# Long-range Cooper pair splitting by chiral Majorana edge states

Oscar Casas-Barrera  Escuela de Física, Universidad Pedagógica y Tecnológica
de Colombia (UPTC), Tunja, Colombia  Departamento de Física, Universidad
Nacional de Colombia, Bogotá, Colombia  Shirley Gómez Páez  Departamento de
Física, Universidad Nacional de Colombia, Bogotá, Colombia  Departamento de
Física, Universidad el Bosque, Bogotá, Colombia  William J. Herrera
Departamento de Física, Universidad Nacional de Colombia, Bogotá, Colombia
Escuela de Física, Universidad Pedagógica y Tecnológica de Colombia (UPTC),
Tunja, Colombia  Departamento de Física, Universidad Nacional de Colombia,
Bogotá, Colombia  Departamento de Física, Universidad el Bosque, Bogotá,
Colombia  Departamento de Física, Universidad Nacional de Colombia, Bogotá,
Colombia

###### Abstract

We analyze the transport properties of a Cooper pair splitter device made up
of two-point electrodes in contact with a ferromagnetic/superconductor (F/S)
junction built on the surface of a topological insulator. For the pair
potential of the S region we consider $s$\- and $d$-wave symmetries, and for
the F region, we regard a magnetization vector normal to the TI surface. The
non-local transport along the F/S interface is mediated by chiral Majorana
edge states, with chirality controlled by the magnetization vector
polarization. We show that the crossed Andreev reflection amplitude slowly
decays with the separation of the electrodes for standard clean samples. Our
system would reach a maximum Cooper pair-splitting efficiency of 80% for a
symmetrical voltage configuration, even in high-temperature superconductors
devices.

##  I Introduction

In recent years, various solid-state systems with entangled electrons have
been proposed for potential applications in quantum teleportation and
computing [1]. Among these standout Cooper pairs splitter devices in which
electrons in pairs are stretched and extracted from the superconductor through
non-local processes known as crossed Andreev reflections (CAR) [2, 3, 4]. The
basic structure of such devices consists of a superconducting region (S)
connected to two normal electrodes (N) separated by a distance of the order of
the superconducting coherence length $\xi_{0}$ [2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29].
Various designs include intermediate quantum dots [30, 31, 32, 33, 34, 35, 36,
37, 38, 39], anisotropic superconductivity [2, 11, 40, 41, 42], and some
devices have been realized experimentally [14, 15, 17, 19, 32, 22, 33, 34].
For subgap voltages, CARs compete with local Andreev reflections (AR) and
elastic co-tunneling (EC) between electrodes. Then, the pair-splitting
efficiency is highly dependent on suppressing these secondary processes by
adequately setting the system parameters. Although some systems could reach an
efficiency of 100% under ideal conditions, the impurities or defects present
in samples give rise to quantum noise and decoherence processes [43, 44].

Topological superconductors (TS) constitute a possible solution to this
problem by presenting surface Andreev bound states (SABS) topologically
protected against perturbations that preserve the discrete symmetries of the
system [45, 46, 47, 48, 49, 50, 51, 52]. Zero-energy SABS are Majorana modes,
quasiparticles that are their antiparticle and exhibit non-trivial statistics,
the reason why they could be implemented in topological quantum computing
devices [53, 54, 55, 56, 57, 58]. Additionally, placing a conventional
superconductor in contact with the surface of a topological insulator(TI)
gives rise to an artificial TS phase [59, 60, 61, 62, 63, 64, 51, 65]. The
resulting superconducting order parameter has spinless $p$-wave chiral
symmetry, giving rise to chiral Majorana edge states (CMES) at the interface
with a ferromagnetic region (F) [59, 61, 62, 63, 64, 51].

In the last decade, some Cooper pairs splitter devices on the surface of a TI
have been proposed, most of them F/SF planar junctions where exchange fields
in the F regions control the CAR processes efficiency, their doping levels,
the interfaces transparency, or the pair potential symmetry ($s$ o $d$) [66,
67, 68, 69, 70]. However, transport across interfaces in this configuration
gives rise to an oscillating CAR transmission coefficient that decays rapidly
over distances of the order of $\xi_{0}$ due to the mediation of evanescent
states. Nevertheless, it has been found by numerical calculations that the
presence of a TS phase with chiral $p$-wave symmetry in Sr2RuO4 may give rise
to a non-local, unidirectional transport independent of the electrode spacing
[71].

In this work, we investigate the spatial dependence and Cooper pair-splitting
efficiency of CMES-mediated CAR processes at the interface of an F/S planar
junction on the surface of a TI. In the F region, the TI is in contact with a
ferromagnetic insulator with a magnetization vector normal to the surface. In
the S region, the TI is proximitized by an intrinsic s- or d-wave
superconductor. For $s$ and $d_{x^{2}-y^{2}}$ symmetries, it is observed that
the CMES gives rise to a non-oscillating long-range unidirectional CAR
transport between the electrodes with a decay length of the order of
$10^{2}\xi_{0}$ for samples with a typical degree of impurities, similar to
that found in [71]. However, in our system, the chirality can be controlled by
the magnetization polarization of the F region, and the superconductivity
would be more robust to impurities than Sr2RuO4 [72, 73]. For a Cooper pairs
splitter configuration, we suppress EC processes by setting symmetrical bias
voltages in the electrodes, thus reaching a maximum splitting efficiency of
$80\%$.

This article is structured as follows: Section II discusses the model of the
system and contains a derivation of each region’s Hamiltonian and transport
observables. Section III studies a toy model for infinite magnetization,
analyzes the dispersion relations of CMES, and finds expressions for the
transmission coefficients and the splitting efficiency as a function of the
separation between electrodes. Section IV presents the numerical calculations
of observables for a system with finite magnetization and analyzes the
dependence of the CAR transmission coefficient on the separation between
electrodes and the magnetization value. Finally, section V presents the
conclusions of this work.

##  II Theoretical framework and transport observables

The characteristic of the Cooper pairs splitter device under analysis in this
work can be seen in Fig. 1. The system consists of an F/S planar junction
constructed on the surface of a TI in contact with two thin metallic
electrodes. The F/S junction is built by proximity effect on the TI surface as
follows: the left region (L) is put in contact with a ferromagnetic insulator
(F), and the right region (R) with a spin-singlet intrinsic superconductor
(S), either conventional ($s$-wave) or high-temperature ($d$-wave), so that
the induced superconducting order parameter has the form
$\hat{\Delta}=\Delta_{0}$cos$\left[\beta\left(2\theta+\alpha\right)\right]i\sigma_{y}$,
where $\theta$ is the polar angular coordinate of the wave vector
$\mathbf{k}=(k,q)$ and $\sigma_{y}$ is the second Pauli matrix in spin space
($\beta=0$ for $s$-wave symmetry, $\beta=1$ for $d$-wave symmetry, $\alpha=0$
for $d_{x^{2}-y^{2}}$ and $\alpha=\pi/4$ for $d_{xy}$). The two thin metal
electrodes denoted by $a$ and $b$ are subjected to a voltage bias $V_{a(b)}$
and are separated by a certain distance $d$ on the TI surface in region F. For
subgap voltages, an electron incident on an electrode can be: reflected as a
hole in the same electrode (AR process), transmitted as an electron to the
second electrode (EC process), or reflected as a hole at the second electrode
(CAR process). The last process corresponds to Cooper pair-splitting under
time reversal.

Figure 1: Cooper pairs splitter device formed by a F/S junction on the surface
of a TI and two metallic electrodes separated by a distance $d$. The F/S
interface is built by proximity effect on the TI. CAR transport between
electrodes is mainly mediated by the CMES at the F/S interface.

The elementary excitations of the superconducting state in the S region are
described by the Bogoliubov-de Genes (BdG) Hamiltonian [59, 47, 48]

$\mathcal{H}_{BdG}\left(\mathbf{r}\right)=\left(\begin{array}[]{cc}\mathcal{H}_{s}\left(\mathbf{r}\right)-E_{F}&\hat{\Delta}\\\
\hat{\Delta}^{{\dagger}}&E_{F}-\mathcal{H}_{s}^{T}\left(-\mathbf{r}\right)\end{array}\right)\text{,}$
(1)

where
$\mathcal{H}_{s}\left(\mathbf{r}\right)=v_{F}\left(\boldsymbol{\sigma}\times\mathbf{\hat{p}}\right)_{z}$
is the effective Hamiltonian of the TI surface around the $\Gamma$ point,
$v_{F}$ is the speed of the charge carriers at the Fermi level,
$\mathbf{\hat{p}}=-i\mathbf{\hbar\boldsymbol{\nabla_{\mathbf{r}}}}$ is the
momentum operator, and
$\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the Pauli vector
in the spin subspace [74, 75]. The associated excitation spectrum is given by

$E_{n}\left(\mathbf{k}\right)=\sqrt{\left(n\hbar
v\left|\mathbf{k}\right|-E_{F}\right)^{2}+|\hat{\Delta}|^{2}}\text{,}$ (2)

where $n=\pm 1$ is a band index and
$|\hat{\Delta}|=\textrm{Tr}(\hat{\Delta}^{\dagger}\hat{\Delta})/2$ [76]. At
the weak coupling limit ($E_{F}\gg\Delta_{0}$), the properties of the system
are mainly determined by the states near the Fermi surface ($n=+1$ and
$\left|\mathbf{k}\right|\simeq k_{F}$). When projecting
$\mathcal{H}_{BdG}\left(\mathbf{k}\right)$ in this base of states
($\psi_{e}=\left(1,-i\mathrm{e}^{i\theta},0,0\right)^{T}/\sqrt{2}$ and
$\psi_{h}=\left(0,0,-1,i\mathrm{e}^{-i\theta}\right)^{T}/\sqrt{2}$), an
effective spinless $\Delta$ with mixed symmetry is obtained

$\Delta_{eff}=\left\langle\psi_{e}\left|\mathcal{H}_{BdG}\left(\mathbf{k}\right)\right|\psi_{h}\right\rangle=i\Delta_{0}\cos\left[\beta\left(2\theta+\alpha\right)\right]\mathrm{e}^{-i\theta}\text{.}$
(3)

For the normal region F, we have $\hat{\Delta}\rightarrow 0$ in (1) and the
exchange field perpendicular to the surface $\mathbf{M}=M\mathbf{\hat{z}}$
introduces an additional Zeeman-type term in the $\mathcal{H}_{s}$ Hamiltonian

$\displaystyle\mathcal{H}_{s}$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{H}_{s}+\mathcal{H}_{Z}\text{,}$ (4)
$\displaystyle\mathcal{H}_{Z}$ $\displaystyle=$
$\displaystyle\mathbf{M}\cdot\boldsymbol{\sigma}=M\sigma_{z}\text{,}$

which breaks the time reversal symmetry of (1) at the F/S interface and
induces topologically protected CMES [59, 60, 61]. It will be assumed that
$E_{F}=0$ for region F to ensure that transport takes place only on the TI
surface [61].

The transport properties of this type of nanostructure can be described by the
Hamiltonian approximation, where each region of the system is coupled to
adjacent regions through a tight-binding Hamiltonian [77, 78] (see appendix
D). In this formalism, the stationary transport properties can be expressed in
terms of the equilibrium Green functions of the system, which are calculated
by applying a Dyson equation to the Green functions of adjacent regions, and
for planar junctions, these can be obtained analytically by the asymptotic
solutions method [79, 80] (see appendices A to C).

For low voltages ($eV_{a(b)}\ll\Delta$), the local current $I_{i}$ through
electrode $i$ involves the contribution of AR processes

$\displaystyle I_{i,AR}$ $\displaystyle=$
$\displaystyle\frac{2e}{h}\int\mathrm{d}E\left(\tilde{n}_{i,e}-\tilde{n}_{i,h}\right)R_{i,AR}\text{,}$
(5) $\displaystyle R_{i,AR}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left(\bar{\rho}_{i,e}\hat{G}_{ii,eh}^{r}\bar{\rho}_{i,h}\hat{G}_{ii,eh}^{a}\right)\text{,}$
(6)

where $R_{i,AR}$ is the local Andreev reflection coefficient,
$\hat{G}_{ij}^{a/r}$ are the advanced/retarded Green functions of the system
between electrodes $i$ and $j$,
$\bar{\rho}_{i}=2\pi\hat{t}_{i}^{{\dagger}}\hat{\rho}_{i}\hat{t}_{i}$ with
$\hat{t}_{i}$ the hopping matrix between electrode $i$ and junction F/S,
$\hat{\rho}_{i}$ the density of states matrix of the electrode $i$ in spin
space and
$\tilde{n}_{i,\mu}(E)=n_{i,\mu}(E)\tilde{\sigma}_{0}=\mathrm{diag}(n_{i,\mu\uparrow}(E),n_{i,\mu\downarrow}(E))$
is the quasiparticle occupancy matrix in the electrode $i$, with
$n_{i,e/h,\uparrow\downarrow}\left(E\right)=f\left(E\mp eV_{m}\right)$ the
Fermi-Dirac distribution function. On the other hand, the non-local current
$I_{ij}$ includes the contribution of the CAR and EC processes

$I_{ij}=I_{CAR}+I_{EC}\text{,}$ (7) $\displaystyle I_{CAR}$ $\displaystyle=$
$\displaystyle\frac{2e}{h}\int\mathrm{d}E\left(n_{i,e}-n_{j,h}\right)T_{CAR}\text{,}$
(8) $\displaystyle I_{EC}$ $\displaystyle=$
$\displaystyle\frac{2e}{h}\int\mathrm{d}E\left(n_{i,e}-n_{j,e}\right)T_{EC}\text{,}$
(9)

where $T_{CAR}$ and $T_{EC}$ are the respective transmission coefficients of
these processes

$\displaystyle T_{CAR}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left(\bar{\rho}_{i,e}\hat{G}_{ij,eh}^{r}\bar{\rho}_{j,h}\hat{G}_{ji,eh}^{a}\right)\text{,}$
(10) $\displaystyle T_{EC}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left(\bar{\rho}_{i,e}\hat{G}_{ij,ee}^{r}\bar{\rho}_{j,\mu}\hat{G}_{ji,ee}^{a}\right)\text{.}$
(11)

The EC current cancels for a symmetric electrodes voltages configuration
($V_{a}=V_{b}=V$), and the device works in Cooper pairs splitter mode [33, 24,
38], hence there is only competition between CAR and AR processes. The total
differential conductance across the electrode $i$ is $G_{i}=dI_{i}/dV$ and is
given by

$G_{i}\left(V\right)=G_{0}\left(T_{CAR}\left(V\right)+T_{CAR}\left(-V\right)+2R_{i,AR}\left(V\right)\right)\text{,}$
(12)

where $G_{0}=2e^{2}/h$, then, the Cooper pair-splitting efficiency is defined
as

$\eta=\frac{T_{CAR}\left(V\right)+T_{CAR}\left(-V\right)}{T_{CAR}\left(V\right)+T_{CAR}\left(-V\right)+R_{AR,a}\left(V\right)+R_{AR,b}\left(V\right)}\text{.}$
(13)

##  III Toy Model

First, we consider the system in Fig. 1 at the high-doping limit of the S
region ($E_{FR}\gg\Delta_{0}$,
$\mathrm{e}^{i\theta_{e}}\simeq\mathrm{e}^{i\theta_{h}}=\mathrm{e}^{i\theta}$)
and high magnetization of F region ($M\rightarrow-\infty$, for details, see
appendix A). Under these conditions, the Green function evaluated at the point
of contact of the electrodes with the F/S interface is given by (44) con
$x_{a}=x_{b}=0$, $y_{b}-y_{a}=d$, where the integrand
$\hat{G}_{RR}^{r,a}\left(E,0,0,q\right)=\hat{G}_{S0}^{r,a}\left(E,q\right)$ is
calculated by the expression (25)

$\hat{G}_{S0}^{r,a}\left(E,q\right)=\frac{-1}{\hbar
v_{F}}\left(\begin{array}[]{cccc}M&0&N\mathrm{e}^{-i\phi_{+}}&0\\\ 1&0&0&0\\\
N\mathrm{e}^{i\phi_{-}}&0&M&0\\\ 0&0&-1&0\end{array}\right)\text{,}$ (14)

with

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{i}{D}\left(1-\gamma^{2}\mathrm{e}^{-i\Delta\varphi}\right)\text{,
\ \ \
}N=-\frac{\gamma}{D}\left(\mathrm{e}^{i\theta}+\mathrm{e}^{-i\theta}\right)\text{,}$
$\displaystyle D$ $\displaystyle=$
$\displaystyle\mathrm{e}^{i\theta}+\gamma^{2}\mathrm{e}^{-i\theta}\mathrm{e}^{-i\Delta\varphi}\text{,}$
(15) $\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\sqrt{\frac{E-\Omega}{E+\Omega}}\text{,\ \ \ and \ \ \
}\Omega=\sqrt{E^{2}-\left|\Delta\left(\theta\right)\right|^{2}}\text{.}$ (16)

For the case of a strong magnetic, the dispersion relation of the SABS states
barrier is obtained from the poles of the Green function (14) and is given by

$\displaystyle E$ $\displaystyle=$
$\displaystyle\pm\left|\Delta_{eff}\right|\text{{cos}}\left(\Delta\varphi/2\right)\text{,}$
(17) $\displaystyle\Delta_{+}$ $\displaystyle=$
$\displaystyle\Delta_{eff}\left(\theta\right)\text{, \ \ \ \
}\Delta_{-}=\Delta_{eff}\left(\pi-\theta\right)\text{,}$ (18)
$\displaystyle\Delta\varphi$ $\displaystyle=$
$\displaystyle\varphi_{+}-\varphi_{-}\text{, \ \ \ \
}\varphi_{\pm}=\text{{arg}}\left(\Delta_{\pm}\right)\text{,}$

with $\Delta_{eff}$ given by (3), which matches the general expression for
SABS [81] and with those found for planar F/S junctions in TIs for finite
magnetizations [59, 60, 61]. In Fig. 2 a), CMES dispersion curves are shown
for $s$-wave, $d_{x^{2}-y^{2}}$ and $d_{xy}$ symmetries. These states can also
be seen in panels (b)-(c) of Fig. 2, where spectral density $\mathcal{A}$ maps
evaluated at the interface of an F/S junction with infinite magnetization are
shown for different symmetries of the pair potential, with
$\mathcal{A}(q)=\textrm{Tr}(\hat{\rho}_{F/S}(q))$.

Figure 2: a) Surface Andreev bound states curves at the F/S interface with
infinite magnetization obtained from expression (17) ($M$
$\rightarrow-\infty$). For the other polarization functions change its parity.
Panels (b) - (d) spectral density $\mathcal{A}$ evaluated at the F/S junction
interface for a superconductor with b) $s$-wave, c) $d_{x^{2}-y^{2}}$, and d)
$d_{xy}$ symmetry. In all three cases, the presence of chiral Majorana edge
states is observed. Figure 3: a) Crossed Andreev reflection coefficient
$T_{CAR}\left(V=0\right)$ along the F/S interface as a function of distance
$d$ for $s$-wave symmetry for infinite magnetization and different values of
$\varepsilon$. All curves in panel a) have been normalized to $T_{CAR}(d=0)$ y
$E_{F}=13$ $\Delta_{0}$. b) Curves of $T_{CAR}(V=0)$ for the different
symmetries with infinite magnetization, $E_{FR}=13$ $\Delta_{0}$ and
$\varepsilon=0.002$ $\Delta_{0}$. The dashed black line represents the fit
with the analytic function (19) for $s$-wave symmetry. All curves are
normalized to the reflection coefficient $e-e$ of the junction evaluated at
the origin for $M=0$ and $\Delta_{0}=0$.

As can be seen, the dispersion relation is linear in $q$ for $s$-wave symmetry
while undulated for $d$-wave symmetries. Note that the density of states for
$s$-wave and $d_{x^{2}-y^{2}}$ symmetries presents a high value and the same
chirality in the neighborhood of $q=0$, while for $d_{xy}$ becomes symmetrical
and diffuse. In all cases, the dispersion relations are odd functions of $q$,
and the polarization of magnetization at the F region defines their chirality.
The following section shows that these states are responsible for the solid
chiral character of non-local transport processes along the interface.

For wave $s$-wave symmetry ($\beta=0$) y $V=0+i\varepsilon$
($\varepsilon\ll\Delta_{0}$) the calculation of (10) can be carried out
analytically, resulting in

$\displaystyle T_{CAR}\left(d\right)$ $\displaystyle=$ $\displaystyle
C\left[\Theta\left(sd\right)\left(1+\frac{\varepsilon}{\Delta_{0}}\right)\right]^{2}\mathrm{e}^{-d/\lambda}\text{,}$
(19) $\displaystyle C$ $\displaystyle=$
$\displaystyle\tilde{\rho}_{e,b}\tilde{\rho}_{h,a}\left(2\pi iL/\hbar
v_{F}\right)^{2}\left(k_{F}/\pi\right)^{6}\text{,}$ (20)

where $\Theta(x)$ is the Heaviside step function, $s=-\mathrm{sgn}(M)$, and
$\lambda=\Delta_{0}/2\varepsilon k_{F}=\xi_{0}(\hbar v_{F}/2\pi\varepsilon
k_{F})$ is the characteristic decay length of CAR processes along the
interface, where $\xi_{0}=\pi\Delta_{0}/\hbar v_{F}$ is the coherence length
of the conventional superconductor. The curves in Fig. 3 (a) show the behavior
of $T_{CAR}\left(d\right)$ for different values of $\varepsilon$.

Figure 4: Maps of crossed Andreev reflection coefficient $T_{CAR}(V=0)$ as a
function of the $b$ electrode coordinates over the S region with the $a$
electrode fixed at the origin (F/S interface), for infinite magnetization of
the F region and a) $s$-wave, b) $d_{x^{2}-y^{2}}$ c) $d_{xy}$ symmetry. A
logarithmic scale is used for $T_{CAR}$.

Note that the electron-hole component of the Green function has a
unidirectional behavior (odd with distance) and approaches the step function
in the limit $\varepsilon\rightarrow 0$. Since $\varepsilon\sim\hbar/2\tau$
with $\tau$ the quasiparticle lifetime, it is expected that for weak-disorder
junctions, these processes have a long-range (decay length of the order of
$10^{2}\xi_{0}$), analogous to that obtained in a 2D intrinsic $p$-wave chiral
superconductor (appendix E) or a 3D one [71]. In this case, the propagating
CMES present along the F/S interface favorably mediates the CAR processes and
confers their chirality and topological protection. It is essential to
highlight that this behavior with distance contrasts with that obtained for
conventional superconductors with $s$-wave ($T_{CAR}\left(d\right)\propto
e^{2d/\pi\xi_{0}}/d^{3}$) or $d$-wave ($T_{CAR}\left(d\right)\propto 1/d^{2}$)
superconductivity [41].

Although the Green function amplitude is proportional to the Fermi level of
the S region, the probability of AR processes is a quarter of that of CAR
processes, and according to the formula (13), the percentage splitting
efficiency would be at most $80\%$ for symmetrical voltages ($\Theta(0)=1/2$).
In Fig. 3 (b), the behavior of $T_{CAR}(V=0)$ for infinite magnetization and
different symmetries of the pair potential is observed. For $d_{x^{2}-y^{2}}$
symmetry case, the result is qualitatively similar to that of the $s$-wave
symmetry due to the mediation of CMES around $q=0$ [Fig. 2 (c)], and because
$\Delta\varphi=0$ for most of the integration interval (except in the
neighborhood of $\theta=\pm\pi/4$ nodes), although this presents some small
additional oscillations due to the angular dependence of
$\Delta\left(\theta\right)$. Both symmetries, $s$-wave and $d_{x^{2}-y^{2}}$
are close to the analytical result (19).

For the $d_{xy}$ case, the overall amplitude is lower than in the previous
cases, and $T_{CAR}(V=0)$ is an even function of $d$ with two small maxima
flanking $d=0$. This is attributed to the two low-density counter-propagating
CMES present around $q=0$ [Fig. 2(d)], and mathematically because
$\Delta\varphi=\pi$ for the integration interval, which leads to real roots
for $D$ and the impossibility of analytical integration. Furthermore,
numerical computation shows an even behavior with the distance that decreases
proportionally to $\sim 1/d^{2}$ as in the conventional $d_{xy}$ case [41].

All these behaviors can also be visualized in Fig. 4, where maps of
$T_{CAR}\left(V=0\right)$ are shown for the S region as a function of
$\mathbf{r}_{b}\mathbf{=}(x,y)$ with $\mathbf{r}_{a}$ fixed at the F/S
interface with infinite magnetization. As can be seen in panels a) and b) for
$s$-wave and $d_{x^{2}-y^{2}}$ symmetries respectively, the strong chirality
of the non-local transport is due to the presence of the CMES around
$\mathbf{k}=0$, while for $d_{xy}$ case a symmetrical interference pattern is
observed due to the low spectral density of these states at $q=0$.

##  IV Cooper pairs splitter device with finite magnetization in the F region

Figure 5: Crossed Andreev reflection coefficient $T_{CAR}(V=0)$ for $s$-wave
symmetry a) as a function of $x$ for $y=1.7$ $\xi_{0}$, and b) as a function
of $y$ for the points labeled in panel a). In both figures $M=-10$
$\Delta_{0}$.

In this section, we consider the system of Fig. 1 for finite magnetization of
region F. In panel a) of Fig. 5, it can be seen that $T_{CAR}$ decays in an
oscillatory manner towards the interior of the region S over distances of the
order of $5\xi_{0}$. In contrast, it decays smoothly towards the region F
($x<0$) with a characteristic length that decreases with the value of
$\left|M\right|$. Along the interface, $T_{CAR}$ exhibits the behavior
discussed above along the F/S interface for $s$-wave, as can be seen in panel
b) for the maxima of panel a) in Fig. 5. These results for $s$-wave symmetry
show that propagating CMES give rise to long-range CAR transport along the F/S
interface, and for $d_{x^{2}-y^{2}}$ symmetry similar results were obtained
despite the presence of nodes in $\Delta(\theta)$ since longitudinal transport
is carried out mainly along the lobes of $\Delta(\theta)$, which opens the
possibility of obtaining long-range CAR in high-temperature superconductors
(HTS).

Figure 6: Crossed Andreev reflection coefficient $T_{CAR}(V=0)$ for a)
$s$-wave symmetry and b) graphs for $d_{x^{2}-y^{2}}$ and different values of
magnetization ($E_{FR}=13$ $\Delta_{0}$, $\varepsilon=0.002$ $\Delta_{0}$). In
both figures the curves are normalized to $R_{ee}$ at the interface for
$\Delta_{0}=M=0$. Figure 7: Cooper pair-splitting efficiency $\eta$ as a
function of a) the electrode separation $d$ for $s$-wave symmetry and
different values of $M$; b) as function of the magnetization for $s$\- and
$d$-wave symmetries and $d=1.6$ $\xi_{0}$.

In Fig. 6, we analyze the effect of the magnitude of $\mathbf{M}$ on the CAR
processes probability at the interface. Panel a) shows the curves of
$T_{CAR}(V=0)$ for $s$-wave symmetry as a function of $d$ for some
magnetization values. As seen in this figure, the maximum probability of CAR
processes increases with the value of $\left|M\right|$, but in a decelerated
manner until reaching a limit value for $\left|M\right|=\infty$. Similar
behavior were observed for $d_{x^{2}-y^{2}}$ symmetry in panel b), although
with the undulations described above for Fig. 3. Another important aspect is
that the decay length of $T_{CAR}$ decreases with magnetization until reaching
the minimum value of $\lambda=\Delta_{0}/2\varepsilon k_{F}$ for the infinite
case.

Additionally, the Fig. 6 a) shows the Cooper pair-splitting efficiency as a
function of the distance along the F/S interface for various values of $M$. As
can be seen, the efficiency becomes increasingly chiral as the magnetization
increases, at the same time as its maximum value between $0<d<\xi_{0}$
gradually increases from $0.5$ to $\sim$ $0.8$. As can be appreciated in panel
b), the efficiency rapidly reaches its maximum for magnetization values on the
order of $\Delta_{0}$, and from this value, it remains constant for $s$-wave
and $d_{x^{2}-y^{2}}$ symmetries.

##  V Conclusions

We have examined the efficiency of crossed Andreev reflection processes in a
Cooper pairs splitter device made up of two thin metal electrodes in contact
with an F/S junction built on the surface of a TI by proximity effect. We have
considered a magnetization vector normal to the TI surface for the F region
and an induced $s$\- and $d$-wave superconducting order parameter for the S
region. The compelling spinless chiral $p$-wave symmetry at the F/S interface
presents chiral Majorana edge states, whose chirality can be controlled by the
magnetization polarization. With a simple analytical model, we have shown that
the probability of crossed Andreev reflection processes mediated by chiral
Majorana edge states along the F/S interface does not oscillate with the
electrode separation and decays slowly in low-disorder samples (long-range
behavior). Under symmetrical voltages configuration, our system would reach
maximum splitting efficiency of 80% and is stable with the electrode
separation for $M\sim\Delta_{0}$. Our results are also valid for the
$d_{x^{2}-y^{2}}$ symmetry and could be easily extrapolated to finite
temperatures, which allows its possible experimental realization in High Tc
supercondutors.

##  Appendix A Asymptotic Solutions Method for Green functions en TIs [80]

For a planar junction with translational invariance along the $y$ axis, the
advanced ($a$) and retarded ($r$) Green functions are given by

$\hat{g}^{r,a}\left(E,x,x^{\prime},y-y^{\prime}\right)=\int
dq\mathrm{e}^{iq\left(y-y^{\prime}\right)}\hat{g}^{r,a}\left(E,x,x^{\prime},q\right)/2\pi\text{,}$
(21)

where the Fourier transform on the integrand satisfies the inhomogeneous
equation

$\left[\left(E\pm
i\varepsilon\right)-\mathcal{H}_{i}\left(x,q\right)\right]\hat{g}^{r/a}\left(E,x,x^{\prime},q\right)=\delta\left(x-x^{\prime}\right)\text{,}$
(22)

with $E$ the excitation energy of the system, $q$ the conserved wave vector
along the interface and $\varepsilon$ an infinitesimal scalar.

The surface of a TI has no open boundaries, so it is always possible to choose
artificial boundary conditions as long as the corresponding $\hat{H}_{T}$
guarantees a transparent coupling between the regions on the IT surface. In
this work, we adopt the boundary conditions
$\Psi_{<,\downarrow}^{\mu}\left(x_{L}\right)=\Psi_{>,\uparrow}^{\mu}\left(x_{R}\right)=0$
for the spin components of spinors at the left $x_{L}$ and right $x_{R}$ edge
of each region, so the coupling Hamiltonian $\hat{H}_{T}$ assumes the specific
form

$\hat{H}_{T}=t\int\mathrm{d}q\hat{c}_{q,L\downarrow}^{\dagger}\hat{c}_{q,R\uparrow}\text{+h.c},$
(23)

with $t=\hbar v_{F}$ the transparent hopping amplitude between adjacent
regions. Thus, the equilibrium Green functions of the coupled system
$\hat{G}_{kl}=\hat{G}^{r/a}(x_{k},x_{l}^{\prime})$ ($x_{k}$ in region $k$) can
be determined by a Dyson equation of the form

$\hat{G}_{kl}=\hat{g}_{k}\delta_{kl}+\hat{g}_{k}\delta_{km}\hat{\Sigma}_{mn}\hat{G}_{nl}\text{,}$
(24)

where the self-energies
$\hat{\Sigma}_{LR}=\hat{\Sigma}_{RL}^{T}=t\tau_{z}(\sigma_{x}-i\sigma_{y})\equiv\hat{t}$
correspond to the matrix form of $\hat{H}_{T}$. This boundary condition
physically corresponds to a high-intensity magnetic barrier ($M$
$\rightarrow\pm\infty$), where the spinors components with spin projection
opposite to the magnetization direction become null. Then, to model large
magnetization values and doping levels for the left region, it is enough to
omit the coupling with the F region and only consider the right semi-infinite
region with an open boundary condition.

The Green function of a given region of a planar junction with translational
invariance in the transverse direction can be expressed in terms of the
asymptotic solutions of the system as

$\hat{g}(x,x^{\prime})=\left\\{\begin{array}[]{cc}\sum\limits_{\mu,\nu=e,h}\hat{C}_{\mu\nu}\Psi_{<}^{\mu}\left(x\right)\Psi_{>}^{\nu
T}\left(x^{\prime}\right)&x<x^{\prime}\\\
\sum\limits_{\mu,\nu=e,h}\hat{C}_{\mu\nu}^{\prime}\Psi_{>}^{\nu}\left(x\right)\Psi_{<}^{\mu
T}\left(x^{\prime}\right)&x>x^{\prime}\end{array}\right.\text{,}$ (25)

where the $\hat{C}_{\mu\nu}$ ($\mu,\nu=e,h$) are matrix coefficients
determined by equation (22), and the $\Psi_{<,>}^{\mu}\left(x\right)$ are
asymptotic solutions of $\mathcal{H}_{i}$, that is, solutions that obey
certain boundary conditions at the left ($<$) and right ($>$) edge of region
$i$. For the right semi-infinite superconducting region S, these solutions
involve processes as conventional reflections ($\mu=\nu$) and branch exchange
$e-h$ ($\mu\neq\nu$) at the interface

$\displaystyle\Psi_{<}^{e}\left(x\right)$ $\displaystyle=$
$\displaystyle\psi_{-}^{e}\left(x\right)+r_{L}^{ee}\psi_{+}^{e}\left(x\right)+r_{L}^{eh}\psi_{-}^{h}\left(x\right)\text{,}$
(26) $\displaystyle\Psi_{<}^{h}\left(x\right)$ $\displaystyle=$
$\displaystyle\psi_{+}^{h}\left(x\right)+r_{L}^{hh}\psi_{-}^{h}\left(x\right)+r_{L}^{he}\psi_{+}^{e}\left(x\right)\text{,}$
$\displaystyle\Psi_{>}^{e}\left(x\right)$ $\displaystyle=$
$\displaystyle\psi_{+}^{e}\left(x\right)\text{, \
}\Psi_{>}^{h}\left(x\right)=\psi_{-}^{h}\left(x\right)\text{,}$ (27)

where $\psi_{\eta}^{\mu}\left(x\right)$ are eigensolutions of
$\mathcal{H}_{BdG}\left(x,q\right)$ that propagate in the
$\eta\mathbf{\hat{x}}$ ($\eta=\pm 1$) direction, and the $r_{i}^{\mu\nu}$ are
the reflection coefficients on the left ($L$) or right ($R$) edge defined by
the chosen boundary conditions. For the left magnetic normal region F there
are no $e-h$ conversion processes ($r_{R}^{eh}=r_{R}^{he}=0$)

$\displaystyle\Psi_{>}^{e}\left(x\right)$ $\displaystyle=$
$\displaystyle\psi_{+}^{e}\left(x\right)+r_{R}^{ee}\psi_{-}^{e}\left(x\right)\text{,}$
(28) $\displaystyle\Psi_{>}^{h}\left(x\right)$ $\displaystyle=$
$\displaystyle\psi_{-}^{h}\left(x\right)+r_{R}^{hh}\psi_{+}^{h}\left(x\right)\text{,}$
$\displaystyle\Psi_{<}^{e}\left(x\right)$ $\displaystyle=$
$\displaystyle\psi_{-}^{e}\left(x\right)\text{, \
}\Psi_{<}^{h}\left(x\right)=\psi_{+}^{h}\left(x\right)\text{,}$ (29)

##  Appendix B Parameters of the magnetic region’s Green function

The spectrum of Hamiltonian (1) for a magnetic region in a TI surface is

$E_{e/h}=\pm\left(\sqrt{\left(\hbar
v_{F}\left|\mathbf{k}\right|\right)^{2}+M^{2}}-E_{F}\right)\text{,}$ (30)

with eigenspinors

$\psi_{\varepsilon}^{e}\left(\mathbf{r}\right)=\mathrm{e}^{iqy}\mathrm{e}^{\varepsilon
ik_{e}x}\left(\hat{\varphi}_{\varepsilon}^{e},0\right)^{T}\text{, \ \ \
}\psi_{\varepsilon}^{h}\left(\mathbf{r}\right)=\mathrm{e}^{iqy}\mathrm{e}^{\varepsilon
ik_{h}x}\left(0,\hat{\varphi}_{\varepsilon}^{h}\right)^{T}\text{,}$ (31)

where

$\displaystyle\hat{\varphi}_{\varepsilon}^{e}$ $\displaystyle=$
$\displaystyle\left(M_{+}^{e},-\varepsilon iM_{-}^{e}\mathrm{e}^{\varepsilon
i\alpha_{e}}\right)^{T}/\sqrt{2}\text{,}$ (32)
$\displaystyle\hat{\varphi}_{\varepsilon}^{h}$ $\displaystyle=$
$\displaystyle\left(\varepsilon iM_{+}^{h}\mathrm{e}^{\varepsilon
i\alpha_{h}},M_{-}^{h}\right)^{T}/\sqrt{2}\text{,}$ (33) $\displaystyle
M_{\pm}^{e}$ $\displaystyle=$ $\displaystyle\sqrt{E+E_{F}\pm
M}/\sqrt{E+E_{F}}\text{,}$ (34) $\displaystyle M_{\pm}^{h}$ $\displaystyle=$
$\displaystyle\sqrt{E_{F}-E\pm M}/\sqrt{E_{F}-E}\text{,}$ (35)
$\displaystyle\mathrm{e}^{i\alpha_{\mu}}$ $\displaystyle\equiv$
$\displaystyle\hbar
v_{F}\left(k_{\mu}+iq\right)/\left|\mathbf{k}\right|\text{,}$ (36)

($\mu=e,h$) and with wave vector in $x$ direction

$k_{e/h}=\mathrm{sgn}\left(E_{F}\pm E\right)\sqrt{\frac{\left(E_{F}\pm
E\right)^{2}-M^{2}}{\left(\hbar v_{F}\right)^{2}}-q^{2}}\text{,}$ (37)

where $sgn$ sets the correct sign for the valence band. For the chosen
boundary conditions, the reflection coefficients are

$r_{R}^{ee}=-1\text{, \ \ }r_{R}^{hh}=\mathrm{e}^{-2i\alpha_{h}}\text{.}$ (38)

Integrating equation (22) between $x^{\prime}-0^{+}$ and $x^{\prime}+0^{+}$,
we get the following constraint relation

$\hat{g}\left(x^{\prime}+0^{+},x^{\prime}\right)-\hat{g}\left(x^{\prime}-0^{+},x^{\prime}\right)=\frac{i}{\hbar
v_{F}}\left(\tau_{z}\otimes\sigma_{y}\right)\text{.}$ (39)

From this expression, we obtain the matrix coefficients of (25) for this
region

$\displaystyle\hat{C}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\hat{C}_{\mu\mu}\delta_{\mu\nu}\text{, \ \
}\hat{C}_{\mu\mu}=\hat{C}_{\mu\mu}^{\prime}=\hat{C}_{ee}\text{,}$ (40)
$\displaystyle\hat{C}_{ee}$ $\displaystyle=$ $\displaystyle\frac{-i}{\hbar
v_{F}}\left(\frac{N_{e}}{\text{{cos}}\alpha_{e}}\frac{\tau_{0}+\tau_{z}}{2}+\frac{N_{h}}{\text{{cos}}\alpha_{h}}\frac{\tau_{0}-\tau_{z}}{2}\right)\text{,}$
(41) $\displaystyle N_{e/h}$ $\displaystyle=$ $\displaystyle\frac{E_{F}\pm
E}{\sqrt{\left(E_{F}\pm E\right)^{2}-M^{2}}}\text{.}$ (42)

For the studied system, the contact points of the electrodes with the F region
are modeled as heavily doped magnetic leads

$\hat{g}_{i}^{r(a)}\simeq\left(\begin{array}[]{cc}\tilde{g}&0\\\
0&-\tilde{g}^{\ast}\end{array}\right)\text{, \ \ \ \ \ \
}\tilde{g}=\left(\begin{array}[]{cc}0&1\\\ 0&i\end{array}\right)\text{.}$

The effect of finite width $L$ of electrodes is modeled by a weight factor
$f\left(q\right)$ in the Fourier transform (21) of the equilibrium Green
functions of the junction [41, 82]

$\displaystyle\hat{G}_{ij}^{r,a}\left(E\right)$ $\displaystyle\rightarrow$
(43) $\displaystyle\int$ $\displaystyle\frac{dq}{2\pi}$
$\displaystyle\left|f\left(q\right)\right|^{2}\mathrm{e}^{iq\left(y_{j}-y_{i}\right)}\hat{G}^{r,a}\left(E,x_{j},x_{i},q\right)\text{,}$
(44) $\displaystyle f\left(q\right)$ $\displaystyle=$
$\displaystyle\left\langle k_{1}|q\right\rangle\sqrt{k_{F}^{2}-q^{2}}\text{, \
\ }$ (45) $\displaystyle\left\langle k_{1}|q\right\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\pi}{L_{y}^{3}}}\frac{\text{{cos}}(qL/2)}{k_{1}^{2}-q^{2}}\text{.}$
(46)

This factor is proportional to the transverse wave vector of states at the
superconducting region and depends only on the first transverse mode of the
electrode. ($k_{1}=\pi/L$).

##  Appendix C Parameters of the superconducting region’s Green function

The eigenspinors of Hamiltonian (1) are given by

$\displaystyle\psi_{\varepsilon}^{e}\left(x\right)$
$\displaystyle=\mathrm{e}^{\varepsilon
ik_{e}x}\left(u_{0}\hat{\varphi}_{\varepsilon}^{e},-iv_{0}\text{{e}}^{-i\phi_{\varepsilon}}\sigma_{y}\hat{\varphi}_{\varepsilon}^{e}\right)^{T}\text{,}$
(47) $\displaystyle\psi_{\varepsilon}^{h}\left(x\right)$
$\displaystyle=\mathrm{e}^{\varepsilon
ik_{h}x}\left(-iv_{0}\sigma_{y}\hat{\varphi}_{\varepsilon}^{h},u_{0}\text{{e}}^{-i\phi_{\varepsilon}}\hat{\varphi}_{\varepsilon}^{h}\right)^{T}\text{,}$

where the coherence factors are defined as

$u_{0}=\sqrt{\frac{1}{2}\left(1+\frac{\Omega}{E}\right)}\text{,}\,\,v_{0}=\sqrt{\frac{1}{2}\left(1-\frac{\Omega}{E}\right)}\text{,}\,\,\Omega=\sqrt{E^{2}-\left|\Delta\right|^{2}}\text{,}$
(48)

and the spinors $\hat{\varphi}_{\varepsilon}^{\mu}$ ($\mu=e,h$) by the
expressions

$\displaystyle\hat{\varphi}_{\varepsilon}^{e}$ $\displaystyle=$
$\displaystyle\left(1,-\varepsilon i\mathrm{e}^{\varepsilon
i\alpha_{e}}\right)^{T}/\sqrt{2}\text{,}$ (49)
$\displaystyle\hat{\varphi}_{\varepsilon}^{h}$ $\displaystyle=$
$\displaystyle\left(\varepsilon i\mathrm{e}^{\varepsilon
i\alpha_{h}},1\right)^{T}/\sqrt{2}\text{,}$ (50)

with wave vector $x$

$k_{e/h}=\sqrt{\frac{\left(E_{F}\pm\Omega\right)^{2}}{\hbar^{2}v_{F}^{2}}-q^{2}}\text{.}$
(51)

For the chosen boundary conditions, the reflection coefficients are:

$\displaystyle r_{L}^{ee}$ $\displaystyle=$
$\displaystyle\frac{1}{Y}\left(\mathrm{e}^{-i\alpha_{e}}-\gamma^{2}\mathrm{e}^{-i\alpha_{h}}\right)\mathrm{e}^{-i\phi_{-}}\text{,}$
$\displaystyle r_{L}^{eh}$ $\displaystyle=$
$\displaystyle-\frac{1}{Y}\gamma\left(\mathrm{e}^{-i\alpha_{e}}\mathrm{e}^{-i\phi_{+}}+\mathrm{e}^{i\alpha_{e}}\mathrm{e}^{-i\phi_{-}}\right)\text{,}$
$\displaystyle r_{L}^{he}$ $\displaystyle=$
$\displaystyle-\frac{1}{Y}\gamma\left(\mathrm{e}^{-i\alpha_{h}}\mathrm{e}^{-i\phi_{+}}+\mathrm{e}^{i\alpha_{h}}\mathrm{e}^{-i\phi_{-}}\right)\text{,}$
$\displaystyle r_{L}^{hh}$ $\displaystyle=$
$\displaystyle-\frac{1}{Y}\left(\mathrm{e}^{i\alpha_{e}}-\gamma^{2}\mathrm{e}^{i\alpha_{h}}\right)\mathrm{e}^{-i\phi_{+}}\text{,}$
$\displaystyle Y$ $\displaystyle=$
$\displaystyle\gamma^{2}\mathrm{e}^{-i\alpha_{h}}\mathrm{e}^{-i\phi_{+}}+\mathrm{e}^{i\alpha_{e}}\mathrm{e}^{-i\phi_{-}}\text{,
\ \ }\gamma=v_{0}/u_{0}\text{.}$

In this case the matrix coefficients of (25) are given by

$\displaystyle\hat{C}_{ee}$ $\displaystyle=$ $\displaystyle-\frac{1}{\hbar
v}\frac{1}{C^{2}+BF-AH}\left(\begin{array}[]{cccc}H&0&-C&B\\\ 0&H&-F&-C\\\
C&B&-A&0\\\ -F&C&0&-A\end{array}\right)\text{,}$ $\displaystyle\hat{C}_{hh}$
$\displaystyle=$ $\displaystyle\lambda\hat{C}_{ee}\text{, \ \
}\hat{C}_{eh}=\hat{C}_{he}=0\text{,}$

where

$\displaystyle A$ $\displaystyle=$ $\displaystyle
i\left(\mathrm{e}^{-i\alpha_{e}}+\mathrm{e}^{i\alpha_{e}}\right)-\lambda
i\gamma^{2}\left(\mathrm{e}^{-i\alpha_{h}}+\mathrm{e}^{i\alpha_{h}}\right)\text{,}$
$\displaystyle B$ $\displaystyle=$ $\displaystyle
i\gamma\lambda\left(\mathrm{e}^{i\alpha_{h}}\mathrm{e}^{-i\phi_{+}}+\mathrm{e}^{-i\alpha_{h}}\mathrm{e}^{-i\phi_{-}}\right)$
$\displaystyle-$ $\displaystyle
i\gamma\left(\mathrm{e}^{i\alpha_{e}}\mathrm{e}^{-i\phi_{+}}+\mathrm{e}^{-i\alpha_{e}}\mathrm{e}^{-i\phi_{-}}\right)\text{,}$
$\displaystyle C$ $\displaystyle=$
$\displaystyle\gamma\left(\mathrm{e}^{-i\phi_{+}}-\mathrm{e}^{-i\phi_{-}}\right)\left(\lambda-1\right)\text{,}$
$\displaystyle F$ $\displaystyle=$ $\displaystyle
i\gamma\lambda\left(\mathrm{e}^{-i\alpha_{h}}\mathrm{e}^{-i\phi_{+}}+\mathrm{e}^{i\alpha_{h}}\mathrm{e}^{-i\phi_{-}}\right)$
$\displaystyle-$ $\displaystyle
i\gamma\left(\mathrm{e}^{-i\alpha_{e}}\mathrm{e}^{-i\phi_{+}}+\mathrm{e}^{i\alpha_{e}}\mathrm{e}^{-i\phi_{-}}\right)\text{,}$
$\displaystyle H$ $\displaystyle=$ $\displaystyle
i\gamma^{2}\left(\mathrm{e}^{-i\alpha_{e}}+\mathrm{e}^{i\alpha_{e}}\right)\mathrm{e}^{-i\phi_{+}}\mathrm{e}^{-i\phi_{-}}$
$\displaystyle-$ $\displaystyle\lambda
i\left(\mathrm{e}^{-i\alpha_{h}}+\mathrm{e}^{i\alpha_{h}}\right)\mathrm{e}^{-i\phi_{+}}\mathrm{e}^{-i\phi_{-}}\text{,}$
$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle
r_{L}^{eh}/r_{L}^{he}\text{.}$

##  Appendix D Hamiltonian approach and calculation of the non-local current
of the system

For a superconducting system with two contacts (as in a Cooper pairs-splitter
device), the Hamiltonian has the form [77, 78, 79, 83]

$H=H_{a}+H_{b}+H_{s}+H_{as}+H_{bs}\text{,}$ (53)

with $H_{a,b,s}$ the Hamiltonians of normal electrodes $a$, $b$ and the
superconducting region $s$, and $H_{is}$ the hopping Hamiltonians between the
point electrode $i$ and region $s$

$H_{is}=t_{i}\sum_{\sigma,\sigma^{\prime}}\mathrm{e}^{i\phi_{i}(\tau)/2}\hat{c}_{i\sigma}^{\dagger}\hat{b}_{i\sigma^{\prime}}+\text{h.c.,}$
(54)

where $\sigma,\sigma^{\prime}=\uparrow,\downarrow$ are the spin projection
indices in the $z$ direction,
$\phi_{i}(\tau)=\phi_{0}+2(\mu_{i}-\mu_{s})\tau/\hbar$ are the gauge phases
induced by the gradient of chemical potential in the vicinity of the electrode
contact point $i=a,b$ with region S, $t_{i}$ is the hopping amplitude at this
contact point, the $\hat{c}_{i\sigma}$ are the annihilation operators for the
electrode $i$, and $\hat{b}_{i\sigma}$ are the annihilation operators at the
point of the region S in contact with the electrode $i$. In the Heisenberg
picture the average current in contact $i$ is given by

$\displaystyle
I_{i}(\tau)=-e\left\langle\frac{d}{d\tau}N_{i}(\tau)\right\rangle$ (55)
$\displaystyle=it_{i}\frac{e}{\hbar}\sum_{\sigma,\sigma^{\prime}}\left[\left\langle\hat{c}_{i\sigma}^{{\dagger}}(\tau)\hat{b}_{i\sigma^{\prime}}(\tau)\right\rangle-\left\langle\hat{b}_{i\sigma^{\prime}}^{\dagger}(\tau)\hat{c}_{i\sigma}(\tau)\right\rangle\right]\text{,}$

which can be expressed in terms of the Keldysh Green functions as

$\displaystyle\hat{G}_{ijkl}^{\alpha\beta}\left(\tau_{\alpha},\tau_{\beta}^{\prime}\right)=-i\left\langle
T_{c}\left[\hat{D}_{q,ki}\left(\tau_{\alpha}\right)\hat{D}_{q,lj}^{{\dagger}}\left(\tau_{\beta}^{\prime}\right)\right]\right\rangle\text{,}$
(56)
$\displaystyle\hat{D}_{ki}\left(\tau\right)=\left(\hat{d}_{ki\uparrow}\left(\tau\right),\hat{d}_{ki\downarrow}\left(\tau\right),\hat{d}_{ki\uparrow}^{{\dagger}}\left(\tau\right),\hat{d}_{ki\downarrow}^{{\dagger}}\left(\tau\right)\right)^{T}\text{,}$
(57)

with $i,j=(a,b)$ the contact indices, $k,l=N,S$ the region indices
($\hat{d}_{ni\sigma}=\hat{c}_{i\sigma}$ ,
$\hat{d}_{si\sigma}=\hat{b}_{i\sigma}$), $\alpha,\beta=(+,-)$ the indices of
the Keldysh temporal contour and $T_{c}$ is the Keldysh temporal ordering
operator. Assuming normal electrodes on the surface of the TI, the hopping
Hamiltonian 54 takes the form 23, and considering the two possible choices of
the boundary conditions, the average current acquires a factor of $2$ [83,
80]. In a stationary situation, the average current in the electrode $i$ can
be expressed in the energy space as

$I_{i}=\frac{e}{h}\int\mathrm{d}E\mathrm{Tr}\left(\tau_{z}\left[\hat{t}_{i}\hat{G}_{iSN}^{+-}(E)-\hat{t}_{i}^{{\dagger}}\hat{G}_{iNS}^{+-}(E)\right]\right)\text{,}$
(58)

with $\hat{t}_{i}$ the self-energy matrix associated with 23 and where we have
used the following shorthand notation for repeated indices ($ii\rightarrow
i$). Using Dyson equations [78]

$\displaystyle\hat{G}_{iSN}^{+-}(E)$ $\displaystyle=$
$\displaystyle\hat{G}_{iS}^{+-}(E)\hat{t}_{i}^{T}\hat{g}_{iN}^{a}(E)+\hat{G}_{iS}^{r}(E)\hat{t}_{i}^{T}\hat{g}_{iN}^{+-}(E)\text{,}$
(59) $\displaystyle\hat{G}_{iNS}^{+-}(E)$ $\displaystyle=$
$\displaystyle\hat{g}_{iN}^{+-}(E)\hat{t}_{i}\hat{G}_{iS}^{a}(E)+\hat{g}_{iN}^{r}(E)\hat{t}_{i}\hat{G}_{iS}^{+-}(E)\text{,}$
(60)

the average current can be written in terms of Green functions evaluated over
a single type of region as

$\displaystyle I_{i}$ $\displaystyle=$
$\displaystyle\frac{e}{2h}\int\mathrm{d}E\mathrm{Tr}\left(\tau_{z}\hat{t}_{i}^{{\dagger}}\left[\hat{g}_{iN}^{+-}\left(E\right)\hat{t}_{i}\hat{G}_{iS}^{-+}\left(E\right)\right.\right.$
$\displaystyle\left.\left.-\hat{g}_{iN}^{-+}\left(E\right)\hat{t}_{i}\hat{G}_{iS}^{+-}\left(E\right)\right]\right)\text{,}$

where the unperturbed Keldysh Green functions of the electrodes are given by

$\displaystyle\hat{g}_{iN}^{+-}\left(E\right)$ $\displaystyle=$ $\displaystyle
2\pi i\hat{\rho}_{iN}\left(E\right)\hat{n}_{iN}\left(E\right)\text{,}$ (62)
$\displaystyle\hat{g}_{iN}^{-+}\left(E\right)$ $\displaystyle=$
$\displaystyle-2\pi
i\hat{\rho}_{iN}\left(E\right)\left(\tau_{0}-\hat{n}_{iN}\left(E\right)\right)\text{,}$
(63)

with $\hat{n}_{iN}\left(E\right)$ the occupation matrix and
$\hat{\rho}_{iN}=\mp\mathrm{Im}(\hat{g}_{i}^{r(a)})/\pi$ the density-of
-states matrix of electrode $i$, being $\hat{g}_{i}^{r(a)}$ its equilibrium
Green function.

Considering the following Dyson equation between the equilibrium and non-
equilibrium Green functions involving the equilibrium local and non-local
Green functions of the system ($\gamma=+-,-+$) [83, 80]

$\hat{G}_{q,iS}^{\gamma}=\hat{G}_{q,iS}^{r}\hat{t}_{i}^{{\dagger}}\hat{g}_{q,N}^{\gamma}\hat{t}_{i}\hat{G}_{q,iS}^{a}+\hat{G}_{q,ijS}^{r}\hat{t}_{i}\hat{g}_{q,jS}^{\gamma}\hat{t}_{i}^{{\dagger}}\hat{G}_{q,jiS}^{a}\text{,}$
(64)

the total average current induced in the electrode $i$ takes the form
$I_{i}=I_{ii}+I_{ij}$, where the local $I_{ii}$ and not local $I_{ij}$
contribution ($i\neq j$) are given by

$\displaystyle I_{ij}$ $\displaystyle=$
$\displaystyle\frac{e}{h}\int\mathrm{d}E\mathrm{Tr}\tau_{z}\bar{\rho}_{i}\left[\hat{n}_{i}\hat{G}_{ij}^{r}\bar{\rho}_{j}-\hat{G}_{ij}^{r}\bar{\rho}_{j}\hat{n}_{j}\right]\hat{G}_{ij}^{a}\text{,}$
(65)

and expressing (65) in terms of matrices in the Nambu subspace we obtain the
expressions (5) and (7).

##  Appendix E CAR processes in conventional chiral $p$-wave superconductivity

For a conventional superconductor, the element $eh$ of the Green function
evaluated at the edge of a semi-infinite region is given by

$\displaystyle\hat{G}_{S0,eh}^{r,a}\left(E,q\right)=\frac{-2im}{\hbar^{2}D}\gamma\mathrm{e}^{i\varphi_{-}}$
(66)
$\displaystyle\times\left(\frac{1}{k_{e}}+\frac{1}{k_{h}}+\frac{D\left(1-\mathrm{e}^{i\Delta\varphi}\right)}{2\left(1-\gamma^{2}\right)}\left(\frac{1}{k_{e}}-\frac{1}{k_{h}}\right)\right)\text{,}$
(67) $\displaystyle
D=\left(1-\gamma^{2}\mathrm{e}^{-i\Delta\varphi}\right)\text{,}$ (68)

with $m$ the mass of the electron. For chiral $p$-wave symmetry
$\Delta\left(\theta\right)=\Delta_{0}e^{i\theta}$, then
$e^{i\varphi_{+}}=e^{i\theta}=-e^{-i\varphi_{-}}$, and for $E=0+i\varepsilon$
and $E_{F}\gg\Delta_{0}$ ($k_{e}\approx k_{h}$) we get

$\displaystyle\hat{G}_{S0,eh}^{r,a}\left(0,\theta\right)$
$\displaystyle\approx$
$\displaystyle-\frac{4m}{\hbar^{2}}\frac{\delta\mathrm{e}^{-i\theta}}{\left(1-\delta^{2}\mathrm{e}^{-2i\theta}\right)}\frac{1}{k_{F}\text{{cos}}\theta}\text{,}$
(69) $\displaystyle\delta$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\eta-\varepsilon}{\eta+\varepsilon}}\text{, \ \ \
}\eta=\sqrt{\varepsilon^{2}+\Delta_{0}^{2}}\text{.}$

Expressing the corresponding integral (44) in terms of $q\approx
k_{F}\mathrm{sin}\theta$, we obtain for (10)

$T_{CAR}\left(d\right)=\tilde{\rho}_{e,b}\tilde{\rho}_{h,a}\left(\frac{4mL}{\hbar^{2}}\right)^{2}\left(\frac{k_{F}}{\pi}\right)^{4}\Theta^{2}\left(sd\right)\mathrm{e}^{-d/\lambda}\text{,}$

which has a dependence with the electrode separation $d$ identical to that of
the case studied above for a TI in contact with a conventional $s$-wave
superconductor.

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# Analog Cavity Emulators to Support LLRF Development††thanks: This work was
supported by the LCLS-II and ALS-U Projects and the U.S. Department of Energy,
Contract DE-AC02-76SF00515

S.D. Murthy<EMAIL_ADDRESS>L. Doolittle  LBNL  Berkeley  CA 94720  USA
A. Benwell  SLAC  Stanford  CA 94309  USA

###### Abstract

The goal of a LLRF system is to control an actual RF cavity with beam. While
digital simulations have a place, having an analog circuit to stand in for the
cavity can be tremendously helpful in validating hardware+firmware+software
under development. A wide range of cavity emulators have been developed in
collaboration with SLAC, and LBNL. Cavity emulators are typically based on
quartz crystals and frequency conversion hardware. The choice of crystal
frequency and coupling mechanism depends in part on the bandwidth and coupling
of the cavity it’s intended to emulate. Examples of bandwidth range from 800
Hz (SLAC) as a stand-in for a SRF cavity, to 31 kHz (LBNL) for a room-
temperature accumulator-ring cavity. An external LO is used to tune the
emulated cavity frequency. The coupling properties are also of interest if the
scope includes emulating reverse power waveforms. LLRF system checks such as
closed-loop bandwidth, and determining cavity detuning can be performed
interactively and as part of a Continuous Integration (CI) process. This paper
describes the design, implementation, and performance of the cavity emulators.

##  1 INTRODUCTION

At most of the accelerator facilities, the real RF cavities will still be
under construction when the LLRF system is being developed. Even when the
cavities become available, setting up the whole high-power system (e.g.,
cryomodule) just to validate the LLRF system under development is difficult to
justify. People also worry about damaging the cavities with an unproven
controller. Digital simulations are a great tool at the beginning of the
development process to understand and test the basic functionality of the
firmware [1]. In light sources, requirements flow from the quality of the
X-ray beams to the electron beam quality and then to requirements on the
stability of the accelerating cavity fields. Examples for the stability
requirement range from $0.01^{\circ}$ in phase and $0.01\%$ in amplitude for
SLAC’s Linear Coherent Light Source Linac (LCLS-II) system [2] and
$0.1^{\circ}$ in phase and $0.1\%$ in amplitude for LBNL’s Advanced Light
Source Upgrade (ALS-U) system. Careful engineering of the entire LLRF system
is required to attain this stability requirement.

To support the development process, cavity emulators have been developed at
various accelerator labs over the years to validate the LLRF system before
controlling a real RF cavity. A collaboration between SLAC and LBNL has
resulted in the design of cavity emulators for SRF cavities with 800 Hz
bandwidth for LCLS-II and room temperature accumulator-ring cavities with 31
kHz bandwidth.

##  2 Cavity Emulator Design

An analog cavity emulator consists of two components: a crystal resonator, and
frequency conversion chains as shown in Fig. 1. They can either be constructed
on Printed Circuit Boards (PCBs), or with componentized parts, or a
combination of both.

Figure 1: Cavity emulator components.

###  2.1 Resonator hardware

The resonator consists of a commercial quartz crystal, often with additional
passive components. A quartz crystal is normally modeled as a series RLC
circuit with a parallel capacitor. The former represents the mechanical
vibrations of the crystal, and the latter represents the electrodes attached
to the crystal[3]. A crystal has two resonance modes, series and parallel. The
frequency at which the series inductance and series capacitance combination
exhibit a phase angle of zero is defined as series resonance frequency
($f_{s}$). A parallel resonance ($f_{p}$) is created when series inductance
and capacitance interact with parallel capacitance.

All crystal parameters, such as Equivalent Series Resistance (ESR), Shunt
Capacitance ($C_{0}$), Motion Inductance ($L_{m}$), Motion Capacitance
($C_{m}$), and Quality factor ($Q_{L}$) need to be characterized using the
measured transfer function ($S_{21}$) to achieve the desired bandwidth and
minimize broadband coupling [4].

$ESR=2\cdot Z_{0}\cdot(1/|S_{21}|-1)$ (1)

$R_{\rm eff}=2\cdot Z_{0}+ESR$ (2)

$\Delta f=f_{h}-f_{l}$ (3)

$C_{m}=\frac{\Delta f}{2\cdot\pi\cdot f_{s}^{2}\cdot R_{\rm eff}}$ (4)

$L_{m}=\frac{R_{\rm eff}}{2\cdot\pi\cdot\Delta f}$ (5)

$Q_{L}=\frac{2\cdot\pi\cdot f_{s}\cdot L_{m}}{ESR}$ (6)

Here $Z_{0}$ is the transmission line impedence, $R_{\rm eff}$ is the
effective resistance across the crystal, $f_{h}$ and $f_{l}$ are the
frequencies at $+45^{\circ}$ and $-45^{\circ}$ $S_{21}$ phase respectively.

The complex $S_{21}$ of the crystal is given by equation (7). When
$\beta_{0}$, $\beta_{1}$ coefficients are zero, it becomes a pure complex
Lorentzian function equation (8). If there were no parasitics, and the network
analyzer calibration was perfect (including correcting for the test set),
there would be no additional phase shift or offset required.
$\beta_{0}+j\beta_{1}$ are broadband coupling coefficients,
$\beta_{2}+j\beta_{3}$ are scaling and rotation coefficients due to Lorentzian
[5]. A wide range of commercially available crystals have been characterized,
using NumPy and Jupyter-Notebook to curve-fit the acquired $S_{21}$ data.

$S_{21}=\beta_{0}+j\beta_{1}+(\beta_{2}+j\beta_{3})\cdot a$ (7)

$a=\frac{1}{1+2j\cdot Q_{L}\cdot(\frac{f}{f_{s}}-1)}$ (8)

One way to reduce the broadband coupling is to add a compensating inductance
in parallel with the crystal. That inductance cancels the crystal’s parasitic
shunt capacitance. For trimming purposes, one can either use a variable
inductor (SLAC), or use a fixed inductor with an additional variable capacitor
(LBNL) . A series resistance can also be added to adjust the circuit’s
bandwidth. If this load resistance increases, the signal strength decreases
and the bandwidth increases. But some loss in the signal strength is
acceptable, because the upconversion chains need to be designed to handle
forward and reverse signals attenuated by the directional coupler.

Fig. 2 shows the frequency response measurement of the series resonance for
the LBNL cavity emulator. The measured loaded $Q_{L}$ is 1341 (which
corresponds to a full-bandwidth of about 31 kHz), and the insertion loss is 23
dB. The complex frequency response $S_{21}$ of the curve-fitted data is Fig.
3. Similar frequency response curves can be derived for any available
commercial crystals.

Figure 2: Crystal resonance frequency (LBNL) (span = 600 kHz). Figure 3:
Complex frequency response with curve fitting.

###  2.2 Frequency conversion hardware

The frequency conversion hardware consists of a single down conversion chain
and three up conversion chains. The down converter is used to convert the
incoming RF cavity drive signal to the crystal resonance frequency. The
emulated cavity output is then up converted to RF. Forward and reverse signals
are also be produced with a directional coupler at IF, and then up converted.
Special care is taken to select a well-defined bandpass filter to remove any
spurs/leakage after the up-conversion chains. An external signal source is
used as a LO to tune the emulated system’s resonance frequency.

##  3 Performance

The cavity emulators designed to emulate SLAC’s SRF cavity and LBNL’s
accumulator-ring were integrated with the LLRF system as shown in Fig. 4,
along with picture of the PCB (LBNL) is shown in Fig. 5 and the picture of the
componentized cavity emulator (SLAC) in Fig. 6.

Figure 4: Cavity emulator integrated with LLRF system. Figure 5: Cavity
emulator PCB at LBNL. Figure 6: Componentized cavity emulator at SLAC.

Several emulators were deployed for Continuous Integration (CI) purposes. A
server runs automated tests to ensure that any changes to
hardware/firmware/software don’t break the existing codebase. By extension,
the codebase is always checked on the test stand and can be deployed to the
rest of the LLRF system without any issues. These automatic tests include
bringing up the cavity emulator and closing the RF feedback loop. The closed-
loop bandwidth of the controllers is also measured along with the total group
delay. The in-loop stability analysis for both the SLAC and LBNL system also
meets the requirement. The cavity detuning function was also characterized by
changing the LO frequency.

Complete characterization of a LLRF system cannot be done using the cavity
emulator; this includes the out-of-loop stability analysis. The Signal-to-
Noise Ratio (SNR) of a cavity emulator is worse than real cavities, and there
might be spurs caused by LO leakage.

##  4 CONCLUSION

Cavity emulators were designed and built in collaboration with SLAC, and LBNL.
They continue to offer a good benchtop evaluation for testing LLRF systems
including the hardware+firmware+software and train system engineers to
understand the loop parameters better. Emulation of different RF cavities can
also be achieved with the existing system by using a different crystal,
coupling mechanism, and up conversion bandpass filters.

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