# Nonreciprocity of supercurrent along applied magnetic field

Filippo Gaggioli<EMAIL_ADDRESS>Department of Physics, Massachusetts
Institute of Technology, Cambridge, MA-02139,USA  Yasen Hou  Francis Bitter
Magnet Laboratory & Plasma Science and Fusion Center, Massachusetts Institute
of Technology, Cambridge, MA-02139, USA  Jagadeesh S. Moodera  Francis Bitter
Magnet Laboratory & Plasma Science and Fusion Center, Massachusetts Institute
of Technology, Cambridge, MA-02139, USA  Department of Physics, Massachusetts
Institute of Technology, Cambridge, MA-02139,USA  Akashdeep Kamra  Condensed
Matter Physics Center (IFIMAC) and Departamento de Física Teórica de la
Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain

###### Abstract

Nonreciprocal current responses arise in a broad range of systems, from
magnons and phonons to supercurrents, due to an interplay between spatial and
temporal symmetry breakings. These find applications in devices, such as
circulators and rectifiers, as well as in probing the interactions and states
that underlie the nonreciprocity. An established symmetry argument anticipates
emergence of nonreciprocal currents along a direction perpendicular to the
applied magnetic field that breaks the time-reversal symmetry. Here, motivated
by recent experiments, we examine the emergence of nonreciprocity in vortex-
limited superconducting critical currents along an applied magnetic field.
Employing London’s equations for describing the Meissner response of a
superconducting film, we find that an additional symmetry breaking due to a
preferred vortex axis enables nonreciprocal critical currents along the
applied magnetic field, consistent with the so far unexplained experimental
observation. Building on our concrete theoretical model for supercurrents, we
discuss a possible generalization of the prevailing symmetry consideration to
encompass nonreciprocal currents along the time-reversal symmetry breaking
direction.

Introduction.—When forward and backward directions can be distinguished in a
system, nonreciprocal behavior can manifest via, for example, different
resistances or currents along these opposite directions [1, 2]. A prototypical
example is a p-n junction diode in which a simple identification of p-type and
n-type subsystems admits different backward and forward current flows. Thus,
in such systems divisible into two “lumped” subsystems, the spatial-inversion
breaking along a direction $\hat{\boldsymbol{n}}$ alone can be sufficient to
realize nonreciprocal behavior along this same axis [3, 4, 5, 6]. In extended
systems, one may distinguish between forward and backward transport along the
axis $\hat{\boldsymbol{n}}\times\hat{\boldsymbol{h}}$, where
$\hat{\boldsymbol{h}}$ is the direction of time-reversal symmetry breaking via
an applied magnetic field or similar [7, 8, 9]. Consistent with this
principle, nonreciprocal responses of magnons [10, 11, 12, 13], phonons [14,
15], and Cooper pairs [16, 17, 18, 19, 20] perpendicular to an applied
magnetic field have been observed using a wide range of systems and mechanisms
[1, 2]. Such responses enable useful devices, such as rectifiers and
circulators. At the same time, they offer a convenient probe for
unconventional interactions and states of quantum matter [21, 22, 23, 24].
Both these tasks, namely design of devices and effective probing, strongly
rely on a general understanding of the symmetries that allow for nonreciprocal
responses [2, 25, 26].

It therefore came as a surprise when unequal superconducting critical currents
in the forward and backward directions, a phenomenon dubbed the
superconducting diode effect (SDE) [20, 16], were observed in thin film
superconductors subjected to a magnetic field parallel to the current flow
direction [19], because the observation defied the symmetry-based expectation
stated above. The phenomenon of SDE has gained a renewed interest [9, 27, 28,
29] and focus in the recent years due to its observation via a broad range of
systems and mechanisms [16, 17, 18, 19, 20]. We here focus on thin films of a
nominally centrosymmetric superconductor, where the time-reversal and spatial-
inversion symmetries are broken, respectively, by an applied magnetic field
and by inequivalent vortex surface barriers on the two sides [30, 27, 31, 32],
possibly due to defects and geometric features introduced during the
lithography process [29, 19]. The critical current in such thin films is
typically determined by the Bean-Livingston vortex surface barrier [33]. The
current which is large enough to exert sufficient Lorentz force on the
vortices to overcome the weakest surface barrier becomes the critical value
[34, 35, 36, 31] (Fig. 1). The consequent vortex-mediated SDE has been
investigated and understood for applied magnetic fields perpendicular to the
current flow direction [29, 19, 37, 38, 39], consistent with the prevailing
symmetry argument above.

Figure 1: Schematic depiction of the vortex mechanism underlying the SDE in an
in-plane magnetic field $\mathbf{H}$ parallel to the applied bias current
$\mathbf{I}$. $(a)$ For $\mathbf{H}=0$, vortices penetrate the superconductor
edges with their axes out-of-plane. For a sample with asymmetry along the
$\hat{\boldsymbol{n}}$ direction, the right surface barrier, for example, is
weaker and the vortices preferably penetrate from this edge with their axis
(anti-)parallel to the $z-$axis for negative (positive) currents, thereby
determining $I^{\pm}_{c}(0)$. $(b)$ At finite $\mathbf{H}$, the vortices tilt
in the direction of the magnetic field and form an angle $\theta$
($\pi-\theta$) with the $z-$axis for negative (positive) currents. When the
surface barrier is angular dependent and strongest (or weakest) along
$\mathbf{v}_{0}$ ($\tilde{\mathbf{v}}_{0}$), e.g., because of the lithographic
process (white stripes), the different barriers experienced by the titled
vortices result in unequal critical current densities for positive and
negative current flows, giving rise to the SDE. $(c)$ The current density
across a $xz-$section of the thin film close to the right edge. This Meissner
response to $\mathbf{H}$ is given by Eq. (5) (for the $z-$component).

In this Letter, we examine the critical currents and SDE in a conventional
centrosymmetric superconductor film subjected to a magnetic field parallel to
the current flow direction, thereby going beyond the usual symmetry
consideration. Considering the surface barrier mechanism typical of
superconducting thin films [32], we find that the critical current is
determined by the penetration of vortices whose axis is (weakly) tilted in the
applied field direction. A surface barrier that is maximal for vortices with a
finite tilt angle then gives rise to different critical currents in the
forward and backward directions [Fig. 1$\,(b)$]. Such an angular dependence
for the vortex barrier may result from, for example, (controllable)
geometrical defects induced by lithographic preparation of the film. Our
theoretical results agree with the experimental data [19] semi-quantitatively
and thus offer a plausible explanation. We further discuss a possible
generalization of the prevailing symmetry argument and find nonreciprocity
along the direction $\hat{\boldsymbol{n}}\times V_{0}\hat{\boldsymbol{h}}$,
where $V_{0}$ is a tensor characterizing the vortex barrier angular
anisotropy. This reduces to the standard
$\hat{\boldsymbol{n}}\times\hat{\boldsymbol{h}}$ framework when the vortex
barrier has no angular dependence.

Vortex-limited critical current.— In a wide range of superconducting films,
the critical supercurrent is determined by the presence of a steep surface
barrier preventing the penetration of vortices inside the sample [34, 35, 36,
40, 41, 32]. This barrier is first overcome when the average Lorentz force
density $\bar{\boldsymbol{F}}_{\scriptscriptstyle L}$ acting on the vortices –
whose axis is not necessarily parallel to the out-of-plane direction [31] –
reaches a critical value, associated to an equivalent critical current density
$j_{s}$. For a thin film of width $-W/2\leq x\leq W/2$ that extends
indefinitely along $y$, this critical condition reads

$\bar{\boldsymbol{F}}_{\scriptscriptstyle
L}\left(\pm\frac{W}{2}\right)=\left(\frac{\phi_{0}}{c}\right)\bar{\boldsymbol{\j}}\times\boldsymbol{\hat{e}}_{v}=\mp\left(\frac{\phi_{0}}{c}\right)j_{s}\,\hat{\boldsymbol{x}},$
(1)

for the vortex axis $\boldsymbol{\hat{e}}_{v}=(0,\sin\theta,\cos\theta)$ that
maximizes the Lorentz force working against the surface barrier [34, 31].
Here, we denote the current density $\boldsymbol{j}$ averaged over the
coherence length $\xi$ by $\bar{\boldsymbol{\j}}$, $\phi_{0}=hc/2e$ is the
flux quantum. The task at hand is therefore to calculate the average current
density $\bar{\boldsymbol{\j}}$ flowing at distances $\lesssim\xi$ from the
superconductor edges. Vortices with axis
$\boldsymbol{\hat{e}}_{v}\perp\bar{\boldsymbol{\j}}$ will then be the first to
fulfill the condition (1), thereby determining $I_{c}(H)$.

In the absence of trapped vortices, the current density $\boldsymbol{j}$
inside the superconductor is determined self-consistently by the interplay of
the external field $\boldsymbol{H}$ and the self-field produced by the
current. In thin films with thickness $d$ much smaller than the London length
$\lambda$, the self-field is negligible [32] and the magnetic field is
approximately constant inside and outside the superconductor. In this case,
the current density distribution is found from the London equation [34, 32]

$\boldsymbol{\nabla}\times\boldsymbol{j}=-\frac{c}{4\pi\lambda^{2}}\boldsymbol{H},$
(2)

with the additional condition that the integral
$\int\\!j_{y}\,\mathrm{d}x\,\mathrm{d}z$ is equal to the applied bias current
$\boldsymbol{I}\parallel\hat{\boldsymbol{y}}$.

The solution to Eq. (2) depends on the magnetic field, the bias current and,
via the vanishing of the current density $j_{\perp}$ at the boundaries, on the
precise film geometry. For a perpendicular magnetic field along the z axis,
Eq. (2) yields $x$-dependent current density along $y$,

$j_{y}(x)=\frac{I}{dW}-\frac{c}{4\pi\lambda^{2}}Hx.$ (3)

Following the principle described via Eq. (1), vortices enter the
superconductor when the condition $j_{y}(x)=j_{s}$ is first met on either side
of the film ($x=\pm W/2$): at zero magnetic field (external bias), this
happens when the external bias (magnetic field) reaches [32]

$I_{0}\equiv j_{s}\,dW,\quad H_{s}\equiv\frac{8\pi\lambda^{2}}{cW}j_{s}.$ (4)

In the parallel field scenario presented in Fig. 1$\,(b)$, on the other hand,
the bias current determines $j_{y}\\!=\\!\bar{\j}_{y}\\!=\\!I/d\,W$, while the
screening currents $\propto H$ flow in the $xz$-plane [Fig. 1$\,(c)$] with
boundary conditions $j_{x}(x\\!=\\!\pm W/2,z)\\!=\\!j_{z}(x,z\\!=\\!\pm
d/2)=0$. The evaluation of these Meissner currents is detailed in the
Appendix. For a small aspect ratio $d/W\ll 1$, we then find that

$\displaystyle
j_{z}(x,z)\approx\pm\frac{c}{4\pi\lambda^{2}}Hd\left(\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(k_{n}d/2)^{2}}e^{-k_{n}\Delta
x}\cos{k_{n}z}\right),$ (5)

with $k_{n}=(n+1/2)\,2\pi/d$ and $\Delta x=|W/2\mp x|$ the distance from the
edges. As expected, $j_{z}$ grow rapidly at distances $\lesssim d/2$ from the
edges, where the effect of the boundary conditions
$\boldsymbol{j}\perp\hat{\boldsymbol{x}}$ is important.

To average $j_{z}$ over the size $\xi$ of the vortex core, we take advantage
of the rapid decay of the sum in Eq. (5) and consider only the contribution of
the first (dominant) term to $\bar{\j}_{z}$. For the typical thin films where
$\pi\xi/d\gg 1$, this gives

$\displaystyle\bar{\j}_{z}\approx\pm\frac{2d}{\xi}\left(\frac{2}{\pi^{2}}\right)^{2}\\!\frac{c}{4\pi\lambda^{2}}Hd.$
(6)

Using that $\hat{\boldsymbol{e}}_{v}\\!\perp\\!\bar{\boldsymbol{\j}}$ for
maximal Lorentz force and inserting the expressions for
$\bar{\j}_{y},\,\bar{\j}_{z}$, the condition (1) then provides an equation for
$I_{c}(H)$,

$\sqrt{\left(\frac{I_{c}(H)}{dW}\right)^{2}+\left(\frac{2d}{\xi}\left(\frac{2}{\pi^{2}}\right)^{2}\\!\frac{c}{4\pi\lambda^{2}}Hd\right)^{2}}=j_{s}.$
(7)

This finally yields the field dependence of the critical current

$\displaystyle
I_{c}(H)=I_{0}\sqrt{1-\left(\frac{H}{H^{\parallel}_{s}}\right)^{2}},$ (8)

with the characteristic field scale

$\displaystyle H_{s}^{\parallel}=\left(\frac{\pi}{2}\right)^{4}\left(\frac{\xi
W}{d^{2}}\right)H_{s},$ (9)

that is directly related to the sample penetration field $H_{s}$ in a
perpendicular field [32]. Equations (8) and (9) show two interesting features
that are in marked constrast with the $I_{c}(H)$ of the superconducting thin
film in a perpendicular magnetic field. First, the field dependence is
quadratic and not linear as for the perpendicular case [36, 32, 30]. Second,
the magnetic scale $H_{s}^{\parallel}$ is much larger than the corresponding
$H_{s}$, as the prefactor in Eq. (9) is of the order of $\sim 10^{4}$ for
$\xi/d\sim 1$ and a typical aspect ratio $d/W\sim 10^{-3}$. It then follows
that an experiment measuring changes in the critical current for perpendicular
fields in the order of Gauss should observe similar variations in $I_{c}(H)$
for parallel fields in the range of Tesla.

Having found the critical current (8), we fix $\bar{\j}_{x}=I_{c}(H)$ and use
that $\hat{\boldsymbol{e}}_{v}\\!\perp\\!\bar{\boldsymbol{\j}}$ to evaluate
the vortex tilt angles $\theta^{\scriptscriptstyle(L,R)}$ on the left and
right edge

$\left(\sin\theta^{\scriptscriptstyle(L,R)},\cos\theta^{\scriptscriptstyle(L,R)}\right)=\left(\frac{H}{H^{\parallel}_{s}},\pm\,\frac{I}{|I|}\sqrt{1-\left(\frac{H}{H^{\parallel}_{s}}\right)^{2}}\right).$
(10)

As shown in Fig. 1$(b)$ and in agreement with the pseudovector properties of
the magnetic field, the tilt angles at the two edges are exchanged upon
switching the direction of the bias current $I$, such that
$\theta^{\scriptscriptstyle R}\to\theta^{\scriptscriptstyle
L}=\pi-\theta^{\scriptscriptstyle R}$.

Superconducting diode effect.— Let us now discuss the nonreciprocal transport
properties of the parallel field setup discussed above [19]. We first
recapitulate the SDE in a perpendicular field, as this provides a useful
comparison. The SDE, in the perpendicular case, is realized when the critical
current densities $j_{s}^{\scriptscriptstyle(L,R)}$ on the left and right edge
of the superconductor are not identical,

$j_{s}^{\scriptscriptstyle(L,R)}=j_{s,0}\mp\Delta
j_{s}\left(\hat{\boldsymbol{n}}\cdot\hat{\boldsymbol{x}}\right),$ (11)

with $\Delta j_{s}$ assumed positive, without a loss of generality, and the
asymmetry vector $\hat{\boldsymbol{n}}=\left(-1,0,0\right)$ shown in Fig.
1$\,(a)$. As a result, $I_{c}(H)$ reaches its maximum at the peak field [32]

$\frac{H_{\mathrm{max}}}{H_{s,0}}=\frac{\Delta
j_{s}}{j_{s,0}}\left[(\hat{\boldsymbol{n}}\times\hat{\boldsymbol{h}})\cdot\hat{\boldsymbol{I}}\right],$
(12)

which in turn determines the magnitude of the SDE.

In parallel in-plane fields, the magnitude of the current density
$\bar{\boldsymbol{\j}}$ at criticality is constant and determined by the
weakest surface barrier. Vortices then always enter from the same edge,
suggesting that no SDE can be realized in agreement with the fact that
$\hat{\boldsymbol{n}}\times\hat{\boldsymbol{h}}=0$. To understand the
parallel-field SDE, we consider an additional kind of symmetry breaking, this
time at the level of the individual surface barriers, that allows the system
to distinguish between opposite signs of the bias current. We take into
account a dependence of the surface barrier on the vortex tilt angle $\theta$

$j_{s}^{\scriptscriptstyle(L,R)}(\theta)=j_{s}^{\scriptscriptstyle(L,R)}+\delta
j_{s}\cos 2\left(\theta_{0}-\theta\right),$ (13)

that may result, for example, from columnar tracks left by the litographic
process, represented as white stripes in Fig. 1. In assuming the form of this
angular dependence, time-reversal symmetry requires the surface barrier to be
the same for vortices with opposite fluxes. Considering reflections across the
$xz-$plane, we then expect that the vortices will experience a different
surface barrier (13) as $\theta\to\pi-\theta$ and $I\to-I$, giving rise to the
SDE as long as $\theta_{0}\neq 0,\pi/2$. In agreement with the time-reversal
symmetry, SDE vanishes for $H=0$ as the $\cos 2\theta$ dependence ensures that
$j_{s}(\theta)$ remains the same when $\theta$ is changed from $0$ to $\pi$.

In evaluating the critical condition (1) with $j_{s}(\theta)$ given by Eq.
(13), we assume that the angular dependence is weak, i.e., $\delta j_{s}\ll
j^{\scriptscriptstyle(L,R)}_{s}$, and much weaker than the difference between
$j_{s}^{\scriptscriptstyle L}$ and $j_{s}^{\scriptscriptstyle R}$, i.e.,
$\delta j_{s}\ll\Delta j_{s}$. This corresponds to the limit where vortex
penetration happens from the weak (assumed right here) edge only, such that
$j_{s}^{\scriptscriptstyle R}(\theta)$ alone determines the critical current
$I_{c}(H)$ while $j_{s}^{\scriptscriptstyle L}(\theta)$ does not play any
role. In what follows, we will therefore neglect the $(L,R)$ indices unless
necessary.

With the right tilt angle $\theta$ given by Eq. (10), the field dependence of
the surface barrier (13) can be immediately found (we neglect terms
$\propto(\delta j_{s}/j_{s})^{2}$),

$\displaystyle j_{s}(H)$ $\displaystyle=j_{s}\left[1+\frac{\delta
j_{s}}{j_{s}}\cos\left(2\theta_{0}\pm 2H/H_{s}^{\parallel}\right)\right],$
(14)

where the plus and minus signs refer to positive and negative bias currents.
To determine $I_{c}(H)$ from Eq. (1), we now plug in (14) and expand the
cosine term around $2\theta_{0}$ while neglecting terms $\propto(\delta
j_{s}/j_{s})^{2}$ and $\propto(\delta j_{s}/j_{s})(H/H^{\parallel}_{s})^{2}$
leading to

$I^{\pm}_{c}(H)\approx
I_{\max}\sqrt{1-\left(\frac{H-H^{\pm}_{\max}}{H^{\parallel}_{s}}\right)^{2}},$
(15)

where we introduced the peak field

$H_{\mathrm{max}}^{\pm}=\mp\left(2\sin 2\theta_{0}\frac{\delta
j_{s}}{j_{s}}\right)H^{\parallel}_{s},$ (16)

and the peak current

$\displaystyle
I_{\max}=I_{0}\sqrt{1+\cot(2\theta_{0})\frac{|H^{\pm}_{\max}|}{H_{s}^{\parallel}}}.$
(17)

Figure 2: Experimental data from Ref. 19 (circles and solid lines) for the
critical current $I^{\pm}_{c}(H)$ (orange and blue) and the SDE efficiency
$|\eta(H)|$ (pink) for a superconducting vanadium thin film in a parallel in-
plane magnetic field. Fitting Eqs. (15) and (18) from our theoretical model
(dashed lines), we find very good agreement for the indicated choice of
parameters. The field $H_{s}^{\parallel}\approx 7.30\,\text{T}$ obtained from
fitting agrees well with the prediction $H_{s}^{\parallel}\sim 9\,\text{T}$
from Eq. (9), using the experimental values for $H_{s},\xi,d$ and $W$ from
Ref. 19.

The shifted field dependence (15) results from the two-fold contributions of
the magnetic field to vortex penetration: on the one hand, to enhance the
Meissner current (6) and, on the other, to reinforce the surface barrier (14).
These effects compensate at the peak field $H_{\mathrm{max}}$, which does not
depend on the left-right asymmetry $\Delta j_{s}/j_{s,0}$, as in Eq. (12), but
is fixed entirely by the parameters $\delta j_{s}/j_{s}$ and $\theta_{0}$ in
the limit where $I_{c}(H)$ is determined by the right edge alone.

As in the case of a perpendicular field, the peak field $H_{\mathrm{max}}$
determines the efficiency $\eta$ of the superconducting diode device
$\eta(H)\equiv(I_{c}^{+}(H)-I_{c}^{-}(H))/(I_{c}^{+}(H)+I_{c}^{-}(H))$. Using
Eq. (15) and neglecting terms $\propto(H_{\max}/H^{\parallel}_{s})^{2}$ and
$\propto(H_{\max}/H^{\parallel}_{s})(H/H_{s}^{\parallel})^{3}$, this is found
to read

$\eta(H)\approx\left(\frac{H^{+}_{\max}}{H_{s}^{\parallel}}\right)\frac{H}{H_{s}^{\parallel}}.$
(18)

Equations (15) and (18) lend themselves conveniently to a comparison with the
experimental results on the parallel-field SDE reported in Ref. 19. Fitting
$I_{c}^{\pm}(H)$ and $\eta(H)$ to the experimental data, we obtain the dashed
curves in Fig. 2. These reproduce the critical currents and the efficiency
curve very well for the choice of $I_{\max},\,H_{\max},\,H_{s}^{\parallel}$
reported in the plot. The small differences can result from the various
approximations employed in our analytic simplifications. This combination of
parameters is consistent with the assumptions underlying Eqs. (15) and (18),
as $(\delta j_{s}/j_{s})\sim(H_{\mathrm{max}}^{\pm}/H^{\parallel}_{s})\ll 1$.

Finally, we compare the field value $H_{s}^{\parallel}=7.30\,\text{T}$
estimated from the fit in Fig. 2 to our theoretical prediction (9). Using the
experimental value $H_{s}\approx 11\,\text{Oe}$ for the perpendicular
penetration field and taking $\xi\approx 11\,\text{nm},\,d\approx
8\,\text{nm}$ and $W=8\,\mu\text{m}$ from Ref. 19, we find that
$H_{s}^{\parallel}\sim 9\,\text{T}$, in satisfactory agreement with the result
of our fit.

Generalized symmetry indicator for nonreciprocity.— Building upon the insights
gained from our theoretical model of the SDE in a magnetic field applied along
the current flow direction [19], we now discuss an appropriate generalization
of $\hat{\boldsymbol{n}}\times\hat{\boldsymbol{h}}$ as the direction of
nonreciprocity. The crucial addition comes from the angular dependence of the
vortex surface barrier. Our model suggests that we should find SDE whenever
the vortex tilting due to the applied magnetic field causes the vortices to
experience different surface barriers for forward and backward current flows.
This general idea supersedes the simple case of applied magnetic field along
the current direction, as we now formulate.

As discussed below Eq. (13), the surface barrier needs to satisfy time-
reversal symmetry and thus remain the same when $\theta\to\theta+\pi$. For
this reason, the symmetry-breaking associated with angular dependence
$j_{s}(\theta)$ cannot be parametrized by a simple vector, as was the case for
the left-right asymmetry (11), but requires the introduction of a tensor
quantity
$V_{0}=\boldsymbol{v}_{0}\boldsymbol{v}_{0}^{T}-\tilde{\boldsymbol{v}}_{0}\tilde{\boldsymbol{v}}_{0}^{T}$,
built from the vectors $\boldsymbol{v}_{0}=(0,\sin\theta_{0},\cos\theta_{0})$
and $\tilde{\boldsymbol{v}}_{0}=(0,\cos\theta_{0},-\sin\theta_{0})$ that
constitute the symmetry axes of the angular dependence [see Eq. (13) and Fig.
1$\,(b)$]. In terms of $V_{0}$, the surface barrier (13) reads

$j_{s}^{\scriptscriptstyle(L,R)}(\theta)=j_{s}^{\scriptscriptstyle(L,R)}+\delta
j_{s}\left(\hat{\boldsymbol{e}}_{v}\,V_{0}\,\hat{\boldsymbol{e}}_{v}\right),$
(19)

such that Eqs. (16) and (18) take the simple form

$\eta(H)\propto\frac{H_{\mathrm{max}}}{H_{s}^{\parallel}}=\frac{2\delta
j_{s}}{j_{s}}\left[(\hat{\boldsymbol{n}}\times
V_{0}\,\hat{\boldsymbol{h}})\cdot\hat{\boldsymbol{I}}\right].$ (20)

The peak field (20) is maximal when $\hat{\boldsymbol{h}}$ is parallel to
$\hat{\boldsymbol{I}}$, and vanishes when they are orthogonal. The structure
of Eqs. (19) and (20), however, suggests that this is just a special case.
Considering now edges that are tilted by an angle $\phi$ away from the
$z$-axis in the $xz$-plane, such that
$\left(\boldsymbol{v}_{0},\tilde{\boldsymbol{v}}_{0}\right)\to
R_{y}(\phi)\left(\boldsymbol{v}_{0},\tilde{\boldsymbol{v}}_{0}\right)$ with
the matrix $R_{y}(\phi)$ describing rotations about the $y$-axis, we find that
the tensor $V_{0}$ transforms as $R_{y}(\phi)V_{0}R^{T}_{y}(\phi)$. Inserting
this into Eq. (20) then allows to evaluate $H_{\mathrm{max}}$ for an arbitrary
direction of the in-plane field $\hat{\boldsymbol{h}}=(h_{x},h_{y},0)$

$\frac{H_{\mathrm{max}}}{H_{s}^{\parallel}}=\frac{2\delta
j_{s}}{j_{s}}\left[h_{y}\sin 2\theta_{0}+h_{x}\sin\phi\cos
2\theta_{0}\right]\cos\phi,$ (21)

with the SDE efficiency $\eta(H)$ again $\propto
H_{\mathrm{max}}/H^{\parallel}_{s}$. This further admits finite SDE when
magnetic field is applied along an arbitrary in-plane direction and is
consistent with the experiments [19] similar to those discussed in Fig. 2 when
the field is applied along the $x$-axis.

Conclusion.— Our theoretical treatment of vortex-limited superconducting
critical current offers a possible explanation for, and a good agreement with,
the experimentally observed SDE in superconducting thin films subjected to in-
plane magnetic fields, especially along the current flow direction. It further
suggests new avenues for a lithographic control of the SDE via the angle
$\theta_{0}$. Our consequent generalization of the symmetry arguments for
observing nonreciprocity offers an important supplement to the existing
understanding and may guide similar search for nonreciprocal responses in
other non-superconductor systems.

###### Acknowledgements.

Acknowledgements.— F.G. is grateful for the financial support from the Swiss
National Science Foundation (Postdoc.Mobility Grant No. 222230) and the
support of the EU Cost Action CA16218 (NANOCOHYBRI). A.K. acknowledges
financial support from the Spanish Ministry for Science and Innovation – AEI
Grant CEX2018-000805-M (through the “Maria de Maeztu” Programme for Units of
Excellence in R&D) and grant RYC2021-031063-I funded by
MCIN/AEI/10.13039/501100011033 and “European Union Next Generation EU/PRTR”.
The work at J.S.M Lab was supported by Air Force Office of Sponsored Research
(FA9550-23-1-0004 DEF), Office of Naval Research (N00014-20-1-2306), National
Science Foundation (NSF-DMR 1700137 and 2218550); the Army Research Office
(W911NF-20-2-0061 and DURIP W911NF-20-1-0074) and the Center for Integrated
Quantum Materials (NSF-DMR 1231319).

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##  Appendix A Meissner currents distribution

In this Appendix, we determine the distribution of the screening currents
$j_{x},j_{z}$ induced by an in-plane magnetic field applied parallel to the
direction of the bias current. To do so, we treat the $xz-$section of the thin
film as a two-dimensional superconductor of length $W$ and width $d$, threaded
by a perpendicular magnetic field in the $y-$direction, see Fig. 1$\,(c)$ in
the main text.

We then write the London equation for the current density $\boldsymbol{j}$ in
terms of the vector potential $\boldsymbol{A}$ and the order paramter phase
$\varphi$,

$\boldsymbol{j}=-\frac{c}{4\pi\lambda^{2}}\left[\boldsymbol{A}-\frac{\phi_{0}}{2\pi}\boldsymbol{\nabla}\varphi\right]$
(22)

with boundary conditions $j_{x}(x\\!=\\!\pm W/2,z)\\!=\\!j_{z}(x,z\\!=\\!\pm
d/2)=0$. Taking the curl of Eq. (22) reproduces Eq. (2) in the main text.

Having chosen the gauge $\boldsymbol{A}=-Hz\,\hat{\boldsymbol{x}}$, we use the
incompressibility condition $\boldsymbol{\nabla}\cdot\boldsymbol{j}=0$ to
obtain the Laplace equation $\nabla^{2}\varphi=0$ with boundary conditions
$\partial_{z}\,\varphi(x,z=\pm d/2)=0$ and $\partial_{x}\,\varphi(x=\pm
W/2,z)=+(2\pi/\phi_{0})Hz$. The solution is then obtained by the method of
separation of variables [42] and reads (cf. Ref. 40)

$\varphi(x,z)=\frac{8\pi
H}{\phi_{0}d}\sum_{n=0}^{\infty}\frac{(-1)^{n}\cosh\left(k_{n}x\right)\sin\left(k_{n}z\right)}{k_{n}^{3}\cosh\left(k_{n}W/2\right)}$
(23)

with $k_{n}=(n+1/2)\,2\pi/d$. Eq. (23) satisfies the boundary conditions
above, as can be proven with the help of Poisson’s summation formula.

Taking the gradient of the phase (23) and inserting into Eq. (22), we obtain
the screening current distribution shown in Fig. 1$(c)$ in the main text. For
a film with small aspect ratio $d/W\ll 1$, we expand the hyperbolic cosine and
find

$\displaystyle j_{x}$
$\displaystyle\approx-\frac{c}{4\pi\lambda^{2}}Hd\left(\frac{z}{d}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(k_{n}d/2)^{2}}e^{-k_{n}\Delta
x}\sin{k_{n}z}\right),$ (24) $\displaystyle j_{z}$
$\displaystyle\approx\pm\frac{c}{4\pi\lambda^{2}}Hd\left(\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(k_{n}d/2)^{2}}e^{-k_{n}\Delta
x}\cos{k_{n}z}\right),$ (25)

where the plus and minus signs refer to the right and left halves of the
superconductor and $\Delta x=|W/2\mp x|$ is the distance from the closest
edge.


# Fermi surfaces of the topological semimetal CaSn3 probed through de Haas van
Alphen oscillations

K A M Hasan Siddiquee  Riffat Munir  Charuni Dissanayake  Department of
Physics, University of Central Florida, Orlando, Florida 32816, USA  Xinzhe Hu
Swapnil Yadav  Yasumasa Takano  Department of Physics, University of Florida,
Gainesville, Florida 32611, USA  Eun Sang Choi  National High Magnetic Field
Laboratory, Florida State University, Tallahassee, Florida 32816, USA  Duy Le
Talat S Rahman  Yasuyuki Nakajima∗ Department of Physics, University of
Central Florida, Orlando, Florida 32816, USA

###### Abstract

In the search of topological superconductors, nailing down the Fermiology of
the normal state is as crucial a prerequisite as unraveling the
superconducting pairing symmetry. In particular, the number of time-reversal-
invariant momenta in the Brillouin zone enclosed by Fermi surfaces is closely
linked to the topological class of time-reversal-invariant systems, and can
experimentally be investigated. We report here a detailed study of de Haas van
Alphen quantum oscillations in single crystals of the topological semimetal
CaSn3 with torque magnetometry in high magnetic fields up to 35 T. In
conjunction with density functional theory based calculations, the observed
quantum oscillations frequencies indicate that the Fermi surfaces of CaSn3
enclose an odd number of time-reversal-invariant momenta, satisfying one of
the proposed criteria to realize topological superconductivity. Nonzero Berry
phases extracted from the magnetic oscillations also support the nontrivial
topological nature of CaSn3.

Topological superconductors (TSCs) hosting Majorana fermions on the boundaries
have recently attracted much attention because of potential application in
quantum computing and other areas [1]. Although extensive studies to explore
TSCs, such as metal-intercalated Bi2Se3 Hor et al. (2010); Kriener et al.
(2011); Liu et al. (2015); Shruti et al. (2015) and half Heusler systems Butch
et al. (2011); Nakajima et al. (2015), have been conducted, unambiguous
experimental identification of topological superconductivity is still lacking.

According to a theory proposed by Fu and Berg Fu and Berg (2010), a time-
reversal-invariant centrosymmetric superconductor is a topological
superconductor, if it possesses the following properties: (1) odd-parity
pairing symmetry with a full superconducting gap and (2) an odd number of
time-reversal-invariant momenta (TRIM) in the Brillouin zone, enclosed by its
Fermi surfaces. Together with the study of the superconducting gap structure
in the superconducting state, detailed investigation of Felmiology in the
normal state is crucial for identifying topological superconductivity in
topological materials.

Recently, the binary stannide semimetal CaSn3 has been proposed to be a
promising candidate for realizing topological superconductivity, as it is
predicted to be a topologically nontrivial semimetal Gupta et al. (2017). The
nontrivial electronic band structure harbors topological nodal lines in the
absence of spin orbit coupling (SOC). Upon turning on SOC, the nodal lines
evolve into topological point nodes Gupta et al. (2017). More notably,
superconductivity has experimentally been confirmed Luo et al. (2015); Zhu et
al. (2019), and the possible nontrivial Berry phase associated with the
topological nature of CaSn3 has been obtained from a recent quantum
oscillation study in the normal state. However, it is still unclear that this
system satisfies the proposed criterion for topological superconductors, i.e.
an odd number of Fermi surfaces enclosing TRIM in the Brillouin zone.

Here we present a detailed study of de Haas van Alphen (dHvA) oscillations in
CaSn3 with torque magnetometry in high magnetic fields up to 35 T. We observe
four fundamental dHvA frequencies associated with the Fermi surfaces as well
as strong Zeeman splitting. With the aid of our band structure calculations,
we assign the measured frequencies to the Fermi surfaces of CaSn3, unveiling
that an odd number of TRIM is enclosed by the Fermi surfaces. Combined with
the observation of nontrivial Berry phases, this finding suggests that CaSn3
is a promising material to realize topological superconductivity.

Single crystals of CaSn3 were grown by a Sn self flux method. The starting
elements with a ratio of Ca:Sn = 1:9 were placed in an alumina crucible, which
in turn was sealed in a quartz tube. The mixture was heated up to 800∘C, kept
for 24 hours, and cooled slowly down to 300∘C at a rate of 2∘C/h. The excess
of Sn flux was decanted by centrifugation. We confirmed the cubic AuCu3-type
structure with space group $Pm\bar{3}m$ with the lattice constant of $a$ =
4.7331(5) Å via powder x-ray diffraction. No Sn impurities were observed
unlike the previous reports Luo et al. (2015); Zhu et al. (2019). Torque
magnetometry was performed using a capacitive cantilever in a 35 T resistive
magnet at the National High Magnetic Field Laboratory (NHMFL), Tallahassee,
FL.

The band structure and Fermi surfaces are calculated by means of density
functional theory (DFT) employing the projector-augmented wave (PAW)
pseudopotential method Blöchl (1994); Kresse and Joubert (1999) and a plane-
wave basis set as implemented in the Vienna Ab initio Simulation Package
Kresse and Furthmüller (1996); Kresse and Hafner (1993). Exchange correlation
effects are included via the generalized-gradient approximation (GGA) in the
form of Perdew-Berke-Enzerhoff (PBE) exchange-correlation functionals Perdew
et al. (1996, 1997). We set a cutoff energy of 500 eV for plane-wave
expansion, and adopt the DFT+U approach Dudarev et al. (1998) to mitigate the
lack of electron correlation in the PBE functionals. The effective Hubbard $U$
parameter is chosen to be 2.55 eV to produce a lattice parameter 4.741 Å,
close to a value of 4.742 Å that was determined experimentally Luo et al.
(2015) and not far from our measured value, $a$ = 4.7331(5) Å. All electronic
iterations converged with 0.01 meV threshold. We use a Gaussian smearing of
0.1 eV and sample the Brillouin Zone with a 15×15×15 $\Gamma$-centered grid.
SOC is incorporated in the calculations of electronic band structure of the
system. We calculate eigenvalues for all $k$-points of a
41$\times$41$\times$41 uniform grid.

Figure 1: Temperature dependent oscillatory magnetic torque in CaSn3.
Oscillatory part of magnetic torque as a function of $1/B$ at $\theta$ =
-1.7∘, where $\theta$ is the angle between the magnetic field and the [100]
axis while rotating the field from [100] to [010] in the (001) plane as
illustrated in the lower inset. The oscillation amplitudes decrease with
increasing temperatures. Upper inset: Magnetic torque as a function of the
magnetic field $B$ for $\theta$ = 8.3∘ at $T$ = 1.5 K.

Our torque magnetometry data exhibit clear dHvA oscillations. Upon applying
magnetic field at $\theta=8.3^{\circ}$, where $\theta$ is the angle between
the applied magnetic field and the [100] axis in the (001) plane, the
oscillations in the magnetic torque $\tau$ are discernible above $\sim$ 5 T at
1.5 K (fig.1 inset). The magnitudes of the oscillatory part of the magnetic
torque $\Delta\tau$ decrease rapidly with increasing temperature (fig.1).
While the fast oscillations are completely suppressed above 20 K, the slow
oscillations can be observed up to at least 80 K.

Figure 2: Fast Fourier Transform (FFT) spectrum of oscillatory magnetic torque
and peak splitting in CaSn3. (a) FFT spectrum of oscillatory magnetic torque
for $\theta$ = -1.7∘ at $T$ = 1.5 K (field range 5–35 T). Four fundamental
frequencies, $F_{\alpha}$, $F_{\beta}$, $F_{\gamma}$, and $F_{\delta}$, with
splittings are observed. Inset: Peak splitting in the $\alpha$ orbit at $T$ =
1.5 K. We observe beating due to two oscillatory components associated with
(b) the $\beta$ orbit with the periodicity of 0.0027 T-1 ($F\sim$ 370 T) for
$\theta$ = -1.7∘ at $T$ = 1.5 K, and (c) the $\alpha$ orbit with the
periodicity of 0.0195 T-1 ($F\sim$ 50 T) for $\theta$ = 38.3∘ at $T$ = 1.5 K.
Arrows indicate nodes in the beating patterns. Plus and minus signs indicate
the positions of the maxima and minima of the oscillations, respectively.

Fast Fourier transform (FFT) allows us to determine the frequencies of
oscillation components measured at $\theta$ = -1.7∘ (fig. 2a). We find four
fundamental frequencies $F_{\alpha}$, $F_{\beta}$, $F_{\gamma}$, and
$F_{\delta}$. Each of them in fact consists of two frequencies close to each
other (fig. 2a and the inset), identified to be $F_{\alpha 1}$ = 56 T,
$F_{\alpha 2}$ = 63 T, $F_{\beta 1}$ = 380 T, $F_{\beta 2}$ = 422 T,
$F_{\gamma 1}$ = 655 T, $F_{\gamma 2}$ = 729 T, $F_{\delta 1}$ = 2845 T, and
$F_{\delta 2}$ = 2890 T. This paring of close frequencies is corroborated by
clear beating patterns in the magnetic torque as a function of $1/B$. As
indicated by arrows in figs. 2b and c, nodes in the beating patterns produce
$\pi$ phase shifts in the oscillations. While the frequencies of $F_{\alpha}$,
$F_{\beta}$, and $F_{\gamma}$ are in good agreement with those previously
reportedZhu et al. (2019), $F_{\delta}$ that is prominent above 30 T (or
$1/B<$ 0.033 T-1) is newly identified by the present work.

The observed frequencies $F$ are associated with extremal cross sectional
areas, $S$, of the Fermi surfaces in the momentum space by the Onsager
relation $F=(\hbar/2\pi e)S$. Tracking the extremal cross sectional areas in
rotating magnetic fields about the crystallographic axes, we can trace the
Fermi surface topology of CaSn3. The angular dependence of dHvA oscillations
in CaSn3 about the [001] axis is plotted in fig.3a. As expected for the cubic
structure of CaSn3, we observe symmetric behavior in the magnetic torque with
respect to $\theta\sim 45^{\circ}$, similar to the previous report Zhu et al.
(2019). The dHvA frequencies extracted from the FFT analysis exhibit nearly
isotropic angular dependence for the $\alpha$, $\beta$, $\gamma$, and $\delta$
orbits, as shown in fig.3b.

To compare with the experimental observations, we show the calculated band
structure of CaSn3 in fig.4a. The predicted bands consist of a hole band (band
25) and an electron band (band 26). Band 25 comprises a large hole pocket
around the $\Gamma$ point in the Brillouin zone, surrounded by numerous small
hole pockets (fig.4b), and band 26 has cross-shaped electron pockets around
the X points and triangle-shaped electron pockets located between the $\Gamma$
and $R$ points, along with tiny electron pockets around the M points (fig.4c).
Figure 3c shows theoretical quantum oscillation frequencies generated from the
DFT calculations using the supercell K-space extremal area finder (SKEAF) code
Rourke and Julian (2012).

Figure 3: Angular dependence of dHvA oscillation frequencies. (a) Angular
dependence of the oscillatory part of magnetic torque as a function of $1/B$.
Reflecting the cubic crystal structure, the oscillations are symmetric with
respect to $\theta\sim$ 45∘. (b) Angular dependence of dHvA oscillation
frequencies. The observed frequencies $F_{\alpha}$, $F_{\beta}$, $F_{\gamma}$,
and $F_{\delta}$ are nearly independent of angle $\theta$. (c) Theoretical
results obtained from DFT calculations using the SKEAF code for band 25 (blue)
and band 26 (red).  Figure 4: Electronic band structure and Fermi surfaces of
CaSn3. (a) Band structure of CaSn3 calculated by DFT. The local minimum of the
electron band (shown in orange) at the M point is very close to the Fermi
energy $E_{F}$, suggesting that the sizes of Fermi pockets at the M points are
extremely sensitive to $E_{F}$. Inset: high symmetry points in the Brillouin
zone. (b) Fermi surfaces of band 25 (hole). A large hole pocket is located
around the $\Gamma$ point, surrounded by numerous tiny hole pockets. (c) Fermi
surfaces of band 26 (electron), consisting of cross-shaped electron pockets
around the X points, triangle-shaped electron pockets between the $\Gamma$ and
R points, and small electron pockets around the M points. Calculated dHvA
orbits for (d) $\delta$, (e) $\beta$, (f) $\gamma,\gamma^{\prime}$, and
$\gamma^{\prime\prime}$. Corresponding frequencies are plotted in fig.3c.

On comparing the theoretical oscillation frequencies with the observed ones,
we can allocate the frequencies to corresponding Fermi surfaces of CaSn3. The
$\delta$ orbit with $F_{\delta}\sim$ 2900 T corresponds well to the large hole
pocket around the $\Gamma$ point (fig.4d). The $\beta$ orbit with
$F_{\beta}\sim$ 400 T is close to the frequency branch of triangle-shaped
electron pockets as shown in fig.4e, and the $\gamma$ orbit with
$F_{\gamma}\sim$ 690 T is comparable to the frequency associated with the
cross-shaped electron pockets around the X points (fig.4f).

However, there are important discrepancies between the theoretical results and
the measured dHvA frequencies. We observe nearly isotropic angular dependence
of dHvA frequencies, and the splittings of frequencies are also isotropic. In
the $\delta$ orbit, the observed splitting is $\Delta F_{\delta}^{obs}\sim 45$
T, but the theoretical calculations predict no splitting in the branch near
$\sim$2900 T. In the $\beta$ and $\gamma$ orbits, the observed splittings are
$\Delta F_{\beta}^{obs}\sim$ 40 T and $\Delta F_{\delta}^{obs}\sim$ 70 T.
However, the theoretical calculations indicate that the frequencies for the
$\beta^{\prime}$ and $\gamma^{\prime}$ orbits, stemming from the triangle-
shaped and cross-shaped electron pockets (figs.4e and f), respectively, depend
strongly on the angle $\theta$ as shown in fig.3c, yielding angular dependent
splitting in the oscillation frequencies. Moreover, the splittings of
calculated frequencies $\Delta
F_{\beta}^{calc}=F_{\beta^{\prime}}^{calc}-F_{\beta}^{calc}$ = 150 T at
$\theta=45^{\circ}$ and $\Delta
F_{\gamma}^{calc}=F_{\gamma^{\prime}}^{calc}-F_{\gamma}^{calc}$ = 1200 T at
$\theta=0^{\circ}$ and 90∘ are much larger than the observed $\Delta
F_{\beta}^{obs}$ and $\Delta F_{\gamma}^{obs}$. We therefore conclude that the
experimentally detected $\beta$ and $\gamma$ orbits represent only parts of
the corresponding theoretical orbits. The absence of these angle-dependent
branches in our measurements can be attributed to the suppression of
oscillation amplitudes by the large curvature factor
$\Delta\tau\propto|\partial^{2}S(k)/\partial k_{\parallel}^{2}|^{-1/2}$,
suggested by the shape of calculated Fermi surfaces.

Figure 5: Spin-splitting of Fermi surface due to nonlinear Zeeman effect. (a)
Magnetic field dependence of the extremal cross-sectional areas for spin-up
and down Fermi surfaces $S_{up}$ and $S_{down}$ due to the linear Zeeman
effect. The observed extremal cross-sectional area $S^{obs}=S(B)-BdS(B)/dB$ is
identical to the zero-field limit of cross-sectional area $S(0)$, leading to
$S_{up}^{obs}=S_{down}^{obs}$. (b) Magnetic field dependence of the extremal
cross-sectional areas for spin-up and spin-down Fermi surfaces due to the
nonlinear Zeeman effect. The observed extremal cross-sectional area is given
by the extrapolation of $dS(B)/dB$ to $B=0$, yielding $S_{up}^{obs}\neq
S_{down}^{obs}$. Figure 6: Comparison between experimental and calculated dHvA
frequencies. Angular dependence of average dHvA frequencies for (a) the
$\delta$ orbit, (b) the $\gamma$ orbit, and (c) the $\alpha$ and $\beta$
orbits. Black and red dashed lines represent theoretical dHvA frequencies
without and with Fermi-energy shifts, respectively. Fermi surfaces with
shifting the Fermi energy by (d) +40 meV for the hole band aound the $\Gamma$
point, (e) +110 meV for electron pockets around the M points, (f) +15 meV for
electron pockets located between the R and $\Gamma$ points, and -20 meV for
electron pockets around the X points. Green lines represent extremal orbits
when the field is applied parallel to the [100] axis.

The slight splitting of oscillation frequencies observed in CaSn3 is
reminiscent of the spin splitting of energy bands. For instance, in
noncentrosymmetric systems with strong SOC, asymmetric SOC can induce the
splitting of Fermi surfaces even in the absence of a magnetic field Mineev and
Samokhin (2005); Butch et al. (2011). However, since CaSn3 is a
centrosymmetric system, asymmetric SOC is absent. Instead, the observed
splitting in CaSn3 can be attributed to spin-up and down Fermi surfaces with
field-dependent extremal cross sectional areas $S(B)$ due to the Zeeman
effect. Taking the Zeeman effect into account, the Onsager relation can be
rewritten as $F=\hbar(S(B)-BdS(B)/dB)/(2\pi e)$. Hence, the measured $F$
represents a back projection of field-dependent cross-sectional area $S(B)$,
i.e. an extrapolation of the tangent at $B$ to $B=0$ (fig.5) Shoenberg (2011);
McCollam et al. (2005); Rourke et al. (2008); Mercure et al. (2009). In the
conventional linear Zeeman effect, the back projection provides field-
independent $S^{obs}=S(0)$, or field-independent dHvA frequencies $F^{obs}$
(fig.5a). In this case, the observed frequencies for spin-up and down Fermi
surfaces are identical $(F^{obs}_{up}=F^{obs}_{down})$. On the other hand, the
nonlinear Zeeman effect can give rise to nonlinear magnetic field dependence
of extremal areas $S_{up}(B)$ and $S_{down}(B)$, yielding the disparity
between the observed extremal areas $S_{up}^{obs}$ and $S_{down}^{obs}$, i.e.
$F^{obs}_{up}\neq F^{obs}_{down}$ (fig.5b). In this situation, averaging two
splitting frequencies provides the zero-field limit for each orbit:
$F_{\alpha}^{ave}$ = 60 T, $F_{\beta}^{ave}$ = 401 T, $F_{\gamma}^{ave}$ = 692
T, and $F_{\delta}^{ave}$ = 2868 T. Shifting the Fermi energy upward by 15 meV
for the $\beta$ orbit, downward by 20 meV for the $\gamma$ and
$\gamma^{\prime}$ orbits, and upward by 40 meV for the $\delta$ orbit fully
reproduce their absolute frequencies (figs.6a-c). Field-dependent dHvA
frequencies for spin-up and down Fermi surfaces have also been reported for
PrPb3 with the same AuCu3-type cubic structure Endo et al. (2002).

Whereas the Fermi pockets corresponding to the measured orbits $\beta$,
$\gamma$, and $\delta$ are uniquely identified, the $\alpha$ orbit with
$F_{\alpha}^{ave}=60$ T remains unassigned. We note that the hole pockets
around the $\Gamma$ points and triangle-shaped electron pockets between
$\Gamma$ and R points provide no extremal orbits other than $F\sim 3000$ T and
$\sim 400$ T at $\theta$ = 0, respectively (figs.4d, 4e, 6d and 6f). On the
other hand, the cross-shaped electron pockets around the X points has extra
extremal orbits $\gamma^{\prime\prime}$ with the frequency of $\sim$ 190 T
(150 T) for the unshifted (shifted) Fermi pockets (figs.4f and 6g) at $\theta$
= 0. Since the calculated frequencies for the $\gamma^{\prime\prime}$ orbit,
with or without the Fermi-energy shift, are larger by a factor of $\sim$3 than
the measured frequency of 60 T, the $\alpha$ orbit cannot be ascribed to the
cross-shaped electron pockets. Moreover, the isotropy of the dHvA frequencies
for the $\alpha$ orbit suggests that it is quite unlikely that the $\alpha$
orbit originates from the small nonspherical hole pockets located off the high
symmetry points in band 25.

We therefore attribute the $\alpha$ orbit to the electron pockets around the M
points. In the band structure with the unshifted Fermi energy, however, the
cross sectional areas of the electron pockets around the M points are very
small (fig.4c). Nevertheless, as shown in the calculated band structure
(fig.4a), the sizes of the electron pockets are extremely sensitive to the
Fermi energy $E_{F}$. Indeed, an upward shift of $E_{F}$ by 110 meV yields
perfect agreement in the absolute frequency and its angular dependence
(fig.6c) just like the other Fermi pockets in CaSn3. This assignment to the
$\alpha$ orbit, with a shifted Fermi energy, leads to the intriguing
conclusion that the three Fermi pockets in CaSn3 surround the TRIM—namely, the
$\Gamma$, X, and M points—satisfying one of the criteria theoretically
proposed for the realization of topological superconductivity.

Figure 7: Effective masses and Berry phases of CaSn3. Effective masses of (a)
the $\alpha$ and $\beta$ orbits and (b) the $\gamma$ and $\delta$ orbits in
CaSn3, obtained from temperature dependence of FFT amplitudes at $\theta$ =
-1.7∘ (fig.2a) using the Lifshitz-Kosevich model with the average inverse
field $1/{\bar{B}}=(1/B_{min}+1/B_{max})/2$, where $B_{min}$ = 5 T and
$B_{max}$ = 35 T. A two component LK fit to the band-pass filtered data for
(c) the $\gamma$ and (d) the $\delta$ orbits. The extracted Berry phase
$\phi_{B}$—0.7$\pi$ for the $\gamma$ orbit and 1.0$\pi$ for the$\delta$
orbit—indicate nontrivial topological nature of CaSn3.

To take a closer look at the effective mass and topological nature of each
band, we utilize the Lifshitz-Kosevich (LK) model that explains the
oscillatory part of magnetic torque in metals Shoenberg (2011),

$\displaystyle\Delta\tau^{osc}=M_{\perp}^{osc}\times B,$ (1) $\displaystyle
M_{\perp}^{osc}=-\frac{1}{F}\frac{dF}{d\theta}M_{\parallel}^{osc},$ (2)
$\displaystyle
M_{\parallel}^{osc}\propto-{B}^{\frac{1}{2}}\left|\frac{\partial^{2}S(k)}{\partial
k_{\parallel}^{2}}\right|^{-\frac{1}{2}}R_{T}R_{D}\sin\left[2\pi\left(\frac{F}{B}+\phi^{\sigma}\right)\right].$
(3)

Here, $R_{T}=x/\sinh x$—with $x=am^{\ast}T/m_{e}B$ and
$a=2\pi^{2}k_{B}m_{e}/e\hbar=14.69$ T/K—is the thermal damping factor,
$R_{D}=\exp(-am^{\ast}T_{D}/B)$ is the Dingle damping factor, $m^{\ast}$ is
the carrier effective mass, $m_{e}$ is the free electron mass, and $T_{D}$ is
the Dingle temperature. The phase shift $\phi^{\sigma}$ ($\sigma$ = up, down)
for a spin-up/down Fermi surface is given by
$\phi^{\sigma}=-1/2+\phi_{B}^{\sigma}/2\pi\pm\phi_{Z}^{\sigma}/2+\phi_{3D}$,
where $\phi_{B}^{\sigma}$ is the Berry phase and
$\phi_{Z}^{\sigma}=gm^{\ast}/2m_{e}$ ($g$: g-factor) is a phase shift due to
the linear Zeeman effect. The phase shift determined by dimensionality of the
Fermi surface, $\phi_{3D}$, is 1/8 for extreme minima (maxima) for electron
(hole) pockets and -1/8 for extreme maxima (minima) for electron (hole)
pockets. Note here that instead of using the spin damping factor
$R_{S}=\cos(\pi gm^{\ast}/2m_{e})$ in the conventional LK formula, we adopt
the Zeeman phase shift $\phi_{Z}^{\sigma}$ to describe the effect of spins due
to the spin-dependent dHvA frequencies $(F_{up}\neq F_{down})$. To extract a
reliable Berry phase from dHvA oscillations, $\phi_{Z}^{\sigma}$, together
with the sign of $dF/d\theta$, should be considered in the LK analysis.

To this goal, we first determine the effective masses $m^{\ast}$ through the
thermal damping factor $R_{T}$ by fitting the observed FFT amplitudes to the
LK formula Shoenberg (2011). We find the effective masses $m^{\ast}$ are
$0.040m_{e}$, $0.042m_{e}$, $0.10m_{e}$, $0.10m_{e}$, $0.21m_{e}$,
$0.25m_{e}$, $0.17m_{e}$, and $0.18m_{e}$ for the $\alpha_{1}$, $\alpha_{2}$,
$\beta_{1}$, $\beta_{2}$, $\gamma_{1}$, $\gamma_{2}$, $\delta_{1}$, and
$\delta_{2}$ orbits, respectively (figs.7a and b). These light masses are
consistent with the previously reported values Zhu et al. (2019).

Nontrivial topological nature of CaSn3 is confirmed by the nonzero Berry
phase. To determine the Berry phase, we fit a two-component LK formula to
band-pass filtered data for the $\gamma$ and $\delta$ orbits for $\theta$ =
-1.7∘ at $T$ = 1.5 K. We use the oscillation frequencies obtained from the FFT
analysis and the effective masses extracted from the temperature dependence of
FFT amplitudes as fixed parameters. We assume that the spin-up and down Fermi
surfaces have the same Berry phase, i.e.
$\phi_{B}=\phi_{B}^{up}=\phi_{B}^{down}$. Taking into account the signs of
$dF/d\theta$ ($<0$ around $\theta=0$) for the $\gamma$ and $\delta$ orbits
(figs.6a and b), we find the extracted Berry phases $\phi_{B}=0.7\pi$ for the
$\gamma$ orbit and $\phi_{B}=1.0\pi$ for the $\delta$ orbit, which indicate
the nontrivial topological nature of these bands. The lower limits of
effective $g$-factors, obtained from the Zeeman phase shift, are 8.61 for the
$\gamma$ orbit and 11.6 for the $\delta$ orbit. Similar, sizable enhancement
of $g$ has been observed in various topological semimetals, including ZrSiS Hu
et al. (2017), ZrTe5 Liu et al. (2016), Cd3As2 Cao et al. (2015), and PtBi2-x
Xing et al. (2020), all consistent with the large Zeeman splitting.

In summary, we have observed four fundamental dHvA oscillation frequencies in
CaSn3 via torque magnetometry in magnetic fields up to 35 T, determined to be
$F_{\alpha}^{ave}$ = 60 T, $F_{\beta}^{ave}$ = 401 T, $F_{\gamma}^{ave}$ = 692
T, and $F_{\delta}^{ave}$ = 2868 T. We have identified the correspondence
between the experimental quantum oscillation frequencies and theoretically
calculated orbits in this topological superconductor candidate, revealing that
an odd number of Fermi pockets enclose the TRIM, prerequisite to one of the
theoretical criteria for topological superconductivity. The nonzero Berry
phases are also confirmed by using the LK model, supporting the nontrivial
topological nature of this system. These findings provide a new avenue to
investigate topological superconducting states stabilized in topological
semimetals.

###### Acknowledgements.

The authors thank H. Shishido for helpful discussions. The experimental work
was supported in part by the start-up fund from the University of Central
Florida. H.S and Y.N. were supported by NSF CAREER DMR-1944975, and X.H. and
Y.T. by the NHMFL UCGP program. Theoretical work (D.L. and T.S.R,) was
supported by DOE Grant DE‐FG02‐07ER46354. The NHMFL is supported by the
National Science Foundation through NSF/DMR-1644779 and the State of Florida.

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$*$ Correspondence and requests for materials should be addressed to Y.N.
(Email: Yasuyuki.Nakajima@ucf.edu).