# Andreev reflection spectroscopy in strongly paired superconductors

Cyprian Lewandowski  Department of Physics, California Institute of
Technology, Pasadena CA 91125, USA  Institute for Quantum Information and
Matter, California Institute of Technology, Pasadena CA 91125, USA
<EMAIL_ADDRESS>Étienne Lantagne-Hurtubise  Department of Physics,
California Institute of Technology, Pasadena CA 91125, USA  Institute for
Quantum Information and Matter, California Institute of Technology, Pasadena
CA 91125, USA  Alex Thomson  Department of Physics, California Institute of
Technology, Pasadena CA 91125, USA  Institute for Quantum Information and
Matter, California Institute of Technology, Pasadena CA 91125, USA  Walter
Burke Institute for Theoretical Physics, California Institute of Technology,
Pasadena, California 91125, USA  Department of Physics, University of
California, Davis, California 95616, USA  Stevan Nadj-Perge  T. J. Watson
Laboratory of Applied Physics, California Institute of Technology, 1200 East
California Boulevard, Pasadena, California 91125, USA  Institute for Quantum
Information and Matter, California Institute of Technology, Pasadena CA 91125,
USA  Jason Alicea  Department of Physics, California Institute of Technology,
Pasadena CA 91125, USA  Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena CA 91125, USA

###### Abstract

Motivated by recent experiments on low-carrier-density superconductors,
including twisted multilayer 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 superconductor 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 superconductors. 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, 2, 3, 4, 5]. In the BCS regime, superconductivity arises
from weakly bound Cooper pairs with a characteristic size
$\xi_{\mathrm{pair}}$ that far exceeds the mean inter-particle spacing $d$; in
the BEC regime, by contrast, fermions form tightly bound Cooper pairs with
size $\xi_{\mathrm{pair}}\ll d$. BCS- and BEC-type superconductors can be
separated either by a crossover (e.g., for $s$-wave pairing [6, 7, 8, 9, 10,
11]; Fig 1a), a Lifshitz-type phase transition (e.g., for nodal pairing [12,
13, 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, 17, 18, 19, 20,
21], provide a controlled experimental platform for exploring the evolution
from weak to strong pairing. While most solid-state superconductors reside
firmly in the BCS regime, certain strongly correlated materials such as
cuprates have been rationalized in terms of the BCS-BEC paradigm [22].
Further, a new generation of experiments probing low-carrier-density materials
including iron-based compounds [23, 24, 25, 26], LixZrNCl [27, 28] and moiré
graphene systems [29, 30, 31, 32, 33, 34] has revealed signatures consistent
with proximity to a BEC state—opening a new experimental frontier for
unconventional superconductivity. Developing probes that can unambiguously
identify BEC superconductors and distinguish possible competing phases
therefore poses a pressing problem.

Figure 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: Andreev (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.

In solid-state contexts, BEC superconductivity can manifest in various ways.
Saturation of the Ginzburg-Landau coherence length $\xi_{\rm GL}$ at the
interparticle spacing [27, 29] and a critical temperature approaching
theoretical bounds [35, 36] both constitute indirect evidence for proximity to
the strong pairing regime. Energy-momentum resolved probes [24] can also
reveal the evolution of the quasiparticle dispersion from BCS-like to BEC-like
(Fig. 1a,b). Superconductors with a nodal order 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 transition [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
superconductors from competing insulators since, unlike their BCS
counterparts, they feature less prominent and particle-hole asymmetric
coherence peaks [37, 31].

Figure 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 $v_{L}\sqrt{m_{\rm SC}/\Delta}$ set to (a) 3, (b) 1.5, and
(c) 0.5. Black dashed lines trace $Z=-v_{\rm R}/2v_{\rm L}$, 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 $Z\rightarrow-Z$ symmetric,
pronounced asymmetry emerges in the BEC regime, with enhanced Andreev
reflection possible at negative $Z$ [see panel (c)].

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 analysis of BEC-BEC Josephson junctions in Ref. 39. We
show that this suppression is a consequence of a duality that interchanges the
wavefunctions of BEC and BCS superconductors. Second, we establish that
Andreev reflection can be controllably revived in the BEC regime by tunneling
between the lead and superconductor through an effective potential _well_ —as
opposed to the barrier employed in conventional treatments [40]. This feature
sharply distinguishes 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]. Consider
a 1D setup (Fig. 1c) with an interface at position $x=0$ separating a spinful
normal lead (at $x<0$) with dispersion $\xi_{q}=\frac{q^{2}}{2m_{\rm
L}}-\mu_{\rm L}$ from a singlet superconductor (at $x>0$) with dispersion
$\xi_{k}=\frac{k^{2}}{2m_{\rm SC}}-\mu_{\rm SC}$ and $s$-wave pairing
potential $\Delta$. We take $\mu_{\rm L}>0$ throughout but allow $\mu_{\rm
SC}$ to take either sign to capture the BCS-BEC crossover that occurs as
$\mu_{\rm SC}$ crosses zero. A delta-function tunnel barrier,
$\lambda\delta(x)$, interpolates between the tunneling limit with large
$\lambda>0$ and the QPC limit $\lambda=0$.

Figure 1c illustrates the processes available to an incident electron at the
interface: Andreev (A) or normal (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 appropriate group velocities [41]. In the standard BTK formalism
applied to weakly paired BCS superconductors, the limit $\Delta,E\ll\mu_{\rm
L},\mu_{\rm SC}$ combined with the equal-mass assumption $m_{\rm L}=m_{\rm
SC}$ drastically simplifies the problem, yielding the Andreev approximation in
which $q_{h}=q_{e}=k_{\pm}=q_{F}$ with $q_{F}=\sqrt{2m_{\rm L}\mu_{\rm L}}$
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-Büttiker formalism by

$G(E)=\frac{2e^{2}}{h}\left[1+A(E)-B(E)\right],$ (1)

where $A(E)$ and $B(E)$ denote the Andreev and normal reflection
probabilities. Maximal conductance of $4e^{2}/h$ corresponds to perfect
Andreev reflection $A(E)=1$.

_Zero-bias conductance._ For gapped superconductors, the absence of
transmission processes (C and D) considerably simplifies the analysis of the
sub-gap conductance: employing the normalization condition $A(E)+B(E)=1$
returns $G(E)=\frac{4e^{2}}{h}A(E)$. As shown in the Supplement [41], the
zero-bias Andreev reflection probability, $A_{0}\equiv A(E=0)$, then reduces
to

$A_{0}=\frac{4v_{\rm L}^{2}v_{\rm I}^{2}}{\left[\left(2v_{\rm L}Z+v_{\rm
R}\right)^{2}+v_{\rm I}^{2}+v_{\rm L}^{2}\right]^{2}},$ (2)

where $Z=\lambda/v_{\rm L}$ quantifies the barrier transparency, $v_{\rm
L}=q_{F}/m_{\rm L}$ is the Fermi velocity of the lead, and $v_{\rm
R,I}=\kappa_{\rm R,I}/m_{\rm SC}$ are two characteristic velocities of the
superconductor. The momenta $\kappa_{\rm R,I}$ are defined through

$\kappa e^{i\varphi}=\kappa_{\rm R}+i\kappa_{\rm I}=\sqrt{2m_{\rm
SC}(-\mu_{\rm SC}+i\Delta)},$ (3)

and the length scales $\kappa_{\rm R}^{-1}$ and $\kappa_{\rm I}^{-1}$
respectively control the evanescent and oscillatory behavior of the
wavefunction in the superconductor at $x>0$, $\psi_{\mathrm{SC}}\sim
e^{-(\kappa_{\rm R}+i\kappa_{\rm I})x}$.

Sending $\mu_{\rm SC}\rightarrow-\mu_{\rm SC}$, which connects the BCS and BEC
regimes, yields $\varphi\rightarrow\pi/2-\varphi$ and hence swaps $\kappa_{\rm
R}\leftrightarrow\kappa_{\rm I}$; cf. Eq. (3). This remarkable property
reveals a duality between the BCS and BEC superconductor wavefunctions. As an
instructive limiting case, deep in either the BEC or BCS regimes where
$|\mu_{\mathrm{SC}}|/\Delta\gg 1$, $\kappa_{\rm R,I}$ can be expanded as

$\displaystyle\kappa_{\rm R}+i\kappa_{\rm I}$
$\displaystyle\approx\sqrt{\mathrm{sgn}(\mu_{\mathrm{SC}})}\left(\frac{\Delta}{v_{F}}+i\mathrm{sgn}(\mu_{\mathrm{SC}})k_{F}\right).$
(4)

Here the definition of the Fermi momentum
$k_{F}=\sqrt{2m_{\mathrm{SC}}|\mu_{\mathrm{SC}}|}$ and velocity
$v_{F}=k_{F}/m_{\mathrm{SC}}$ is extended into the BEC regime. In the BCS
limit Eq. (4) aligns with conventional understanding: The wavefunction’s
oscillatory part is set by $k_{F}$, whereas evanescent decay is controlled by
$\Delta/v_{F}$—i.e., the inverse of the BCS pair coherence length $\xi_{\rm
pair}$. The situation flips in the BEC regime, where $k_{F}$ controls the
decay length and $\Delta/v_{F}$ dictates the oscillatory behavior.

This duality has immediate implications for the conductance. In the QPC limit,
$Z=0$, the Andreev reflection probability in Eq. (2) is maximized when the
lead Fermi velocity satisfies $v_{\rm L}=\sqrt{v_{\rm R}^{2}+v_{\rm I}^{2}}$,
yielding the upper bound $A_{0}\leq v_{\rm I}^{2}/(v_{\rm I}^{2}+v_{\rm
R}^{2})$ valid for any $\mu_{\rm SC}$. Near-perfect Andreev reflection thus
requires $v_{\rm R}\ll v_{\rm I}$, implying many oscillation periods over the
decaying envelope of the superconductor wavefunction. This requirement exactly
translates to the BCS limit, $\mu_{\mathrm{SC}}\gg\Delta$, as originally
pointed out by Andreev [42]. By contrast, the deep BEC regime is characterized
by the converse inequality, $v_{\rm R}\gg v_{\rm I}$, an the upper bound above
accordingly reduces to $A_{0}\lesssim v_{\rm I}^{2}/v_{\rm
R}^{2}=\Delta^{2}/\mu_{\mathrm{SC}}^{2}$. Physically, the fast decay of the
superconducting wavefunction relative to its oscillation period suppresses
Andreev reflection from the plane wave electrons of the lead. At the “self-
dual” point $\mu_{\rm SC}=0$, the superconductor exhibits a single length
scale, with $v_{\rm R}=v_{\rm I}=\sqrt{\Delta/m_{\rm SC}}$, yielding an
Andreev reflection probability upper-bounded by $1/2$.

_Andreev revival._ The preceding bound on Andreev reflection in the BEC regime
($\mu_{\rm SC}<0$) implies that the zero-bias conductance in the QPC limit
cannot exceed the value $2e^{2}/h$ 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
probability $A_{0}$. Interestingly, however, Andreev reflection can be
enhanced by exploiting a pronounced asymmetry in the sign of $Z$ that emerges
in the BEC regime.

Writing $v_{\rm I}=\alpha v_{\rm L}$ and $v_{\rm R}=\beta v_{\rm L}$, one can
re-express Eq. (2) as $A_{0}=1/(1+2\tilde{Z}^{2})^{2}$ in terms of a
renormalized transparency parameter,

$\displaystyle\tilde{Z}^{2}$
$\displaystyle=\frac{(Z+\beta/2)^{2}}{\alpha}+\frac{(\alpha-1)^{2}}{4\alpha},$
(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 dimensionless parameter $\beta/\alpha=v_{\rm R}/v_{\rm I}$
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 $v_{\rm I}\gg v_{\rm R}$. In the BEC regime,
by contrast, $v_{\rm R}>v_{\rm I}$ 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 $\tilde{Z}=0$, i.e., when $Z=-v_{\rm R}/(2v_{\rm L})$ and $v_{\rm
L}=v_{\rm I}$. These trends are illustrated in Fig. 2, which plots the zero-
bias conductance versus $Z$ across the BCS-BEC evolution tuned by $\mu_{\rm
SC}/\Delta$, for different lead velocities $v_{\rm L}$ in panels (a)-(c). Note
that the standard BTK result for a BCS superconductor is recovered for large
$v_{\rm L}$ [Fig. 2(a)].

Figure 3: Sketch of the effective lead formalism. A lead tunnels electrons
onto a gapped superconductor (left). Integrating out the gapped degrees of
freedom generates an effective lead-only “impurity” problem (right) featuring
a shifted local potential $U_{\rm SC}$ and a local pairing term $W_{\rm SC}$
mediated by the superconductor.  Figure 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 $v_{\rm L}=v_{\rm I}$ for each $\mu_{\rm SC}$
value. Black dashed lines at energies $E=\pm\mu_{\mathrm{L}}$ show the
corresponding optimal lead chemical potential $\mu_{\rm L}$, and green dashed
lines denotes the quasiparticle gap $E_{\rm gap}$. (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 $Z=\pm 1$ are similar in the BCS regime, showing a gap
surrounded by coherence peaks at $E=\pm\Delta$. In the BEC regime, sub-gap
conductance enhancement occurs _only_ for tunneling through potential wells,
as in (c).

_Effective lead formalism._ For complementary insight, we now examine normal
metal-superconductor tunneling via an effective lead formalism (ELF). Consider
a 1D lead at positions $x\leq 0$ with linearized kinetic energy and Fermi
velocity $v_{\mathrm{L}}=q_{F}/m_{\mathrm{L}}$. At its endpoint, the lead
exhibits a local potential $U_{0}\delta(x)$ 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

$U_{\rm
SC}=t^{2}\int\frac{d^{d}\bm{k}}{(2\pi)^{d}}\frac{\xi_{\bm{k}}}{\xi_{\bm{k}}^{2}+\Delta^{2}}$
(6)

and generates a local singlet pairing term with amplitude

$\displaystyle W_{\rm SC}$
$\displaystyle=t^{2}\int\frac{d^{d}\bm{k}}{(2\pi)^{d}}\frac{\Delta}{\xi_{\bm{k}}^{2}+\Delta^{2}}.$
(7)

Explicit evaluation of the integrals reveals that, for $d=1$, $U_{\rm SC}$ and
$W_{\rm SC}$ exactly swap under $\mu_{\rm SC}\rightarrow-\mu_{\rm SC}$,
manifesting the duality uncovered above within the BTK formalism (for details
and extensions see [41]).

Extracting the wavefunctions from the effective lead-only Hamiltonian yields a
zero-bias Andreev reflection probability [41]

$A_{0}^{\mathrm{ELF}}=\frac{4v_{\rm L}^{2}(W_{\rm
SC}/2)^{2}}{\left[(U_{\mathrm{eff}}/2)^{2}+(W_{\mathrm{SC}}/2)^{2}+v_{\rm
L}^{2}\right]^{2}},$ (8)

with $U_{\rm eff}=U_{0}+U_{\rm SC}$. Eq. (8) reproduces Eq. (2) upon
identifying the ELF/BTK correspondence $U_{0}/2\leftrightarrow 2v_{\rm L}Z$,
$U_{\rm SC}/2\leftrightarrow v_{\rm R}$, and $W_{\rm SC}/2\leftrightarrow
v_{\rm I}$. Thus negative values of $U_{0}$ enable the cancellation of the
‘large’ potential $U_{\rm SC}$ generated by BEC superconductors—in turn
allowing Andreev processes mediated by $W_{\rm SC}$ to dominate.

_Extensions._ Figure 4 presents the finite-bias conductance $G(E)$ for an
$s$-wave superconductor. Data were obtained numerically from the full BTK
analysis, including 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 $v_{\rm L}=v_{\rm I}$
across the BCS-BEC evolution; that is, for each $\mu_{\rm SC}$ 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 $G(E)=4e^{2}/h$ 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 coherence 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 kinematic constraints imposed by the lead
chemical potential $\mu_{\rm L}$ (the required backwards propagating hole
state does not exist for $|E|>\mu_{\rm L}$).

Our 1D analysis extends to the experimentally relevant scenario of a multi-
channel lead normally incident on a 2D superconductor [41]. In the limit of
many ballistic channels, the total conductance $G(E)$ follows from an angular
average over the contributions from all momenta in the plane of the interface
[43, 38, 44, 41]. 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 distinguishing feature is that sub-gap conductance generated
by Andreev reflection in the BCS regime acquires an inverted 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
differentiates BEC phenomenology from the familiar properties of BCS
superconductors. In particular, the emergence of a pronounced $Z\rightarrow-Z$
asymmetry in the sub-gap conductance provides a striking signature of BEC
superconductivity. This feature stands in stark contrast to the above-gap
conductance $G(E\gg\Delta)$ that is manifestly symmetric with respect to the
sign of $Z$ [41]. Possible experimental setups for probing the negative-$Z$
regime include (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
coating 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 superconductivity arising from a
single parabolic band. Extending these ideas to BEC physics in multiband
superconductors, believed to be important to understanding recent experiments
[24], is thus warranted. The limit of narrow-band superconductivity, where
quantum geometry plays an important role in determining the superfluid
stiffness [48, 49, 50, 51, 32, 52, 53], is another interesting avenue for
future work with possible applications to moiré materials [29, 31]. Finally,
self-consistent treatments [54] and extensions beyond mean-field [11] might
lead to further insights into BEC superconductors and associated pseudogap
physics [10, 55] at temperatures above $T_{c}$.

##  I Acknowledgments

We thank Mohit Randeria, Hyunjin Kim, Michał Papaj, 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 É.L.-H.); the Army
Research Office under Grant Award W911NF-17-1-0323; the Caltech Institute for
Quantum Information and Matter, an NSF Physics Frontiers Center with support
of the Gordon and Betty Moore Foundation through Grant GBMF1250; and the
Walter Burke Institute for Theoretical Physics at Caltech.

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##  Appendix 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
analogous 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
scattering 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 conventional BTK analysis, however, the Andreev approximation [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, imagine carving a hole into a 2D
superconductor (with mass $m_{\rm SC}$, chemical potential $\mu_{\rm SC}$) and
filling in the hole with a disc-shaped lead (with mass $m_{\rm L}$, chemical
potential $\mu_{\rm L}$). An incident electron from the lead can then scatter
into the adjacent 2D superconductor in any in-plane direction, with a given
in-plane direction defining an effective 1D scattering problem as shown Fig.
1c.

Denoting the in-plane direction characterized by an angle $\theta$ as the $x$
axis for concreteness, we take our scattering states in the lead region
($x\leq 0$) to be

$\displaystyle\varphi^{L}_{e}(x)$
$\displaystyle=e^{iq_{e}x}+a_{e}e^{-iq_{e}x}\,,$
$\displaystyle\varphi^{L}_{h}(x)$ $\displaystyle=a_{h}e^{iq_{h}x}\,.$ (9)

Here $q_{e}=\sqrt{2m_{\rm L}(\mu_{\rm L}+E)}$ and $q_{h}=\sqrt{2m_{\rm
L}(\mu_{\rm L}-E)}$ are the momenta of the electron and hole, respectively,
with $E$ the incident electron energy. We normalized the amplitude 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

$\displaystyle\varphi^{S}_{e}(x)$
$\displaystyle=b_{+}u_{+}e^{ik_{+}x}+b_{-}u_{-}e^{-ik_{-}x}$
$\displaystyle\varphi^{S}_{h}(x)$
$\displaystyle=b_{+}v_{+}e^{ik_{+}x}+b_{-}v_{-}e^{-ik_{-}x}\,,$ (10)

where
$k_{\pm}=\sqrt{2m_{\mathrm{SC}}\left(\mu_{\mathrm{SC}}\pm\sqrt{E^{2}-\Delta^{2}}\right)}$
are the momenta corresponding to the two solutions ($C$ and $D$) shown in Fig.
1c. Coefficients $b_{\pm}$ correspond to the transmission of electrons into
the superconductor without ($k_{+}$, process $D$) and with ($k_{-}$, process
$C$) branch crossing. Possible angular dependence of the superconductor’s
order parameter enters through the pairing amplitude $\Delta(\theta)$, with
$\theta$ the polar angle in the 2D plane. We note that in other geometries
[43, 44] the angular dependence of the pairing amplitude $\Delta(\theta)$ 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 $u_{\pm}$, $v_{\pm}$, evaluated at their respective branches, that
here follow from:

$\displaystyle u_{+}^{2}$
$\displaystyle=\frac{1}{2}\left(1+\frac{\sqrt{E^{2}-\Delta^{2}}}{E}\right)=v_{-}^{2}\,,$
(11) $\displaystyle v_{+}^{2}$
$\displaystyle=\frac{1}{2}\left(1-\frac{\sqrt{E^{2}-\Delta^{2}}}{E}\right)=u_{-}^{2}\,.$
(12)

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 dependence drops out,
while for the nodal (e.g., $d$-wave) case angular dependence enters only via
the pairing potential $\Delta(\theta)$.

To determine the scattering coefficients we include a $\lambda\delta(x)$
potential as in the main text and employ standard boundary conditions at
$x=0$,

$\displaystyle\varphi^{L}_{e}(x=0)=\varphi^{S}_{e}(x=0)\,,\varphi^{L}_{h}(x=0)=\varphi^{S}_{h}(x=0)\,,$
(13)
$\displaystyle\frac{1}{2m_{\mathrm{SC}}}\left.\frac{d\varphi^{S}_{e}}{dx}\right\rvert_{x=0}-\frac{1}{2m_{\mathrm{L}}}\left.\frac{d\varphi^{L}_{e}}{dx}\right\rvert_{x=0}=\lambda\varphi^{S}_{e}(x=0)\,,$
(14)
$\displaystyle\frac{1}{2m_{\mathrm{SC}}}\left.\frac{d\varphi^{S}_{h}}{dx}\right\rvert_{x=0}-\frac{1}{2m_{\mathrm{L}}}\left.\frac{d\varphi^{L}_{h}}{dx}\right\rvert_{x=0}=\lambda\varphi^{S}_{h}(x=0)\,.$
(15)

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 discontinuous change in mass and $\delta$-function
potential at $x=0$. The resulting system of equations is solved numerically
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
$a_{e},a_{h},b_{\pm}$ with factors 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 justifying use of the expression Eq. (1) for the total
junction conductance. We thereby obtain

$\displaystyle
A=|a_{h}|^{2}\left\lvert\frac{v_{\mathrm{A}}}{v_{\mathrm{IN}}}\right\rvert\,,$
$\displaystyle
B=|a_{e}|^{2}\left\lvert\frac{v_{\mathrm{B}}}{v_{\mathrm{IN}}}\,\right\rvert,$
(16) $\displaystyle
C=|b_{-}|^{2}\left\lvert\frac{v_{\mathrm{C}}}{v_{\mathrm{IN}}}\right\rvert\,,$
$\displaystyle
D=|b_{+}|^{2}\left\lvert\frac{v_{\mathrm{D}}}{v_{\mathrm{IN}}}\,\right\rvert,$
(17)

with $v_{\mathrm{A/B/C/D}}$ the respective group velocities of carriers at
momenta $q_{h},q_{e},k_{\pm}$. Note that the incoming group velocity is
$v_{\mathrm{IN}}=v_{\mathrm{B}}$, and in the sub-gap regime $C,D=0$ is
enforced by the vanishing group velocity for the evanescent solutions. The
conductance then follows as

$\displaystyle G(E)$
$\displaystyle=\int_{0}^{2\pi}\frac{d\theta}{2\pi}G(E,\theta)$ (18)
$\displaystyle=\int_{0}^{2\pi}\frac{d\theta}{2\pi}[1+A(E,\theta)-B(E,\theta)]\,,$
(19)

where we made the angular dependence of the probability coefficients evident.
For the case of an isotropic pairing where $\Delta(\theta)=\Delta$, 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
$E\rightarrow E+i\Gamma$ [56]. This factor accounts for a finite lifetime due
to thermal broadening/impurity scattering, while also regularizing the
solution at $E=\Delta$ [where the BTK scattering problem, cf. Eqs. (13),
becomes underdetermined].

Lastly, we comment on the behavior of the conductance for $E\geq\mu_{\rm L}$
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 probability ($A\to 0$) for
$E_{\mathrm{gap}}\geq E\geq\mu_{\rm L}$. Here
$E_{\mathrm{gap}}=\sqrt{\Delta^{2}+\Theta(-\mu_{\mathrm{SC}})\mu_{\mathrm{SC}}^{2}}$
is the quasi-particle gap shown in Fig. 4.

###  A.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 $|E|<\Delta$ where we can safely neglect quasiparticle
transmission across the N-S interface. The only remaining processes are normal
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 ($x\leq 0$) as in Eqs. (9). In the superconducting region $x>0$ we now
specialize to the sub-gap regime and replace Eqs. (10) with evanescent
solutions of the form

$\displaystyle\varphi^{S}_{e}(x)$ $\displaystyle=b_{e}e^{-\kappa x},$ (20)
$\displaystyle\varphi^{S}_{h}(x)$ $\displaystyle=b_{h}e^{-\kappa x},$ (21)

where normalizability requires $Re[\kappa]>0$. Solving the eigenvalue equation
$H_{\rm BdG}\Psi=E\Psi$ with

$\displaystyle H_{\rm BdG}(x)=\begin{pmatrix}-\frac{1}{2m_{\rm
SC}}\partial_{x}^{2}-\mu_{\rm SC}&\Delta\\\ \Delta^{*}&\frac{1}{2m_{\rm
SC}}\partial_{x}^{2}+\mu_{\rm SC}\end{pmatrix}$ (22)

identifies two possible solutions for $\kappa$, denoted $\kappa_{\pm}$, which
satisfy

$\frac{\kappa_{\pm}^{2}}{2m_{\rm SC}}=-\mu_{\rm SC}\pm
i\sqrt{|\Delta|^{2}-E^{2}}.$ (23)

Upon finding the corresponding eigenvectors as well, the general solution for
the electron and hole wavefunctions in the superconductor region reads

$\displaystyle\varphi^{S}_{e}(x)=$ $\displaystyle
b_{e}^{+}e^{-\kappa_{+}x}+b_{e}^{-}e^{-\kappa_{-}x},$ (24)
$\displaystyle\varphi^{S}_{h}(x)=$
$\displaystyle\frac{E+i\sqrt{|\Delta|^{2}-E^{2}}}{\Delta}b_{e}^{+}e^{-\kappa_{+}x}$
$\displaystyle+$
$\displaystyle\frac{E-i\sqrt{|\Delta|^{2}-E^{2}}}{\Delta}b_{e}^{-}e^{-\kappa_{-}x}.$
(25)

Note that the structure of the prefactors of $b_{e}^{\pm}$ differs from those
in Eqs. (10) and (11) in a manner that streamlines analysis of the sub-gap
conductance. In particular, in the present form the $E\to 0$ limit can be
taken straightforwardly.

Repeating the analysis of Eq. (13) and specializing to the zero-bias $E=0$
limit, we find an Andreev reflection amplitude

$a_{h}=\frac{v_{\rm L}(v_{-}-v_{+})}{v_{\rm L}^{2}+v_{+}v_{-}+2v_{\rm
L}Z(v_{+}+v_{-})+4v_{\rm L}^{2}Z^{2}}\,.$ (26)

Here we defined velocities $v_{\rm L}=q_{F}/m_{\rm L}$,
$v_{\pm}=\kappa_{\pm}/m_{\rm SC}$ and barrier strength parameter $Z=m_{\rm
L}\lambda/q_{F}=\lambda/v_{\rm L}$. We also used the fact that at zero energy
$q_{e}=q_{h}=\sqrt{2m_{\rm L}\mu_{\rm L}}\equiv q_{F}$. Equation (23) reduces
to $\kappa_{\pm}^{2}/2m=-\mu_{\rm SC}\pm i\Delta$ at zero energy. Upon
decomposing $\kappa_{\pm}=\kappa_{\rm R}\pm i\kappa_{\rm I}$ as in the main
text, Eq. (26) takes the more convenient form

$a_{h}=\frac{-2iv_{\rm L}v_{\rm I}}{(v_{\rm R}+2v_{\rm L}Z)^{2}+v_{\rm
I}^{2}+v_{\rm L}^{2}},$ (27)

which leads to the zero-bias Andreev reflection probability
$A_{0}=|a_{h}|^{2}$ quoted in Eq. (2). Unlike the case of finite-bias
tunneling, here the group velocity of the incident electrons and Andreev
reflected holes is identical; hence the group velocity factors in Eqs. (16)
drop out.

###  A.2 Effective barrier parameter $\tilde{Z}$ and tunneling in the
$E\gg\Delta$ 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 effective barrier
parameter $\tilde{Z}$ [57]:

$\displaystyle\tilde{Z}^{2}$
$\displaystyle=\frac{(Z+\beta/2)^{2}}{\alpha}+\frac{(\alpha-1)^{2}}{4\alpha},$
(28)

such that the zero-energy Andreev reflection probability reads

$A_{0}=\frac{1}{\left(1+2\tilde{Z}^{2}\right)^{2}}.$ (29)

Here, as in the main text, $\alpha=v_{\rm I}/v_{\rm L}$ and $\beta=v_{\rm
R}/v_{\rm L}$. 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 general form Eq. (29) 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
parameters [57].

Maximizing Andreev reflection translates to making $\tilde{Z}$ as small as
possible, which in turn requires that each contribution to $\tilde{Z}$ needs
to be _individually_ small because they add in quadrature. In the BCS limit
where $\beta$ can be neglected, optimal Andreev reflection occurs for $Z=0$
and $\alpha=1$. In the BEC limit $Z=-\beta/2$ is optimal—but we still need
$\alpha\sim 1$ which in BEC superconductors translates to $v_{\rm L}\sim
v_{\rm I}\approx\Delta/(m_{\rm SC}v_{F})$. Velocity 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
$v_{\rm L}=q_{F}/m_{\rm L}=\sqrt{(2\mu_{\rm L})/(m_{\rm L})}$ (for a parabolic
band). Commercially available nickel-based STM tips with $m_{L}\sim 28m_{e}$
could serve as promising leads in this regard. As shown in App. D, however,
the condition for observing the optimal tunneling regime $v_{\rm L}\sim v_{\rm
I}$ becomes somewhat relaxed in the presence of multiple scattering channels
(see Fig. S4).

The above generalized barrier parameter $\tilde{Z}$ manifests in the high-bias
$G(E\gg\Delta)$ 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

$G(E\gg\Delta)\approx\frac{1}{1+\tilde{Z}_{N}^{2}}.$ (30)

The effective tunneling barrier parameter for normal-normal scattering,
$\tilde{Z}_{N}$, follows from

$\displaystyle\tilde{Z}_{N}^{2}$
$\displaystyle=\frac{Z^{2}}{\alpha_{N}}+\frac{(\alpha_{N}-1)^{2}}{4\alpha_{N}},$
(31)

where $\alpha_{N}=v_{\rm I,N}/v_{\rm L,N}$ and $v_{\rm
I,N}=k_{+}/m_{\mathrm{SC}}$, $v_{\rm L,N}=q_{e}/m_{\mathrm{L}}$ (see App. A).
The expression (30) is manifestly $Z\to-Z$ symmetric and maximal at $Z=0$,
demonstrating that the $Z\to-Z$ asymmetry is solely a property of the sub-gap
conductance in the BEC regime (at least within our theoretical framework).

##  Appendix B Generalization to nodal pairing

Figure S1:  Dependence of the BTK zero-bias tunneling conductance on the
barrier parameter $Z$ along the BCS-BEC evolution for a $d$-wave
superconductor ($\Delta(\theta)=\Delta\cos(2\theta)$), with $v_{\rm
L}\sqrt{m_{\rm SC}/\Delta}$ set to (a) 3, (b) 1.5, and (c) 0.5. Black dashed
lines trace $Z=-v_{\rm R}/2v_{\rm L}$, 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 $Z\rightarrow-Z$ symmetric, pronounced asymmetry
emerges in the BEC regime, with enhanced Andreev reflection possible at
negative $Z$ [see panel (c)]. Note the qualitative similarity with Fig. 2,
which displays analogous results for $s$-wave superconductors. Figure S2:
Finite-bias tunneling spectroscopy across the BCS-BEC evolution for a $d$-wave
superconductor ($\Delta(\theta)=\Delta\cos(2\theta)$) with (a) $Z=0$, (b)
$Z=1$ and (c) $Z=-1$, assuming an optimal lead with $v_{\rm L}=v_{\rm I}$ for
each $\mu_{\rm SC}$ value. Black dashed lines at energies
$E=\pm\mu_{\mathrm{L}}$ show the corresponding optimal lead chemical potential
$\mu_{\rm L}$, and green dashed lines denote the quasiparticle gap
$E_{\mathrm{gap}}$ in the anti-nodal direction. (a) The quantum point contact
limit $Z=0$ exhibits a sub-gap conductance enhancement due to Andreev
reflection in the BCS regime, but not in the BEC regime. In contrast to
$s$-wave superconductors [see Fig. 4] the sub-gap conductance has an “inverted
V-shape” profile reflecting the presence of gapless excitations. (b,c)
Tunneling spectra for $Z=\pm 1$ are similar in the BCS regime, showing a gap
surrounded by coherence peaks at $E=\pm\Delta$. In the BEC regime, sub-gap
conductance enhancement occurs _only_ for tunneling through potential wells,
as in (c). Note the qualitative similarity with Fig. 4, which displays
analogous results for $s$-wave superconductors.

For superconductors with anisotropic pairing gaps, we can compute the
tunneling conductance—assuming a ballistic point contact such that the in-
plane components of the incident electron momentum is conserved—by averaging
over the in-plane momentum components as captured in Eq. (18). (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 pairing potential
$\Delta(\bm{k})\sim\Delta_{0}\cos(\nu\theta)$ with $\nu=1,2,3,...$ 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 conductance shown in Fig. S1a-c is qualitatively
similar to that of Fig. 2a-c, with the locus of perfect Andreev reflection
tracing the $Z=-v_{\mathrm{R}}/2v_{\mathrm{L}}$ condition as before. As
expected, distinctions are present in the finite-bias tunneling 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 $Z\to-Z$ 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.

##  Appendix C ELF model and duality

Here we explore the Effective Lead Model (ELF) model that complements the BTK
analysis and provides an independent 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

$\displaystyle H$
$\displaystyle=H_{\mathrm{Lead}}+H_{\mathrm{SC}}+H_{\mathrm{Lead}\text{-}{\mathrm{SC}}},$
$\displaystyle H_{\mathrm{Lead}}$
$\displaystyle=\int_{x}\psi^{\dagger}(x)\left(-iv_{\mathrm{L}}\partial_{x}\right)\psi(x)+U_{0}\psi^{\dagger}(0)\tau^{z}\psi(0)$
$\displaystyle H_{\mathrm{SC}}$
$\displaystyle=\int_{\bm{r}}d^{\dagger}(\bm{r})\Bigg{[}\left(-\frac{\bm{\partial}^{2}}{2m_{\mathrm{SC}}}-\mu_{\mathrm{SC}}\right)\tau^{z}+\Delta\tau^{x}\Bigg{]}d(\bm{r})$
$\displaystyle H_{\mathrm{Lead}\text{-}\mathrm{SC}}$
$\displaystyle=t[\psi^{\dagger}(x=0)\tau^{z}d(\bm{r}=0)+h.c.]\,.$ (32)

Here, $\psi(x)$ and $d(\bm{r})$ are Nambu spinors respectively associated with
the lead and superconductor, i.e.,

$\displaystyle\begin{pmatrix}d_{1}(\bm{r})\\\ d_{2}(\bm{r})\end{pmatrix}$
$\displaystyle=\begin{pmatrix}c_{\uparrow}(\bm{r})\\\
c_{\downarrow}^{\dagger}(\bm{r})\end{pmatrix},$
$\displaystyle\begin{pmatrix}\psi_{1}(x)\\\ \psi_{2}(x)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\chi_{\uparrow}(x)\\\
\chi_{\downarrow}^{\dagger}(x)\end{pmatrix},$ (33)

with $c_{\alpha}(\bm{r})$ and $\chi_{\alpha}(x)$ the associated annihilation
operators for electrons with spin $\alpha$. Pauli matrices $\tau^{0,x,y,z}$
act on the Nambu indices. The second line of Eq. (C) incorporates the lead
kinetic energy with velocity $v_{\rm L}$ as well as a potential $U_{0}$ at the
interface with the superconductor. The third describes the $d$-dimensional
superconductor, which we assume has mass $m_{\rm SC}$, chemical potential
$\mu_{\rm SC}$, and $s$-wave pairing amplitude $\Delta$. The final line
incorporates spin-conserving tunneling between the lead at $x=0$ and the
superconductor at ${\bf r}=0$ with strength $t$.

Since the superconductor is fully gapped, we can integrate out its degrees of
freedom to obtain the effective lead-only Hamiltonian

$\displaystyle H_{\mathrm{ELF}}$
$\displaystyle=\int_{z}\psi^{\dagger}(x)\left(-iv_{\mathrm{L}}\partial_{x}\right)\psi(x)$
$\displaystyle\quad+\psi^{\dagger}(0)\left[(U_{0}+U_{\mathrm{SC}})\tau^{z}+W_{\mathrm{SC}}\tau^{x}\right]\psi(0).$
(34)

The new couplings introduced by the superconductor are given by

$\displaystyle U_{\mathrm{SC}}$
$\displaystyle=t^{2}\int_{\bm{k}}\frac{\xi(\bm{k})}{\xi^{2}(\bm{k})+\Delta^{2}},$
$\displaystyle W_{\mathrm{SC}}$
$\displaystyle=t^{2}\int_{\bm{k}}\frac{\Delta}{\xi^{2}(\bm{k})+\Delta^{2}},$
(35)

where $\xi(\bm{k})=\bm{k}^{2}/(2m_{\mathrm{SC}})-\mu_{\mathrm{SC}}$ is the
normal state dispersion of the superconductor. Note that here we ignore all
frequency dependence of these coefficients as well as any wavefunction
renormalization, an approximation that is justified by the low energies that
we focus on. Although we assumed $s$-wave pairing for convenience, a similar
effective lead-only Hamiltonian follows also with nodal pairing symmetry in
the BEC phase (where the superconductor is also gapped). The sole difference
is that we would then include angular dependence of the pairing amplitude in
Eq. (C) via $\Delta\to\Delta(\bm{k})$.

Next we solve $H_{\rm ELF}$ to quantify the zero-bias conductance predicted by
this formulation. A wavefunction $\varphi(x)$ of energy $E$ must satisfy

$\displaystyle E\varphi(x)$
$\displaystyle=-iv_{\mathrm{L}}\partial_{x}\varphi(x)+\delta(x)\left(U_{\mathrm{eff}}\tau^{z}+W_{\mathrm{SC}}\tau^{x}\right)\varphi(x),$
(36)

where we have followed the main text and defined $U_{\rm eff}=U_{0}+U_{\rm
SC}$. Away from $x=0$, the solutions are plane waves with momentum
$k=E/v_{\mathrm{L}}$. The boundary conditions at $x=0$ are

$\displaystyle\varphi(0^{+})-\varphi(0^{-})$
$\displaystyle=-\frac{i}{2v_{\mathrm{L}}}\left({U}_{\mathrm{eff}}\tau^{z}+{W}_{\mathrm{SC}}\tau^{x}\right)$
$\displaystyle\times\left[\varphi(0^{+})+\varphi(0^{-})\right],$ (37)

where $0^{\pm}$ indicates $0\pm\varepsilon$ in the limit $\varepsilon\to 0$,
$\varepsilon>0$ and on the right-hand side we have regularized $\varphi(0)$
symmetrically as $\left[\varphi(0^{+})+\varphi(0^{-})\right]/2$. The incoming
and outgoing states for our semi-infinite lead of interest accordingly take
the form

$\displaystyle\varphi_{\mathrm{in}}(x)$ $\displaystyle=\begin{pmatrix}a_{e}\\\
a_{h}\end{pmatrix}e^{ikx},\qquad x<0,$
$\displaystyle\varphi_{\mathrm{out}}(x)$
$\displaystyle=\begin{pmatrix}b_{e}\\\ b_{h}\end{pmatrix}e^{ikx},\qquad
x>0\,,$ (38)

with $a_{e/h}$ the incoming electron/hole amplitudes and $b_{e/h}$ the
outgoing electron/hole amplitudes. For our purposes it suffices to consider
incoming electrons only and set $a_{h}=0$. In that case, the boundary
conditions reduce to

$\displaystyle\begin{pmatrix}b_{e}-a_{e}\\\ b_{h}\end{pmatrix}$
$\displaystyle=-\frac{i}{2v_{\mathrm{L}}}\begin{pmatrix}{U}_{\mathrm{eff}}&{W}_{\mathrm{SC}}\\\
{W}_{\mathrm{SC}}&-{U}_{\mathrm{eff}}\end{pmatrix}\begin{pmatrix}b_{e}+a_{e}\\\
b_{h}\end{pmatrix}.$ (39)

Solving the system of equations returns

$\displaystyle\frac{b_{e}}{a_{e}}$
$\displaystyle=\frac{4v_{\mathrm{L}}(2v_{\mathrm{L}}-i{U}_{\mathrm{eff}})}{{U}_{\mathrm{eff}}^{2}+{W}_{\mathrm{SC}}^{2}+(2v_{\mathrm{L}})^{2}}-1,$
$\displaystyle\frac{b_{h}}{a_{e}}$
$\displaystyle=-\frac{4iv_{\mathrm{L}}{W}_{\mathrm{SC}}}{{U}_{\mathrm{eff}}^{2}+{W}_{\mathrm{SC}}^{2}+(2v_{\mathrm{L}})^{2}},$
(40)

yielding an Andreev reflection coefficient

$\displaystyle A^{\mathrm{ELF}}$
$\displaystyle=\left|\frac{b_{h}}{a_{e}}\right|^{2}=\frac{4(2v_{\mathrm{L}})^{2}{W}_{\mathrm{SC}}^{2}}{\left[{U}_{\mathrm{eff}}^{2}+{W}_{\mathrm{SC}}^{2}+(2v_{\mathrm{L}})^{2}\right]^{2}},$
(41)

which can be rewritten in the form of Eq. (8) in the main text. Note that Eq.
(41) is independent of energy within the sub-gap regime where the effective
lead-only Hamiltonian is applicable. Energy independence reflects the fact the
perturbations on the second line of Eq. (34) are marginal. Generically,
additional perturbations at $x=0$ involving derivatives of $\psi$ would be
present and would yield nontrivial energy dependence not captured in our
treatment. Such corrections drop out, however, at zero energy—which we
specialized to in the main text Eq. (8).

Returning to the specific form of the perturbations in Eq. (C), for $d=1$ the
integrals evaluate to

$\displaystyle{U}_{\mathrm{SC}}$
$\displaystyle=t^{2}\frac{\sqrt{m_{\mathrm{SC}}}}{2}\sqrt{\frac{-\mu_{\rm
SC}+\sqrt{\Delta^{2}+\mu_{\rm SC}^{2}}}{\Delta^{2}+\mu_{\rm SC}^{2}}}$ (42)
$\displaystyle{W}_{\mathrm{SC}}$
$\displaystyle=t^{2}\frac{\sqrt{m_{\mathrm{SC}}}}{2}\sqrt{\frac{\mu_{\rm
SC}+\sqrt{\Delta^{2}+\mu_{\rm SC}^{2}}}{\Delta^{2}+\mu_{\rm SC}^{2}}}.$ (43)

The duality between BCS and BEC superconductors is manifest here, as sending
$\mu_{\rm SC}\rightarrow-\mu_{\rm SC}$ swaps $U_{\rm SC}\leftrightarrow W_{\rm
SC}$. Moreover, the explicit expressions above yield a ratio
${U}_{\mathrm{SC}}/{W}_{\mathrm{SC}}=v_{\mathrm{R}}/v_{\mathrm{I}}$—in
agreement with the ELF/BTK correspondence provided in the main text below Eq.
(8). (Interestingly, these relations persist despite the fact that the ELF
model considers a slightly different geometry from BTK: the former tunnels
into the center of an infinite 1D superconductor, while the latter tunnels
into the end of a semi-infinite 1D superconductor.)

##  Appendix D Tunneling with multiple channels

Throughout the main text and the preceding supplemental material sections, we
worked within the simplifying approximation of a single ballistic one-
dimensional channel (or a sum of one-dimensional ballistic channels along the
azimuthal directions in the case of 1D to 2D BTK geometry). Here we generalize
our results to multiple scattering channels, showing that all conclusions of
the main text remain valid. Moreover, we demonstrate that the condition for
observing the optimal tunneling regime $v_{\rm L}\sim v_{\rm I}$—which
requires a lead with a correspondingly small chemical potential and/or a large
effective electron mass—becomes relaxed in the presence of multiple scattering
channels.

Figure S3: Normalized zero-bias conductance
$G_{\mathrm{W}}(E=0)/G_{\mathrm{W}}(E\gg\Delta)$ of an $s$-wave superconductor
as a function of lead width $W$ and tunneling parameter $Z$ for (a) BCS, (b)
crossover, and (c) BEC regimes. Here the lead chemical potential is fixed for
all three panels at $\mu_{\mathrm{L}}=3\Delta$ — i.e., in contrast to Figs. 4
and S2 it is not tuned to its optimal value. Figure S4: Dependence of the
multi-channel BTK zero-bias tunneling conductance on the barrier parameter $Z$
along the BCS-to-BEC crossover, for a one-dimensional $s$-wave superconductor
with $v_{{\rm L},0}\sqrt{m_{\rm SC}/\Delta}$ set to (a) 3, (b) 1.5, and (c)
0.5. A larger number of channels (Here we set
$W=40\sqrt{2}/m_{\mathrm{L}}v_{\mathrm{L,0}}$) enlarges the parameter space
where a non-zero signal is seen compared to Fig. 2, but overall suppresses the
maximum response as in Fig. S3. Black dashed lines indicate the condition
$Z=-v_{\rm R}/2v_{\rm L}$ necessary for perfect Andreev reflection. In panels
(d)-(f) we compute the difference between the panels of Fig. 2 (normalized by
the appropriate above-gap conductance) and panels (a)-(c). In the BCS limit
there is an overall reduction in the conductance, but in the BEC regime the
range of $(Z,\mu_{\mathrm{SC}})$ parameters over which non-zero signal is seen
is broadened.

To develop a model of multiple-channel scattering, we consider a quasi-1D lead
with width $W$ along $y$ and semi-infinite extent along $x$. Assuming an
isotropic 2D dispersion $\xi_{\bm{q}}=q^{2}/2m_{\rm L}$ for the lead, the band
structure of the quasi-1D problem exhibits sub-bands associated with the
quantized transverse momenta $q_{y}=n\pi/W$:

$\xi_{n}(q_{x})=\frac{q_{x}^{2}}{2m_{\rm L}}+\frac{n^{2}\pi^{2}}{2m_{\rm
L}W^{2}}.$ (44)

The sub-band index $n$ runs from 1 to $n_{c}$, where $n_{c}$ is the largest
integer that admits a momentum on the Fermi surface; that is, $n_{c}=\lfloor
q_{F}W/\pi\rfloor$.

The total conductance is then taken as a sum over the contributions from all
such sub-bands (or channels),

$G_{W}(E=0)=\frac{4e^{2}}{h}\sum_{n=1}^{n_{c}}A_{n}$ (45)

with (at zero bias)

$A_{n}=|a^{(n)}_{h}|^{2}=\frac{4v_{{\rm L},n}^{2}v_{\rm
I}^{2}}{\left[\left(2\lambda+v_{\rm R}\right)^{2}+v_{\rm I}^{2}+v_{{\rm
L},n}^{2}\right]^{2}}.$ (46)

Compared to Eq. (2) we have replaced $v_{\rm L}\rightarrow v_{{\rm L},n}$ to
describe the Fermi velocity along the $x$ direction (perpendicular to the
interface) of each sub-band $n$ of the lead; the sub-band dispersion relation
in Eq. (44) yields

$v_{{\rm L},n}=\frac{q^{x}_{\rm F,n}}{m_{\rm L}}=\sqrt{\frac{2}{m_{\rm
L}}\left(\mu_{\rm L}-\frac{n^{2}\pi^{2}}{W^{2}}\right)}.$ (47)

Additionally, we replaced the factor $v_{\mathrm{L}}Z$ with $\lambda$ to
explicitly highlight lack of dependence of the tunnel barrier on the sub-band
index $n$.

In Fig. S3a-c we show the zero-bias conductance $G_{W}(E=0)$ of a lead with
fixed chemical potential, in (a) the BCS regime, (b) at the crossover, and (c)
in the BEC regime. As the total conductance grows linearly with the number of
channels, we normalize $G_{W}(E=0)$ by the above-gap (i.e., normal-state)
conductance

$G_{W}(E\gg\Delta)=\frac{2e^{2}}{h}\sum_{n=1}^{n_{c}}\frac{1}{1+\tilde{Z}_{N,n}^{2}}\,,$
(48)

where $\tilde{Z}_{N,n}$ for normal-normal scattering is defined by Eq. (31),
and now depends explicitly on the sub-band index $n$ through $v_{\rm
L}\rightarrow v_{{\rm L},n}$.

We recover all qualitative conclusions of the single-channel problem examined
in the main text, including suppression of Andreev reflection for $Z=0$ in the
BEC regime and its revival for $Z<0$. The large number of channels manifests
in an enlarged parameter range ($Z$, $\mu_{\mathrm{SC}}$) over which revival
of Andreev reflection is seen, see Fig. S4 and the comparison with Fig. 2 for
the same parameter range.

###  D.1 Scattering in 3D setups

Finally, as a last generalization of our analysis we briefly comment on the
experimentally relevant case of a 3D material in a quasi-1D lead
geometry—e.g., a semi-infinite wire (or STM tip) along the $x$ direction with
_two_ quantized transverse directions along $y$ and $z$, with $q_{y}=n\pi/W$
and $q_{z}=m\pi/W$. We imagine tunneling into the surface of a 3D
superconductor.

Taking the “continuum” limit where the number of sub-bands is large ($W\gg
1/q_{F}$), we can transform the sum over sub-band indices $m,n$ into an
integral over the subset of the lead Fermi surface with positive $q_{x}$
components (a half-sphere of forward-propagating modes) using

$\displaystyle\sum_{m,n=1}^{n_{c}}$
$\displaystyle=\left(\frac{W}{\pi}\right)^{2}\int_{\rm FS}dq_{y}dq_{z}$ (49)
$\displaystyle=\frac{W^{2}q_{F}^{2}}{2\pi^{2}}\int_{0}^{\pi/2}d\theta\sin\left(2\theta\right)\int_{0}^{2\pi}d\varphi,$
(50)

with $\theta$ the angle measured from the $q_{x}$ axis and $\varphi$ the
azimuthal angle. The non-standard Jacobian $\frac{1}{2}q_{F}^{2}\sin(2\theta)$
arises from our hybrid coordinate system $(q_{x},\theta,\varphi)$.

To compute the conductance we substitute $v_{{\rm L},n}\rightarrow v_{\rm
L}\cos\theta$ in the expression for the Andreev reflection coefficient. For an
s-wave superconductor with an isotropic Fermi surface, $A(\theta)$ therefore
does not depend on the azimuthal angle. Taking into account the conservation
of momentum in the plane of the interface, the pairing gap $\Delta$ entering
the Andreev reflection coefficient can however become $\varphi$ dependent, as
in the case of nodal superconductors. We therefore have

$G(E)=\frac{4e^{2}}{h}\frac{W^{2}q_{F}^{2}}{2\pi^{2}}\int_{0}^{\pi/2}d\theta\sin\left(2\theta\right)\int_{0}^{2\pi}d\varphi
A(\theta,\varphi)\,.$ (51)

Even in the case of an isotropic Fermi surface in the superconductor, $A$ can
depend on $\varphi$ and $\theta$ via the possible angular and momentum
dependence of the pairing amplitude $\Delta$, respectively.

The $\varphi$ integral is analogous to the angular average used to compute the
Andreev reflection for two-dimensional nodal superconductors in Appendix A.
The presence of an additional dimension however manifests through the $\theta$
integral, which will modulate the numerical results shown in Figs. S1 and S2.
A more detailed analysis of the conductance in such setups will thus depend on
the details of the Fermi surfaces of both the lead and superconductor, as well
as on the precise geometry of junction [44].