunneling channel is
available in the system. If $V_{0}=0$ and $s=s^{\prime}=s^{\prime\prime}$ for
an intra-spin-channel tunneling, we have $q_{x}=k_{x}=k_{x}^{\prime}$. In this
case, we find from Eqs. (14a), (14b), (15a) and (15b) that $t\to 1$, $r\to 0$,
${\cal
T}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})\to
1$ and ${\cal
R}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})\to
0$ simultaneously. If $V_{0}=0$ but $s\neq s^{\prime\prime}$ and $s\neq
s^{\prime}$ for forward and backward spin processions, on the other hand, we
get $q_{x}\neq k_{x}$ and $k_{x}^{\prime}\neq k_{x}$.

As a special gapless case with $\Delta_{Z}=0$, we get directly from Eqs.
(16a), (16b) and (16c) that

$\displaystyle k^{(0)}_{\pm}$
$\displaystyle\equiv\sqrt{k_{x}^{2}+k_{y}^{2}}=\sqrt{2}k_{R}\left\\{1+\frac{E_{K}+E_{\rm
min}}{2E_{R}}\pm\sqrt{1+\frac{E_{K}+E_{\rm min}}{E_{R}}}\ \right\\}^{1/2}\ ,$
(21a) $\displaystyle q^{(0)}_{\pm}$
$\displaystyle\sqrt{q_{x}^{2}+k_{y}^{2}}=\sqrt{2}k_{R}\left\\{1+\frac{E_{K}-V_{0}+E_{\rm
min}}{2E_{R}}\pm\sqrt{1+\frac{E_{K}-V_{0}+E_{\rm min}}{E_{R}}}\
\right\\}^{1/2}\ ,$ (21b) $\displaystyle k^{\prime(0)}_{\pm}$
$\displaystyle\equiv\sqrt{k_{x}^{\prime
2}+k_{y}^{2}}=\sqrt{2}k_{R}\left\\{1+\frac{E_{K}+E_{\rm
min}}{2E_{R}}\pm\sqrt{1+\frac{E_{K}+E_{\rm min}}{E_{R}}}\ \right\\}^{1/2}\ ,$
(21c)

which always present two nonzero intersection points for wave numbers $k$,
$k^{\prime}$ and $q$ except for a degeneracy case with $E_{K}=-E_{\rm
min}=E_{R}$ or $E_{K}=V_{0}-E_{\rm min}=V_{0}+E_{R}$. Explicitly, from Eqs.
(15a) and (15b) we get

$\displaystyle{\cal
R}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})$
$\displaystyle=$
$\displaystyle\left[\frac{\Gamma_{s}(k)}{\Gamma_{s^{\prime}}(k^{\prime})}\right]^{2}\,\left[\frac{k\cos\theta_{\bf
k}-q\cos\theta^{\prime\prime}_{\bf q}}{k^{\prime}\cos\theta_{{\bf
k}^{\prime}}+q\cos\theta^{\prime\prime}_{\bf
q}}\right]^{2}\left|\frac{[\alpha\hbar
k^{\prime}+s^{\prime}\Gamma_{s^{\prime}}(k^{\prime})\alpha_{R}]\cos\theta_{{\bf
k}^{\prime}}}{[\alpha\hbar k+s\Gamma_{s}(k)\alpha_{R}]\cos\theta_{\bf
k}}\right|\ ,$ (22) $\displaystyle{\cal
T}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$}\,|s,s^{\prime},s^{\prime\prime})$
$\displaystyle=$
$\displaystyle\left[\frac{\Gamma_{s}(k)}{\Gamma_{s^{\prime\prime}}(q)}\right]^{2}\,\left[\frac{k^{\prime}\cos\theta_{{\bf
k}^{\prime}}+k\cos\theta_{\bf k}}{k^{\prime}\cos\theta_{{\bf
k}^{\prime}}+q\cos\theta^{\prime\prime}_{\bf
q}}\right]^{2}\left|\frac{[\alpha\hbar
q+s^{\prime\prime}\Gamma_{s^{\prime\prime}}(q)\alpha_{R}]\cos\theta^{\prime\prime}_{\bf
q}}{[\alpha\hbar k+s\Gamma_{s}(k)\alpha_{R}]\cos\theta_{\bf k}}\right|\ ,$
(23)

where $\Gamma_{s}(k)=\gamma_{s}(k)/[1+\gamma_{s}^{2}(k)]$ and
$\gamma_{s}(k)=\alpha_{R}\hbar
k/[\sqrt{\alpha^{2}_{R}\hbar^{2}k^{2}+\Delta^{2}_{Z}}+s\Delta_{Z}]$.

Next, we further include a $\delta$-function potential with an amplitude
$d_{0}$ (in unit of $\hbar\alpha_{R}=2k_{R}\hbar^{2}\alpha$) at the interface
and write down $V_{B}(x)=V_{0}\Theta_{0}(x)+d_{0}\delta(x)$. This leads to a
discontinuity in the derivative of a wave function. As a result, the boundary
conditions in Eqs. (13a) and (13b) for $\alpha=\alpha^{\prime\prime}$ are
generalized to

$\displaystyle-
is^{\prime\prime}\Gamma_{s^{\prime\prime}}(q)\,\texttt{e}^{i\theta^{\prime\prime}_{\bf
q}}$ $\displaystyle=-is\Gamma_{s}(k)\,\texttt{e}^{i\theta_{\bf
k}}+is^{\prime}\Gamma_{s^{\prime}}(k^{\prime})\,\texttt{e}^{-i\theta_{{\bf
k}^{\prime}}}\,r\ ,$ (24a) $\displaystyle
q_{x}s^{\prime\prime}\Gamma_{s^{\prime\prime}}(q)\,\texttt{e}^{i\theta_{\bf
q}^{\prime\prime}}\,t$
$\displaystyle=k_{x}s\Gamma_{s}(k)\,\texttt{e}^{i\theta_{\bf
k}}+k^{\prime}_{x}s^{\prime}\Gamma_{s^{\prime}}(k^{\prime})\,\texttt{e}^{-i\theta_{{\bf
k}^{\prime}}}\,r+\left(\frac{d_{0}}{\hbar^{2}\alpha}\right)is^{\prime\prime}\Gamma_{s^{\prime\prime}}(q)\,\texttt{e}^{i\theta^{\prime\prime}_{\bf
q}}\,t\ ,$ (24b)

This produces the solutions that

$\displaystyle
t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime};d_{0})$
$\displaystyle=\left[\frac{s\Gamma_{s}(k)}{s^{\prime\prime}\Gamma_{s^{\prime\prime}}(q)}\right]\,\texttt{e}^{i(\theta_{\bf
k}-\theta^{\prime\prime}_{\bf q})}\,\left[\frac{k^{\prime}\cos\theta_{{\bf
k}^{\prime}}+k\cos\theta_{\bf k}}{k^{\prime}\cos\theta_{{\bf
k}^{\prime}}+q\cos\theta^{\prime\prime}_{\bf
q}-i(d_{0}/\hbar^{2}\alpha)}\right]\ ,$ (25a) $\displaystyle
r(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime};d_{0})$
$\displaystyle=-\left[\frac{s\Gamma_{s}(k)}{s^{\prime}\Gamma_{s^{\prime}}(k^{\prime})}\right]\,\texttt{e}^{i(\theta_{\bf
k}+\theta_{{\bf k}^{\prime}})}\,\left[\frac{k\cos\theta_{\bf
k}-q\cos\theta^{\prime\prime}_{\bf
q}+i(d_{0}/\hbar^{2}\alpha)}{k^{\prime}\cos\theta_{{\bf
k}^{\prime}}+q\cos\theta^{\prime\prime}_{\bf
q}-i(d_{0}/\hbar^{2}\alpha)}\right]\ .$ (25b)

In this case, however, we find $r\neq 0$ and $|t|<1$ even for $V_{0}=0$,
$q=k^{\prime}=k$ and $\theta_{\bf k}=\theta_{{\bf
k}^{\prime}}=\theta^{\prime\prime}_{\bf q}$ due to the presence of a
$\delta$-function potential ($d_{0}\neq 0$).

From the physics perspective, for non-magnetic materials, the electrical
control of inter-spin-channel tunneling by Rashba interaction demonstrates a
very interesting and unique mechanism and becomes ideal for applications in
spintronics. It is believed that such a phenomenon depends sensitively on the
selections of branch indexes $s$ and $s^{\prime\prime}$, incident kinetic
energy $E_{K}$, incident angle $\theta_{\bf k}$, inverse effective mass
$\alpha$, Zeeman gap $\Delta_{Z}$, barrier height $V_{0}$, Rashba spin-orbit
parameter $\alpha_{R}$ and even the amplitude $d_{0}$ of a $\delta$-function
potential.

More interestingly, for a specifically selected incident kinetic energy
$E_{K}$ of an electron and its corresponding $\\{s,k\\}$ electronic state, two
relevant $\\{s^{\prime\prime},\mbox{\boldmath$q$}_{1,2}\\}$ states of the
lower branch $s^{\prime\prime}=-1$ (or the lower and upper branches
$s^{\prime\prime}=\pm 1$) exist within the barrier region as $V_{0}\neq 0$,
e.g., $s=-s^{\prime\prime}$ in Fig. 1D. Therefore, for a given incident angle
$\theta_{\bf k}$ or $k_{y}$ and a fixed $k$ by $E_{K}$ as well, two equivalent
electron tunneling processes can coexist simultaneously in the barrier region
with different diffraction angles $\theta^{\prime\prime}_{{\bf q}_{1}}$ and
$\theta^{\prime\prime}_{{\bf q}_{2}}$. Consequently, these diffracted electron
beams with different exit angles are expected to interference with each other,
similar to a double-slit experiment for a normally-incident light wave. Here,
the spatial distribution of a total probability density $\left|\Psi_{\rm
tot}^{\rm(t)}(\mbox{\boldmath$r$})\right|^{2}$ of electrons on the step-
barrier side takes the form

$\displaystyle{\cal A}\left|\Psi_{\rm
tot}^{\rm(t)}(\mbox{\boldmath$r$})\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2N_{T}}\,\left|t_{1}(k_{x},k^{\prime}_{x},q_{1x}\,|s,s^{\prime},s^{{}^{\prime\prime}}_{1})\,\frac{\texttt{e}^{iq_{1x}x}}{\sqrt{1+\gamma_{s_{1}^{\prime\prime}}^{2}(q_{1})}}\left[\begin{array}[]{c}-is_{1}^{\prime\prime}\gamma_{s_{1}^{\prime\prime}}(q_{1})\texttt{e}^{i\theta^{\prime\prime}_{{\bf
q}_{1}}}\\\ \\\ 1\end{array}\right]\right.$ (29) $\displaystyle+$
$\displaystyle\left.t_{2}(k_{x},k^{\prime}_{x},q_{2x}\,|s,s^{\prime},s^{{}^{\prime\prime}}_{2})\,\frac{\texttt{e}^{iq_{2x}x}}{\sqrt{1+\gamma_{s_{2}^{\prime\prime}}^{2}(q_{2})}}\left[\begin{array}[]{c}-is_{2}^{\prime\prime}\gamma_{s_{2}^{\prime\prime}}(q_{2})\texttt{e}^{i\theta^{\prime\prime}_{{\bf
q}_{2}}}\\\ \\\ 1\end{array}\right]\right|^{2}\ ,$ (33)

which includes the unique spin-state quantum interference between two
correlated coherent electron waves. Similar conclusion can be drawn for the
total reflection probability density $\left|\Psi_{\rm
tot}^{\rm(r)}(\mbox{\boldmath$r$})\right|^{2}$ of electrons outside the step
region, written as

$\displaystyle{\cal A}\left|\Psi_{\rm
tot}^{\rm(r)}(\mbox{\boldmath$r$})\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2N_{T}}\,\left|r_{1}(k_{x},k^{\prime}_{1x},q_{x}\,|s,s_{1}^{\prime},s^{{}^{\prime\prime}})\,\frac{\texttt{e}^{-ik^{\prime}_{1x}x}}{\sqrt{1+\gamma_{s_{1}^{\prime}}^{2}(k^{\prime}_{1})}}\left[\begin{array}[]{c}-is_{1}^{\prime}\gamma_{s_{1}^{\prime}}(k^{\prime}_{1})\texttt{e}^{i(\pi-\theta_{{\bf
k}^{\prime}_{1}})}\\\ \\\ 1\end{array}\right]\right.$ (37) $\displaystyle+$
$\displaystyle\left.r_{2}(k_{x},k^{\prime}_{2x},q_{x}\,|s,s_{2}^{\prime},s^{{}^{\prime\prime}})\,\frac{\texttt{e}^{-ik^{\prime}_{2x}x}}{\sqrt{1+\gamma_{s_{2}^{\prime}}^{2}(k^{\prime}_{2})]}}\left[\begin{array}[]{c}-is_{2}^{\prime}\gamma_{s_{2}^{\prime}}(k^{\prime}_{2})\texttt{e}^{i(\pi-\theta_{{\bf
k}^{\prime}_{2}})}\\\ \\\ 1\end{array}\right]\right|^{2}\ .$ (41)

It is interesting to note from Eqs. (33) and (41) that the degree of such
spin-state quantum interference can be controlled electrically by their
tunneling amplitudes $t_{1,2}$ and diffraction angles
$\theta^{\prime\prime}_{{\bf q}_{1,2}}$ or reflection amplitudes $r_{1,2}$ and
angles $\theta_{{\bf k}^{\prime}_{1,2}}$, respectively. Such a unique property
is expected acquiring a lot of implications for spintronics and quantum-
computation devices.

##  IV Numerical Results and Discussion

Table 1: Parameters used for calculations Parmeter param1 ; param2  | Description | Value | Units
---|---|---|---
$k_{R}^{c}$ | Rashba wave number | 0.015 | Å-1
$k_{R}^{v}$ | Rashba wave number | 0.015 | Å-1
$E^{c}_{R}$ | Rashba energy | 118 | meV
$E^{v}_{R}$ | Rashba energy | 235 | meV
$2\Delta_{Z}$ | bandgap |  $50$ param2  | meV
$d_{0}$ | amplitude | 2 | $2E_{R}^{c}/k^{c}_{R}$

The parameters used in numerical computations are listed in Table 1. Other
parameters, such as $V_{0}$, $E_{K}$, $\theta_{\bf k}$ and $d_{0}$, will be
given directly in figures. In Sec. IV, we start with discussing various cases
without a step-potential barrier (i.e. $V_{0}\to 0$) in Sec. IV.1, from which
we demonstrate the spin-procession dynamics for both forward and backward
motions of a spin-polarized electron due to Rashba-Zeeman coupling and spin-
mixing effects in the system. In Sec. IV.2, we further study the dynamics for
tunneling and reflection of a spin-polarized incident electron under a step-
potential barrier in the system, which exhibits spin-state interference in
either reflected or transmitted pair of mixed-spin-state electrons with
different reflection or diffraction angles.

###  IV.1 Spin-Procession Dynamics For $V_{0}\to 0$

For numerical computations, we first divide the incident energy range of
$E_{K}$ into three sub-ranges, (I), (II) and (III) for energy dispersion
$E_{s}(k)$, as illustrated in Fig. 1. In range I, we have $0\leq
E_{K}\leq|E_{\rm min}|-\Delta_{Z}$ below the lower edge of the Zeeman gap,
where $|E_{\rm min}|=E_{R}(1+\Delta_{Z}^{2}/E_{R}^{2})$. In range II, on the
other hand, we have $|E_{\rm min}|-\Delta_{Z}\leq E_{K}\leq|E_{\rm
min}|+\Delta_{Z}$ within the Zeeman gap. Finally, in range III, we have
$E_{K}\geq|E_{\rm min}|+\Delta_{Z}$ above the upper edge of the Zeeman gap.
Next, for each range of $E_{K}$, we select different values for the step-
barrier height $V_{0}$ within the interval of $0\leq V_{0}\leq E_{K}$.
Moreover, for given values of $E_{K}$ and $V_{0}$, we choose different
incident angles within $-\pi/2\leq\theta_{\bf k}\leq\pi/2$, corresponding to
various values of $k_{y}$. Furthermore, for fixed $E_{K}$ and $V_{0}$ values,
there always exist either one or two wave-number intersection points with two
Rashba-Zeeman split energy bands in the incident and barrier regions,
respectively. This gives rise to values of $q_{\pm}$ for tunneling in the
barrier side while $k_{\pm}$ and $k^{\prime}_{\pm}$ for incidence and
reflection in the incident side, as calculated by Eqs. (16a), (16b) and (16c).

Table 2: Configurations for $E_{K}=50\,$meV within range-I and $V_{0}=0$ Case Number | Incidence | Reflection | Transmission | Tunneling Status
---|---|---|---|---
$1$ | $a$ | $a$ | $a^{\prime}$ | $\checkmark$
$2$ | $a$ | $a$ | $b^{\prime}$ | $\checkmark$
$3$ | $a$ | $b$ | $a^{\prime}$ | $\checkmark$
$4$ | $b$ | $a$ | $a^{\prime}$ | $\times$
$5$ | $a$ | $b$ | $b^{\prime}$ | $\times$
$6$ | $b$ | $a$ | $b^{\prime}$ | $\checkmark$
$7$ | $b$ | $b$ | $a^{\prime}$ | $\checkmark$
$8$ | $b$ | $b$ | $b^{\prime}$ | $\checkmark$
Figure 2: (a) Calculated split Rashiba-Zeeman energy bands as a function of
scaled electron wave number k/$k_{R}$, where the incident energy $E_{K}$ is
indicated by a black-dashed line. Symbols $a$, $a^{\prime}$, $b$, $b^{\prime}$
denote four wave-number intersection points for incident energy $E_{K}$ = 50
meV of electrons within range-I and $V_{0}$ = 0 or in situation-A of Fig. 1.
(b-c) Polar plots of partial tunneling coefficients
${\mathcal{T}}_{s\rightarrow s^{\prime\prime}}$ as a function of incident
angle $\theta_{k}$ for fixed $E_{K}$ = 50 meV and $V_{0}$ = 0 in NT = 8
different cases listed in Table 2.

Our numerically computed results are displayed in Fig. 2 for all $N_{\rm T}=8$
cases listed in Table 2. As seen in Table 2, for Case-4 we have $k=k_{+}$,
$k^{\prime}=q=k_{-}\ll k$, and therefore $\theta_{\bf k}\ll\theta_{{\bf
k}^{\prime}}=\theta^{\prime\prime}_{\bf q}$ since
$k_{y}=k_{y}^{\prime}=q_{y}$. As a result, we know from Eq. (14a) that
$|t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})|/\sqrt{N_{\rm
T}}\gg 1$ in Case-4, which implies that such a forward partial spin procession
should be excluded. On the other hand, for Case-5 we have $k=k_{-}$,
$k^{\prime}=q=k_{+}\gg k$, leading to $\theta_{\bf k}\gg\theta_{{\bf
k}^{\prime}}=\theta^{\prime\prime}_{\bf q}$ due to
$k_{y}=k_{y}^{\prime}=q_{y}$. Consequently, we find ${\cal T}_{s\to
s^{\prime\prime}}/N_{\rm T}\gg 1$ from Eq. (15a), which indicates that such a
forward partial spin procession should also be ruled out.

For Case-1 in Fig. 2, we have a simple situation with $k=k^{\prime}=q$ or
$\theta_{\bf k}=\theta_{{\bf k}^{\prime}}=\theta^{\prime\prime}_{\bf q}$. In
this case, it is clear from Eqs. (14a) and (15a) that ${\cal T}_{s\to
s^{\prime\prime}}\equiv 1$ but with no spin procession involved. Similar
conclusion can be drawn for Cases-3 & 8 without having a spin procession. For
Case-6, although we have ${\cal T}_{s\to s^{\prime\prime}}\equiv 1$ with no
spin procession, it is restricted by $\theta_{\bf
k}<\sin^{-1}(k_{-}/k_{+})\equiv\theta_{\rm c}$ due to the constraint
$k_{x}^{\prime 2}=k_{-}^{2}-k_{y}^{2}=k_{-}^{2}-k_{+}^{2}\sin^{2}\theta_{\bf
k}\geq 0$ for the incident angle $\theta_{\bf k}$, and then, no forward motion
of electrons is expected in this case if $\theta_{\bf k}>\theta_{\rm c}$.
Finally, for Case-2 we acquire $k=k^{\prime}=k_{-}$ but $q=k_{+}\gg k$. This
leads to $\theta_{\bf k}=\theta_{{\bf
k}^{\prime}}\gg\theta^{\prime\prime}_{\bf q}$. Consequently, from Eq. (14a) we
arrive at a conclusion that
$|t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})|<1$, or a
forward partial spin procession should occur without a critical angle
$\theta_{\rm c}$ in this case. In a similar Case-7, we get
$k=k^{\prime}=k_{+}$ while $q=k_{-}\ll k$, and in this case we still have
${\cal T}_{s\to s^{\prime\prime}}<1$ from Eq. (15a) for a forward partial spin
procession but with an angle restriction due to $\theta^{\prime\prime}_{\bf
q}\gg\theta_{\bf k}=\theta_{{\bf k}^{\prime}}$.

Figure 3: (a) Calculated split Rashiba-Zeeman energy bands as a function of
scaled electron wave number k/$k_{R}$, where the incident energy $E_{K}$ is
indicated by a black-dashed line. Symbols $c$, $c^{\prime}$ denote two wave-
number intersection points for incident energy $E_{K}$ = 130 meV of electrons
within range-II and $V_{0}$ = 0 or in situation-A of Fig. 1. (b) Polar plots
of partial tunneling coefficients ${\mathcal{T}}_{s\rightarrow
s^{\prime\prime}}$ as a function of incident angle $\theta_{k}$ for fixed
$E_{K}$ = 130 meV and $V_{0}$ = 0 in a single (NT = 1) case.

Figure 3 displays numerical result for a single case with $E_{K}$ fallen into
an energy gap within range II. Here, with $k=k^{\prime}=q=k_{+}$, we always
acquire ${\cal T}_{s\to s^{\prime\prime}}=1$ for all incident angles
$\theta_{\bf k}$. Clearly, no spin procession is involved in this case for
both forward and backward motions of an incident electron.

Table 3: Configurations for $E_{K}=200\,$meV in range-III and $V_{0}=0$ Case Number | Incidence | Reflection | Transmission | Tunneling Status
---|---|---|---|---
$1$ | $d$ | $d$ | $d^{\prime}$ | $\checkmark$
$2$ | $d$ | $d$ | $e^{\prime}$ | $\checkmark$
$3$ | $d$ | $e$ | $d^{\prime}$ | $\checkmark$
$4$ | $e$ | $d$ | $d^{\prime}$ | $\times$
$5$ | $d$ | $e$ | $e^{\prime}$ | $\times$
$6$ | $e$ | $d$ | $e^{\prime}$ | $\checkmark$
$7$ | $e$ | $e$ | $d^{\prime}$ | $\checkmark$
$8$ | $e$ | $e$ | $e^{\prime}$ | $\checkmark$
Figure 4: (a) Calculated split Rashiba-Zeeman energy bands as a function of
scaled electron wave number k/$k_{R}$, where the incident energy $E_{K}$ is
indicated by a black-dashed line. Symbols $d$, $d^{\prime}$, $e$, $e^{\prime}$
denote four wave-number intersection points for incident energy $E_{K}$ = 200
meV of electrons within range-III and $V_{0}$ = 0 or in situation-A of Fig. 1.
(b-c) Polar plots of partial tunneling coefficients
${\mathcal{T}}_{s\rightarrow s^{\prime\prime}}$ as a function of incident
angle $\theta_{k}$ for fixed $E_{K}$ = 200 meV and $V_{0}$ = 0 in NT = 8
different cases listed in Table 3.

Compared with the results in Fig. 2, for an incident energy $E_{K}$ sitting
within range III in Fig. 4 as well as for all $N_{\rm T}=8$ cases listed in
Table 3, we find that quite similar analysis could be performed. From Fig. 4
and Table 3, we further know that Cases-3 & 6 are associated with no spin
procession while Cases-2 & 7 relate to a forward partial spin procession,
quite similar to those displayed in Fig. 2.

Discussions on Figs. 2-4 are only limited to a forward partial spin-procession
process. For a backward partial spin-procession process, similar analysis can
also be performed in the same way.

###  IV.2 Tunneling-Dynamics For $V_{0}>0$

In previous Sec. IV.1, we have explored both forward and backward spin-
procession physics for a spin-polarized incident electron in the absence of a
potential barrier ($V_{0}\to 0$). Now, we would consider dynamics for intra-
spin-channel (without a spin procession) and inter-spin-channel (with a spin
procession) tunnelings of a spin-polarized incident electron in the presence
of a step-potential barrier ($V_{0}>0$) in the system.

Table 4: Configurations for $E_{K}=130\,$meV in range-II and $V_{0}=50\,$meV Case Number | Incidence | Reflection | Transmission | Tunneling Status
---|---|---|---|---
$1$ | $c$ | $c$ | $a^{\prime}$ | $\checkmark$
$2$ | $c$ | $c$ | $b^{\prime}$ | $\checkmark$
Figure 5: (a) Calculated split Rashiba-Zeeman energy bands as a function of
scaled electron wave number k/$k_{R}$, where the incident energy $E_{K}$ is
indicated by a black-dashed line while dispersion curves on the incident and
barrier sides are represented by blue and red curves. Symbols $c$,
$a^{\prime}$, $b^{\prime}$ denote three wave-number intersection points for
incident energy $E_{K}$ = 130 meV of electrons within range-II and $V_{0}$ =
50 meV or in situation-B of Fig. 1. (b) Polar plots of partial tunneling
coefficients ${\mathcal{T}}_{s\rightarrow s^{\prime\prime}}$ as a function of
incident angle $\theta_{k}$ for fixed $E_{K}$ = 130 meV and $V_{0}$ = 50 meV
in NT = 2 different cases listed in Table 4.

For $V_{0}>0$ in Fig. 5, we take a step-potential barrier height
$V_{0}=50\,$meV and a kinetic energy $E_{K}=130\,$meV for spin-polarized
incident electrons within the energy range II. From Fig. 5 and Table 4, we
realize that both inter-spin-channel ($c\to a^{\prime}$ with spin procession)
and intra-spin-channel ($c\to b^{\prime}$ without spin procession) partial
tunnelings of electrons appear in Cases-1 & 2, respectively. However, no spin-
procession process occurs for the reflection of electrons in these two cases.
Furthermore, due to $k=k^{\prime}\geq q$ and $\theta_{\bf k}=\theta_{{\bf
k}^{\prime}}\leq\theta^{\prime\prime}_{\bf q}$, a resonant partial intra-spin-
channel tunneling in Case-2 suffers from an angle restriction beyond which
tunneling coefficients ${\cal T}_{s\to s^{\prime\prime}}$ drops to zero.
Meanwhile, for Case-1, on the other hand, we have $k=k^{\prime}\gg q$ and
$\theta_{\bf k}=\theta_{{\bf k}^{\prime}}\ll\theta^{\prime\prime}_{\bf q}$,
and then, only a relatively small and non-resonant partial inter-spin-channel
tunneling with ${\cal T}_{s\to s^{\prime\prime}}<1$ is seen very close to
$\theta_{\bf k}=0$, which is quite different from the observation of ${\cal
T}_{s\to s^{\prime\prime}}\equiv 1$ for all $\theta_{\bf k}$ presented in Fig.
3 when $V_{0}=0$.

Table 5: Configurations for $E_{K}=180\,$meV in range-III and $V_{0}=50\,$meV Case Number | Incidence | Reflection | Transmission | Tunneling Status
---|---|---|---|---
$1$ | $d$ | $d$ | $c^{\prime}$ | $\checkmark$
$2$ | $d$ | $e$ | $c^{\prime}$ | $\times$
$3$ | $e$ | $d$ | $c^{\prime}$ | $\checkmark$
$4$ | $e$ | $e$ | $c^{\prime}$ | $\checkmark$
Figure 6: (a) Calculated split Rashiba-Zeeman energy bands as a function of
scaled electron wave number k/$k_{R}$, where the incident energy $E_{K}$ is
indicated by a black-dashed line. Symbols $d$, $e$, $c^{\prime}$ denote three
wave-number intersection points for incident energy $E_{K}$ = 180 meV of
electrons within range-III and $V_{0}$ = 50 meV or in situation-B of Fig. 1.
(b) Polar plots of partial tunneling coefficients ${\mathcal{T}}_{s\rightarrow
s^{\prime\prime}}$ as a function of incident angle $\theta_{k}$ for fixed
$E_{K}$ = 180 meV and $V_{0}$ = 50 meV in NT = 4 different cases listed in
Table 5.

For the same barrier height $V_{0}=50\,$meV, numerical results are presented
in Fig. 6 as we increase the incident energy $E_{K}=180\,$meV from range II to
range III. As found from Table 5, for Case-1 we have $k=k^{\prime}\ll q$, and
$\theta^{\prime\prime}_{\bf q}\ll\theta_{\bf k}=\theta_{{\bf k}^{\prime}}$ as
well due to $k_{y}=k_{y}^{\prime}=q_{y}$. Then, we know from Eq. (14a) that
$|t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})|<1$ for
partial inter-spin-channel tunneling of electrons. For Case-2, on the other
hand, we get $k\ll q<k^{\prime}$ as well as $\theta_{\bf
k}\gg\theta^{\prime\prime}_{\bf q}>\theta_{{\bf k}^{\prime}}$. In such a case,
we find ${\cal T}_{s\to s^{\prime\prime}}/N_{\rm T}>1$ from Eq. (15a), and
therefore, we expect that this partial inter-spin-channel tunneling of
electrons should be excluded. Finally, for partial intra-spin-channel
tunneling of electrons in Cases-3 & 4, we have $k>q\gg k^{\prime}$ and
$\theta_{\bf k}<\theta^{\prime\prime}_{\bf q}\ll\theta_{{\bf k}^{\prime}}$ as
well as $k=k^{\prime}>q$ and $\theta_{\bf k}=\theta_{{\bf
k}^{\prime}}<\theta^{\prime\prime}_{\bf q}$. As a result, by using Eqs. (14a)
and (15a), we reach the conclusions that
$|t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})|/\sqrt{N_{\rm
T}}>1$ is disallowed while ${\cal T}_{s\to s^{\prime\prime}}\leq 1$ is allowed
for Cases-3 & 4, separately, where both Case-3 and Case-4 are further limited
by a critical angle $\theta_{\rm c}$ beyond which ${\cal T}_{s\to
s^{\prime\prime}}$ reduces to zero.

Table 6: Configurations for $E_{K}=180\,$meV in range-III and $V_{0}=100\,$meV Case Number | Incidence | Reflection | Transmission | Tunneling Status
---|---|---|---|---
$1$ | $d$ | $d$ | $a^{\prime}$ | $\checkmark$
$2$ | $d$ | $e$ | $a^{\prime}$ | $\checkmark$
$3$ | $e$ | $d$ | $a^{\prime}$ | $\times$
$4$ | $e$ | $e$ | $a^{\prime}$ | $\checkmark$
$5$ | $d$ | $d$ | $b^{\prime}$ | $\checkmark$
$6$ | $d$ | $e$ | $b^{\prime}$ | $\times$
$7$ | $e$ | $d$ | $b^{\prime}$ | $\checkmark$
$8$ | $e$ | $e$ | $b^{\prime}$ | $\checkmark$
Figure 7: (a) Calculated split Rashiba-Zeeman energy bands as a function of
scaled electron wave number k/$k_{R}$, where the incident energy $E_{K}$ is
indicated by a black-dashed line. Symbols $d$, $e$, $a^{\prime}$, $b^{\prime}$
denote four wave-number intersection points for incident energy $E_{K}$ = 180
meV and $V_{0}$ = 100 meV or in situation-D of Fig. 1. (b-c) Polar plots of
partial tunneling coefficients ${\mathcal{T}}_{s\rightarrow s^{\prime\prime}}$
as a function of incident angle $\theta_{k}$ for fixed $E_{K}$ = 180 meV and
$V_{0}$ = 100 meV in NT = 8 different cases listed in Table 6.

As the last situation considered in Sec. IV.2, we would keep the incident
kinetic energy $E_{K}=180\,$meV in range III while lift the barrier height
$V_{0}$ from $50\,$meV to $100\,$meV, as seen in Fig. 7. In this way, the
kinetic energy of electrons on the barrier side moves down from range II in
Fig. 6 into range I in Fig. 7. Compared with Fig. 6 and Table 5 with $N_{\rm
T}=4$, we have found that the increase of $V_{0}$ in Fig. 7 has doubled the
configuration space with $N_{\rm T}=8$ as listed in Table 6. Here, we have
$k\gg q\sim k^{\prime}$ or $\theta_{\bf k}\ll\theta^{\prime\prime}_{\bf
q}\sim\theta_{{\bf k}^{\prime}}$ for Case-3, which leads to an excluded
partial inter-spin-channel tunneling with
$|t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})|/\sqrt{N_{\rm
T}}\gg 1$ according to Eq. (14a) and is further accompanied by a critical
angle beyond which ${\cal T}_{s\to s^{\prime\prime}}$ drops to zero.
Similarly, for Case-6, we find $k\ll q<k^{\prime}$ or $\theta_{\bf
k}\gg\theta^{\prime\prime}_{\bf q}>\theta_{{\bf k}^{\prime}}$, and then we
arrive at ${\cal T}_{s\to s^{\prime\prime}}/N_{\rm T}\gg 1$ by using Eq. (15a)
for an excluded partial inter-spin-channel tunneling. On the other hand, we
get $k=k^{\prime}>q$ or $\theta_{\bf k}=\theta_{{\bf
k}^{\prime}}<\theta^{\prime\prime}_{\bf q}$ for Case-8. Therefore, from Eq.
(15a) we know the partial intra-spin-channel tunneling ${\cal T}_{s\to
s^{\prime\prime}}\leq 1$ but it is still limited by a critical angle
$\theta_{\rm c}$ for this case. A similar conclusion can be drawn for Case-1,
where $k\sim q\sim k^{\prime}$ or $\theta_{\bf
k}\sim\theta^{\prime\prime}_{\bf q}\sim\theta_{{\bf k}^{\prime}}$ which gives
rise to the partial intra-spin-channel
$|t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})|>1$ based on
Eq. (14a) but it is still allowed physically since
$|t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})|/\sqrt{N_{\rm
T}}<1$. However, ${\cal T}_{s\to s^{\prime\prime}}$ in Case-1 reduces to zero
once $\theta_{\bf k}$ becomes bigger than the critical angle $\theta_{\rm c}$
.

For Cases-2 & 5 in Fig. 7, we acquire $k^{\prime}\gg q\sim k$ or $\theta_{{\bf
k}^{\prime}}\ll\theta^{\prime\prime}_{\bf q}\sim\theta_{\bf k}$ and
$k=k^{\prime}\ll q$ or $\theta_{\bf k}=\theta_{{\bf
k}^{\prime}}\gg\theta^{\prime\prime}_{\bf q}$, respectively. As a result, we
expect partial intra-spin-channel tunneling coefficient ${\cal T}_{s\to
s^{\prime\prime}}<1$ from Eq. (15a) for Case-3 while partial inter-spin-
channel tunneling coefficient
$|t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})|\ll 1$ from
Eq. (14a) for Case-5. For these two considered cases, angle restriction only
applies to Case-2 but not to Case-5. Finally, for Cases-4 & 7, we have
$k=k^{\prime}\gg q$ and $k^{\prime}\ll q<k$ separately, which corresponds to
$\theta_{\bf k}=\theta_{{\bf k}^{\prime}}\ll\theta^{\prime\prime}_{\bf q}$ and
$\theta_{\bf k}<\theta^{\prime\prime}_{\bf q}\ll\theta_{{\bf k}^{\prime}}$.
Then, for the former we get partial inter-spin-channel tunneling coefficient
${\cal T}_{s\to s^{\prime\prime}}\ll 1$ from Eq.(15a), while we find partial
intra-spin-channel tunneling coefficient ${\cal T}_{s\to s^{\prime\prime}}>1$
from the same Eq.(15a) for the latter case, where physical condition ${\cal
T}_{s\to s^{\prime\prime}}/N_{\rm T}<1$ is satisfied. Moreover, an angle
restriction $\theta_{\bf k}\leq\theta_{\rm c}$ has been further added to
Case-7.

Discussions on Figs. 5-7 are only limited to intra- and inter-spin-channel
tunnelings of electrons. For cases of intra- and inter-spin-channel
reflections of electrons, an analysis can be done in a similar way.

###  IV.3 Tunneling Energy Spectra and Effect of Interface $\delta$-Function

Figure 8: (a) Calculated partial tunneling coefficients
${\mathcal{T}}_{s\rightarrow s^{\prime\prime}}$ as a function of the kinetic
energy $E_{K}$ of incident electrons for $\theta_{k}$ = 0 and cases 3, 4, 5, 8
selected from Table VI. (b-c) Polar plots of partial tunneling coefficients
${\mathcal{T}}_{s\rightarrow s^{\prime\prime}}$, taken from Figs. 7(b) and
7(c) as a reference here, for fixed $E_{K}$ = 50 meV and $V_{0}$ = 100 meV in
all cases listed in Table 6.

In Sec. IV.2, we only deal with the incidence-angle dependence of partial
intra- and inter-spin-channel tunneling coefficients ${\cal T}_{s\to
s^{\prime\prime}}$. Here, we present in Fig. 8 the dependence of ${\cal
T}_{s\to s^{\prime\prime}}$ on the kinetic energy $E_{K}$ of incident spin-
polarized electrons at $\theta_{\bf k}=0$ and $V_{0}=50\,$meV. For specific,
we select Cases-3, 4, 5, 8 as listed in Table 6 for comparisons and
discussions.

For fixed $V_{0}=100\,$meV and $\theta_{\bf k}=0$ (or $k_{y}=0$), Cases-4 and
8 in Fig. 8 display a first threshold $E^{(1)}_{\rm T}=V_{0}=100\,$meV for
tunneling coefficients ${\cal T}_{s\to s^{\prime\prime}}$, below which ${\cal
T}_{s\to s^{\prime\prime}}$ remains to be zero due to lack of available states
for tunneling. Moreover, Cases-2 and 5 present a second threshold
$E^{(2)}_{\rm T}=2\Delta_{Z}-E_{\rm min}$ for tunneling coefficients ${\cal
T}_{s\to s^{\prime\prime}}$ since $2\Delta_{Z}-E_{\rm min}\sim
150\,\mbox{meV}>V_{0}$. As $E_{K}>E^{(1)}_{\rm T}$, ${\cal T}_{s\to
s^{\prime\prime}}$ quickly approach unity for Case-8, while ${\cal T}_{s\to
s^{\prime\prime}}$ for Case-4 further experiences a tunneling-forbidden region
for $E_{K}$ within which ${\cal T}_{s\to s^{\prime\prime}}$ is suppressed to
zero. The presence of such a tunneling-forbidden region with respect to
incident energy $E_{K}$ is related to the fact that $E_{K}$ enters into a
Zeeman gap on the barrier side and then the tunneling-ending
$a^{\prime}$-state, illustrated in Fig. 7, becomes inaccessible. However, such
a tunneling-ending $a^{\prime}$-state appears once again as $E_{K}$ becomes
above the Zeeman gap. As a result, the same tunneling-forbidden region is
expected to show up for Case-2 based on the same reason. Finally, when $E_{K}$
goes beyond this tunneling-forbidden region, ${\cal T}_{s\to
s^{\prime\prime}}$ for all Cases-2, 4, 5, 8 will gradually increase with
$E_{K}$ until a unity value is reached.

Figure 9: Comparison of two calculated polar plots of partial tunneling
coefficients ${\mathcal{T}}_{s\rightarrow s^{\prime\prime}}$ as seen in Figs.
2(b) and 2(c) with $d_{0}$ = 0 (Left) and with $d_{0}>$ 0 (Right),
respectively, for fixed $E_{K}$ = 50 meV and $V_{0}$ = 0, where all $N_{T}$ =
8 cases considered here are listed in Table 2.

Now, we turn to studying the effect of an interface $\delta$-function
($d_{0}>0$) on spin-procession processes in two different situations with
either $V_{0}=0$ or $V_{0}>0$. From Eqs. (25a) and (25b), we know that one
always obtain $|r|\neq 0$ and $|t|<1$ even in the case of $V_{0}=0$ if
$d_{0}\neq 0$. By directly comparing two situations with ($d_{0}>0$) and
without ($d_{0}=0$) an interface $\delta$-function in Fig. 9 and Fig. 2, we
find that previous Cases-4 and 5 in Fig. 2 with no spin-procession processes
change to opposite situations with weak but finite forward spin-procession
processes between $b\to a^{\prime}$ and $a\to b^{\prime}$ states due to
introduced discontinuity in their wave-function derivatives by a finite
$d_{0}$ value, as can be verified from Eqs. (24a) and (24b). On the other
hand, previous three Cases-1, 3, 8 without spin-procession process in Fig. 2
are greatly weakened in Fig. 9 for the first two but only slightly decreased
for the last one. Meanwhile, the angle-restricted partial forward spin-
procession process in Case-6 of Fig. 2 has been fully suppressed in Fig. 9.
Finally, for the rest Cases-2 and 7 with partial spin-procession processes in
Fig. 2, the existence of an interface $\delta$-function in Fig. 9 seems to
have no influence on situation in Case-2 but significantly broadens the
critical angle $\theta_{\rm c}$ for Case-7.

Figure 10: Comparison of two calculated polar plots of partial tunneling
coefficients ${\mathcal{T}}_{s\rightarrow s^{\prime\prime}}$ as seen in Figs.
7(b) and 7(c) with $d_{0}$ = 0 (Left) and with $d_{0}>$ 0 (Right),
respectively, for fixed $E_{K}$ = 180 meV and $V_{0}$ = 100 meV, where all
$N_{T}$ = 8 cases considered here are listed in Table 6.

In order to reveal variation in intra- and inter-spin-channel tunnelings of
electrons in Fig. 7 under an interface $\delta$-function, we compare
calculated partial tunneling coefficients ${\cal T}_{s\to s^{\prime\prime}}$
with $E_{K}=180\,$meV and $V_{0}=100\,$meV in Fig. 10 for all listed $N_{\rm
T}=8$ cases in Table 6.

Under an interface $\delta$-function, the previous excluded partial inter-
spin-channel tunneling coefficients ${\cal T}_{s\to s^{\prime\prime}}/N_{\rm
T}\gg 1$ in Case-6, as well as
$|t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime})|/\sqrt{N_{\rm
T}}\gg 1$ in Case-3, of Fig. 7 switch to two weakened allowed ones in Fig. 10
with and without an angle restriction, respectively. On the contrary, an
excluded partial inter-spin-channel tunneling coefficient ${\cal T}_{s\to
s^{\prime\prime}}/N_{\rm T}\gg 1$ in Case-5 of Fig. 10 corresponds to an
allowed one in Fig. 7 with no angle restriction. Additionally, previous full
angle-restricted partial intra-spin-channel tunneling coefficients ${\cal
T}_{s\to s^{\prime\prime}}$ for Cases-8 & 1 of Fig. 7 decrease greatly in Fig.
10 free with an angle restriction. Furthermore, under the same interface
$\delta$-function, the angle-restricted partial intra-spin-channel tunneling
coefficient ${\cal T}_{s\to s^{\prime\prime}}$ for Case-2 of Fig. 7 changes
into a quite extraordinary one acquiring ${\cal T}_{s\to s^{\prime\prime}}\neq
0$ in Fig. 10 even at $\theta_{\bf k}=\pm\pi/2$. Finally, for Cases-4 & 7, the
sharply angle-restricted partial inter-spin-channel tunneling coefficient
${\cal T}_{s\to s^{\prime\prime}}$ for the former in Fig. 7 remains largely
unchanged in Fig. 10, however the angle-restricted intra-spin-channel
tunneling coefficient ${\cal T}_{s\to s^{\prime\prime}}$ for the latter in
Fig. 7 is reduced significantly in Fig. 10.

###  IV.4 Quantum Interference of Correlated-Pair of Spin States of Electrons

In Sec. IV.4, we would utilize the calculated transmission amplitude
$t_{1,2}(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s_{1,2}^{\prime\prime};d_{0})$
and reflection amplitude
$r_{1,2}(k_{x},k^{\prime}_{x},q_{x}\,|s,s_{1,2}^{\prime},s^{\prime\prime};d_{0})$
from Eqs. (25a) and (25b), and then apply them to Eqs. (33) and (41) for
calculating the total tunneling probability density $|\Psi^{({\rm t})}_{\rm
tot}(\mbox{\boldmath$r$})|^{2}$ and the total reflection probability density
$|\Psi^{({\rm r})}_{\rm tot}(\mbox{\boldmath$r$})|^{2}$. In this way, we are
able to quantify the quantum-interference effects between pair of correlated
spin states of electrons on either tunneling or reflection side of an
interface in the system, respectively.

Table 7: Configuration for $E_{K}=180\,$meV in range-III and $V_{0}=100\,$meV Incidence | Transmission | Reflection-1 | Reflection-2
---|---|---|---
$d$ | $a^{\prime}$ | $d$ | $e$
Figure 11: Plots for total reflection-wavefunction density ${\cal
A}|\Psi^{({\rm r})}_{\rm tot}(\mbox{\boldmath$r$})|^{2}/N_{T}$ in Eq. (41) at
the interface $x=0$ as a function of incident angle $\theta_{\bf k}$ in $(a)$
as well as a function of position $x$ on the reflection side in $(b)$ for
three different incident angles $\theta_{\bf k}=0^{\rm o},\,30^{\rm
o},\,58^{\rm o}$. Here, we set $E_{K}$ = 180 meV, $V_{0}$ = 100 meV and
$N_{T}$ = 8 for the case listed in Table 7.

For this purpose, let us first study the case listed in Table 7, corresponding
to cases 1 & 2 in Table 6, for a pair of correlated spin states ($d$ & $e$) of
electrons on the reflection side under a single incident spin state $d$ of
electron, i.e., $d\to d$ (without a spin flip) and $d\to e$ (with a spin flip)
in the absence of an interface $\delta$-function or $d_{0}=0$.

Table 8: Configuration for $E_{K}=180\,$meV in range-III and $V_{0}=100\,$meV Incidence | Reflection | Transmission-1 | Transmission-2
---|---|---|---
$d$ | $d$ | $a^{\prime}$ | $b^{\prime}$

From results in Fig. 11, we know that whenever the incident angle $\theta_{\bf
k}$ becomes far away from its angular-distribution boundaries at $\pm 60^{\rm
o}$, the superposition of a pair of correlated electron spin states $d$ & $e$
on the reflection side presents no sign of quantum interference and also
becomes very small due to their similar amplitudes and nearly out-of-phase
cancellation between them, as can be easily verified from the right panel of
Fig. 11. However, as $\theta_{\bf k}=58^{\rm o}$, the quantum-interference
effect is greatly enhanced and gives rise to a periodically oscillating
function of the position $x$ on the reflection side away from the interface at
$x=0$ with a spatial period around $200\,$Å. Physically, such a quantum
interference results from the unique band structure associated with Rashba
spin-orbital coupling, and meanwhile provides us with a tool for controlling
reflection of an incident electron with a fixed kinetic energy $E_{K}$ and a
step-barrier height $V_{0}$ with or without a spin flip.

Next, we consider the case in Table 8, which is associated with cases 1 & 5 in
Table 6, for another pair of correlated spin states ($a^{\prime}$ &
$b^{\prime}$) of electrons on the tunneling side under a single incident spin
state $d$, i.e., $d\to a^{\prime}$ (without a spin flip) and $d\to b^{\prime}$
(with a spin flip) under $d_{0}=0$.

Figure 12: Plots for total tunneling-wavefunction density ${\cal
A}|\Psi^{({\rm t})}_{\rm tot}(\mbox{\boldmath$r$})|^{2}/N_{T}$ in Eq. (33) at
the interface $x=0$ as a function of incident angle $\theta_{\bf k}$ in $(a)$
as well as a function of position $x$ on the tunneling side in $(b)$ for three
different incident angles $\theta_{\bf k}=0^{\rm o},\,30^{\rm o},\,58^{\rm
o}$. Here, we set $E_{K}$ = 180 meV, $V_{0}$ = 100 meV and $N_{T}$ = 8 for the
case listed in Table 8.

From results presented in Fig. 12, we observe that the superposition of a pair
of correlated electron spin states $a^{\prime}$ & $b^{\prime}$ on the
tunneling side leads to a strong quantum interference between them and, in
contrast to those in Fig. 11, no cancellation between them are found, as seen
clearly from the right panel of Fig. 12. Specifically, for $\theta_{\bf
k}=0^{\rm o}$ and $30^{\rm o}$, the resulting interference patterns are
strong, similar to each other but with a relative phase sift between them, and
acquire an enlarged oscillation period around $300\,$Å. For $\theta_{\bf
k}=58^{\rm o}$, on the other hand, the spatially-oscillating feature is found
slightly weakened as a function of the position $x$ on the tunneling side away
from the interface at $x=0$ although the total amplitude is increased due to
nearly in-phase superposition of two correlated electron spin states
$a^{\prime}$ & $b^{\prime}$. Furthermore, we also find from the left panel of
Fig. 12 that a quantum-interference pattern within a limited angle region
$|\theta_{\bf k}|<60^{\rm o}$ occurs even at the interface $x=0$ due to
different diffraction angles $\theta^{\prime\prime}_{{\bf q}_{1,2}}$ for
electron tunneling as a function of incident angle $\theta_{\bf k}$, instead
of position $x$ dependence in the right panel of Fig. 12.

##  V Summary

In conclusion, we have established a theory for calculating the transmission
$\mbox{\boldmath$J$}^{({\rm
t})}(s^{\prime\prime},\mbox{\boldmath$q$};\,|t|^{2})$ and reflection
$\mbox{\boldmath$J$}^{({\rm
r})}(s^{\prime},\mbox{\boldmath$k$}^{\prime};\,|r|^{2})$ probability currents
of a charged particle across a steepness-enhanced potential step
$V_{B}(x)=V_{0}\Theta(x)+d_{0}\delta(x)$ within a quantum well in the presence
of a Rashba spin-orbit interaction. By using calculated energy eigenstate
$E_{s}(k)$ for Rashba-Zeeman coupled two-dimensional conduction electrons in
the presence of a spin-split gap $\Delta_{Z}$, both reflection and transmitted
probability currents are explicitly computed for electrons under the potential
step $V_{B}(x)$ within a quantum well. Moreover, by matching the spin-
dependent boundary conditions at the interface $x=0$ for a barrier step, both
reflection
$r(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime};d_{0})$ and
transmission
$t(k_{x},k^{\prime}_{x},q_{x}\,|s,s^{\prime},s^{\prime\prime};d_{0})$
coefficients are determined as functions of incident kinetic energy $E_{K}$,
selected incident-electron spin stets $s=\pm 1$ and angle of incidence
$\theta_{\bf k}$ for different step-barrier heights $V_{0}$ and steepness
$d_{0}$, as well as of various spin directions
$s^{\prime},\,s^{\prime\prime}=\pm 1$ for reflection and tunneling of
electrons, Zeeman gaps $\Delta_{Z}$, and Rashba parameter values $\alpha_{R}$.

In particular, our model system contains multiple spin channels for both
reflection and tunneling of an incident electron. For this situation, we have
introduced average transmission ${\cal T}_{\rm
av}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$})$ and
reflection ${\cal R}_{\rm
av}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$})$
coefficients by means of the total number $N_{T}$ of inequivalent cases for
reflection or tunneling so that ${\cal T}_{\rm
av}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$})+{\cal
R}_{\rm
av}(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},\mbox{\boldmath$q$})\equiv
1$ can be maintained. By varying $E_{K}$ and $s=\pm 1$ for incident electrons
or $V_{0}$ and $d_{0}$, different features in tunneling and reflection of
electrons have been demonstrated with the inclusion of inter-channel electron
tunneling and reflection processes. Importantly, these spin-dependent unique
features are further accompanied by spin-state quantum interference for either
reflected or transmitted two spin-correlated electrons with the same $E_{K}$
but unequal reflection $\theta_{{\bf k}^{\prime}_{1,2}}$ or diffraction
$\theta^{\prime\prime}_{{\bf q}_{1,2}}$ angles. The distinctive property
predicted in this paper is expected to acquire a lot of applications in both
non-magnetic spintronics and quantum-computation devices.

## Acknowledgments

P.-H. Shih would like to thank the Ministry of Science and Technology of
Taiwan for the support through Grant No. MOST 110-2636-M-006-002. G.G. would
like to acknowledge the support from the Air Force Research Laboratory (AFRL)
through Grant No. FA9453-21-1-0046. D.H. would like to thank the Air Force
Office of Scientific Research (AFOSR) for support.

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# Use large language models to promote equity

Emma Pierson1,2,14, Divya Shanmugam3,14, Rajiv Movva1,14, Jon Kleinberg4,14,
Monica Agrawal5, Mark Dredze6, Kadija Ferryman6, Judy Wawira Gichoya7,
Dan Jurafsky8, Pang Wei Koh9, Karen Levy4, Sendhil Mullainathan10,
Ziad Obermeyer11, Harini Suresh12, Keyon Vafa13

1Cornell Tech, New York, NY, USA. 2Weill Cornell Medical College, New York,
NY, USA. 3Massachusetts Institute of Technology, Cambridge, MA, USA. 4Cornell
University, Ithaca, NY, USA. 5Duke University, Durham, NC, USA. 6Johns Hopkins
University, Baltimore, MD, USA. 7Emory University, Atlanta, GA, USA. 8Stanford
University, Palo Alto, CA, USA. 9University of Washington, Seattle, WA, USA.
10University of Chicago, Chicago, IL, USA. 11University of California,
Berkeley, Berkeley, CA, USA. 12Brown University, Providence, RI, USA.
13Harvard University, Cambridge, MA, USA. 14Denotes co-first-authorship.

Advances in large language models (LLMs) have driven an explosion of interest
about their societal impacts. Much of the discourse around how they will
impact social equity has been cautionary or negative (Figure 1), focusing on
questions like “how might LLMs be biased and how would we mitigate those
biases?” This is a vital discussion: the ways in which AI generally, and LLMs
specifically, can entrench biases have been well-documented (?). But equally
vital, and much less discussed, is the more opportunity-focused counterpoint:
“what promising applications do LLMs enable that could _promote_ equity?” If
LLMs are to enable a more equitable world, it is not enough just to play
defense against their biases and failure modes. We must also go on offense,
applying them positively to equity-enhancing use cases to increase
opportunities for underserved groups and reduce societal discrimination. There
are many choices which determine the impact of AI, and a fundamental choice
very early in the pipeline is the _problems we choose to apply it to_. If we
focus only later in the pipeline — making LLMs marginally more fair as they
facilitate use cases which intrinsically entrench power — we will miss an
important opportunity to guide them to equitable impacts.

A long tradition of efforts to promote equitable AI shows it is essential for
both researchers and policymakers to take this broader view (?, ?), seeking
out equity-promoting use cases from the outset. Rapid recent improvements in
LLMs mean that the set of possible equity-promoting use cases involving
language or text data has expanded dramatically. Here, we highlight the
emerging potential of LLMs to promote equity by presenting four newly
possible, promising research directions, while keeping risks and cautionary
points in clear view. We then turn our focus to concrete implementation
strategies: how can we incentivize this type of research and ultimately effect
societal change? We draw on extensive conversations with researchers, real-
world practitioners, and policymakers who are excited by tremendous new
potential in this area. We focus on health as a concrete domain with many
high-impact applications of LLMs both broadly and, as we discuss in detail
below, to health equity specifically; however, all the applications we discuss
have analogues in non-health domains as well.

Figure 1: Many more LLM papers focus on equity-related harms than
opportunities (see SI for methodology). We discuss four important
opportunities for using LLMs to promote equity.

## Equity-promoting applications of LLMs

### LLMs can improve detection of bias.

Human biases contribute to inequity: in healthcare, for example, doctors
describe Black patients as “difficult” more often than white patients (?),
which can lead to a lower quality of care. LLMs can improve our ability to
detect human bias from text data. For example, previous research has found
evidence of biased and stigmatizing language in clinical notes (?) but has
often relied on relatively rudimentary linguistic analysis methods, like
searching for specific words. LLMs can more precisely characterize subtle
linguistic biases (e.g., in sentiment, tone, and stereotypes), enabling more
comprehensive characterization of clinical biases (?). (Importantly, LLMs
themselves have biases (?, ?); nonetheless, they remain a useful diagnostic
for human bias because they are much better at capturing linguistic subtleties
than past keyword search methods.) In addition to _identifying_ biases, LLMs
can also be used to suggest ways to rewrite clinical notes to _mitigate_
biases, preventing them from adversely affecting downstream care. Similar LLM-
based techniques could be used to accelerate the detection of biases in
medical textbooks, which often involves time-consuming manual annotation by
domain experts. The overall approach is to treat text data as an “artifact”
(?) created by a biased society which can be examined to understand and
mitigate biased practices. This echoes calls for researchers to “study up” —
that is, focus their analysis on the powerful actors, structures and
institutions which perpetuate disparities (?, ?). Analogous examples exist in
non-healthcare domains: in law, LLMs could be applied to detect bias in trial
transcripts, and in hiring, in letters of recommendation. All these
applications flow from the fact that human behavior is often documented by
rich text datasets — clinical notes, medical textbooks, trial transcripts, and
letters of recommendation — and LLMs can characterize, and suggest ways to
mitigate, biases in behavior detectable from text.

### LLMs can create structured datasets of equity-relevant information.

Structured datasets, which track a standardized set of fields in a consistent
way — for example, the race and health insurance provider of a patient — allow
us to quantify disparities and develop policies to mitigate them. However,
creating these datasets often requires substantial human effort to extract
necessary information from raw text data. Within healthcare, for example,
researchers manually extract measures of employment, housing, and insurance
from clinical notes to understand the impact of these factors on patient
outcomes. Activists scour news articles to create databases tracking violence
that disproportionately impacts marginalized populations—for example, police
shootings or gender-based violence. Journalists collate legislative texts to
produce structured databases of restrictions on women’s and reproductive
healthcare, or restrictions on gender-affirming surgeries. LLMs have
demonstrated impressive performance on extracting structured information from
unstructured datasets (?), and could thus be used in service of each of these
use cases. As with all applications we discuss here, it is essential for
researchers to center the communities they seek to support: for example, LLM
tools which track gender-based violence should be co-designed with the
activists who have long been doing this work (?). Again, the potential
benefits of LLMs are not limited to healthcare; extraction of structured
information is widely relevant to domains which produce a large volume of
unstructured text data (for example, police reports in criminal justice or
research articles in academic disciplines). LLMs thus present enormous
potential to alleviate the burden of creating and maintaining structured
datasets used to study inequity in multiple domains.

### LLMs can improve equity of access to information.

Large inequities persist in access to specialized information in healthcare,
law, finance, government benefits, and other domains. Within healthcare, we
see multiple promising applications of LLMs to mitigate these disparities. For
patients who lack easy access to a medical provider or cannot afford to see
one, LLMs can be used to provide accurate and understandable answers to
patient questions, empowering them to advocate for their health needs. There
are ongoing efforts to develop and deploy LLM-based systems in developing
countries to alleviate gaps in patient access to health information. Prior
deployments of medical chatbots have struggled with parsing medical
terminology and personalizing medical recommendations, attributes along which
LLMs have markedly improved (?). LLMs are still not a replacement for
clinicians, and should not be used as a magic bullet displacing existing
efforts to improve access to care (?), but rather as a supplement on top of
these efforts. For patients who do interact with the healthcare system, LLMs
can be applied to translate clinical jargon (for example, diagnostic reports,
clinical notes, or treatment plans) into easy-to-understand language, a task
of longstanding interest to the medical natural language processing community
which complements existing efforts to improve patient access to their own
clinical notes. It is also important to create systems which perform well in
multiple languages, not just English, to ensure access for all patients.
Beyond healthcare, there are promising analogous applications of chatbots to
improve equity of access to specialized information in other domains; e.g.,
chatbots which explain to people what government benefits they are eligible
for and how to apply for them.

### LLMs can improve equity in matching systems.

In many domains, people must navigate specialized matching systems to get
access to resources — for example, finding clinical trials they are eligible
for, jobs they are qualified for, or schools that match their preferences.
Often, the design of these matching systems disfavors underserved populations.
For example, clinical trials can have complex eligibility criteria which can
be both difficult to parse and inadvertently exclusionary towards patients of
particular demographics. The consequent lack of diversity among clinical trial
participants reinforces health disparities by making it challenging to
understand treatment effects in underserved populations. LLMs can be applied
to interpret complex inclusion and exclusion criteria and suggest eligible
patients from underrepresented groups for screening (?). Making this
information directly available to patients — as opposed to merely notifying
existing decision-makers — could empower patients to seek out trials they are
eligible for. LLMs can also be applied to parse eligibility criteria and
understand how modifications to these criteria impact patient diversity, and
inform the selection of equity-promoting inclusion rules. Finally, LLMs can be
used to automatically detect violations of equity-driven reporting standards
(e.g., when institutions fail to report the demographics of trial participants
or fail to separate results by demographic) by parsing public documentation of
clinical trials. Similar patterns of inequity emerge in other matching
systems, including hiring or housing — eligible candidates can be
inadvertently excluded due to unnecessarily strict, or difficult to parse,
criteria (for example, required qualifications in a job posting). LLMs could
be used to both study these criteria and identify eligible candidates to
promote more equitable hiring and housing practices.

## Steps towards implementation

As the research community’s discussion of the impact of AI on equity has
centered risks, the policy landscape has largely followed suit: AI research
has been framed as having significant potential for growth, but potentially
harmful effects on marginalized groups. As such, policy has focused on audits
and regulation with an eye for possible harms. Mitigating these harms is
necessary for implementing many of the applications in this proposal: a
medical chatbot must avoid providing unsound medical advice; LLMs that
aggregate datasets should maintain accuracy across all demographic groups.
However, this focus on harm mitigation should not overshadow the potential for
AI to actively promote equity. In light of this, we have three concrete
recommendations for policymakers.

First and foremost, policymakers should not only seek to mitigate equity-
related LLM harms, but also to incentivize equity-promoting LLM use cases.
Markets alone will not adequately reward the creation of tools which benefit
underserved populations. To provide adequate incentives, it is imperative to
employ a range of policy instruments: research funding, targeted procurement,
and health insurance reimbursement policies which encourage the development of
AI-assisted care. For example, in the United States, clarifying how health
chatbots will be reimbursed by Medicare and Medicaid could foster innovation.

Incentivizing equity-promoting use cases is an essential first step, but LLMs
must also be deployed responsibly in service of those use cases. Our second
recommendation is thus for policymakers to support efforts to ensure
responsible and equitable deployments of LLMs. This includes basic research
funding to increase the safety and reliability of LLMs: for example, ensuring
they do not provide false information, detecting and mitigating their biases,
and supporting open-source models which are thoroughly vetted by the research
community and are not controlled by a handful of powerful private actors.
Policymakers should also support outreach and education initiatives to inform
the public about benefits and limitations of LLMs, mitigating disparities in
awareness and access. Finally, evaluating LLM performance in realistic
settings remains a critical open challenge, so policymakers should support the
creation of evaluations which center equity. For example, chatbots need to be
tested on real queries from diverse patient populations, including underserved
patient populations; perform well in multiple languages; and avoid
caricaturing or stereotyping patients (?). Evaluation metrics should be
developed and vetted using a stakeholder-inclusive process that recognizes and
aims to minimize power asymmetries between participants (e.g., doctors vs.
nurses and patients). Past AI development efforts have been hugely accelerated
by the creation of standardized, widely available evaluation datasets and
infrastructure — e.g., the ImageNet dataset in computer vision, or the MIMIC
dataset in medical machine learning — and by government-funded datasets like
the All of Us and UK Biobank datasets. Drawing lessons from these past
efforts, policymakers should provide research funding to support development
of datasets, metrics, and infrastructure for proper model evaluation, and to
encourage the collaboration across institutions — e.g., between academic
researchers and health organizations — necessary to support these efforts.

Third, LLMs should be treated as one tool in the toolbox rather than as a
panacea to health inequity, which stems from myriad socioeconomic,
environmental, and structural factors (?) defying simple technical fixes. LLM-
enabled interventions should be coupled with non-AI interventions to address
the structural causes of disparities. LLMs can also be used to illuminate
structural inequities rather than as a bandaid fix to avoid confronting them
(?, ?): for example, a medical chatbot could ask a patient to list barriers to
receiving health information, and these barriers could be assessed as
potential targets for non-AI interventions.

The use cases we highlight suggest LLMs have important equity-promoting
applications in healthcare and more broadly: improving detection of bias,
creating structured databases of equity-relevant information, increasing
equity of access to information, and improving equity in matching systems. We
stand to gain enormously if we are able to recognize these applications by
intervening in the early stages of design and acting to advance the interests
of underserved populations.

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