# Andreev Spectroscopy of Nonhelical Spin Textures in Topological Insulators

David J. Alspaugh  Current address: Department of Physics and Astronomy,
California State University, Northridge, CA 91330<EMAIL_ADDRESS>Department of Physics and Astronomy, Louisiana State University, Baton Rouge,
LA 70803-4001  Mahmoud M. Asmar<EMAIL_ADDRESS>Department of Physics and
Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001  Department
of Physics, Kennesaw State University, Marietta, GA 30060  Daniel E. Sheehy
<EMAIL_ADDRESS>Department of Physics and Astronomy, Louisiana State
University, Baton Rouge, LA 70803-4001  Ilya Vekhter<EMAIL_ADDRESS>Department of Physics and Astronomy, Louisiana State University, Baton Rouge,
LA 70803-4001

###### Abstract

We study how nonhelical spin textures affect the proximity-induced
superconductivity of topological insulator (TI)-superconductor (SC) interface
states. In particular we calculate the conductance of lateral heterojunctions
which comprise a TI surface that is only partially covered by a
superconducting material. Interface potentials at the TI-SC interface may lead
to a Fermi velocity and spin texture mismatch between the two regions of the
lateral heterojunction. By enforcing the hermiticity of the total Hamiltonian,
we derive the boundary conditions and calculate the conductance of the
structure in both the normal and superconducting state. The total Andreev
conductance is calculated for both $s$-wave and spin-triplet parent SCs, and
for several examples of nonhelical spin textures which lead to different Fermi
surface mismatches between the two planar regions of the heterojunction. We
find that for spin-triplet SCs, nonzero conductance signatures only appear for
certain combinations of nonhelical spin textures and parent superconducting
material.

##  I Introduction

Topological Insulators (TIs) are a class of materials which admit linearly
dispersing surface states and preserve time reversal (TR) symmetry. Hasan and
Kane (2010); Qi and Zhang (2011) At pristine vacuum terminations these surface
states have isotropic dispersions and are perfectly helical, such that the
spin of the propagating state is perpendicular to the direction of its
momentum and is confined within the plane of the termination. The common
assumption is that at interfaces of TIs with other materials, the interface
states of the TI exhibit these same properties. However, the properties of the
interfacial boundary may be qualitatively different than that of a vacuum
terminated surface, and the available set of symmetries which the interface
states must obey is lower than that of the bulk TI material. Effects from
charge redistribution, dangling bonds, and lattice mismatch can introduce TR-
preserving interface potentials which lower the symmetry and consequently
alter the spin structure and dispersion of the interface states. These states
have been shown to exhibit elliptical energy contours and nonhelical spin-
momentum locking. Asmar  _et al._ (2017); Alspaugh  _et al._ (2018); Zhang
_et al._ (2012); Thareja  _et al._ (2020)

Figure 1: Schematic of the superconducting lateral heterojunction. The device
is created by placing only half of a TI surface in contact with a
superconducting material. Region I on the $x<0$ half-space defines the TI-
vacuum surface, and region II on the $x>0$ half-space defines the TI-SC
interface. Arrows indicate the possible scattering events of normal and
Andreev reflection, which determine the conductance of the device. For an
incoming electron with angle $\theta$ towards the $x=0$ boundary, these
possibilities include Andreev reflection as a hole ($A$), specular reflection
($B$), transmission as an electron-like quasiparticle ($C$), and transmission
as a hole-like quasiparticle ($D$).

Some of the most exciting potential applications of TIs focus on the
possibility that they may be useful as platforms for performing fault-tolerant
quantum computations. For instance, If these materials are placed in contact
with superconductors (SCs), the interface state of the TI experiences
superconductivity due to the proximity effect. When vortices are present or
when placed alongside ferromagnetic systems, these junctions are predicted to
host Majorana zero modes. Fu and Kane (2008) These objects are non-abelians
anyons which have the ability to encode quantum information as they are
braided around one-another in space. Fu and Kane (2008); Read and Green
(2000); Ivanov (2001) However, the realization of Majorana zero modes in
experiment has proved elusive. Zhang  _et al._ (2018, 2021) These difficulties
emphasize the need for a detailed understanding of the physics of topological
interfaces. In the case of TI-SC heterostructures, the properties of the
induced superconductivity strongly depend on the spin structure of the
interface state as well as the properties of the parent SC. Existing
conclusions about the interface superconductivity, including the prediction
that some parent spin-triplet SCs do not induce superconductivity at the
interface at all, have been reached assuming helical TI surface states. In
Ref. Alspaugh  _et al._ , 2018, it was demonstrated that nonhelical spin
textures are required in order to observe proximity-induced superconductivity
for several classes of parent SCs.

To study the consequences of junctions on the TI interface state, in this work
we analyze the conductance signatures of lateral heterojunctions of TI-vacuum
and TI-SC interfaces. As shown schematically in Fig. 1, such lateral
heterojunctions can be constructed by covering only part of a TI surface with
a parent SC, where here we denote the TI-vacuum interface on the $x<0$ half
space as region I and the TI-SC interface on the $x>0$ half space as region
II. While the TI-vacuum interface has perfectly helical surface states,
interface potentials between the TI and SC materials result in nonhelical
interface states. These new states with their altered spin textures have
different anisotropic Fermi velocities compared to the helical states, making
the analysis of their junctions nontrivial. In Sec. II we introduce the model
for the lateral heterojunction, and by enforcing the hermiticity of the
Hamiltonian across both regions I and II in Fig. 1, we develop a formalism
allowing us to calculate the normal state conductance across the device. In
Sec. III we introduce superconductivity by placing an $s$-wave SC in region II
and calculate the superconducting Andreev conductance across the lateral
heterojunction. In Sec. IV we analyze these results for several different
experimental setups. We compare how the scattering across the heterojunction
described by our formalism compares to the effects of a mismatch in the
chemical potential between regions I and II, and we analyze the Andreev
conductance arising from several different nonhelical spin textures within
region II. In Sec. V we modify our model from Sec. III by introducing spin-
triplet SCs within region II, and we analyze the Andreev conductance for
parent spin-triplet SCs from the $D_{4\rm h}$ point group. Critically, we find
that for certain types of spin-triplet SCs, nonzero conductance signatures
exist only if nonhelical spin textures are present in region II. In Sec. VI we
summarize our main results and discuss the key experimental consequences of
nonhelical spin textures.

##  II Model of the Lateral Heterojunction

We begin with a description of the lateral heterojunction in the normal state.
As shown in Fig. 1, this device can be created by covering half of a TI
surface with a superconducting material. Here we define the $x<0$ half-space
with the TI-vacuum surface as region I, and the $x>0$ half-space with the TI-
SC interface as region II. While region I will host a helical TI surface state
with surface state spins confined to the surface (i.e. in the $x$-$y$ plane of
Fig. 1), the material junction in region II can result in interface states
with nonhelical spin textures, giving rise to elliptical energy dispersions
and spins with components that point out of the plane of the interface (i.e.,
in the $z$-direction of Fig. 1). Asmar  _et al._ (2017); Alspaugh  _et al._
(2018) To study the heterojunction we therefore write a helical surface
Hamiltonian in region I and include the most general effective linear
Hamiltonian that is TR invariant in region II,

$H=\begin{cases}\hbar v_{\rm
F}(\bm{\sigma}\times-i\bm{\nabla})_{z}-\mu_{L},&\text{for}\ x<0\\\
\bm{c}\cdot\bm{\sigma}-\mu_{R},&\text{for}\ x>0\end{cases}.$ (1)

Here $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})^{T}$ is a vector of Pauli
matrices in spin-space, $v_{\rm F}$ is the Fermi velocity, $\mu_{L}$ and
$\mu_{R}$ are the chemical potentials in regions I and II respectively, and
$\bm{c}$ is a three-component vector defined by
$c_{i}=-i\sum_{j=x,y}c_{ij}\partial_{j}$. We notice that in the case
$\bm{c}_{\rm D}=-i\hbar v_{F}(\partial_{y},-\partial_{x},0)^{T}$, the
Hamiltonians of regions I and II equal each other and we recover a single TI
surface over all space. The presence of nonhelical spin textures in the TI-SC
interface of region II are can then be encoded in the choice of $c_{ij}$
coefficients. Asmar  _et al._ (2017) This prescription is valid so long as the
transmission between the TI and SC materials is low, so that the TI interface
state is still well-defined. Zhao  _et al._ (2010); Grein  _et al._ (2012) We
also note that from the construction of the vector $\bm{c}$, the form of the
Hamiltonian in region II is identical to that of antisymmetric spin-orbit
coupling in noncentrosymmetric metals, whose consequences on superconductivity
have been extensively studied. Smidman  _et al._ (2017); Samokhin (2015) What
is different however is that the usual quadratic energy term is absent,
placing us in a regime of infinite spin-orbit coupling strength. Pesin and
MacDonald (2012)

###  II.1 Normal State Conductance

In this section we calculate the normal state conductance of the lateral
heterojunction described by Eq. (1). To do this, we must solve the scattering
problem in which incoming electrons in region I of Fig. 1 may either
specularly reflect at the $x=0$ boundary or transmit into region II. Matching
the wave functions at the boundary is nontrivial, however, as the Hamiltonians
of the two regions have different Fermi velocities. Our goal then is to match
the wave functions in a way that is consistent with the hermiticity of the
total Hamiltonian in Eq. (1).

The effective long-wavelength description cannot account for the rapid
variations of the wave functions in the vicinity of the potential edge at
$x=0$. Therefore, due to the fact that we only have a linear description, in
this approach the envelope wave function is not continuous. Tanhayi Ahari  _et
al._ (2016); Bonneau  _et al._ (2001); Gitman  _et al._ (2012); Stone and
Goldbart (2009) Instead it satisfies

$\psi_{\rm I}(0,y)=\mathcal{M}\psi_{\rm II}(0,y),$ (2)

for some matrix $\mathcal{M}$ and wave functions $\psi_{\rm I}(\bm{r})$ and
$\psi_{\rm II}(\bm{r})$ in regions I and II respectively. In order for this to
be a valid boundary value relation for our system, and hence valid for all
wave functions of the Hilbert space, $\mathcal{M}$ must respect the
hermiticity of the Hamiltonian $H$ expressed in Eq. (1). This ensures particle
conservation and therefore continuity of the current normal to the boundary.
That is, if both $\psi_{1}(\bm{r})$ and $\psi_{2}(\bm{r})$ are wave functions
for all $x$ that belong to $H$, we must have the inner product relations
$\langle\psi_{1}|H\psi_{2}\rangle=\langle H\psi_{1}|\psi_{2}\rangle$.
Inserting Eq. (2) into this condition for hermiticity and integrating by
parts, we find

$-\hbar v_{\rm
F}\mathcal{M}^{\dagger}\sigma_{y}\mathcal{M}=\sum_{i=x,y,z}c_{ix}\sigma_{x}.$
(3)

To explicitly solve for each element in $\mathcal{M}$, it is helpful to
restrict the form of $\mathcal{M}$ by requiring that it must also respect the
discrete symmetries of the system. We let $\psi(\bm{r})$ be a wave function
over all space such that

$\psi(\bm{r})=\begin{cases}\psi_{\rm I}(\bm{r})&\text{for}\ x<0\\\ \psi_{\rm
II}(\bm{r})&\text{for}\ x>0\end{cases},$ (4)

and write $\mathcal{T}=i\sigma_{y}K$ as the TR symmetry operator where $K$ is
complex conjugation. Because the system preserves TR symmetry, we know that
$\mathcal{T}\psi(\bm{r})$ is also a wave function of our system. Eq. (2) holds
for all wave functions in the Hilbert space, which implies both $\psi_{\rm
I}(0,y)=\mathcal{M}\psi_{\rm II}(0,y)$ and $\mathcal{T}\psi_{\rm
I}(0,y)=\mathcal{M}\mathcal{T}\psi_{\rm II}(0,y)$. However, by applying
$\mathcal{T}$ on the left of both sides of Eq. (2), we have
$\mathcal{T}\psi_{\rm I}(0,y)=\mathcal{T}\mathcal{M}\psi_{\rm II}(0,y)$. This
demonstrates that the commutator $[\mathcal{T},\mathcal{M}]=0$. This condition
implies that $\mathcal{M}$ must take the form
$\mathcal{M}=\gamma_{0}\sigma_{0}+i(\gamma_{x}\sigma_{x}+\gamma_{y}\sigma_{y}+\gamma_{z}\sigma_{z})$,
where all the $\gamma_{i}$ are purely real. With this restriction, Eq. (3) can
be solved to explicitly find each component of $\mathcal{M}$,

$\mathcal{M}(\beta)=\sqrt{\dfrac{v}{v_{\rm
F}}}\bigg{[}e^{i\sigma_{y}\beta}+\dfrac{i}{2\hbar
v}(c_{xx}\sigma_{z}-c_{zx}\sigma_{x})e^{-i\sigma_{y}\beta}\bigg{]}.$ (5)

Here $v=(\sqrt{\sum_{i}c_{ix}^{2}}-c_{yx})/2\hbar$, and $\beta$ is a free
parameter such that $\beta\in[0,2\pi)$.

Now that we can match the wave functions at the $x=0$ boundary between regions
I and II, the next step is to solve the scattering problem to obtain the
normal state conductance of the lateral heterojunction. We consider an
incoming electron in region I of Fig. 1 with momentum
$\bm{k}_{1}=(k_{x},k_{y})^{T}$ and in-plane momentum angle
$\theta=\tan^{-1}k_{y}/k_{x}$, a specularly reflected electron with momentum
$\bm{k}_{2}=(-k_{x},k_{y})^{T}$, and a transmitted electron in region II with
momentum $\bm{k}_{1}^{\prime}=(k_{x}^{\prime},k_{y})^{T}$. The outgoing angle
$\theta^{\prime}$ may be solved for in terms of the incoming angle due to the
conservation of normal state energy and the conservation of the $k_{y}$
momentum. The wave function in region I is given by

$\psi_{\rm
I}(\bm{r})=\dfrac{e^{i\bm{k}_{1}\cdot\bm{r}}}{\sqrt{2}}\begin{pmatrix}1\\\
-ie^{i\theta}\end{pmatrix}+r\dfrac{e^{i\bm{k}_{2}\cdot\bm{r}}}{\sqrt{2}}\begin{pmatrix}1\\\
ie^{-i\theta}\end{pmatrix}.$ (6)

In region II the wave function is

$\psi_{\rm
II}(\bm{r})=te^{i\bm{k}_{1}^{\prime}\cdot\bm{r}}\begin{pmatrix}\cos(\vartheta_{\bm{c}(\bm{k}_{1}^{\prime})}/2)\\\
e^{i\varphi_{\bm{c}(\bm{k}_{1}^{\prime})}}\sin(\vartheta_{\bm{c}(\bm{k}_{1}^{\prime})}/2)\end{pmatrix}.$
(7)

Here $\vartheta_{\bm{c}(\bm{k}_{1}^{\prime})}$,
$\varphi_{\bm{c}(\bm{k}_{1}^{\prime})}$ are the polar and azimuthal angles of
the vector $\bm{c}(\bm{k}_{1}^{\prime})$ respectively, where $\bm{c}(\bm{k})$
is defined by $c_{i}(\bm{k})=\sum_{j=x,y}c_{ij}k_{j}$, and $r$ and $t$ are the
coefficients for the reflected and transmitted parts of the wave function
respectively. Matching the wave functions at the $x=0$ boundary by using Eq.
(2), we solve for the reflection coefficient $r$ and define the normal state
conductance as

$\sigma_{N}(\theta,\beta)=1-|r(\theta,\beta)|^{2}.$ (8)

To demonstrate the consequences of the free parameter $\beta$ on the normal
state conductance, we analyze $\sigma_{N}$ in the case when both regions I and
II are described by helical Hamiltonians and set $\bm{c}=\bm{c}_{\rm D}$ in
Eq. (1). When $\beta$ is zero, the conductance is always unity (in this case
the heterojunction of Fig. 1 is equivalent to one single slab of a TI
surface). However, when $\beta$ is nonzero the conductance decreases for
nonzero angles. As shown in Fig. 2, the most dramatic changes occur for
$\beta=\pi/2$. We see that $\beta$ acts as an angle-dependent scattering
amplitude localized at $x=0$ between the two planar regions. Incoming
electrons with $\theta=0$ experience full transmission into region II as Dirac
particles are incapable of back-scattering from normal incidence due to Klein
tunneling. Castro Neto  _et al._ (2009); Katsnelson  _et al._ (2006); Allain
and Fuchs (2011); Asmar and Ulloa (2014) This feature of Dirac particles is
generally true for TR invariant systems as counter-propagating modes form
Kramers’ pairs. We can thus see that the effects of interfacial scattering
potentials can be encoded into the form of the boundary matching matrix
$\mathcal{M}(\beta)$.

Figure 2: Normal state conductance $\sigma_{N}(\theta,\beta)$ as given by Eq.
(8) for a lateral heterojunction between two helical surface Hamiltonians with
the same Fermi velocity. We see that $\beta$ acts as an angle-dependent
scattering amplitude at the one dimensional interface between regions I and II
of the lateral heterojunction.

##  III Superconducting Lateral Heterojunctions

Having discussed the normal state of the lateral heterojunction, we now
analyze the superconducting state. To calculate the conductance of the device
shown in Fig. 1, we first focus on region II and derive the proximity-induced
superconducting order parameter of the TI-SC interface. We then use the
Andreev equations to solve for the wave functions and find the conductance of
the normal-superconducting junction of regions I and II. Snelder  _et al._
(2015); Blonder  _et al._ (1982); Bruder (1990); Kashiwaya  _et al._ (1996) To
obtain the conductance we solve the Andreev reflection scattering problem in
which an incoming electron from region I may either be specularly reflected at
the $x=0$ boundary, or be retroreflected as a hole, corresponding to
scattering events $B$ and $A$ in Fig. 1 respectively. Once again matching the
wave functions is nontrivial, and we employ the boundary conditions derived
above in the form suitable to treat superconductivity.

Figure 3: Conductance of the superconducting lateral heterojunction for a
parent $s$-wave SC. (a) Normalized conductance $\sigma_{R}(E,\theta,\beta)$
for several incident electron angles, such that $\bm{c}(\bm{k})=\bm{c}_{\rm
D}(\bm{k})$ and $\beta=\pi/2$. (b) Total conductance $\sigma_{T}(E,\beta)$ of
the heterojunction such that $\bm{c}(\bm{k})=\bm{c}_{\rm D}(\bm{k})$. The
normalized conductance $\sigma_{R}(E,\theta,\beta)$ and total conductance
$\sigma_{T}(E,\beta)$ are both defined in Eq. (17).

Focusing on region II, we can obtain the form of the proximity-induced order
parameter of the TI-SC interface by projecting the Cooper pair structure of
the parent SC onto the eigenstates of the interface Hamiltonian. Writing
$(b_{\bm{k}\uparrow},b_{\bm{k}\downarrow})$ as the electron annihilation
operators and introducing the Nambu-spinor
$\Psi=(b_{\bm{k}\uparrow},b_{\bm{k}\downarrow},b_{-\bm{k}\uparrow}^{\dagger},b_{-\bm{k}\downarrow}^{\dagger})^{T}$,
the Bogoliubov-de Gennes (BdG) Hamiltonian of the interface may be written as

$\mathcal{H}=\dfrac{1}{2}\sum_{\bm{k}}\Psi^{\dagger}\begin{pmatrix}H(\bm{k})&\hat{\Delta}\\\
\hat{\Delta}^{\dagger}&-H^{T}(-\bm{k})\end{pmatrix}\Psi.$ (9)

Here $H(\bm{k})=\bm{c}(\bm{k})\cdot\bm{\sigma}-\mu_{R}$ is the interface
Hamiltonian in the normal state, and the proximity-induced order parameter is
given by
$\hat{\Delta}=[\psi(\bm{k})+\bm{d}(\bm{k})\cdot\bm{\sigma}](i\sigma_{y})$. The
spin-singlet and spin-triplet components of the proximity-induced order are
described by $\psi(\bm{k})$ and $\bm{d}(\bm{k})$ respectively. In this
section, we shall focus on the case that the parent material of the
heterostructure is an $s$-wave SC by setting $\psi(\bm{k})=\Delta_{0}$ and
$\bm{d}(\bm{k})=0$.

We then take the weak coupling limit to obtain the BdG wave functions.
Alspaugh  _et al._ (2018) In the weak coupling limit where
$\mu_{R}\gg\Delta_{0}$ interband pairing can be ignored. Writing $a_{\bm{k}}$
as the annihilation operator of the conduction band eigenstate of $H(\bm{k})$,
we find

$\mathcal{H}\approx\dfrac{1}{2}\sum_{\bm{k}}\Phi^{\dagger}\begin{pmatrix}\xi(\bm{k})&\Delta(\bm{k})\\\
\Delta^{*}(\bm{k})&-\xi(\bm{k})\end{pmatrix}\Phi.$ (10)

Here $\Phi=(a_{\bm{k}},a_{-\bm{k}}^{\dagger})^{T}$ and
$\Delta(\bm{k})=-e^{-i\varphi_{\bm{c}(\bm{k})}}\Delta_{0}$, with
$\varphi_{\bm{c}(\bm{k})}$ being the azimuthal angle of the vector
$\bm{c}(\bm{k})$. The normal state energy is given as
$\xi(\bm{k})=|\bm{c}(\bm{k})|-\mu_{R}$. In the basis of Eq. (10), the
eigenfunction of $\mathcal{H}$ has the form
$\phi(\bm{r})=(u(\bm{r}),v(\bm{r}))^{T}$.

We may then invoke the Andreev approximation to model the boundary between
regions I and II. For a spatially inhomogeneous order parameter, the BdG
equations are given by Bruder (1990)

$\displaystyle Eu(\bm{r}_{1})$ $\displaystyle=\xi u(\bm{r}_{1})+\int
d\bm{r}_{2}\Delta(\bm{r}_{1},\bm{r}_{2})v(\bm{r}_{2}),$ (11) $\displaystyle
Ev(\bm{r}_{1})$ $\displaystyle=-\xi v(\bm{r}_{1})+\int
d\bm{r}_{2}\Delta^{*}(\bm{r}_{1},\bm{r}_{2})u(\bm{r}_{2}).$

Here $\Delta(\bm{r}_{1},\bm{r}_{2})$ is the two-point correlation function. In
the Andreev approximation, we write $\phi(\bm{r})=e^{i\bm{k}_{\rm
F}\cdot\bm{r}}[U(\bm{r}),V(\bm{r})]^{T}$, where $\bm{k}_{\rm F}$ lies on the
Fermi surface. By writing $\bm{R}=(\bm{r}_{1}+\bm{r}_{2})/2$ and
$\bm{s}=\bm{r}_{2}-\bm{r}_{1}$ as the center of mass and relative coordinates
respectively, defining
$\Delta(\bm{R},\bm{k})=\int\Delta(\bm{R},\bm{s})e^{i\bm{k}\cdot\bm{s}}d\bm{s}$,
and writing $\xi(\bm{k}_{\rm
F}-i\bm{\nabla})\approx(\partial\xi/\partial\bm{k})|_{\bm{k}_{\rm
F}}\cdot(-i\bm{\nabla})$, we obtain the Andreev equations Bruder (1990)

$\displaystyle EU(\bm{r}_{1})=-i\bm{v}_{\rm F}(\bm{k}_{\rm
F})\cdot\bm{\nabla}U(\bm{r}_{1})+\Delta(\bm{r}_{1},\bm{k}_{\rm
F})V(\bm{r}_{1}),$ (12) $\displaystyle EV(\bm{r}_{1})=i\bm{v}_{\rm
F}(\bm{k}_{\rm
F})\cdot\bm{\nabla}V(\bm{r}_{1})+\Delta^{*}(\bm{r}_{1},\bm{k}_{\rm
F})U(\bm{r}_{1}).$

Here $\bm{v}_{\rm F}(\bm{k}_{\rm
F})=(\partial\xi/\partial\bm{k})|_{\bm{k}_{\rm F}}$, and
$\Delta(\bm{r}_{1},\bm{k}_{\rm F})=-e^{-i\varphi_{\bm{c}(\bm{k}_{\rm
F})}}\Delta_{0}\Theta(x)$. By solving these equations we can obtain the wave
functions for our system.

To solve the Andreev scattering problem, we assume a helical surface
Hamiltonian in region I and consider an incoming incident electron with
momentum $\bm{k}_{1}$ and in-plane angle $\theta$. At the boundary, the
electron may be retroreflected as a hole with momentum $\bm{k}_{1}$ or
specularly reflected as an electron with momentum $\bm{k}_{2}$, corresponding
to scattering events $A$ and $B$ in Fig. 1 respectively. In the basis of Eq.
(9) and setting $H(\bm{k})=\hbar v_{\rm
F}(\bm{\sigma}\times\bm{k})_{z}-\mu_{L}$, the wave function is given by

$\displaystyle\psi_{\rm I}(\bm{r})$
$\displaystyle=\dfrac{e^{i\bm{k}_{1}^{+}\cdot\bm{r}}}{\sqrt{2}}\begin{pmatrix}1\\\
-ie^{i\theta}\\\ 0\\\
0\end{pmatrix}+A\dfrac{e^{i\bm{k}_{1}^{-}\cdot\bm{r}}}{\sqrt{2}}\begin{pmatrix}0\\\
0\\\ 1\\\ -ie^{-i\theta}\end{pmatrix}$ (13) $\displaystyle\
+B\dfrac{e^{i\bm{k}_{2}^{+}\cdot\bm{r}}}{\sqrt{2}}\begin{pmatrix}1\\\
ie^{-i\theta}\\\ 0\\\ 0\end{pmatrix}.$

Note that we choose to use the spin basis of Eq. (9) since the helicity basis
of Eq. (10) is momentum dependent. Here $\bm{k}^{\pm}=(k_{x}^{+},k_{y})^{T}$,
where $k_{x}^{\pm}=k_{x}\pm E/v_{Fx}(\bm{k})$. In region II the incident
electron can either be transmitted as an electron-like quasiparticle with
momentum $\bm{k}_{1}^{\prime}$ and in-plane angle $\theta^{\prime}$ or a hole-
like quasiparticle with momentum
$\bm{k}_{2}^{\prime}=(-k_{x}^{\prime},k_{y})^{T}$, corresponding to scattering
events $C$ and $D$ in Fig. 1 respectively. Here we note that due to the
translational invariance along the $y$-direction, the $k_{y}$ momentum is
conserved. The nonhelical wave function in region II is then

$\displaystyle\psi_{\rm II}(\bm{r})$
$\displaystyle=C\dfrac{e^{i\bm{k}_{1}^{\prime+}\cdot\bm{r}}}{\sqrt{N_{-}}}\begin{pmatrix}\cos(\vartheta_{\bm{c}(\bm{k}_{1}^{\prime})}/2)\\\
e^{i\varphi_{\bm{c}(\bm{k}_{1}^{\prime})}}\sin(\vartheta_{\bm{c}(\bm{k}_{1}^{\prime})}/2)\\\
f_{-}(\bm{k}^{\prime}_{1})\sin(\vartheta_{\bm{c}(\bm{k}_{1}^{\prime})}/2)\\\
-f_{-}(\bm{k}^{\prime}_{1})e^{-i\varphi_{\bm{c}(\bm{k}_{1}^{\prime})}}\cos(\vartheta_{\bm{c}(\bm{k}_{1}^{\prime})}/2)\end{pmatrix}$
(14) $\displaystyle\
+D\dfrac{e^{i\bm{k}_{2}^{\prime-}\cdot\bm{r}}}{\sqrt{N_{+}}}\begin{pmatrix}\cos(\vartheta_{\bm{c}(\bm{k}_{2}^{\prime})}/2)\\\
e^{i\varphi_{\bm{c}(\bm{k}_{2}^{\prime})}}\sin(\vartheta_{\bm{c}(\bm{k}_{2}^{\prime})}/2)\\\
f_{+}(\bm{k}^{\prime}_{2})\sin(\vartheta_{\bm{c}(\bm{k}_{2}^{\prime})}/2)\\\
-f_{+}(\bm{k}^{\prime}_{2})e^{-i\varphi_{\bm{c}(\bm{k}_{2}^{\prime})}}\cos(\vartheta_{\bm{c}(\bm{k}_{2}^{\prime})}/2)\end{pmatrix}.$

Here $N_{\pm}^{-1/2}=\Delta_{0}/\sqrt{2E(E\pm\sqrt{E^{2}-\Delta_{0}^{2})}}$
and
$f_{\pm}(\bm{k})=(E\pm\sqrt{E^{2}-\Delta_{0}^{2})}/(-e^{-i\varphi_{\bm{c}(\bm{k})}}\Delta_{0})$.
Once again $\vartheta_{\bm{c}(\bm{k})}$ and $\varphi_{\bm{c}(\bm{k})}$ are the
polar and azimuthal angles of the vector $\bm{c}(\bm{k})$ respectively. We
have that $\bm{k}^{\prime\pm}=(k_{x}^{\prime\pm},k_{y})^{T}$ where
$k_{x}^{\prime\pm}=k_{x}^{\prime}\pm\sqrt{E^{2}-\Delta_{0}^{2}}/v_{Fx}(\bm{k}^{\prime})$.
As in the previous section the outgoing angle $\theta^{\prime}$ may be solved
for in terms of the incoming angle $\theta$ due to the conservation of the
$k_{y}$ momentum and the conservation of the normal state energy, noting that
$\xi_{L}(\bm{k})=\hbar v_{\rm F}|\bm{k}|-\mu_{L}$ in region I and
$\xi_{R}(\bm{k})=|\bm{c}(\bm{k})|-\mu_{R}$ in region II.

To solve the scattering problem we then must match the wave functions at the
$x=0$ boundary. Once again we can generally write

$\psi_{\rm I}(0,y)=\mathcal{M}_{S}\psi_{\rm II}(0,y).$ (15)

Figure 4: (a) Total conductance $\sigma_{T}(E,\beta)$ as given by Eq. (17) of
the superconducting lateral heterojunction for a parent $s$-wave SC as
$\bm{c}(\bm{k})=\bm{c}_{\rm D}(\bm{k})$. Here it can be observed that a
mismatch in the chemical potentials can introduce a peak at the gap edge, but
not as large as the consequences of the angle-dependent scattering amplitude
controlled by $\beta$ localized at the $x=0$ boundary. (b) Total conductance
$\sigma_{T}(E,\beta)$ of the superconducting lateral heterojunction for a
parent $s$-wave SC in the case that the Fermi velocity in region II is
reduced, such that $v_{2}=0.7v_{\rm F}$.

In the mean field BdG description the Hamiltonian $\mathcal{H}$ is written in
a redundant formalism where the hole degrees of freedom are constructed from
the electron Hamiltonian $H(\bm{k})$ by writing $-H^{T}(-\bm{k})$. Following
the methods of Sec. II we find that the boundary value matrix for the hole
Hamiltonian $-H^{T}(-\bm{k})$ is given by $\mathcal{M}^{*}(\beta)$. Because
$-H^{T}(-\bm{k})$ is not independent of $H(\bm{k})$, we see that if
$\mathcal{M}(\beta)$ has a particular value for its free parameter $\beta$,
then $\mathcal{M}^{*}(\beta)$ must also have this same value for its free
parameter. The boundary value matrix for the BdG wave functions is then

$\mathcal{M}_{S}(\beta)=\begin{pmatrix}\mathcal{M}(\beta)&0\\\
0&\mathcal{M}^{*}(\beta)\end{pmatrix}.$ (16)

The block diagonal nature of the matrix $\mathcal{M}_{S}(\beta)$ means that
there is no additional particle hole mixing due to the boundary conditions,
which is a natural assumption.

Matching the wave functions by writing $\psi_{\rm
I}(0,y)=\mathcal{M}_{S}(\beta)\psi_{\rm II}(0,y)$ and solving for the
coefficients $A$ and $B$, we then define the transmission coefficient of the
superconducting lateral heterojunction Blonder  _et al._ (1982); Kashiwaya
_et al._ (1996); Tanaka and Kashiwaya (1995) as
$\sigma_{S}(E,\theta,\beta)=1+|A|^{2}-|B|^{2}$. From here, we can define the
normalized conductance and the total conductance respectively as

$\displaystyle\sigma_{R}(E,\theta,\beta)=\dfrac{\sigma_{S}(E,\theta,\beta)}{\sigma_{N}(\theta,\beta)},$
(17)
$\displaystyle\sigma_{T}(E,\beta)=\dfrac{\int_{-\pi/2}^{\pi/2}\sigma_{S}(E,\theta,\beta)\cos\theta
d\theta}{\int_{-\pi/2}^{\pi/2}\sigma_{N}(\theta,\beta)\cos\theta d\theta},$

where $\sigma_{N}(\theta,\beta)$ is the normal state conductance given in Eq.
(8).

Figure 5: (a) Total conductance $\sigma_{T}(E,\beta)$ as given by Eq. (17) of
the superconducting lateral heterojunction in the case that
$\bm{c}(\bm{k})=\bm{c}_{\perp}(\bm{k})$ in region II, where
$\bm{c}_{\perp}(\bm{k})=\hbar v_{\rm F}(\lambda k_{y},-k_{x},-\lambda
k_{y})^{T}$. (b) Total conductance in the case that
$\bm{c}(\bm{k})=\bm{c}_{\parallel}(\bm{k})$ in region II, where
$\bm{c}_{\parallel}(\bm{k})=\hbar v_{\rm F}(k_{y},-\lambda k_{x},-\lambda
k_{x})^{T}$. Here both $\bm{c}_{\perp}(\bm{k})$ and
$\bm{c}_{\parallel}(\bm{k})$ are defined in Eq. (18), and for both cases we
have a parent $s$-wave SC and $\lambda=2/3$.

##  IV Conductance of the Lateral Heterojunction

Figure 6: Total conductance $\sigma_{T}(E,\beta)$ as given by Eq. (17) for
parent helical spin-triplet SCs according to the $D_{4\rm h}$ point group in
the case that $\bm{c}(\bm{k})=\bm{c}_{\rm D}(\bm{k})$ for (a) a parent $\rm
A_{2\rm u}$ SC (b) a parent $\rm B_{1\rm u}$ SC (c) a parent $\rm B_{2\rm u}$
SC.

In this section we analyze the consequences of nonhelical spin textures, as
encoded in the choice of $c_{ij}$ coefficients, on the conductance of
superconducting lateral heterostructures with parent $s$-wave SCs. We first
analyze the normalized conductance $\sigma_{R}$ in the case that
$\bm{c}(\bm{k})=\bm{c}_{\rm D}(\bm{k})$ in region II, where $\bm{c}_{\rm
D}(\bm{k})=\hbar v_{\rm F}(k_{y},-k_{x},0)^{T}$ so that both sides of the
heterojunction admit helical surface states. We find that when $\beta=0$ there
is no scattering at the junction and all incoming angles experience perfect
Andreev reflection for energies below the gap edge. When $\beta=\pi/2$ the
probability of specular reflection due to scattering increases, resulting in a
suppressed conductance for energies below the gap edge as shown in Fig. 3(a).
In the standard treatment by Blonder et. al., as the scattering at the
junction increases the conductance evolves from the Andreev limit to the
profile reflecting the tunneling density of states. Blonder  _et al._ (1982)
Note that this is not what takes place with increasing $\beta$: the low energy
conductance never vanishes, as seen in the total conductance in Fig. 3(b) and
the black dotted line in Fig. 4(a). This is because the normal incidence angle
always experience perfect Andreev reflection regardless of the strength of
$\beta$, reflecting the inability to confine Dirac particles due to Klein
tunneling. As in the normal state, Fig. 3(a) shows that $\beta$ acts as an
angle-dependent scattering amplitude. In Fig. 3(b) we plot the total
conductance for all $\beta\in[0,\pi/2)$. For $\beta=0$ we see that we have a
constant $\sigma_{T}(E,0)=2$ for all $E<\Delta_{0}$, followed by a decreasing
conductance for larger energies above the superconducting gap. Whenever
$\beta\neq 0$ the conductance instead finds a maximum value at the
$E=\Delta_{0}$ gap edge. For conventional $s$-wave SCs in a normal-
superconducting junction, localized interface potentials at the boundary can
result in divergences of the total conductance at the gap edge.Blonder  _et
al._ (1982) This is in contrast to the Dirac Hamiltonians that describe our
lateral heterojunction, as the total conductance only obtains a maximum which
is largest at the gap edge for $\beta=\pi/2$. The concomitant result is that
the total conductance has a maximum, rather than a singularity, at the gap
edge.

In Fig. 4(a), we plot the total conductance for the case
$\bm{c}(\bm{k})=\bm{c}_{\rm D}(\bm{k})$ and compare the effects of the angle-
dependent scattering controlled by $\beta$ to that of a mismatch of the
chemical potentials between regions I and II. While $\mu_{L}\neq\mu_{R}$
similarly creates a maximum at the gap edge, the effect is not as pronounced
as that arising from setting $\beta=\pi/2$. In Fig. 4(b) we plot the total
conductance in the case that region II has a renormalized Fermi velocity such
that $\bm{c}(\bm{k})=\hbar v_{2}(k_{y},-k_{x},0)$, where $v_{2}=0.7v_{\rm F}$.
The reduced Fermi velocity creates a mismatch in the size of the Fermi
surfaces between regions I and II, leading to extra scattering in addition to
that controlled by $\beta$.

Nonhelical spin textures in region II that result from rotational symmetry
breaking interface potentials within TI-SC heterostructures can introduce
elliptical energy contours and out-of-plane spin textures within the TI
interface states.Alspaugh  _et al._ (2018); Asmar  _et al._ (2017) Two generic
examples of such nonhelical spin textures can be modeled by

$\displaystyle\bm{c}_{\perp}(\bm{k})$ $\displaystyle=\hbar v_{\rm F}(\lambda
k_{y},-k_{x},-\lambda k_{y})^{T},$ (18)
$\displaystyle\bm{c}_{\parallel}(\bm{k})$ $\displaystyle=\hbar v_{\rm
F}(k_{y},-\lambda k_{x},-\lambda k_{x})^{T},$

where $0<\lambda<1$. Alspaugh  _et al._ (2018) The former spin texture
introduces an elliptical Fermi surface with a major axis perpendicular to the
$x$-axis, while the latter spin texture has an elliptical Fermi surface with a
major axis parallel to the $x$-axis. In Figs. 5(a) and 5(b) we plot the total
conductance for each of these two cases respectively. Comparing Fig. 3(b) and
Fig. 5(b), we see that $\bm{c}_{\parallel}(\bm{k})$ has the same total
conductance spectrum as the helical surface states of $\bm{c}_{\rm
D}(\bm{k})$, due to the fact that the conservation of the $k_{y}$ momentum is
unaffected by the ellipticity in this direction. In contrast, the total
conductance of $\bm{c}_{\perp}(\bm{k})$ is modified compared to that of the
helical surface states. However, for $\beta=\pi/8$ in Fig. 5(a) the scattering
of the $x=0$ boundary is reduced and the system experiences perfect Andreev
reflection. It follows that the direction of the ellipticity of the Fermi
surface cannot be distinguished from the consequences of $\beta$ as different
choices of $\beta$ can produce similar conductance spectra between
$\bm{c}_{\perp}(\bm{k})$ and $\bm{c}_{\parallel}(\bm{k})$. In addition, we
find that different ellipse directions require different values of $\beta$ to
maximize the normal state conductance.

##  V Spin-Triplet Superconductors

Next we study the conductance signatures for several examples of parent spin-
triplet SCs. To analyze these systems, we replace the order parameter of the
parent SC in Eq. (9) with
$\hat{\Delta}=[\bm{d}(\bm{k})\cdot\bm{\sigma}](i\sigma_{y})$, where the vector
$\bm{d}(\bm{k})$ defines the spin-triplet part of the pairing field. Carrying
through a similar analysis as before, it has been shown in Ref. Alspaugh  _et
al._ , 2018 that the proximity-induced order parameter of the TI-SC interface
in Eq. (10) now takes the form
$\Delta(\bm{k})=-e^{-i\varphi_{\bm{c}(\bm{k})}}[\hat{\bm{c}}(\bm{k})\cdot\bm{d}(\bm{k})]$,
where $\hat{\bm{c}}(\bm{k})=\bm{c}(\bm{k})/|\bm{c}(\bm{k})|$.

For concreteness we assume a tetragonal crystal symmetry for the parent
superconducting material and classify $\bm{d}(\bm{k})$ according to the
irreducible representations of the $D_{4\rm h}$ point group. The conductance
signatures of the chiral $\rm E^{\pm}_{2\rm u}$ pairing states in which
$\bm{d}_{\rm E^{\pm}_{2\rm u}}(\bm{k})=\Delta_{0}(\hat{k}_{x}\pm
i\hat{k}_{y})\hat{\bm{z}}$ have been studied in Ref. Alspaugh  _et al._ ,
2018, and below we focus on the helical $\rm A_{1\rm u}$, $\rm A_{2\rm u}$,
$\rm B_{1\rm u}$, and $\rm B_{2\rm u}$ pairing states.

Assuming that $\bm{c}(\bm{k})=\bm{c}_{\rm D}(\bm{k})$ in region II and thus
$\Delta_{\rm D}(\bm{k})=-ie^{-i\theta}[\bm{d}(\bm{k})\times\hat{\bm{k}}]_{z}$,
we find that the $\rm A_{2\rm u}$ pairing state with $\bm{d}_{\rm A_{2\rm
u}}(\bm{k})=\Delta_{0}(\hat{k}_{y}\hat{\bm{x}}-\hat{k}_{x}\hat{\bm{y}})$
produces isotropic fully gapped superconductivity at the interface. In
contrast, the $\rm B_{1\rm u}$ pairing state with $\bm{d}_{\rm B_{1\rm
u}}(\bm{k})=\Delta_{0}(\hat{k}_{x}\hat{\bm{x}}-\hat{k}_{y}\hat{\bm{y}})$ and
the $\rm B_{2\rm u}$ pairing state with $\bm{d}_{\rm B_{2\rm
u}}(\bm{k})=\Delta_{0}(\hat{k}_{y}\hat{\bm{x}}+\hat{k}_{x}\hat{\bm{y}})$ both
produce $d$-wave-like nodal gaps. These differences manifest themselves in the
Andreev conductance spectra plotted in Fig. 6. For the $\rm B_{2\rm u}$
pairing state at $\beta=0$ there is a bright spot corresponding to Andreev
reflection near $E=0$. That peak is shifted towards the finite in-gap value as
$\beta$ increases. In contrast, for the $\rm B_{1\rm u}$ pairing state there
is a zero energy peak in the total conductance for all values of $\beta$.

Figure 7: Total conductance $\sigma_{T}(E,\beta)$ as given by Eq. (17) for an
$\rm A_{1\rm u}$ parent helical spin-triplet SC as defined by the $D_{4\rm h}$
point group. In (a) and (c) we plot the total conductance for the nonhelical
spin textures described by $\bm{c}_{\perp}(\bm{k})$ and
$\bm{c}_{\parallel}(\bm{k})$ as given in Eq. (18) respectively. In (b) and (d)
we plot the same conductance spectra for chosen values of $\beta$.

For the $\rm A_{1\rm u}$ pairing state with $\bm{d}_{\rm A_{1\rm
u}}(\bm{k})=\Delta_{0}(\hat{k}_{x}\hat{\bm{x}}+\hat{k}_{y}\hat{\bm{y}})$ we
find that the order parameter vanishes and there is no proximity-induced
superconductivity when $\bm{c}(\bm{k})=\bm{c}_{\rm D}(\bm{k})$. For this
family of spin-triplet SCs the proximity effect thus requires the presence of
nonhelical spin textures in the TI-SC barrier. In Fig. 7 we plot the total
conductance for the nonhelical spin textures described by
$\bm{c}_{\perp}(\bm{k})$ and $\bm{c}_{\parallel}(\bm{k})$ respectively. In
both cases we find that the conductance always obtains its largest value at
zero energy regardless of the value of $\beta$. As before, the orientation of
the elliptical energy contours cannot be distinguished from the effects of
scattering as different values of $\beta$ give rise to similar conductance
spectra. We calculate that at energies comparable to the proximity-induced gap
edge, the conductance tends towards unity. While we cannot test for the
direction of the dispersion’s ellipticity, the existence of the anisotropy
would be proven by the observation of the signal.

##  VI Discussion

In this work we analyzed how nonhelical spin textures present in TI-SC
interface states affect the conductance of lateral heterojunctions, which can
be created by placing a parent SC in contact with only half of a TI surface as
shown schematically in Fig. 1. It has been shown that TI-vacuum terminations
host helical surface states such that the total angular momentum of the
propagating state is perpendicular to its velocity and confined within the
plane of the surface. Hasan and Kane (2010); Qi and Zhang (2011) In contrast,
TIs in contact with nontopological materials host interface states that can
exhibit elliptical energy contours and nonhelical spin textures that point out
of the plane of the interface. Alspaugh  _et al._ (2018); Asmar  _et al._
(2017)

We evaluated the conductance of the system in the normal state and found that
angle-dependent scattering amplitudes localized at the one-dimensional
boundary between the TI-vacuum surface (region I) and the TI-SC interface
(region II), as shown schematically in Fig. 1, lead to a reduction of the
current through the heterojunction. This reduction is largely due to the
scattering of near-grazing angles at the boundary between regions I and II.
Introducing $s$-wave superconductivity, we then found that both the angle-
dependent scattering amplitude and a mismatch of the chemical potentials
between regions I and II can both reduce the Andreev conductance through the
device, with the former leading to a more dramatic suppression. We also
calculated the total conductance for a variety of nonhelical spin textures,
which all lead to qualitatively similar conductance spectra for parent
$s$-wave SCs characterized by a finite, albeit reduced, conductance at $E=0$
and a peak at the gap edge. Finally, we then computed the total conductance
spectra for parent spin-triplet SCs in region II as classified according to
the irreducible representations of the $D_{4\rm h}$ point group. In
particular, we find that for parent helical $\rm A_{1\rm u}$ SCs, nonzero
conductance spectra only exist when nonhelical spin textures are present
within region II.

Our results in general demonstrate that nonhelical spin textures have
important and observable consequences for TI-based devices. TI-SC
heterostructures placed alongside ferromagnetic materials have earlier been
proposed to host Majorana zero modes as a consequence of the spin-momentum
locking intrinsic to topological interface states. In this article we show
that the spin-momentum locking within heterojunctions is more complex than has
been commonly assumed, and leads to a more diverse set of phenomena even for
non-magnetic heterostructures. Our work therefore constitutes a crucial step
towards understanding the role of non-helical spin states in such
heterostructures.

##  VII Acknowledgments

This research was supported by NSF via Grants No. DMR-1410741 (D.J.A., M.M.A.,
and I.V.) and No. DMR-1151717 (M.M.A. and D.E.S.).

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