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# Controllable critical Josepshon current and $0$-$\pi$ transition in
superconductor-insulator-superconductor heterostructures

G. Livanas1 1Department of Physics, National Technical University of Athens,
GR-15780 Athens, Greece

###### Abstract

We investigate Josephson junctions among conventional superconducting wires,
in the presence of externally applied Zeeman fields and supercurrents. We
demonstrate that the critical Josephson current and the current-phase relation
of the junction depends on the relative orientation of the applied Zeeman
fields and supercurrents. The controllability of the Josephson current relies
on the stabilization of conventional superconductivity in high magnetic
fields, due to the restoration of the opposite momentum opposite spin
projection resonance at the Fermi level. We assert that our results can lead
to the development of novel superconducting electronics.

The intriguing electromagnetic properties of superconductors have led to the
development of superconducting electronics [1, 2] and superconducting
spintronics [3], each field related to information processing and storage
devices relying on the manipulation of charge and spin degrees of freedom,
respectively. Although superconducting devices require cryogenic temperatures,
they minimize energy losses and allow for the design of faster and energy-
efficient logic and memory units. Basic element of superconducting devices is
the Josephson junction (JJ), i. e. two superconducting electrodes connected by
a weak link. A dissipationless current, obeying the current-phase relation
$I=I_{c}\sin\phi$, flows in JJs consisting of superconductors locked at
macroscopic quantum mechanical phase difference $\phi$ [4, 5]. Besides the
phase difference, Josephson junctions are characterised also by the critical
current $I_{c}$, beyond which a voltage drop is developed, i.e. the junction
is in the voltage or resistance state, while below this value the charge flow
remains dissipationless.

Controlling the critical Josephson current is the primary objective of
superconducting memory devices based on JJs. Establishing two distinct
critical current levels in a JJ allows us to store information by switching
among the dissipationless and the voltage state. To date, control of the
critical Josephson current has been demonstrated in magnetic Josephson
junctions (mJJs), i.e. superconductor/ferromagnet/superconductor
heterostructures [6, 7, 8, 9, 10]. Interplay among the Fraunhofer effect [11]
and the magnetic hysteresis of the ferromagnetic layer, lead to the emergence
of two distinct levels for the critical current in these heterostructures.
However, the particular junctions demonstrate low normal-state resistance
$R_{N}$ and therefore low characteristic voltage $V_{c}=I_{c}R_{N}$
corresponding to high switching time $\tau_{s}\propto 1/V_{c}$, rendering them
incompatible with rapid flux quantum logic circuits, which are based on
superconductor/insulator/superconductor junctions [12]. Attaching additional
insulating [13, 14, 15, 16] or ferromagnetic layers [17, 18, 19, 20, 21] can
improve technical characteristics, such as the $\tau_{s}$, the size and the
operating magnetic fields, of the corresponding devices. However, they also
increase their complexity.

An additional degree of freedom of JJs is the phase difference among the SC
electrodes. Controllable switches among the 0 and $\pi$ SCs phase difference
states can significantly reduce the size and noise in classical [22, 23] and
quantum [24, 25] computing circuits, respectively. Essentially, a
$\pi$-Josephson junction, is a junction where the Josephson current obeys an
inverse current-phase relation. Magnetic Josephson junctions can switch among
the $0$ and the $\pi$ state based on manipulation of the magnetisation of the
ferromagnetic layers [26, 27, 28] or the interfacial spin-orbit coupling [29].
Moreover, proposals based on superconductor/normal/superconductor (S/N/S)
junctions suggest the use of charge currents [30, 31, 32] and Zeeman fields
[33] for achieving the $0$-$\pi$ transition in a controllable way.
Nevertheless, compatibility with S/I/S junctions hinders the application of
mJJs or S/N/S in logic circuits.

In this article we present a new approach for controlling the critical
Josephson current and the $0$-$\pi$ transition in conventional $S/I/S$
Josephson junctions, circumventing in this way the obstacle of compatibility
with already developed superconducting devices. Our proposal relies on the
combined effect of Zeeman fields and supercurrents in conventional SCs. A
conventional SC is an isotropic quantum state formed by coherent pairs of
electrons with opposite momenta and opposite spin projection, the singlet
Cooper pairs [34]. Zeeman fields, i.e. magnetic fields coupled only to the
electrons spin, and supercurrents have detrimental effect upon conventional
SC. Regarding Zeeman fields, conventional SC survives up to the Chandrasekhar-
Clogston limit [35, 36], $\Delta_{0}\sqrt{2}$, where $\Delta_{0}$ is the
superconducting energy gap in the absence of a Zeeman field. Respectively, the
Cooper pair velocity $v_{s}$, related to the dissipationless current flow that
a SC wire can sustain, does not exceed $\Delta_{0}/{\hbar k_{F}}$, where
$k_{F}$ is the Fermi wavevector [37].

The destructive effect of Zeeman fields and supercurrents on conventional SCs
is revealed by examining the energy bands of the system. In the absence of
these terms and as long as spatial inversion and spin reversal symmetries are
preserved, an opposite momenta, opposite spin projection resonance is present
at the Fermi level, favouring the stabilisation of a conventional SC ground
state. When a Zeeman field or a current is applied, the particular resonance
vanishes. However, their combination can lead to the restoration of the
resonance at the Fermi level and therefore allow for the formation of singlet
Cooper pairs. Nevertheless, the re-establishment of this resonance occurs only
for one of the two possible configurations of the Cooper pairs
$k\uparrow,-k\downarrow$ or $k\downarrow,-k\uparrow$, where $k>0$ is the
momentum of electrons and $\uparrow,\downarrow$ their two possible spin
projections. Therefore, the increase of the current(Zeeman field) in the
presence of a Zeeman field(current), leads to a first order transition to a
gapless conventional SC state with an imbalance among the Cooper pairs of the
two configurations. The prevailing configuration of the Cooper pairs can be
controlled by the relative orientation of the applied Zeeman fields and
supercurrents. We remark that, although a previous work has also demonstrated
the persistence of conventional SC in high Zeeman fields beyond the
Chandrasekhar-Clogston limit, due to applied voltage bias [38], the effect of
supercurrents which drives the particular first order transition was not
investigated.

Remarkably, as we present in Fig. 1, JJs among conventional SCs in the gapless
regime exhibit significantly different critical Josephson current, depending
on the relative orientation of the applied Zeeman field and supercurrents.
Moreover, the current-phase relation is opposite in the two setups. Thus, by
applying Zeeman fields and supercurrents in the SCs, we can control both the
critical current and the $0$-$\pi$ transition of the JJ. We add, that the
Zeeman field can be also applied by depositing the JJ of Fig. 1 on a
ferromagnetic insulator [39, 40], enhancing in this way its controllability
[41, 42]. A similar setup has been proposed for S/N/S and S/I/S junctions [43,
44]. We remark that, due to the directionality of the current, the effect of
re-emergent SC is prominent in 1D SC wires. Therefore, in the present article
we focus on junctions among 1D SC wires. However, the results qualitatively
holds also for JJs among quasi-1D and 2D SCs.

Figure 1: up) Josephson junction among two conventional SC wires in the
presence of a Zeeman field $m_{h}$. Supercurrents $I_{s}$ are applied in both
wires in a parallel (left) and an anti-parallel (right) configuration. When
the $I$ perpendicular to the junction is above the critical Josephson current
$I_{c}$, a voltage drop $V$ develops at the edges of the junction. down)
Josephson current with respect to phase difference $\phi$ among the two
superconductors for the parallel (blue line) and the anti-parallel currents
configuration (red line). The two configurations exhibit different critical
Josephson current and opposite current-phase relations.

Our analysis initiates from the following momentum space Hamiltonian

$\displaystyle{\cal H}=\sum_{k}\Psi_{k}^{{\dagger}}\left[(t\cos J\cos
k\right.$ $\displaystyle-$ $\displaystyle\mu)\tau_{z}+t\sin J\sin k$ (1)
$\displaystyle+$
$\displaystyle\left.m_{h}\tau_{z}\sigma_{z}+\Delta\tau_{y}\sigma_{y}\right]\Psi_{k}\,,$

which incorporates a Zeeman spin-splitting energy $m_{h}$ and a current term
$t\sin J\sin k$. The spinor
$\Psi_{k}^{{\dagger}}=\left(\psi_{k,\uparrow}^{{\dagger}},\psi_{k,\downarrow}^{{\dagger}},\psi_{-k,\uparrow}^{\phantom{{\dagger}}},\psi_{-k,\downarrow}^{\phantom{{\dagger}}}\right)$,
where $\psi_{k,s}^{({\dagger})}$ destroys(creates) an electron with momentum
$k$ and spin projection $s$, defines the four dimensional Nambu space, while
$\mu$ is the chemical potential, $t\cos J\cos k$ is the inversion symmetric
kinetic energy of electrons and $\Delta$ the conventional SC field. We employ
the Pauli matrices $\tau$ and $\sigma$ acting on particle-hole and spin space
respectively. We have introduced the parameter $J$ which corresponds to the
gradient of the superconducting order parameter phase, $\Delta e^{-i2Jx}$.
Regarding this parameter, SC is sustained as long as $J\cdot\xi_{0}<1$, where
$\xi_{0}=\frac{\hbar v_{F}}{\Delta_{0}}$ is the coherence length of the SC
[45].

Upon conducting the gauge transformation $\Psi_{x}\rightarrow
e^{-iJx\tau_{z}}\Psi_{x}$, where
$\Psi_{x}^{{\dagger}}=\left(\psi_{x,\uparrow}^{{\dagger}},\psi_{x,\downarrow}^{{\dagger}},\psi_{x,\uparrow}^{\phantom{{\dagger}}},\psi_{x,\downarrow}^{\phantom{{\dagger}}}\right)$
is the Nambu spinor in coordinate space and Fourier transforming to momentum
space, we derive the term $t\sin J\sin k$ which relates to the dissipationless
current flow $I_{s}\propto J$ and $t\cos J\cos k$ [46]. Nevertheless, the term
$t\sin J\sin k$ which connects to the charge current operator
$\sum_{s}\psi_{k,s}^{{\dagger}}\psi_{k,s}^{\phantom{{\dagger}}}-\psi_{-k,s}^{{\dagger}}\psi_{-k,s}^{\phantom{{\dagger}}}$,
can be added irrespective of whether the system is in the superconducting or
in the normal phase. When the wire is in the normal phase parameter $J$
associates with a voltage bias applied at the edges of the wire. The
eigenenergies of Hamiltonian Eq. 1 take the following form

$\displaystyle E_{k,s,\pm}=sm_{h}+t\sin J\sin
k\pm\sqrt{\varepsilon_{k}^{2}+\Delta^{2}}\,,$ (2)

with $\varepsilon_{k}=t\cos J\cos k-\mu$ and $s=+(\uparrow)$ for spin up and
$s=-(\downarrow)$ for spin down. In Fig. 2 we present the energy bands of the
wire for four distinct cases. We observe that only when both $m_{h}\neq 0$ and
$J\neq 0$, the opposite momentum opposite spin projection resonance can be
restored, leading to the re-emergence of conventional SC. However, by
examining Eq. 2 for $\Delta=0$, we observe that the resonance is restored when
$sm_{h}+t\sin J\sin k=0$, only for one of the two configurations
$k\uparrow,-k\downarrow$ or $k\downarrow,-k\uparrow$, depending on the
relative orientation of the applied Zeeman field $m_{h}$ and the applied
current. In particular, for $ks,-ks^{\prime}$ configuration the resonance is
restored for $\sin J+\frac{sm_{h}}{t\sin k_{F}}=0$, where
$k_{F}=\cos^{-1}(\frac{\mu}{t\cos J})$ is the Fermi momentum (Fig. 2(d)).

Figure 2:  The energy bands of a conventional SC wire in the presence of a
Zeeman field $m_{h}$ and a supercurrent $J$. (a) For $m_{h}\neq 0$ and $J=0$
and (b) for $m_{h}=0$ and $J\neq 0$, the $ks,ks^{\prime}$ resonance breaks.
(c) When both $m_{h}\neq 0$ and $J\neq 0$, the $ks,-ks^{\prime}$ resonance can
be restored only for the $k\uparrow,-k\downarrow$ or the
$k\downarrow,-k\uparrow$ pair, where $k>0$, leading to the re-emergence of
conventional SC. (d) The Zeeman field $m_{h}$ and parameter $J$ for which the
resonance is restored for the $k\uparrow,-k\downarrow$ pair (solid line) and
the $k\downarrow,-k\uparrow$ pair (dashed line). The chemical potential is
$\mu=0.8$ (red line), $\mu=0.5$ (black line) and $\mu=0.2$ (blue line).

Figure 3:  (a) Phase diagram with respect to the applied Zeeman field $m_{h}$
and the charge current term $t\sin J\sin k_{F}$. Purple line denotes a first
order transition among fully-gapped SC state with
$<\psi_{k,\uparrow}\psi_{-k,\downarrow}>=<\psi_{k,\downarrow}\psi_{-k,\uparrow}>$
(yellow color) and gapless SC state with
$<\psi_{k,\uparrow}\psi_{-k,\downarrow}>\neq<\psi_{k,\downarrow}\psi_{-k,\uparrow}>$
(blue color). (b) For $t\sin J\sin k_{F}=m_{h}$, $\Delta$ normalised to the
$m_{h}=J=0$, $\Delta_{0}$ value, with respect to the applied Zeeman field
$m_{h}$ (blue line, blue line in panel (a)). A first order transition is
observed for $m_{h}=\Delta_{0}/2$ and a second order transition to the normal
state for $m_{h}>9\Delta_{0}$. For $t\sin J\sin k_{F}=m_{h}+0.2\Delta_{0}$
(black line) and $t\sin J\sin k_{F}=m_{h}-0.2\Delta_{0}$ (red line) the
transition to the normal state is first order. (c) The normalised SC order
parameter with respect to $t\sin J\sin k_{F}/\Delta_{0}$ for
$m_{h}=0.5\Delta_{0}$ (red line and red line in panel (a)) and for
$m_{h}=1.5\Delta$ (green line and green line in panel (a)). (d) For values of
$t\sin J\sin k_{F}$ denoted by the black and blue circle in panel (c) the
energy bands of the SC wire (without and with rectangles marks, respectively).
We observe that for $t\sin J\sin k_{F}/\Delta_{0}=0.65$, $E_{-,+}(E_{+,-})$
crosses the Fermi level for $k<0(k>0)$.

To verify the aforementioned arguments we determine the SC order parameter
$\Delta$ which corresponds to the minimum of the free energy of a conventional
SC wire in the presence of a Zeeman field and a charge current. The free
energy of the wire derives from the following equation

$\displaystyle
F=\frac{\Delta^{2}}{g}-\frac{1}{\beta}\sum_{k,s,\pm}ln(1+e^{-\beta
E_{k,s,\pm}})\,,$ (3)

where $g$ is the electrons effective interaction potential and $\beta=k_{B}T$.
Results are presented in Fig. 3, considering a realistic value
$\Delta_{0}=0.01t$ for the SC order parameter. As we observe from the phase
diagram, Fig. 3(a), for finite values of the current, superconductivity is
stabilised for Zeeman fields higher than the Chandrasekhar-Clogston limit. In
Fig. 3(b), we present the order parameter with respect to $m_{h}$, for $t\sin
J\sin k_{F}=m_{h}$. Notice that $\Delta$ is finite up to $m_{h}=9\Delta_{0}$.
Moreover, for $m_{h}=\Delta_{0}/2$ a first order transition to a SC state with
$<\psi_{k,\uparrow}\psi_{-k,\downarrow}>\neq<\psi_{k,\downarrow}\psi_{-k,\uparrow}>$,
where $<\psi_{k,s}\psi_{-k,s^{\prime}}>$ the pairing correlations, i.e. the
Cooper pairs, of the $ks,-ks^{\prime}$ configuration, occurs, while for
$m_{h}=9\Delta_{0}$ the transition to the normal phase is second order.
Nevertheless, as we observe in the same figure the transition to the normal
state is first order when $m_{h}\neq t\sin J\sin k_{F}$. Finally, for
$m_{h}>\Delta_{0}/\sqrt{2}$, SC re-emerges, due to the application of a charge
current (Fig. 3(c)). Note, that parameter $t\sin J\sin k_{F}/\Delta_{0}\simeq
J\xi_{0}$. We remark, that following this approach we do not examine the
stabilization of Fulde-Ferrel-Larkin-Ovchinikov states [47, 48].

In order to get an insight into phase diagram Fig. 3(a), we examine the energy
spectrum of the SC wire (Fig. 3(d)). A first order transition occurs when
$E_{k,-,+}(E_{k,+,-})$ (for $m_{h}\cdot J>0$) or $E_{k,+,+}(E_{k,-,-})$ (for
$m_{h}\cdot J<0$) cross the Fermi level for $k<0(k>0)$, i.e. this transition
is from a fully-gapped SC state to a gapless SC state. Another first order
transition from the gapless SC to the normal state, occurs when the energy
bands for both spin projection cross the Fermi level. As a consequence of the
finite energy gap for only one of the two $ks,-ks^{\prime}$ configurations, an
imbalance among the pairing correlations $<\psi_{k,s}\psi_{-k,s^{\prime}}>$,
emerges. Therefore, the particular SC phase can be considered as a mixed $s+p$
state, where $s$ stands for an isotropic s-wave superconducting order
parameter and $p$ for an order parameter with $p-$wave symmetry and $S_{z}=0$
for the spin-z component of the Cooper pairs [49].

In the following, we investigate JJs among conventional SC wires, in the
presence of a Zeeman field $m_{h}$ and a supercurrent. We consider the
following Hamiltonian describing the JJ, ${\cal H}={\cal H}_{l}^{SC}+{\cal
H}_{r}^{SC}+{\cal H}_{c}$, where ${\cal
H}_{w}^{SC}=\sum_{k}\Psi_{w,k}^{{\dagger}}\hat{H}_{k}\Psi_{w,k}$, with

$\displaystyle\hat{H}_{k}=[\varepsilon_{k}\tau_{z}+m_{h,w}\tau_{z}\sigma_{z}+t\sin
J_{w}\sin k+\Delta_{w}\tau_{y}\sigma_{y}]\,,$ (4)

the Hamiltonian of each SC wire, where index $w=l,r$ denotes the wire. We
introduce the extended spinor
$\Psi_{k}^{{\dagger}}=(\Psi_{l,k}^{{\dagger}},\Psi_{r,k}^{{\dagger}})$ to
model the coupling of the two wires via an insulating layer, by the following
expression, ${\cal
H}_{c}=\sum_{k}\Psi_{k}^{{\dagger}}t_{c}\rho_{x}\tau_{z}\Psi_{k}$, where
$\bm{\rho}$ are the Pauli matrices acting in the wire bands space. Initiating
from Hamiltonian Eq. 4, we can determine the current flow across the junction
based on equation,

$\displaystyle
I\propto\sum_{k,i\omega_{n}}Tr{\rho_{y}\hat{G}_{k,i\omega_{n}}}\,,$ (5)

where $\hat{G}_{k,i\omega_{n}}=[i\omega_{n}-\hat{H}_{k}]^{-1}$ is the Green’s
function matrix and $\omega_{n}=(2n+1)\pi/\beta$ the fermionic Matsubara
frequencies. However, to gain insight, we additionally determine the Josephson
current of the junction based on equation
$I=\frac{j}{\beta}\frac{\partial}{\partial\phi}\sum_{i\omega_{n}}^{k,s}e^{i\phi}F^{l,*}_{ks,-ks^{\prime},i\omega_{n}}F^{r}_{ks,-ks^{\prime},i\omega_{n}}+c.c.$,
where $\phi$ is the phase difference among the two SC wires, $j=-2et_{c}^{2}$
and $F$ is the anomalous Green’s function [50]. Conducting the summation over
$\omega_{n}$, we derive the following expression for the Josephson current,
when $J=m_{h}=0$ and $T=0$

$\displaystyle
I_{0}=2et_{c}^{2}\Delta^{2}\sin\phi\sum_{k}\frac{1}{E_{k}^{3}}\,,$ (6)

where $E_{k}=\sqrt{\varepsilon_{k}^{2}+\Delta^{2}}$ [see Appendix A]. For
$J_{l}=J_{r}=\rho$ and $m_{h,r}=m_{h,l}=\rho$, where $\rho>\Delta/2$ the
Josephson current reduces to $I_{par}=I_{0}/2$. This case corresponds to the
currents applied in a parallel configuration (Fig. 1). When the Zeeman field
is uniform across the junction, but the currents in the two superconductors
are applied in an anti-parallel configuration, $J_{l}=m_{h,l}=\rho$ and
$J_{r}=-m_{h,r}=\rho$, where $\rho>\Delta/2$, we get the expression

$\displaystyle
I_{apar}=-4et_{c}^{2}\Delta^{2}\sin\phi\sum_{k}\frac{1}{E_{k}[(2\rho+E_{k})^{2}-E_{k}^{2}]}\,.$
(7)

The same expression holds if the currents are applied in the parallel
configuration, but the Zeeman fields at the two SCs point at opposite
directions. Notice that the Josephson current in the anti-parallel
configuration has opposite direction with respect to the current in the
parallel configuration. Moreover, $\rho$, which corresponds to the magnitude
of the Zeeman field and the current, enters the denominator in the expression
for $I_{apar}$, due to the energy difference among states with the same
momenta in this case. Since $I_{apar}$ is independent of $\rho$, while
$I_{apar}$ reduces rapidly, for large values of the applied Zeeman field and
current, $I_{apar}<<I_{par}$. Thus, by manipulating the charge current and/or
the Zeeman fields applied on the SCs of the JJ, we can control the flow of the
Josephson current. As presented in Fig. 4(b), corresponding to a JJ where we
vary the applied supercurrent on the right SC, for $J_{r}\simeq J_{l}$ and
$J_{r}\simeq-J_{l}$ the critical Josephson current calculated numerically
based on Eq. 5 directly from Hamiltonian Eq. 4, remains relative constant. The
magnitude of the Josephson current in these two cases is in qualitatively
agreement with Eq. 6 and Eq. 7. For small values of $J_{r}$ we observe a
linear behaviour of $I_{c}$, while for $J_{r}=0$, $I_{c,0}\simeq 1.15I_{par}$
[see Appendix A].

Figure 4:  Critical current for the Josephson junction presented in Fig. 1.
(a) For $m_{h}=t\sin J\sin k_{F}$ the critical Josephson current $I_{c}$ with
respect to Zeeman field $m_{h}$ for the parallel (red line) and the anti-
parallel (blue line) configuration. In the parallel configuration, the
critical Josephson current reduces to $I_{c}/2$ beyond $m_{h}=t\sin J\sin
k_{F}=\Delta_{0}/2$ and remains constant until the second order transition to
the normal state. In the anti-parallel configuration, the critical Josephson
current reverse sign for $m_{h}=\Delta_{0}/2$ and vanishes for
$m_{h}>2\Delta_{0}$. Red and blue lines follow from Eq. 6 and Eq. 7,
respectively, while black marks correspond to results based on Eq. 5. (b) The
critical Josephson current with respect to the applied supercurrent $J_{r}$ in
the right SC wire for $m_{h}/\Delta_{0}=0.5$ (blue line) and
$m_{h}/\Delta_{0}=1.5$ (red line). The current in the left wire remains
constant at $t\sin J_{l}\sin k_{F}=0.6\Delta_{0}$ and $t\sin J_{l}\sin
k_{F}=1.7\Delta_{0}$, respectively. We observe that when supercurrent in the
second wire reverses direction, the critical Josephson current reverse sign
and becomes one (blue line) and two (red line) orders of magnitude smaller
that the critical Josephson current in the parallel configuration.

In conclusion, we examined conventional superconductivity in the presence of a
Zeeman field and a supercurrent flow. We observe that superconductivity can
sustain high magnetic fields beyond the Chandrasekhar-Clogston limit in the
presence of a supercurrent. A first order transition to a gapless
superconducting state with an imbalance in pairing correlations with opposite
spin configuration is observed with increasing supercurrent(Zeeman field) in
the presence of a Zeeman field(supercurrent). Depending on the relative
orientation of the applied Zeeman fields and supercurrents, conventional JJs
exhibit different critical Josephson current and current-phase relations.
Since control of the particular junctions can be achieved by means of
dissipationless currents, we assert that our results can lead to the
development of energy-efficient, multi-functional superconducting devices.

##  Appendix A Josephson current

We introduce the following Hamiltonian describing two conventional SC wires
coupled along the transverse to their length direction, in the presence of a
Zeeman field and a supercurrent,

$\displaystyle{\cal H}_{w}=\sum_{k}\Psi_{w,k}^{{\dagger}}$
$\displaystyle\Big{[}$ $\displaystyle m_{h,w}\tau_{z}\sigma_{z}+[t\cos
J_{w}\cos(k)-\mu]\tau_{z}$ (8) $\displaystyle+$ $\displaystyle t\sin
J_{w}\sin(k)+\Delta_{w}\tau_{y}\sigma_{y}\Big{]}\Psi_{l,k}\,,$

where $w=l,r$ denotes the wire and

$\displaystyle{\cal
H}_{t}=\sum_{k,s}t_{c}[\psi_{l,k,s}^{{\dagger}}\psi_{r,k,s}^{\phantom{{\dagger}}}+\psi_{r,k,s}^{{\dagger}}\psi_{l,k,s}^{\phantom{{\dagger}}}]\,,$
(9)

the coupling part. The Green’s function matrix of ${\cal H}_{w}$,
$[i\omega_{n}I-H_{l,k}]\hat{G}_{l,k}=I$, where $\omega_{n}=(2n+1)\pi/\beta$
the fermionic Matsubara frequencies. Equating the elements of the matrices in
the two part of the above equation we get

$[i\omega_{n}-\epsilon_{k,s}]G_{ks,ks}+s\Delta F_{ks,-ks^{\prime}}^{*}=1$
---
$[i\omega_{n}+\epsilon_{-k,s}]G_{-ks,-ks}-s\Delta^{*}F_{ks^{\prime},-ks}=1$
$[i\omega_{n}-\epsilon_{k,s}]F_{ks,-ks^{\prime}}+s\Delta
G_{-ks^{\prime},-ks^{\prime}}=0$
$[i\omega_{n}+\epsilon_{-k,s}]F_{ks^{\prime},-ks}^{*}-s\Delta^{*}G_{ks^{\prime},ks^{\prime}}=0$

where $\epsilon_{k,s}=sm_{h}+tsJsk+tcJck-\mu$ the energy bands of the wires in
the normal phase and $G_{ks,ks^{\prime}}=-\int_{0}^{\beta}d\tau
e^{i\omega_{n}\tau}<\theta(\tau)\psi_{k,s,\tau}^{{\dagger}}\psi_{k,s^{\prime},0}^{\phantom{{\dagger}}}-\theta(-\tau)\psi_{k,s^{\prime},0}^{\phantom{{\dagger}}}\psi_{k,s,\tau}^{{\dagger}}>$,
$F_{ks,-ks^{\prime}}=-\int_{0}^{\beta}d\tau
e^{i\omega_{n}\tau}<\theta(\tau)\psi_{k,s,\tau}^{{\dagger}}\psi_{-k,s^{\prime},0}^{{\dagger}}-\theta(-\tau)\psi_{-k,s^{\prime},0}^{{\dagger}}\psi_{k,s,\tau}^{{\dagger}}>$.
Based on the above equations, we determine the elements of the Green’s
function matrix

$G_{ks,ks}$ | $=$ | $\frac{i\omega_{n}+\varepsilon_{-k,s^{\prime}}}{(i\omega_{n}-E_{k,s,+})(i\omega_{n}-E_{k,s,-})}$
---|---|---

$G_{-ks,-ks}$ | $=$ | $\frac{i\omega_{n}-\varepsilon_{k,s^{\prime}}}{(i\omega_{n}-E_{k,s^{\prime},+})(i\omega_{n}-E_{k,s^{\prime},-})}$
---|---|---

$F^{*}_{ks,-ks^{\prime}}$ | $=$ | $\frac{s\Delta^{*}}{(i\omega_{n}-E_{k,s,+})(i\omega_{n}-E_{k,s,-})}$
---|---|---

where $E_{k,s,\pm}=sm_{h}+tsJsk\pm\sqrt{\Delta^{2}+\varepsilon_{k}^{2}}$, with
$\varepsilon_{k}=(tcJck-\mu)$, the eigenenergies of the SC wires. Based on Eq.
9 for the coupling of the two wires, the Josephson current emerging, due to a
phase difference $\phi$ among the superconductors, derives from equation

$\displaystyle I$ $\displaystyle=$
$\displaystyle-2et_{c}^{2}\frac{1}{\beta}\frac{\partial}{\partial\phi}\left[\sum_{i\omega_{n},k,s}F^{*}_{l,ks,-ks^{\prime}}F_{r,ks,-ks^{\prime}}^{\phantom{{\dagger}}}+c.c.\right]\rightarrow$
$\displaystyle I$ $\displaystyle=$
$\displaystyle-2et_{c}^{2}\frac{1}{\beta}\frac{\partial}{\partial\phi}\left[\sum_{i\omega_{n},k,s}\frac{\Delta_{l}}{(i\omega_{n}-E_{l,k,s,+})(i\omega_{n}-E_{l,k,s,-})}\frac{\Delta_{r}^{*}}{(i\omega_{n}-E_{r,k,s,+})(i\omega_{n}-E_{r,k,s,-})}+c.c.\right]$
$\displaystyle I$ $\displaystyle=$ $\displaystyle
et_{c}^{2}\Delta_{l}\Delta_{r}\sin\phi\left[\sum_{k,w,s,\pm}\frac{\pm
n_{F}(E_{w,k,s,\pm})}{\sqrt{\Delta_{w}^{2}+\varepsilon_{k}^{2}}[(s(m_{h,w}-m_{h,w^{\prime}})+t(sJ_{w}-sJ_{w^{\prime}})sk\pm\sqrt{\Delta_{w}^{2}+\varepsilon_{k}^{2}})^{2}-(\Delta_{w^{\prime}}^{2}+\varepsilon_{k}^{2})]}\right]$
(10)

We examine three distinct cases. First we consider
$tsJ_{l}sk_{F}=tsJ_{r}sk_{F}=m_{h,l}=m_{h,r}=\rho>0$ and
$\Delta_{l}=\Delta_{r}=\Delta$, i.e. the two superconducting wires are
identical. In this case we can drop the $l,r$ index. Thus, Eq. 10 simplifies
to

$\displaystyle I$ $\displaystyle=$ $\displaystyle
4et_{c}^{2}\Delta^{2}\sin\phi\sum_{i\omega_{n},k,s}\frac{1}{[(i\omega_{n}-E_{k,s,+})(i\omega_{n}-E_{k,s,-})]^{2}}\rightarrow$
$\displaystyle I$ $\displaystyle=$
$\displaystyle-8et_{c}^{2}\Delta^{2}\sin\phi\sum_{k,s}\frac{n(E_{k,s,+})-n(E_{k,s,-})}{(E_{k,\uparrow,+}-E_{k,\uparrow,-})^{3}}\rightarrow$
$\displaystyle I$ $\displaystyle=$ $\displaystyle-
et_{c}^{2}\Delta^{2}\sin\phi\sum_{k,s}\frac{n(E_{k,s,+})-n(E_{k,s,-})}{\sqrt{\varepsilon_{k}^{2}+\Delta^{2}}^{3}}$
(11)

where we have calculated the residue of the second order poles using the
formula $Resf(z)_{z=a}=\left.\partial(z-a)^{2}f(z)/\partial z\right|_{z=a}$.
Considering $\Delta/t=0.01$ we can simplify momenta summation $\sum_{k}$, by
keeping only terms for which $\varepsilon_{k}<<\Delta$. This leads to
replacing $\sum_{k}\rightarrow d_{F}c\Delta$, where $d_{F}$ is the density of
states at the Fermi level and $c\Delta$ is the range of energies we take into
account, with $c$ a small parameter. For T=0, $n(E_{k,s,+})=0$, and for
$\rho<\Delta/2$, both spin bands contribute to the current with
$n(E_{k,s,-})=1$ and Eq. 11 simplifies to

$\displaystyle I_{0}$ $\displaystyle=$ $\displaystyle
2et_{c}\Delta^{2}\sin\phi\sum_{k}\frac{1}{\sqrt{\varepsilon_{k}^{2}+\Delta^{2}}^{3}}$
$\displaystyle I_{0}$ $\displaystyle=$ $\displaystyle
2et_{c}\Delta^{2}\sin\phi\sum_{k}\frac{1}{E_{k}^{3}}\rightarrow$
$\displaystyle I_{0}$ $\displaystyle=$ $\displaystyle
2et_{c}^{2}d_{F}c\sin\phi\,,$ (12)

which is Eq. 6 presented in the main article. However, for $\rho>\Delta/2$,
for $k>0$, $n(E_{k,\uparrow,-})=0$ while for $k<0$ $n(E_{k,\downarrow,+})=1$
and cancels the $n(E_{k,\downarrow,-})=1$ term. Therefore, in this case only
one spin band contributes to the Josephson current. Thus, we get

$\displaystyle I_{par}=et_{c}^{2}d_{F}c\sin\phi=I_{0}/2$ (13)

The second case we examine is where $tsJ_{l}sk_{F}=m_{h,l}=\rho>0$ and
$tsJ_{r}sk_{F}=-m_{h,r}=\rho>0$. In this case we get
$E_{l,k,s,\pm}=-E_{r,k,s,\mp}$. For $\rho<\Delta/2$, $E_{w,k,s,-}<0$ while
$E_{w,k,s,+}>0$ and both spin bands contribute to the current for $T=0$.
Conducting the summation over Matsubara frequencies we get

$\displaystyle
I_{apar}=-4et_{c}^{2}\Delta^{2}\sin\phi\sum_{k,\pm}\frac{1}{E_{k}[(2\rho\pm
E_{k})^{2}-E_{k}^{2}]}\,.\quad$ (14)

Parameter $\rho$ enters the expression for the Josephson current since
supercurrents in the SC wires flow in opposite direction. Replacing $\sum_{k}$
with $d_{F}c\Delta$, the above expression simplifies further to

$\displaystyle I_{apar}$ $\displaystyle=$
$\displaystyle-4et_{c}^{2}\Delta^{2}d_{F}c\sin\phi\sum_{\pm}\frac{1}{[(2\rho\pm\Delta)^{2}-\Delta^{2}]}$
$\displaystyle I_{apar}$ $\displaystyle=$
$\displaystyle-2et_{c}^{2}\Delta^{2}d_{F}c\sin\phi\frac{1}{[\rho^{2}-\Delta^{2}]}\,.\quad$
(15)

For $\rho\rightarrow 0$, $I_{apar}\rightarrow I_{0}$ and as $\rho$ increases
$I_{apar}$ increases above $I_{0}$ quadratically. However, for
$\rho>\Delta/2$, $E_{l,k,\uparrow,-}$($E_{r,k,\uparrow,-}$) corresponding to
term $\frac{1}{[(2\rho-E_{k})^{2}-E_{k}^{2}]}$ for $k>0$($k<0$), becomes
positive and does not contribute in the Josephson current. Moreover,
$E_{r,k,\downarrow,+}$($E_{l,k,\downarrow,+}$), which corresponds to term
$\frac{1}{[(2\rho-E_{k})^{2}-E_{k}^{2}]}$ for $k>0$($k<0$), becomes negative
and cancels with the corresponding term
$E_{l,k,\downarrow,-}$($E_{r,k,\downarrow,-}$) which also corresponds to term
$\frac{1}{[(2\rho-E_{k})^{2}-E_{k}^{2}]}$. Thus, the above expression is
reduced to

$\displaystyle I_{apar}$ $\displaystyle=$
$\displaystyle-4et_{c}^{2}\Delta^{2}\sin\phi\sum_{k}\frac{1}{E_{k}[(2\rho+E_{k})^{2}-E_{k}^{2}]}$
$\displaystyle I_{apar}$ $\displaystyle=$
$\displaystyle-4et_{c}^{2}\Delta^{2}d_{F}c\sin\phi\frac{1}{[(2\rho+\Delta)^{2}-\Delta^{2}]}$
$\displaystyle I_{apar}$ $\displaystyle=$
$\displaystyle-4et_{c}^{2}\Delta^{2}d_{F}c\sin\phi\frac{1}{2\rho(\rho+\Delta)}\,.\quad$
(16)

which is Eq. 7 presented in the main article. This equation holds at the
gapless SC state regime. Notice that, $I_{apar}$ decreases as $\rho$ increases
and reverses sign.

The third case we examine is $m_{h,l}=m_{h,r}$ and
$tsJ_{l}sk_{F}=m_{h,l}=\rho$, where $\rho>\Delta_{l}/2$, but
$tsJ_{r}sk_{F}=\kappa$ where $\kappa$ is small parameter. The eigenenergies of
the wires in this case take the form
$E_{l,\uparrow(\downarrow),\pm}=+(-)2\rho\pm\sqrt{\Delta_{l}^{2}+\varepsilon_{k}^{2}}$,
$E_{l,\downarrow(\uparrow),\pm}=\pm\sqrt{\Delta_{l}^{2}+\varepsilon_{k}^{2}}$
for $k>0$($k<0$) and $E_{r,\pm
k,s,\pm}=s\rho\pm\kappa\pm\sqrt{\Delta_{r}^{2}+\varepsilon_{k}^{2}}$. For
$T=0$, only $E_{l,\downarrow(\uparrow),-}<0$ for $k>0$($k<0$) and
$E_{l,\downarrow,\pm}<0$ for $k<0$ while $E_{r,k,\uparrow,-}<0$ and
$E_{r,k,\downarrow,-}<0$ for any $k$. Therefore, from Eq. 10 we get

$\displaystyle I$ $\displaystyle=$ $\displaystyle
et_{c}^{2}\Delta_{l}\Delta_{r}\sin\phi\sum_{k}\left[\frac{1}{\sqrt{\Delta_{l}^{2}+\varepsilon_{k}^{2}}[(\rho-\kappa+\sqrt{\Delta_{l}^{2}+\varepsilon_{k}^{2}})^{2}-(\Delta_{r}^{2}+\varepsilon_{k}^{2})]}\right.$
(17) $\displaystyle+$
$\displaystyle\left.\frac{1}{\sqrt{\Delta_{r}^{2}+\varepsilon_{k}^{2}}[(\rho-\kappa-\sqrt{\Delta_{r}^{2}+\varepsilon_{k}^{2}})^{2}-(\Delta_{l}^{2}+\varepsilon_{k}^{2})]}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{\sqrt{\Delta_{r}^{2}+\varepsilon_{k}^{2}}[(\rho-\kappa+\sqrt{\Delta_{r}^{2}+\varepsilon_{k}^{2}})^{2}-(\Delta_{l}^{2}+\varepsilon_{k}^{2})]}\right]$

Considering also that $\Delta_{l}\simeq\Delta_{r}/2$ and replacing $\sum_{k}$
with $d_{F}c\Delta$ we can further simplify to

$\displaystyle
I=et_{c}^{2}d_{F}c\Delta^{2}\sin\phi\left[\frac{32(\rho-\kappa)}{[9\Delta^{2}-4(\kappa-\rho)^{2}][\Delta-2(\kappa-\rho)]}\right]$
(18)

For $\kappa=0$ and $\rho=0.6\Delta$, we have

$\displaystyle I=1.15et_{c}^{2}d_{F}c\sin\phi=1.15I_{par}$ (19)

while for $\kappa\neq 0$ we get

$\displaystyle I\simeq I_{par}(1-\frac{\kappa}{\Delta})\,,$ (20)

since denominator of Eq. 18 does not change significantly.

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# Controlling the balance between remote, pinhole, and van der Waals epitaxy
of Heusler films on graphene/sapphire

Dongxue Du  Materials Science and Engineering, University of Wisconsin-
Madison, Madison, WI 53706  Taehwan Jung  Materials Science and Engineering,
University of Wisconsin-Madison, Madison, WI 53706  Sebastian Manzo  Materials
Science and Engineering, University of Wisconsin-Madison, Madison, WI 53706
Zachary T. LaDuca  Materials Science and Engineering, University of Wisconsin-
Madison, Madison, WI 53706  Xiaoqi Zheng  Materials Science and Engineering,
University of Wisconsin-Madison, Madison, WI 53706  Katherine Su  Materials
Science and Engineering, University of Wisconsin-Madison, Madison, WI 53706
Vivek Saraswat  Materials Science and Engineering, University of Wisconsin-
Madison, Madison, WI 53706  Jessica L. McChesney  Advanced Photon Source,
Argonne National Lab, Lemont, IL  Michael S. Arnold  Materials Science and
Engineering, University of Wisconsin-Madison, Madison, WI 53706  Jason K.
Kawasaki  Materials Science and Engineering, University of Wisconsin-Madison,
Madison, WI 53706<EMAIL_ADDRESS>
###### Abstract

Remote epitaxy on monolayer graphene is promising for synthesis of highly
lattice mismatched materials, exfoliation of free-standing membranes, and re-
use of expensive substrates. However, clear experimental evidence of a remote
mechanism remains elusive. In many cases, due to contaminants at the
transferred graphene/substrate interface, alternative mechanisms such as
pinhole-seeded lateral epitaxy or van der Waals epitaxy can explain the
resulting exfoliatable single-crystalline films. Here, we find that growth of
the Heusler compound GdPtSb on clean graphene on sapphire substrates produces
a 30 degree rotated epitaxial superstructure that cannot be explained by
pinhole or van der Waals epitaxy. With decreasing growth temperature the
volume fraction of this 30 degree domain increases compared to the direct
epitaxial 0 degree domain, which we attribute to slower surface diffusion at
low temperature that favors remote epitaxy, compared to faster surface
diffusion at high temperature that favors pinhole epitaxy. We further show
that careful graphene/substrate annealing ($T\sim 700^{\circ}C$) and
consideration of the film/substrate vs film/graphene lattice mismatch are
required to obtain epitaxy to the underlying substrate for a variety of other
Heusler films, including LaPtSb and GdAuGe. The 30 degree rotated
superstructure provides a possible experimental fingerprint of remote epitaxy
since it is inconsistent with the leading alternative mechanisms.

##  I Introduction

In remote epitaxy, a thin film is thought to grow on a graphene (or other 2D
material)-covered substrate via remote interactions that permeate through
graphene Kim et al. (2017). This concept is supported by density functional
theory calculations, which suggest that for ideal graphene/substrate slabs,
the lattice potential of the substrate may sufficiently permeate through
graphene to template epitaxial growth Kong et al. (2018); Kim et al. (2017).
The decoupling between film and substrate is promising for synthesis of highly
lattice mismatched materials with reduced dislocation density Bae et al.
(2020); Jiang et al. (2019); Liu et al. (2022), exfoliation of free-standing
membranes for flexible electronics Kim et al. (2017); Ji et al. (2019),
strain-induced properties Du et al. (2021), and re-use of expensive substrates
Kim et al. (2017).

It remains an outstanding challenge, however, to experimentally validate a
remote epitaxy mechanism. Other mechanisms, which are difficult to rule out,
can produce similar results. For example, pinhole-seeded lateral epitaxy can
also produce single-crystalline exfoliatable films Manzo et al. (2022). A
pinhole mechanism occurs when pinholes or other openings in the graphene
selectively nucleate the direct epitaxy of film on substrate Lim et al.
(2022), followed by lateral overgrowth and coalescence Manzo et al. (2022).
These pinholes can appear natively in the graphene or they can be created
during pre-growth annealing due to desorption of native oxides or other
contaminants at the transferred graphene/substrate interface Kim et al.
(2021); Manzo et al. (2022). Van der Waals epitaxy, in which a film grows with
epitaxial registry to the 2D material rather than the underlying substrate,
can also produce exfoliatable single crystalline films. Examples include GaN
on graphene/SiC (0001) Kim et al. (2014) and GaN on hexagonal BN/Al2O3 (0001)
Kobayashi et al. (2012). Finally, interfacial carbides can form at the
interfaces between some rare earth or transition metals and graphene, e.g.
Ni2C Jacobson et al. (2012) and several Gd-carbides Shevelev et al. (2015),
further complicating the growth mechanisms. These examples illustrate that
epitaxy to the substrate and exfoliation are insufficient to prove a remote
mechanism Manzo et al. (2022). Moreover, graphene is not always required for
exfoliation: interfacial strains in thin film heterostructures can also enable
exfoliation without the need for a graphene interlayer Park et al. (2022). New
forms of evidence are needed to experimentally validate a remote epitaxy
mechanism.

Figure 1: Annealing cleans the graphene / sapphire interface. (a,b) Atomic
force microscopy (AFM) topographic images after the 400 °C and 700 °C anneals.
(c,d) AFM line profiles. After the 700 °C anneal, a step terrace profile from
the underlying sapphire substrate is observed. (e,f) Schematics of the
graphene/sapphire interface after annealing at 400 °C and 700 °C. $L$ is the
distance between pinholes.

Here, we discover alternative evidence for remote epitaxy: a 30 degree rotated
(R30) epitaxial superstructure that cannot be explained by the pinhole or van
der Waals mechanisms. Molecular beam epitaxial (MBE) growth of the half
Heusler compound GdPtSb on monolayer graphene/Al2O3 (0001) produces films that
are epitaxial to the underlying sapphire substrate, but rotated in-plane by 30
degrees compared to GdPtSb grown directly on sapphire. Preliminary
photoemission spectroscopy measurements do not detect interfacial carbides as
the origin of the R30 orientation. We show how the growth temperature,
graphene annealing conditions, and relative film/substrate versus
film/graphene lattice mismatch can tune the competing mechanisms of remote
epitaxy, pinhole epitaxy, and van der Waals epitaxy, across a series of cubic
and hexagonal Heusler compounds with varying lattice parameter: GdPtSb,
LaPtSb, and GdAuGe. All three materials can be exfoliated to produce free-
standing Heusler membranes, which are of great interest for their highly
tunable topological and magnetic properties Palmstrøm (2003); Kawasaki (2019);
Wollmann et al. (2017), including flexomagnetism Du et al. (2021). Our
experiments provide a more complete understanding and control of the competing
growth mechanisms on monolayer graphene.

##  II Results and Discussion

GdPtSb, LaPtSb, and GdAuGe films were grown by molecular beam epitaxy (MBE) on
monolayer graphene covered Al2O3 (0001) substrates. Polycrystalline monolayer
graphene was grown by chemical vapor deposition (CVD) on copper foils and wet
transferred to a pre-annealed Al2O3 (0001) surface, following the methods in
Ref. Du et al. (2021). Cubic GdPtSb ($F\bar{4}3m$), hexagonal LaPtSb
($P6_{3}mc$), and hexagonal GdAuGe ($P6_{3}mc$) films with thickness $\sim 20$
nm were grown by MBE via co-deposition of three elemental sources and capped
with amorphous Ge, following procedures similar to Ref. Du et al. (2021,
2019). Fluxes were calibrated by Rutherford Backascattering Spectrometry of
calibration samples. Sample temperatures were measured using a pyrometer that
is calibrated to the native oxide desorption temperatures of GaAs and GaSb.

We first analyze a crucial graphene preparation step: annealing of the
transferred graphene on sapphire to produce a clean interface before Heusler
film growth. This clean interface is crucial for producing Heusler films with
epitaxial registry to the underlying sapphire substrate. Figs. 1(a,c) show an
atomic force microscope (AFM) image and line profile of transferred graphene
on Al2O3 (0001) after a 400 °C anneal in ultrahigh vacuum ($p<10^{-9}$ Torr)
to remove surface adsorbates. After this light anneal, we observe extended
wrinkles and bumps in the graphene, which we attribute to trapped interfacial
contaminants beneath the graphene. GdPtSb growth on these lightly annealed
surfaces tends to produce fiber textured Heusler films that are primarily
$[111]_{c}$ oriented out of plane, but randomly oriented in-plane
(Supplemental Fig. 1) suggestive of van der Waals epitaxy.

In contrast, annealing the graphene/sapphire at 700 °C produces cleaner
surfaces and interfaces in which the underlying atomic step terraces of the
sapphire are observed by AFM (Fig. 1b,d). The 700 °C annealed
graphene/sapphire also displays a much smaller concentration of pinholes than
transferred graphene on III-V substrates after native oxide desorption: $\sim
10/\mu$m2 for graphene on sapphire, compared to $\sim 200/\mu$m2 for graphene
on GaAs that result from amorphous oxide desorption Manzo et al. (2022). We
attribute the reduced graphene pinhole density on sapphire to the fact that
Al2O3 (0001) is an air stable crystalline surface, in contrast with III-V
surfaces that are terminated with an amorphous oxide. The high temperature
annealed graphene/sapphire interfaces provide a cleaner starting point for
investigating the mechanisms for epitaxy on graphene-covered surfaces.

Figure 2: Epitaxy and exfoliation of GdPtSb, LaPtSb, and GdAuGe on
graphene/Al2O3 (0001). (a) Schematic cross sections of the heterostructures,
as viewed along a sapphire $[\bar{1}2\bar{1}0]$ zone axis. Red = (Gd, La),
yellow = (Au, Pt), blue = (Ge, Sb), black = Al, white = O. (b) X-ray
diffraction (Cu $K\alpha$) $2\theta$ scans of films grown on
graphene/sapphire. The films are oriented [001] cubic or [0001] hexagonal out
of plane. Sapphire substrate refelections are marked with *. (c) $2\theta$
scans of the films after exfoliation. (d) Photos of the exfoliated film and
substrate after exfoliation. Substrate dimensions are $10\times 10$ mm. The
regions of the films grown on the graphene-covered region (center) are
exfoliated.

We find that Heusler films can be epitaxially grown and exfoliated from the
clean graphene/sapphire. Fig. 2(a,b) show schematic layer structures and X-ray
diffraction (XRD) measurements for GdPtSb, LaPtSb, and GdAuGe films grown by
MBE on 700 °C annealed graphene/sapphire, at a Heusler film growth temperature
of 650 °C. The $2\theta-\omega$ scans confirm the expected $[111]_{c}$ and
$[0001]_{h}$ out-of-plane orientations with no secondary phases. All films
could be exfoliated by bonding the film to a glass slide with crystalbond and
mechanically exfoliating (Fig. 2c,d), to produce Heusler membranes with
lateral dimensions of a few millimeters.

Figure 3: In-plane orientations. (a) $\phi$ scans of the GdPtSb 220, LaPtSb
$10\bar{1}2$, and GdAuGe $10\bar{1}2$ film reflections, referenced to the
sapphire $10\bar{1}4$. (b) Distribution of in-plane orientations. Each
distribution represents statistics on at least XX samples. (c) In-plane
hexagonal lattice parameters and crystal structures. For cubic GdPtSb, the
hexagonal lattice parameter is $a_{h}=\frac{1}{2}d_{110}$. For graphene, the
lattice parameter of 4.26 Angstrom corresponds to a
$(\sqrt{3}\times\sqrt{3})R30\degree$ supercell (solid black line) with respect
to the conventional unit cell (dotted black line).

Azimuthal $\phi$ scans reveal differences of the in-plane ordering that vary
with lattice mismatch (Fig. 3), indicating different growth mechanisms for the
three materials. For GdAuGe, which has the largest mismatch to sapphire (7.3%
tensile) and smallest mismatch to graphene ($4.0\%$ compressive for a 30
degree rotation with respect to graphene), we observe a distribution of in-
plane orientations corresponding to a van der Waals growth mode (Fig. 3a,b
green curve). Comparison across several GdAuGe samples on graphene/sapphire
reveal a common distribution of domain orientations, implying epitaxial
registry of the GdAuGe to polycrystalline graphene (Supplemental Fig. 2).

For LaPtSb, which has intermediate mismatch to sapphire (4.2% tensile) and
large mismatch to graphene ($7.0\%$ compressive), we observe a sixfold pattern
of $10\bar{1}2$ reflections that are aligned with the sapphire $10\bar{1}4$
reflections (Fig. 3a,b red curves). This corresponds to the expected hexagon-
on-hexagon (R0) epitaxial relationship $\langle
11\bar{2}0\rangle_{LaPtSb}\parallel\langle 11\bar{2}0\rangle_{Al_{2}O_{3}}$,
which is the same orientation that appears for direct epitaxy of LaPtSb on
sapphire (Supplemental Fig. 3). Both remote epitaxy and pinhole epitaxy
provide consistent explanations for the R0 orientation, since the ability to
exfoliate from graphene does not strictly exclude a pinhole growth mechanism
Manzo et al. (2022).

For GdPtSb, which has the smallest mismatch to sapphire (2.7% tensile), $\phi$
scans of the GdPtSb $220$ reflections reveal two epitaxial domain
orientations: R0 and R30. The R0 reflections are aligned with the sapphire
$10\bar{1}4$, corresponding to the expected hexagon-on-hexagon epitaxial
alignment $\langle 10\bar{1}\rangle_{GdPtSb}\parallel\langle
2\bar{11}0\rangle_{Al_{2}O_{3}}$. This R0 orientation is the same as observed
for direct epitaxy of GdPtSb on sapphire (Fig. 4a), and is consistent with
both pinhole and remote mechanisms. The R30 orientation of GdPtSb is rotated
by 30 degrees with respect to the sapphire: $\langle
211\rangle_{GdPtSb}\parallel\langle 2\bar{11}0\rangle_{Al_{2}O_{3}}$. This
orientation is inconsistent with a pinhole mechanism.

Figure 4: R30 orientation for GdPtSb on graphene/Al2O3 (0001). (a) Azimuthal
$\phi$ scan for GdPtSb grown directly on sapphire, corresponding to the
standard hexagon-on-hexagon epitaxial relationship. (b) $\phi$ scan for GdPtSb
on graphene / Al2O3 (0001). The GdPtSb $220$ reflections are shifted by
$\Delta\phi=30$ degrees with respect to the sapphire $10\bar{1}4$. (c) R0
hexagon-on-hexagon orientation. The GdPtSb lattice is shown in blue and the
sapphire lattice in black. The mismatch is $2.7\%$ tensile. (d) R30
orientation. The corresponding $(5\times 5)$ supercell (red) with $5\cdot
a_{sapphire}\approx 3\cdot(\frac{1}{2}d_{210,GdPtSb})$ has a smaller lattice
mismatch of $1.5\%$ compressive.

To emphasize the unique origins of the R30 orientation, Fig. 4 compares $\phi$
scans for a GdPtSb film grown directly on sapphire with another GdPtSb film
grown on graphene/sapphire. For the sample grown directly on sapphire we
observe a three-fold pattern of 220 reflections that are aligned with the
sapphire $10\bar{1}4$ reflections, corresponding to the R0 hexagon-on-hexagon
epitaxial alignment. A weaker set of 220 reflections are shifted by
$\Delta\phi=60$ degrees from the main reflections, corresponding to antiphase
domains. In contrast, for GdPtSb epitaxy on graphene/sapphire
($T_{anneal}=700$, $T_{growth}=600\degree$ C) we observe a six-fold pattern of
220 reflections that are shifted by $\Delta\phi=30$ degrees from the substrate
reflections (Fig. 4b). This corresponds to a 30 degree rotated epitaxial
relationship (R30) (Fig. 4d).

This R30 orientation of GdPtSb provides a possible fingerprint of remote
epitaxy, since it is inconsistent with the leading competing growth
mechanisms. For pinhole-seeded epitaxy, only the R0 domain appears because the
exposed pinholes are sites for direct epitaxy. For van der Waals epitaxy, a
random distribution of in-plane orientations appears because the
polycrystalline graphene has a random distribution of orientations in plane
(Supplemental Fig. 1). Intercalation under the graphene Briggs et al. (2020),
which could in principle stabilize different epitaxial relationships, is
unlikely because the GdPtSb films can generally exfoliated without large scale
spalling marks (Fig. 2d).

Figure 5: Controlling in-plane rotations (a) Azimuthal $\phi$ scan of the
GdPtSb 220 reflections for three films grown at 600, 650, and 700 °C (blue),
on graphene/sapphire that had been annealed to 700 °C. All curves are
referenced to the $10\bar{1}4$ reflections of the sapphire substrate (black).
(b) Out of plane 444 reflection tracking changes in lattice parameter. (c,d)
Possible mechanisms to explain the temperature dependence. (c) Metastability
of the 30 degree domain on graphene. In this picture, both R0 and R30 domains
appear for growth on graphene. Increasing the growth temperature enables the
system to surmount a kinetic barrier between the two domains. (d) Pinhole vs
remote mechanism. Higher growth temperatures favor growth at pinholes, due to
the increased surface diffusion length $\lambda$. Growth from pinholes results
in the R0 domain.

To test the origins of the R30 orientation on graphene we investigate its
growth temperature dependence. Fig. 5 shows $\phi$ scans of three GdPtSb
samples grown on graphene/sapphire. For all three samples, the
graphene/sapphire was first annealed at 700 °C to produce a clean interface.
We find that the balance of R30 vs R0 domains is strongly tuned by the GdPtSb
growth temperature. High growth temperature (700 °C) favors the R0, low growth
temperature (600 °C) favors the R30, and intermediate growth temperature
produces a mixture of the two orientations. In contrast, for GdPtSb growth
directly on sapphire, only the R0 alignment is observed over the same range of
temperatures (Supplemental Fig. 4). These changes of in-plane orientation for
films on graphene coincide with a change in strain state, where the high
temperature R0 sample films are strained, while the low temperature R30 sample
is relaxed to the bulk lattice constant (Fig. 5b).

Two scenarios may explain this temperature dependence on graphene. First,
increasing the growth temperature is expected to tune the balance between
remote epitaxy and pinhole-seeded epitaxy (Fig. 5d). Pinhole epitaxy is
favored at high growth temperatures, in which the surface diffusion length for
Gd, Pt, and Sb adatoms is larger than the distance between graphene pinholes
($\lambda>L$) Manzo et al. (2022). Here, adatoms can diffuse far enough to
find the more chemically reactive pinhole sites, leading to direct nucleation
of R0 domains at pinholes. At lower growth temperatures the shorter surface
diffusion length ($\lambda<L$) favors random nucleation on clean graphene,
leading to the R30 by remote epitaxy. This scenario is also consistent with
the observed changes in lattice parameter (Fig. 5b), where we find that the
high temperature film with R0 orientation is strained (consistent with direct
epitaxy at pinholes) and the low temperature film with R30 is relaxed
(consistent with growth on graphene).

Another scenario is that both the high and low temperature regimes are remote
epitaxy on graphene, and increasing the growth temperature allows the system
to surmount a kinetic barrier between a metastable R30 and a stable R0 domain
(Fig. 5(c)). Further experiments are required to understand the energetics and
kinetics of rotational domain formation on graphene. Regardless of microscopic
mechanism, the appearance of the R30 at low growth temperature ($<700\degree$
C) is highly suggestive of a remote epitaxy mechanism via elimination of the
pinhole, van der Waals, and intercalation mechanisms in this materials system.

##  III Discussion

Our experiments rule out pinholes, van der Waals epitaxy, and intercalation as
origins of the R30 GdPtSb orientation. Does this imply that the R30 is formed
by a remote epitaxy mechanism? Why is the R30 orientation favored over the
standard R0 on graphene? And is the R30 unique to GdPtSb, or would other
compounds form this orientation?

Regarding the first question of the R30 as proof of remote epitaxy, it is
worth considering one more mechanism: interfacial carbide formation. Although
stable in contact with many materials, graphene is known to react with several
transition and rare earth metals to form interfacial carbides, which can seed
new epitaxial relationships for subsequent film growth. For example,
interfacial Ni-carbides form during the CVD growth of graphene on Ni (111) and
are known to produce graphene domains that are rotated from the direct
graphene on Ni alignment Jacobson et al. (2012). Additionally, several Gd-
carbides form at Gd/graphite interfaces at temperatures ranging from below 800
K to 1100 K Shevelev et al. (2015), which is similar to our GdPtSb growth
temperature. Among the many Gd-carbides, the electride Gd2C has a layered
structure Lee et al. (2020), suggesting that exfoliation from Gd2C may be
possible.

To test the possibility of carbides at the GdPtSb/graphene interface, we
performed photoemission spectroscopy measurements of a 2 monolayer GdPtSb film
on graphene/sapphire (Supplemental Fig. 5). Our preliminary measurements did
not detect any carbide components in the C $1s$ core level, compared to the
known GdC2 and Gd2C3 that produce shifts of 1.6 and 3.5 eV, respectively
Shevelev et al. (2015). However, we were not able to find reference data for
Gd2C. While it is early to completely rule out other carbide formation at the
GdPtSb/graphene interface, it is possible that the ternary GdPtSb/graphene
interface is more stable with respect to carbide formation than metal
Gd/graphite interfaces Shevelev et al. (2015), thus explaining the absence of
C 1s core level shifts for GdPtSb/graphene. Raman spectroscopy also confirms
that after GdPtSb exfoliation, there is leftover graphene on the sapphire
substrate (Supplemental Fig. 6). Further experiments are needed to understand
the possible role of carbides at GdPtSb/graphene and other film/graphene
interfaces, in which the film contains transition or rare earth metals.

Why is R30 favored over R0 for GdPtSb on graphene/sapphire? We hypothesize
that for a remote mechanism, the R30 orientation is favored because the weak
interactions across the graphene change the balance between the energy of
interfacial bonding versus the strain energy, favoring small strains via a
lattice rotation Du et al. (2021). Whereas the R0 GdPtSb orientation has a
lattice mismatch with sapphire of 2.7% tensile (Fig. 4c), the R30 orientation
corresponds to a $(5\times 5)$ superstructure with a mismatch of only $1.5\%$
compressive mismatch to sapphire (Fig. 4d). In this supercell, $5\cdot
a_{sapphire}\approx 3\cdot(\frac{1}{2}d_{210,GdPtSb})$. The lattice relaxation
of the R30 grown at 600 °C compared to R0 grown at 700 °C (Fig. 5b) are
consistent with this picture. Further studies are required to understand the
structure and energetics of GdPtSb/graphene/sapphire interfaces.

Finally, is the R30 unique to GdPtSb? So far we have only observed a phase
pure R30 for GdPtSb on graphene/sapphire. Over a similar range of growth
temperatures, for LaPtSb growth on graphene/sapphire we only observe the R0
and for GdAuGe we observe random in-plane orientations. We anticipate the
formation of R30 or other rotated epitaxial superstructures will depend on the
details of film/substrate lattice mismatch, the surface diffusion length vs
pinhole separation, and the possibility of interfacial phases. Rotational
ordering appears in other systems with weak coupling between film and
substrate Novaco and McTague (1977); Doering (1985); Shaw et al. (1978);
Doering and Semancik (1986). We anticipate that a similar framework may apply
to the Heusler/graphene/sapphire system, in which Heusler film and sapphire
substrate are weakly coupled due to the graphene spacer. Controlling the
rotation angle during synthesis via this weak coupling may provide an
alternative route for fabricating and discovering new electronic phases in
moiré heterostructures Cao et al. (2018); Bistritzer and MacDonald (2011).

##  IV Conclusions

We discovered an R30 rotated superstructure consistent with remote epitaxy
that cannot be explained by the competing mechanisms of pinhole-seeded lateral
epitaxy or van der Waals epitaxy. Further studies are required to understand
possible interfacial carbides. We also showed how the balance between these
three mechanisms can be controlled by growth temperature, graphene annealing
conditions, and lattice mismatch. Van der Waals epitaxy, in which the films
are aligned to graphene, occurs when the graphene/substrate interface is
contaminated or when the film has a closer lattice match to the graphene than
to the substrate. Pinhole epitaxy can dominate at high growth temperatures,
where the surface diffusion length is larger than the spacing between
unintentional pinholes. Finally, remote epitaxy may occur on clean graphene at
lower growth temperatures, where surface diffusion is small enough that films
nucleate on clean regions of graphene rather than only at pinholes.

##  V Acknowledgments

We thank Thomas F. Kuech, Chris J. Palmstrøm, and Donald Savage for
discussions. Heusler synthesis and characterization were primarily supported
by the Air Force Office of Scientific Research (FA9550-21-0127). Preliminary
synthesis of LaPtSb and GdPtSb were supported by the Army Research Office
(W911NF-17-1-0254). Graphene transfers and characterization were supported by
the National Science Foundation (DMR-1752797). Graphene synthesis and
characterization are supported by the U.S. Department of Energy, Office of
Science, Basic Energy Sciences, under award no. DE-SC0016007. We gratefully
acknowledge the use of x-ray diffraction and Raman facilities supported by the
NSF through the University of Wisconsin Materials Research Science and
Engineering Center under Grant No. DMR-1720415. This research used resources
of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of
Science User Facility operated for the DOE Office of Science by Argonne
National Laboratory under Contract No. DE-AC02-06CH11357.

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