of 13
Andreev reflection spectroscopy in strongly paired superconductors
Cyprian Lewandowski,
1, 2
́
Etienne Lantagne-Hurtubise,
1, 2
Alex
Thomson,
1, 2, 3, 4
Stevan Nadj-Perge,
5, 2
and Jason Alicea
1, 2
1
Department of Physics, California Institute of Technology, Pasadena CA 91125, USA
2
Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena CA 91125, USA
3
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA
4
Department of Physics, University of California, Davis, California 95616, USA
5
T. J. Watson Laboratory of Applied Physics, California Institute of Technology,
1200 East California Boulevard, Pasadena, California 91125, USA
Motivated by recent experiments on low-carrier-density superconductors, including twisted multi-
layer graphene, we study signatures of the BCS to BEC evolution in Andreev reflection spectroscopy.
We establish that in a standard quantum point contact geometry, Andreev reflection in a BEC su-
perconductor is unable to mediate a zero-bias conductance beyond
e
2
/h
per lead channel. This
bound is shown to result from a duality that links the sub-gap conductance of BCS and BEC super-
conductors. We then demonstrate that sharp signatures of BEC superconductivity, including perfect
Andreev reflection, can be recovered by tunneling through a suitably designed potential well. We
propose various tunneling spectroscopy setups to experimentally probe this recovery.
The evolution from Bardeen-Cooper-Schrieffer (BCS)
to Bose-Einstein condensate (BEC) superconductivity
has been a recurring theme in the study of physical
systems ranging from cold atomic gases to strongly
correlated materials and neutron stars [1–5].
In the
BCS regime, superconductivity arises from weakly bound
Cooper pairs with a characteristic size
ξ
pair
that far ex-
ceeds the mean inter-particle spacing
d
; in the BEC
regime, by contrast, fermions form tightly bound Cooper
pairs with size
ξ
pair

d
. BCS- and BEC-type super-
conductors can be separated either by a crossover (e.g.,
for
s
-wave pairing [6–11]; Fig 1a), a Lifshitz-type phase
transition (e.g., for nodal pairing [12–14]; Fig. 1b), or a
topological phase transition [15].
Ultracold atoms, where the pairing strength between
fermions can be continuously tuned with the aid of a
Feshbach resonance [16–21], provide a controlled exper-
imental platform for exploring the evolution from weak
to strong pairing. While most solid-state superconduc-
tors reside firmly in the BCS regime, certain strongly
correlated materials such as cuprates have been rational-
ized in terms of the BCS-BEC paradigm [22]. Further,
a new generation of experiments probing low-carrier-
density materials including iron-based compounds [23–
26], Li
x
ZrNCl [27, 28] and moir ́e graphene systems [29–
34] has revealed signatures consistent with proximity to a
BEC state—opening a new experimental frontier for un-
conventional superconductivity. Developing probes that
can unambiguously identify BEC superconductors and
distinguish possible competing phases therefore poses a
pressing problem.
In solid-state contexts, BEC superconductivity can
manifest in various ways. Saturation of the Ginzburg-
Landau coherence length
ξ
GL
at the interparticle spac-
ing [27, 29] and a critical temperature approaching theo-
retical bounds [35, 36] both constitute indirect evidence
E
B
A
D
C
c)
a)
s
-wave pairing
b)
d
-wave pairing
BCS
BEC
BCS
BEC
crossover
transition
Lead
Superconductor
FIG. 1. (a,b) Quasiparticle spectra across the BCS-to-BEC
superconductor evolution, which is (a) a crossover for
s
-
wave pairing but (b) a Lifshitz transition for nodal (e.g.,
d
-wave) pairing.
(c) Scattering processes at the normal-
superconductor interface included in the BTK formalism: An-
dreev (A) and normal (B) reflection, and direct (C) and
band-crossing (D) transmission as a quasiparticle. The green
dashed line denotes the incident electron energy.
for proximity to the strong pairing regime.
Energy-
momentum resolved probes [24] can also reveal the evo-
lution of the quasiparticle dispersion from BCS-like to
BEC-like (Fig. 1a,b). Superconductors with a nodal or-
der parameter exhibit a Lifshitz transition from a gapless
BCS to a gapped BEC phase, which leads to a predicted
divergence in electronic compressibility at the transi-
tion [13] and a gap opening that can be observed, e.g.,
via scanning tunneling microscopy (STM) [12, 14, 31].
In STM it is, however, difficult to distinguish BEC su-
perconductors from competing insulators since, unlike
arXiv:2207.09494v1 [cond-mat.supr-con] 19 Jul 2022
2
their BCS counterparts, they feature less prominent and
particle-hole asymmetric coherence peaks [31, 37].
Here we investigate Andreev reflection spectroscopy
across the BCS-BEC evolution and predict striking new
manifestations of BEC superconductivity. First, in a
standard quantum point contact (QPC) geometry [38] we
find that Andreev reflection is suppressed upon passing
from the BCS to BEC regime, consistent with the analy-
sis of BEC-BEC Josephson junctions in Ref. 39. We show
that this suppression is a consequence of a duality that in-
terchanges the wavefunctions of BEC and BCS supercon-
ductors. Second, we establish that Andreev reflection can
be controllably revived in the BEC regime by tunneling
between the lead and superconductor through an effec-
tive potential
well
—as opposed to the barrier employed
in conventional treatments [40]. This feature sharply dis-
tinguishes BEC from BCS superconductors and can be
probed in experimental setups that we propose.
Setup.
We first employ the Blonder-Tinkham-
Klapwijk (BTK) framework, which describes scattering
between a normal lead and a superconductor [40]. Con-
sider a 1D setup (Fig. 1c) with an interface at position
x
= 0 separating a spinful normal lead (at
x <
0) with
dispersion
ξ
q
=
q
2
2
m
L
μ
L
from a singlet superconduc-
tor (at
x >
0) with dispersion
ξ
k
=
k
2
2
m
SC
μ
SC
and
s
-wave pairing potential ∆. We take
μ
L
>
0 throughout
but allow
μ
SC
to take either sign to capture the BCS-
BEC crossover that occurs as
μ
SC
crosses zero. A delta-
function tunnel barrier,
λδ
(
x
), interpolates between the
tunneling limit with large
λ >
0 and the QPC limit
λ
= 0.
Figure 1c illustrates the processes available to an in-
cident electron at the interface: Andreev (A) or nor-
mal (B) reflection to a hole or electron, respectively,
and transmission to an electron- (C) or hole-like (D)
quasiparticle. The probabilities for these processes are
obtained by matching the wavefunctions and their first
derivatives at the interface and normalizing by the ap-
propriate group velocities [41]. In the standard BTK
formalism applied to weakly paired BCS superconduc-
tors, the limit ∆
,E

μ
L
SC
combined with the equal-
mass assumption
m
L
=
m
SC
drastically simplifies the
problem, yielding the Andreev approximation in which
q
h
=
q
e
=
k
±
=
q
F
with
q
F
=
2
m
L
μ
L
the lead’s Fermi
momentum [42]. To describe the BCS-BEC evolution, we
instead treat the problem in full generality (see Ref. 39
for an analogous treatment of Josephson junctions). The
tunneling conductance
G
(
E
) =
dI/dV
at bias energy
E
=
eV
is given in the Landauer-B ̈uttiker formalism by
G
(
E
) =
2
e
2
h
[1 +
A
(
E
)
B
(
E
)]
,
(1)
where
A
(
E
) and
B
(
E
) denote the Andreev and normal
reflection probabilities. Maximal conductance of 4
e
2
/h
corresponds to perfect Andreev reflection
A
(
E
) = 1.
Zero-bias conductance.
For gapped superconduc-
tors, the absence of transmission processes (C and D)
considerably simplifies the analysis of the sub-gap con-
ductance: employing the normalization condition
A
(
E
)+
B
(
E
) = 1 returns
G
(
E
) =
4
e
2
h
A
(
E
). As shown in the
Supplement [41], the zero-bias Andreev reflection proba-
bility,
A
0
A
(
E
= 0), then reduces to
A
0
=
4
v
2
L
v
2
I
[
(2
v
L
Z
+
v
R
)
2
+
v
2
I
+
v
2
L
]
2
,
(2)
where
Z
=
λ/v
L
quantifies the barrier transparency,
v
L
=
q
F
/m
L
is the Fermi velocity of the lead, and
v
R
,
I
=
κ
R
,
I
/m
SC
are two characteristic velocities of the
superconductor. The momenta
κ
R
,
I
are defined through
κe
=
κ
R
+
I
=
2
m
SC
(
μ
SC
+
i
∆)
,
(3)
and the length scales
κ
1
R
and
κ
1
I
respectively control
the evanescent and oscillatory behavior of the wavefunc-
tion in the superconductor at
x >
0,
ψ
SC
e
(
κ
R
+
I
)
x
.
Sending
μ
SC
→ −
μ
SC
, which connects the BCS and
BEC regimes, yields
φ
π/
2
φ
and hence swaps
κ
R
κ
I
; cf. Eq. (3). This remarkable property reveals a
duality between the BCS and BEC superconductor wave-
functions. As an instructive limiting case, deep in either
the BEC or BCS regimes where
|
μ
SC
|
/

1,
κ
R
,
I
can
be expanded as
κ
R
+
I
sgn(
μ
SC
)
(
v
F
+
i
sgn(
μ
SC
)
k
F
)
.
(4)
Here the definition of the Fermi momentum
k
F
=
2
m
SC
|
μ
SC
|
and velocity
v
F
=
k
F
/m
SC
is extended into
the BEC regime. In the BCS limit Eq. (4) aligns with
conventional understanding: The wavefunction’s oscilla-
tory part is set by
k
F
, whereas evanescent decay is con-
trolled by ∆
/v
F
—i.e., the inverse of the BCS pair coher-
ence length
ξ
pair
. The situation flips in the BEC regime,
where
k
F
controls the decay length and ∆
/v
F
dictates
the oscillatory behavior.
This duality has immediate implications for the con-
ductance. In the QPC limit,
Z
= 0, the Andreev reflec-
tion probability in Eq. (2) is maximized when the lead
Fermi velocity satisfies
v
L
=
v
2
R
+
v
2
I
, yielding the up-
per bound
A
0
v
2
I
/
(
v
2
I
+
v
2
R
) valid for any
μ
SC
. Near-
perfect Andreev reflection thus requires
v
R

v
I
, imply-
ing many oscillation periods over the decaying envelope
of the superconductor wavefunction. This requirement
exactly translates to the BCS limit,
μ
SC

∆, as origi-
nally pointed out by Andreev [42]. By contrast, the deep
BEC regime is characterized by the converse inequality,
v
R

v
I
, an the upper bound above accordingly reduces
to
A
0
.
v
2
I
/v
2
R
= ∆
2
2
SC
. Physically, the fast decay
of the superconducting wavefunction relative to its os-
cillation period suppresses Andreev reflection from the
plane wave electrons of the lead. At the “self-dual” point
μ
SC
= 0, the superconductor exhibits a single length
3
0
1
2
3
4
BEC BCS
BEC BCS
BEC BCS
FIG. 2. Dependence of the BTK zero-bias tunneling conductance on the barrier parameter
Z
along the BCS-to-BEC crossover
for a one-dimensional
s
-wave superconductor with
v
L
m
SC
/
∆ set to (a) 3, (b) 1.5, and (c) 0.5. Black dashed lines trace
Z
=
v
R
/
2
v
L
, one of the necessary conditions for perfect Andreev reflection. Incoming electrons tunnel through a potential
barrier at
Z >
0 but a potential well at
Z <
0. While the conductance in the BCS regime is approximately
Z
→−
Z
symmetric,
pronounced asymmetry emerges in the BEC regime, with enhanced Andreev reflection possible at negative
Z
[see panel (c)].
scale, with
v
R
=
v
I
=
/m
SC
, yielding an Andreev
reflection probability upper-bounded by 1
/
2.
Andreev revival.
The preceding bound on Andreev
reflection in the BEC regime (
μ
SC
<
0) implies that
the zero-bias conductance in the QPC limit cannot ex-
ceed the value 2
e
2
/h
characteristic of a perfect metallic
contact. According to Eq. (2), moving away from the
QPC limit by adding a tunnel barrier modeled by
Z >
0
only further suppresses the Andreev reflection probabil-
ity
A
0
. Interestingly, however, Andreev reflection can be
enhanced by exploiting a pronounced asymmetry in the
sign of
Z
that emerges in the BEC regime.
Writing
v
I
=
αv
L
and
v
R
=
βv
L
, one can re-express
Eq. (2) as
A
0
= 1
/
(1 + 2
̃
Z
2
)
2
in terms of a renormalized
transparency parameter,
̃
Z
2
=
(
Z
+
β/
2)
2
α
+
(
α
1)
2
4
α
,
(5)
that incorporates both the intrinsic barrier transparency
Z
and velocity mismatch effects [41]. The linear-in-
Z
contribution in Eq. (5) is proportional to the dimension-
less parameter
β/α
=
v
R
/v
I
controlling the BEC to BCS
evolution. In the BCS limit, the distinction between
Z
positive and negative—i.e., potential barriers and wells—
is correspondingly negligible since
v
I

v
R
. In the BEC
regime, by contrast,
v
R
> v
I
implies sensitive dependence
on the sign of
Z
. Indeed, potential wells characterized by
Z <
0 can
promote
Andreev reflection—which becomes
perfect at zero bias even deep within the BEC regime
when
̃
Z
= 0, i.e., when
Z
=
v
R
/
(2
v
L
) and
v
L
=
v
I
.
These trends are illustrated in Fig. 2, which plots the
zero-bias conductance versus
Z
across the BCS-BEC evo-
lution tuned by
μ
SC
/
∆, for different lead velocities
v
L
in
panels (a)-(c). Note that the standard BTK result for a
BCS superconductor is recovered for large
v
L
[Fig. 2(a)].
Effective lead formalism.
For complementary in-
sight, we now examine normal metal-superconductor tun-
neling via an effective lead formalism (ELF). Consider a
SC
Lead
Lead
e
e
h
Impurity
t
e
e
h
FIG. 3. Sketch of the effective lead formalism. A lead tun-
nels electrons onto a gapped superconductor (left). Integrat-
ing out the gapped degrees of freedom generates an effective
lead-only “impurity” problem (right) featuring a shifted local
potential
U
SC
and a local pairing term
W
SC
mediated by the
superconductor.
1D lead at positions
x
0 with linearized kinetic en-
ergy and Fermi velocity
v
L
=
q
F
/m
L
. At its endpoint,
the lead exhibits a local potential
U
0
δ
(
x
) and couples
to a gapped
d
-dimensional superconductor via electron
hopping of strength
t
; see Fig. 3. Focusing on sub-
gap energies, the superconductor’s degrees of freedom
can be safely integrated out—yielding an effective lead-
only Hamiltonian with additional (marginal) “impurity”
perturbations [41]. Specifically, the superconductor both
shifts the local potential
U
0
by
U
SC
=
t
2
d
d
k
(2
π
)
d
ξ
k
ξ
2
k
+ ∆
2
(6)
and generates a local singlet pairing term with amplitude
W
SC
=
t
2
d
d
k
(2
π
)
d
ξ
2
k
+ ∆
2
.
(7)
Explicit evaluation of the integrals reveals that, for
d
= 1,
U
SC
and
W
SC
exactly swap under
μ
SC
→ −
μ
SC
, mani-
festing the duality uncovered above within the BTK for-
malism (for details and extensions see [41]).
4
0
1
2
3
4
a)
b)
c)
BCS
BEC
FIG. 4. Finite-bias tunneling spectroscopy across the BCS-BEC evolution with (a)
Z
= 0, (b)
Z
= 1 and (c)
Z
=
1, assuming
an optimal lead with
v
L
=
v
I
for each
μ
SC
value. Black dashed lines at energies
E
=
±
μ
L
show the corresponding optimal lead
chemical potential
μ
L
, and green dashed lines denotes the quasiparticle gap
E
gap
. (a) The quantum point contact limit
Z
= 0
exhibits a plateau-like enhancement of sub-gap conductance due to Andreev reflection in the BCS regime, but not in the BEC
regime. (b,c) Tunneling spectra for
Z
=
±
1 are similar in the BCS regime, showing a gap surrounded by coherence peaks at
E
=
±
∆. In the BEC regime, sub-gap conductance enhancement occurs
only
for tunneling through potential wells, as in (c).
Extracting the wavefunctions from the effective lead-
only Hamiltonian yields a zero-bias Andreev reflection
probability [41]
A
ELF
0
=
4
v
2
L
(
W
SC
/
2)
2
[(
U
eff
/
2)
2
+ (
W
SC
/
2)
2
+
v
2
L
]
2
,
(8)
with
U
eff
=
U
0
+
U
SC
. Eq. (8) reproduces Eq. (2) upon
identifying the ELF/BTK correspondence
U
0
/
2
2
v
L
Z
,
U
SC
/
2
v
R
, and
W
SC
/
2
v
I
. Thus negative val-
ues of
U
0
enable the cancellation of the ‘large’ potential
U
SC
generated by BEC superconductors—in turn allow-
ing Andreev processes mediated by
W
SC
to dominate.
Extensions.
Figure 4 presents the finite-bias conduc-
tance
G
(
E
) for an
s
-wave superconductor. Data were
obtained numerically from the full BTK analysis, includ-
ing quasiparticle transmission channels, for (a)
Z
= 0,
(b)
Z
= 1, and (c)
Z
=
1. To clearly illustrate the
revival of Andreev reflection, the lead Fermi velocity is
tuned such that
v
L
=
v
I
across the BCS-BEC evolution;
that is, for each
μ
SC
on the vertical axis we fix the lead
chemical potential to its optimal value (see dashed black
curves). In the QPC limit,
Z
= 0, Fig. 4(a) shows the
familiar sub-gap conductance plateau at
G
(
E
) = 4
e
2
/h
in the BCS regime [40]. Upon entering the BEC regime,
the entire plateau is suppressed in accordance with the
bound on Andreev reflection at zero bias derived above.
In the tunneling limit (
Z >
0), the BCS regime exhibits
sharp coherence peaks at the gap edge that similarly
diminish upon entering the BEC regime, as shown in
Fig. 4(b). Finally, negative
Z
continues to support co-
herence peaks in the BCS regime while also significantly
enhancing the sub-gap conductance in the BEC regime
[Fig. 4(c)]. This enhancement is present within an energy
interval about zero bias whose extent is limited by kine-
matic constraints imposed by the lead chemical potential
μ
L
(the required backwards propagating hole state does
not exist for
|
E
|
> μ
L
).
Our 1D analysis extends to the experimentally rele-
vant scenario of a multi-channel lead normally incident
on a 2D superconductor [41]. In the limit of many bal-
listic channels, the total conductance
G
(
E
) follows from
an angular average over the contributions from all mo-
menta in the plane of the interface [38, 41, 43, 44]. For
isotropic
s
-wave superconductors this procedure leads to
essentially identical results as in our 1D model. Nodal
superconductors also show qualitatively similar “revival”
behavior at negative
Z
in the BEC regime; the key distin-
guishing feature is that sub-gap conductance generated
by Andreev reflection in the BCS regime acquires an in-
verted V-shape [43] due to transmission processes that
suppress Andreev reflection away from zero bias [41].
Discussion.
The suppression of Andreev reflection
in a quantum point contact geometry, and its recovery
by tunneling through potential wells, clearly differenti-
ates BEC phenomenology from the familiar properties
of BCS superconductors. In particular, the emergence
of a pronounced
Z
→ −
Z
asymmetry in the sub-gap
conductance provides a striking signature of BEC super-
conductivity. This feature stands in stark contrast to
the above-gap conductance
G
(
E

∆) that is manifestly
symmetric with respect to the sign of
Z
[41]. Possible ex-
perimental setups for probing the negative-
Z
regime in-
clude (i) STS measurements using a tip with an effective
quantum well at its end, implemented by attaching, e.g.,
a quantum dot (reminiscent of single-electron transistor
probes [45, 46]) or an impurity atom, or else by coat-
ing the tip with a layer of material with a smaller work
function and (ii) 2D electronic transport setups where
the N-S interface is tunable, either by applying a local
gate near the junction or by constructing “via contacts”
etched into different encapsulating insulators [47].
Throughout the manuscript we focused on supercon-
ductivity arising from a single parabolic band. Extend-
ing these ideas to BEC physics in multiband supercon-
ductors, believed to be important to understanding re-
5
cent experiments [24], is thus warranted. The limit of
narrow-band superconductivity, where quantum geome-
try plays an important role in determining the super-
fluid stiffness [32, 48–53], is another interesting avenue
for future work with possible applications to moir ́e ma-
terials [29, 31]. Finally, self-consistent treatments [54]
and extensions beyond mean-field [11] might lead to fur-
ther insights into BEC superconductors and associated
pseudogap physics [10, 55] at temperatures above
T
c
.
Acknowledgments
We thank Mohit Randeria, Hyunjin Kim, Micha l Pa-
paj, and Kevin Nuckolls for insightful discussions. This
work was supported by the Gordon and Betty Moore
Foundation’s EPiQS Initiative, Grant GBMF8682 (C.L.
and
́
E.L.-H.); the Army Research Office under Grant
Award W911NF-17-1-0323; the Caltech Institute for
Quantum Information and Matter, an NSF Physics Fron-
tiers Center with support of the Gordon and Betty Moore
Foundation through Grant GBMF1250; and the Walter
Burke Institute for Theoretical Physics at Caltech.
cyprian@caltech.edu
[1] Q. Chen, J. Stajic, S. Tan, and K. Levin, Bcs–bec
crossover: From high temperature superconductors to ul-
tracold superfluids, Phys. Rep.
412
, 1 (2005).
[2] C. A. S. de Melo, When fermions become bosons: Pairing
in ultracold gases, Physics Today
61
, 45 (2008).
[3] W. Zwerger,
The BCS-BEC crossover and the unitary
Fermi gas
, Vol. 836 (Springer Science & Business Media,
2011).
[4] M. Randeria and E. Taylor, Crossover from bardeen-
cooper-schrieffer to bose-einstein condensation and the
unitary fermi gas, Annu. Rev. Condens. Matter Phys.
5
,
209 (2014).
[5] G. C. Strinati, P. Pieri, G. R ̈opke, P. Schuck, and M. Ur-
ban, The BCS–BEC crossover: From ultra-cold fermi
gases to nuclear systems, Phys. Rep.
738
, 1 (2018).
[6] D. M. Eagles, Possible pairing without superconductivity
at low carrier concentrations in bulk and thin-film super-
conducting semiconductors, Phys. Rev.
186
, 456 (1969).
[7] A. Leggett, Modern trends in the theory of condensed
matter, Modern Trends in the Theory of Condensed Mat-
ter, Proc. XVI Karpacz Winter School of Theoretical
Physics, 1980 (1980).
[8] P. Nozi ́eres and S. Schmitt-Rink, Bose condensation in
an attractive fermion gas: From weak to strong coupling
superconductivity, J. Low Temp. Phys.
59
, 195 (1985).
[9] M. Randeria, J.-M. Duan, and L.-Y. Shieh, Bound states,
cooper pairing, and bose condensation in two dimensions,
Phys. Rev. Lett.
62
, 981 (1989).
[10] C. A. R. S ́a de Melo, M. Randeria, and J. R. Engelbrecht,
Crossover from bcs to bose superconductivity: Transi-
tion temperature and time-dependent ginzburg-landau
theory, Phys. Rev. Lett.
71
, 3202 (1993).
[11] F. Pistolesi and G. C. Strinati, Evolution from bcs su-
perconductivity to bose condensation: Calculation of the
zero-temperature phase coherence length, Phys. Rev. B
53
, 15168 (1996).
[12] M. Randeria, J.-M. Duan, and L.-Y. Shieh, Supercon-
ductivity in a two-dimensional fermi gas: Evolution from
cooper pairing to bose condensation, Phys. Rev. B
41
,
327 (1990).
[13] R. D. Duncan and C. A. R. S ́a de Melo, Thermodynamic
properties in the evolution from bcs to bose-einstein con-
densation for a d-wave superconductor at low tempera-
tures, Phys. Rev. B
62
, 9675 (2000).
[14] L. Borkowski and C. S. de Melo, Evolution from the BCS
to the bose-einstein limit in a d-wave superconductor at
t=0, Acta Phys. Pol. A
99
, 691 (2001).
[15] N. Read and D. Green, Paired states of fermions in
two dimensions with breaking of parity and time-reversal
symmetries and the fractional quantum hall effect, Phys.
Rev. B
61
, 10267 (2000).
[16] C. A. Regal, M. Greiner, and D. S. Jin, Observation of
resonance condensation of fermionic atom pairs, Phys.
Rev. Lett.
92
, 040403 (2004).
[17] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F.
Raupach, A. J. Kerman, and W. Ketterle, Condensation
of pairs of fermionic atoms near a feshbach resonance,
Phys. Rev. Lett.
92
, 120403 (2004).
[18] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl,
S. Jochim, J. H. Denschlag, and R. Grimm, Observation
of the pairing gap in a strongly interacting fermi gas,
Science
305
, 1128 (2004).
[19] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang,
F. Chevy, M. Teichmann, L. Tarruell, S. J. J. M. F.
Kokkelmans, and C. Salomon, Experimental study of the
bec-bcs crossover region in lithium 6, Phys. Rev. Lett.
93
,
050401 (2004).
[20] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W.
Jack, and R. G. Hulet, Molecular probe of pairing in the
bec-bcs crossover, Phys. Rev. Lett.
95
, 020404 (2005).
[21] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H.
Schunck, and W. Ketterle, Vortices and superfluidity in a
strongly interacting fermi gas, Nature
435
, 1047 (2005).
[22] N. Harrison and M. K. Chan, Magic gap ratio for op-
timally robust fermionic condensation and its implica-
tions for High
T
c
superconductivity, Phys. Rev. Lett.
129
, 017001 (2022).
[23] S. Kasahara, T. Watashige, T. Hanaguri, Y. Kohsaka,
T. Yamashita, Y. Shimoyama, Y. Mizukami, R. Endo,
H. Ikeda, K. Aoyama, T. Terashima, S. Uji, T. Wolf,
H. von L ̈ohneysen, T. Shibauchi, and Y. Matsuda, Field-
induced superconducting phase of FeSe in the BCS-BEC
cross-over, PNAS
111
, 16309 (2014).
[24] S. Rinott, K. B. Chashka, A. Ribak, E. D. L. Rienks,
A. Taleb-Ibrahimi, P. L. Fevre, F. Bertran, M. Rande-
ria, and A. Kanigel, Tuning across the bcs-bec crossover
in the multiband superconductor fe
1+
y
se
x
te
x
: An angle-
resolved photoemission study, Sci. Adv.
3
, e1602372
(2017).
[25] T. Hashimoto, Y. Ota, A. Tsuzuki, T. Nagashima,
A. Fukushima, S. Kasahara, Y. Matsuda, K. Matsuura,
Y. Mizukami, T. Shibauchi, S. Shin, and K. Okazaki,
Bose-einstein condensation superconductivity induced by
disappearance of the nematic state, Sci. Avd.
6
, eabb9052
(2020).
6
[26] Y. Mizukami, M. Haze, O. Tanaka, K. Matsuura,
D. Sano, J. B ̈oker, I. Eremin, S. Kasahara, Y. Mat-
suda, and T. Shibauchi, Thermodynamics of transition to
bcs-bec crossover superconductivity in fese
1
x
s
x
(2021),
arXiv:2105.00739 [cond-mat.supr-con].
[27] Y. Nakagawa, Y. Kasahara, T. Nomoto, R. Arita, T. No-
jima, and Y. Iwasa, Gate-controlled BCS-BEC crossover
in a two-dimensional superconductor, Science
372
, 190
(2021).
[28] T. Shi, W. Zhang, and C. A. R. S. de Melo, Density in-
duced bcs-bose evolution in gated two-dimensional super-
conductors: The berezinskii-kosterlitz-thouless transition
as a function of carrier density (2021), arXiv:2106.10010
[cond-mat.supr-con].
[29] J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and
P. Jarillo-Herrero, Tunable strongly coupled supercon-
ductivity in magic-angle twisted trilayer graphene, Na-
ture
590
, 249 (2021).
[30] Z. Hao, A. M. Zimmerman, P. Ledwith, E. Khalaf, D. H.
Najafabadi, K. Watanabe, T. Taniguchi, A. Vishwanath,
and P. Kim, Electric field–tunable superconductivity in
alternating-twist magic-angle trilayer graphene, Science
371
, 1133 (2021).
[31] H. Kim, Y. Choi, C. Lewandowski, A. Thomson,
Y. Zhang, R. Polski, K. Watanabe, T. Taniguchi, J. Al-
icea, and S. Nadj-Perge, Spectroscopic signatures of
strong correlations and unconventional superconductiv-
ity in twisted trilayer graphene (2021), arXiv:2109.12127
[cond-mat.mes-hall].
[32] H. Tian, S. Che, T. Xu, P. Cheung, K. Watanabe,
T. Taniguchi, M. Randeria, F. Zhang, C. N. Lau, and
M. W. Bockrath, Evidence for Flat Band Dirac Super-
conductor Originating from Quantum Geometry, arXiv
e-prints , arXiv:2112.13401 (2021), arXiv:2112.13401
[cond-mat.supr-con].
[33] J. M. Park, Y. Cao, L. Xia, S. Sun, K. Watanabe,
T. Taniguchi, and P. Jarillo-Herrero, Magic-angle mul-
tilayer graphene: A robust family of moir ́e superconduc-
tors (2021), arXiv:2112.10760 [cond-mat.supr-con].
[34] Y. Zhang, R. Polski, C. Lewandowski, A. Thomson,
Y. Peng, Y. Choi, H. Kim, K. Watanabe, T. Taniguchi,
J. Alicea, F. von Oppen, G. Refael, and S. Nadj-
Perge, Ascendance of superconductivity in magic-angle
graphene multilayers (2021), arXiv:2112.09270 [cond-
mat.supr-con].
[35] S. S. Botelho and C. A. R. S ́a de Melo, Vortex-antivortex
lattice in ultracold fermionic gases, Phys. Rev. Lett.
96
,
040404 (2006).
[36] T. Hazra, N. Verma, and M. Randeria, Bounds on the
superconducting transition temperature: Applications to
twisted bilayer graphene and cold atoms, Phys. Rev. X
9
, 031049 (2019).
[37] Y. L. Loh, M. Randeria, N. Trivedi, C.-C. Chang, and
R. Scalettar, Superconductor-insulator transition and
fermi-bose crossovers, Phys. Rev. X
6
, 021029 (2016).
[38] D. Daghero and R. S. Gonnelli, Probing multiband super-
conductivity by point-contact spectroscopy, Supercond.
Sci. Technol.
23
, 043001 (2010).
[39] F. Setiawan and J. Hofmann, Analytic approach to trans-
port in superconducting junctions with arbitrary carrier
density (2021), arXiv:2108.10333 [cond-mat.mes-hall].
[40] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Tran-
sition from metallic to tunneling regimes in supercon-
ducting microconstrictions: Excess current, charge im-
balance, and supercurrent conversion, Phys. Rev. B
25
,
4515 (1982).
[41] See supplementary material for details.
[42] A. Andreev, The thermal conductivity of the intermedi-
ate state in superconductors, Sov. Phys. JETP
19
, 1228
(1964).
[43] S. Kashiwaya, Y. Tanaka, M. Koyanagi, and K. Ka-
jimura, Theory for tunneling spectroscopy of anisotropic
superconductors, Phys. Rev. B
53
, 2667 (1996).
[44] D. Daghero, M. Tortello, P. Pecchio, V. A. Stepanov,
and R. S. Gonnelli, Point-contact andreev-reflection spec-
troscopy in anisotropic superconductors: The importance
of directionality (review article), Low Temp. Phys.
39
,
199 (2013).
[45] M. J. Yoo, T. A. Fulton, H. F. Hess, R. L. Willett,
L. N. Dunkleberger, R. J. Chichester, L. N. Pfeiffer,
and K. W. West, Scanning single-electron transistor mi-
croscopy: Imaging individual charges, Science
276
, 579
(1997).
[46] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H.
Smet, K. von Klitzing, and A. Yacoby, Observation
of electron–hole puddles in graphene using a scanning
single-electron transistor, Nat. Phys.
4
, 144 (2008).
[47] Q. Cao, E. J. Telford, A. Benyamini, I. Kennedy, A. Zan-
giabadi, K. Watanabe, T. Taniguchi, C. R. Dean, and
B. M. Hunt, Tunneling spectroscopy of two-dimensional
materials based on via contacts (2022), arXiv:2203.07394
[cond-mat.mes-hall].
[48] S. Peotta and P. T ̈orm ̈a, Superfluidity in topo-
logically nontrivial flat bands, Nat. Commun.
6
,
10.1038/ncomms9944 (2015).
[49] A. Julku, S. Peotta, T. I. Vanhala, D.-H. Kim, and
P. T ̈orm ̈a, Geometric origin of superfluidity in the lieb-
lattice flat band, Phys. Rev. Lett.
117
, 045303 (2016).
[50] J. S. Hofmann, E. Berg, and D. Chowdhury, Supercon-
ductivity, pseudogap, and phase separation in topological
flat bands, Phys. Rev. B
102
, 201112 (2020).
[51] P. T ̈orm ̈a, S. Peotta, and B. A. Bernevig, Superfluid-
ity and quantum geometry in twisted multilayer systems
(2021), arXiv:2111.00807 [cond-mat.supr-con].
[52] A. Julku, G. M. Bruun, and P. T ̈orm ̈a, Quantum geome-
try and flat band bose-einstein condensation, Phys. Rev.
Lett.
127
, 170404 (2021).
[53] N. Verma, T. Hazra, and M. Randeria, Optical spec-
tral weight, phase stiffness, and t
c
bounds for trivial
and topological flat band superconductors, PNAS
118
,
e2106744118 (2021).
[54] S. Suman, R. Sensarma, and M. Randeria, In preparation
(2022).
[55] V. J. Emery and S. A. Kivelson, Importance of phase fluc-
tuations in superconductors with small superfluid den-
sity, Nature
374
, 434 (1995).
[56] R. C. Dynes, V. Narayanamurti, and J. P. Garno, Di-
rect measurement of quasiparticle-lifetime broadening in
a strong-coupled superconductor, Phys. Rev. Lett.
41
,
1509 (1978).
[57] S. Lee, V. Stanev, X. Zhang, D. Stasak, J. Flowers, J. S.
Higgins, S. Dai, T. Blum, X. Pan, V. M. Yakovenko,
J. Paglione, R. L. Greene, V. Galitski, and I. Takeuchi,
Perfect andreev reflection due to the klein paradox in
a topological superconducting state, Nature
570
, 344
(2019).
[58] L. Fidkowski, J. Alicea, N. H. Lindner, R. M. Lutchyn,
and M. P. A. Fisher, Universal transport signatures
7
of majorana fermions in superconductor-luttinger liquid
junctions, Phys. Rev. B
85
, 245121 (2012).
A. BTK analysis of tunneling into a 2D
superconductor
Here we summarize the Blonder-Tinkham-Klapwijk
(BTK) formalism [40] as applicable to tunneling into a
2D superconductor. The mathematical setup is analo-
gous to that considered in the literature for the theory
of anisotropic superconductors [43, 44], where the barrier
transparency parameter contains no geometrical factors.
Formally, the problem reduces to a series of 1D BTK scat-
tering problems as we explain below; correspondingly,
along the way we will describe the derivation of 1D BTK
results quoted in the main text. Unlike in the conven-
tional BTK analysis, however, the Andreev approxima-
tion [42] is not taken to allow capturing both the BCS
and BEC regimes.
We will consider tunneling onto both gapped
s
-wave
and nodal superconductors. For a physical picture, imag-
ine carving a hole into a 2D superconductor (with mass
m
SC
, chemical potential
μ
SC
) and filling in the hole with
a disc-shaped lead (with mass
m
L
, chemical potential
μ
L
). An incident electron from the lead can then scat-
ter into the adjacent 2D superconductor in any in-plane
direction, with a given in-plane direction defining an ef-
fective 1D scattering problem as shown Fig. 1c.
Denoting the in-plane direction characterized by an an-
gle
θ
as the
x
axis for concreteness, we take our scattering
states in the lead region (
x
0) to be
φ
L
e
(
x
) =
e
iq
e
x
+
a
e
e
iq
e
x
,
φ
L
h
(
x
) =
a
h
e
iq
h
x
.
(A1)
Here
q
e
=
2
m
L
(
μ
L
+
E
) and
q
h
=
2
m
L
(
μ
L
E
) are
the momenta of the electron and hole, respectively, with
E
the incident electron energy. We normalized the am-
plitude of the incoming electron plane wave to 1, while
a
e
and
a
h
are the amplitudes associated with normal and
Andreev reflection.
In the superconducting region (
x >
0) our ansatz takes
the form
φ
S
e
(
x
) =
b
+
u
+
e
ik
+
x
+
b
u
e
ik
x
φ
S
h
(
x
) =
b
+
v
+
e
ik
+
x
+
b
v
e
ik
x
,
(A2)
where
k
±
=
2
m
SC
(
μ
SC
±
E
2
2
)
are the mo-
menta corresponding to the two solutions (
C
and
D
)
shown in Fig. 1c. Coefficients
b
±
correspond to the
transmission of electrons into the superconductor with-
out (
k
+
, process
D
) and with (
k
, process
C
) branch
crossing. Possible angular dependence of the supercon-
ductor’s order parameter enters through the pairing am-
plitude ∆(
θ
), with
θ
the polar angle in the 2D plane.
We note that in other geometries [43, 44] the angular
dependence of the pairing amplitude ∆(
θ
) need not be
identical for the two momenta
k
+
,
k
. Following the
BTK notation [40] we introduced our scattering states
using the standard coherence factors
u
±
,
v
±
, evaluated
at their respective branches, that here follow from:
u
2
+
=
1
2
(
1 +
E
2
2
E
)
=
v
2
,
(A3)
v
2
+
=
1
2
(
1
E
2
2
E
)
=
u
2
.
(A4)
While the above formulation is conventional and is used
for a finite bias analysis, for analytical treatment of the
zero-bias conductance it helps to define the scattering
states without the
E
= 0 nominal divergence; see next
section. We stress that for the
s
-wave case angular de-
pendence drops out, while for the nodal (e.g.,
d
-wave)
case angular dependence enters only via the pairing po-
tential ∆(
θ
).
To determine the scattering coefficients we include a
λδ
(
x
) potential as in the main text and employ standard
boundary conditions at
x
= 0,
φ
L
e
(
x
= 0) =
φ
S
e
(
x
= 0)
L
h
(
x
= 0) =
φ
S
h
(
x
= 0)
,
(A5)
1
2
m
SC
S
e
dx
x
=0
1
2
m
L
L
e
dx
x
=0
=
λφ
S
e
(
x
= 0)
,
(A6)
1
2
m
SC
S
h
dx
x
=0
1
2
m
L
L
h
dx
x
=0
=
λφ
S
h
(
x
= 0)
.
(A7)
The first line ensures continuity of the wavefunction,
while the other two lines impose a discontinuity in the
wavefunction’s first derivative as appropriate for the dis-
continuous change in mass and
δ
-function potential at
x
= 0. The resulting system of equations is solved nu-
merically for a given choice of parameters.
To maintain unitarity of the scattering problem and
correctly obtain probabilities for Andreev reflection and
normal reflection (i.e., probabilities
A
and
B
in the main
text) one must amend the amplitudes
a
e
,a
h
,b
±
with fac-
tors that account for the ratio of group velocities between
the outgoing and incoming channels [40]. This process
ensures probability conservation for the four scattering
processes such that
A
+
B
+
C
+
D
= 1, thus justify-
ing use of the expression Eq. (1) for the total junction
conductance. We thereby obtain
A
=
|
a
h
|
2
v
A
v
IN
,
B
=
|
a
e
|
2
v
B
v
IN
,
(A8)
C
=
|
b
|
2
v
C
v
IN
,
D
=
|
b
+
|
2
v
D
v
IN
,
(A9)
with
v
A
/
B
/
C
/
D
the respective group velocities of carriers
at momenta
q
h
,q
e
,k
±
. Note that the incoming group ve-
locity is
v
IN
=
v
B
, and in the sub-gap regime
C,D
= 0 is
8
enforced by the vanishing group velocity for the evanes-
cent solutions. The conductance then follows as
G
(
E
) =
2
π
0
2
π
G
(
E,θ
)
(A10)
=
2
π
0
2
π
[1 +
A
(
E,θ
)
B
(
E,θ
)]
,
(A11)
where we made the angular dependence of the proba-
bility coefficients evident. For the case of an isotropic
pairing where ∆(
θ
) = ∆, the above angular integral is
trivial, and the expression reduces to the form of Eq. (1)
considered in the main text.
In our modeling we further add a tiny imaginary part
to the energy
E
E
+
i
Γ [56]. This factor accounts
for a finite lifetime due to thermal broadening/impurity
scattering, while also regularizing the solution at
E
=
∆ [where the BTK scattering problem, cf. Eqs. (A5),
becomes underdetermined].
Lastly, we comment on the behavior of the conduc-
tance for
E
μ
L
in the BEC regime as seen in Fig. 4.
Andreev processes are kinematically forbidden since hole
states are unavailable in the lead at those energies given
the parabolic band structure considered in our model.
We thus find vanishing of the Andreev reflection prob-
ability (
A
0) for
E
gap
E
μ
L
. Here
E
gap
=
2
+ Θ(
μ
SC
)
μ
2
SC
is the quasi-particle gap shown in
Fig. 4.
1. BTK analysis of sub-gap conductance for the
s
-wave case
Although the treatment from the preceding section
can be directly applied to the sub-gap conductance for a
gapped
s
-wave superconductor, it is insightful to examine
the analytic structure of the scattering problem up front
in this regime. We start by considering the 1D version of
BTK formalism [40] in the sub-gap limit
|
E
|
<
∆ where
we can safely neglect quasiparticle transmission across
the N-S interface. The only remaining processes are nor-
mal reflection (process
B
in Fig. 1c) as an electron and
Andreev reflection as a hole (process
A
in Fig. 1c).
We take as an ansatz plane-wave solutions for electrons
and holes in the lead region (
x
0) as in Eqs. (A1). In
the superconducting region
x >
0 we now specialize to
the sub-gap regime and replace Eqs. (A2) with evanescent
solutions of the form
φ
S
e
(
x
) =
b
e
e
κx
,
(A12)
φ
S
h
(
x
) =
b
h
e
κx
,
(A13)
where normalizability requires
Re
[
κ
]
>
0. Solving the
eigenvalue equation
H
BdG
Ψ =
E
Ψ with
H
BdG
(
x
) =
(
1
2
m
SC
2
x
μ
SC
1
2
m
SC
2
x
+
μ
SC
)
(A14)
identifies two possible solutions for
κ
, denoted
κ
±
, which
satisfy
κ
2
±
2
m
SC
=
μ
SC
±
i
|
|
2
E
2
.
(A15)
Upon finding the corresponding eigenvectors as well, the
general solution for the electron and hole wavefunctions
in the superconductor region reads
φ
S
e
(
x
) =
b
+
e
e
κ
+
x
+
b
e
e
κ
x
,
(A16)
φ
S
h
(
x
) =
E
+
i
|
|
2
E
2
b
+
e
e
κ
+
x
+
E
i
|
|
2
E
2
b
e
e
κ
x
.
(A17)
Note that the structure of the prefactors of
b
±
e
differs from
those in Eqs. (A2) and (A3) in a manner that streamlines
analysis of the sub-gap conductance. In particular, in the
present form the
E
0 limit can be taken straightfor-
wardly.
Repeating the analysis of Eq. (A5) and specializing to
the zero-bias
E
= 0 limit, we find an Andreev reflection
amplitude
a
h
=
v
L
(
v
v
+
)
v
2
L
+
v
+
v
+ 2
v
L
Z
(
v
+
+
v
) + 4
v
2
L
Z
2
.
(A18)
Here we defined velocities
v
L
=
q
F
/m
L
,
v
±
=
κ
±
/m
SC
and barrier strength parameter
Z
=
m
L
λ/q
F
=
λ/v
L
.
We also used the fact that at zero energy
q
e
=
q
h
=
2
m
L
μ
L
q
F
. Equation (A15) reduces to
κ
2
±
/
2
m
=
μ
SC
±
i
∆ at zero energy. Upon decomposing
κ
±
=
κ
R
±
I
as in the main text, Eq. (A18) takes the more
convenient form
a
h
=
2
iv
L
v
I
(
v
R
+ 2
v
L
Z
)
2
+
v
2
I
+
v
2
L
,
(A19)
which leads to the zero-bias Andreev reflection proba-
bility
A
0
=
|
a
h
|
2
quoted in Eq. (2). Unlike the case of
finite-bias tunneling, here the group velocity of the in-
cident electrons and Andreev reflected holes is identical;
hence the group velocity factors in Eqs. (A8) drop out.
2. Effective barrier parameter
̃
Z
and tunneling in
the
E

limit
In the standard BTK tunneling problem it is assumed
that the masses of the lead and the superconductor are
identical. More generally, as noted in the main text
[Eq. (5)], different masses (or equivalently different group
velocities) can be reabsorbed in the definition of an ef-
fective barrier parameter
̃
Z
[57]:
̃
Z
2
=
(
Z
+
β/
2)
2
α
+
(
α
1)
2
4
α
,
(A20)
9
such that the zero-energy Andreev reflection probability
reads
A
0
=
1
(
1 + 2
̃
Z
2
)
2
.
(A21)
Here, as in the main text,
α
=
v
I
/v
L
and
β
=
v
R
/v
L
. The
zero-bias conductance across the BCS to BEC evolution,
in the presence of both an interface potential and Fermi
velocity mismatch, can thus always be written in the gen-
eral form Eq. (A21) with a renormalized transparency
parameter. These expressions are also at the root of the
data collapse observed in the Andreev reflection spectra
of various high-density (BCS-like) superconductors with
vastly different barrier and velocity mismatch parame-
ters [57].
Maximizing Andreev reflection translates to making
̃
Z
as small as possible, which in turn requires that each
contribution to
̃
Z
needs to be
individually
small be-
cause they add in quadrature. In the BCS limit where
β
can be neglected, optimal Andreev reflection occurs
for
Z
= 0 and
α
= 1. In the BEC limit
Z
=
β/
2
is optimal—but we still need
α
1 which in BEC su-
perconductors translates to
v
L
v
I
/
(
m
SC
v
F
). Ve-
locity matching thus requires that the Fermi velocity of
the lead be small. This condition can be achieved via a
small lead chemical potential and/or a large lead effective
mass since
v
L
=
q
F
/m
L
=
(2
μ
L
)
/
(
m
L
) (for a parabolic
band). Commercially available nickel-based STM tips
with
m
L
28
m
e
could serve as promising leads in this
regard. As shown in App. D, however, the condition for
observing the optimal tunneling regime
v
L
v
I
becomes
somewhat relaxed in the presence of multiple scattering
channels (see Fig. S4).
The above generalized barrier parameter
̃
Z
manifests
in the high-bias
G
(
E

∆) limit as well. For these large
energies, the problem reduces to tunneling from a normal
metal to a normal metal across a potential barrier (or
well), with different group velocities in the two metals.
In such a case the tunneling conductance can be written
in the standard form
G
(
E

∆)
1
1 +
̃
Z
2
N
.
(A22)
The effective tunneling barrier parameter for normal-
normal scattering,
̃
Z
N
, follows from
̃
Z
2
N
=
Z
2
α
N
+
(
α
N
1)
2
4
α
N
,
(A23)
where
α
N
=
v
I
,
N
/v
L
,
N
and
v
I
,
N
=
k
+
/m
SC
,
v
L
,
N
=
q
e
/m
L
(see App. A). The expression (A22) is manifestly
Z
→−
Z
symmetric and maximal at
Z
= 0, demonstrat-
ing that the
Z
→ −
Z
asymmetry is solely a property
of the sub-gap conductance in the BEC regime (at least
within our theoretical framework).
B. Generalization to nodal pairing
For superconductors with anisotropic pairing gaps,
we can compute the tunneling conductance—assuming
a ballistic point contact such that the in-plane compo-
nents of the incident electron momentum is conserved—
by averaging over the in-plane momentum components
as captured in Eq. (A10). (See also further discussion in
Sec. D.) Let us assume an isotropic normal state, such
that the angular dependence of the Andreev and normal
reflection coefficients only arise through the nodal pair-
ing potential ∆(
k
)
0
cos(
νθ
) with
ν
= 1
,
2
,
3
,...
for
p
-wave,
d
-wave,
f
-wave, ... .
Figures S1 and S2 show the
d
-wave analogs of the
s
-
wave plots from the main text. The zero-bias conduc-
tance shown in Fig. S1a-c is qualitatively similar to that
of Fig. 2a-c, with the locus of perfect Andreev reflec-
tion tracing the
Z
=
v
R
/
2
v
L
condition as before. As
expected, distinctions are present in the finite-bias tun-
neling profile shown in Fig. S2a-c. In particular, in the
BCS regime for the QPC limit
Z
= 0 (Fig. S2a), instead
of a plateau profile present in the sub-gap conductance,
here we see an inverted V-shape profile accounting for
the presence of gapless excitations. Crucially, however,
the qualitative features remain the same as in Fig. 4:
suppression of Andreev reflection for the
Z
= 0 case
(Fig. S2a) and a pronounced
Z
→ −
Z
asymmetry with
revival of Andreev reflection in the BEC regime for
Z <
0
(Fig. S2b,c). At the level of this analysis we expect other
pairing symmetries to produce similar tunneling profiles.
C. ELF model and duality
Here we explore the Effective Lead Model (ELF) model
that complements the BTK analysis and provides an in-
dependent verification of the results. The lead is treated
as a single-channel, semi-infinite wire defined along
x >
0, with
x
= 0 proximate to an infinite
d
-dimensional
superconductor (e.g., for
d
= 2 the lead tunnels into the
center of an infinite superconducting sheet oriented in the
y
z
plane as illustrated in Fig. 3). It is convenient to
equivalently view the lead as an infinite
chiral
wire [58]:
x <
0 is then associated with
incoming
states whereas
x >
0 corresponds to the
outgoing
states. With this
viewpoint in mind, our starting point is the Hamiltonian
H
=
H
Lead
+
H
SC
+
H
Lead-SC
,
H
Lead
=
x
ψ
(
x
) (
iv
L
x
)
ψ
(
x
) +
U
0
ψ
(0)
τ
z
ψ
(0)
H
SC
=
r
d
(
r
)
[
(
2
2
m
SC
μ
SC
)
τ
z
+ ∆
τ
x
]
d
(
r
)
H
Lead-SC
=
t
[
ψ
(
x
= 0)
τ
z
d
(
r
= 0) +
h.c.
]
.
(C1)