Transport properties of an electron-hole bilayer/superconductor hybrid junction

TL;DR

This paper studies how electron-hole bilayer systems in contact with superconductors conduct electricity, revealing a conductance minimum at voltages related to the excitonic gap, indicating the presence of an exciton condensate.

Contribution

It introduces a model for transport in electron-hole bilayer/superconductor hybrids, highlighting the role of Andreev and crossed reflections and identifying conductance signatures of exciton condensation.

Findings

Differential conductance shows a minimum at voltages around the excitonic gap.

Transport is governed by Andreev and coherent crossed reflections.

The conductance minimum indicates the presence of an exciton condensate.

Abstract

We investigate the transport properties of a junction consisting of an electron-hole bilayer in contact with normal and superconducting leads. The electron-hole bilayer is considered as a semi-metal with two electronic bands. We assume that in the region between the contacts the system hosts an exciton condensate described by a BCS-like model with a gap $\Gamma$ in the quasiparticle density of states. We first discuss how the subgap electronic transport through the junction is mainly governed by the interplay between two kinds of reflection processes at the interfaces: The standard Andreev reflection at the interface between the superconductor and the exciton condensate, and a coherent crossed reflection at the semi-metal/exciton-condensate interface that converts electrons from one layer into the other. We show that the differential conductance of the junction shows a minimum at voltages of the order of $\Gamma/e$. Such a minimum can be seen as a direct hallmark of the existence of the gapped excitonic state.

Equations

$${\mathcal{H}}_{\text{Hyb}}=$$

$$\Gamma (x)$$

$$\Delta (x)$$

$${\xi }_{\Gamma }^{-1}=\sqrt{\frac{1}{2}\left({\chi }_Q+\sqrt{{\chi }_Q^2+\frac{4m^2}{{\hslash }^4}({\Gamma }^2-{\mathcal{E}}^2)}\right)}.$$

$${\xi }_{\Gamma }=\frac{4\hslash \sqrt{m{G}^3}}{\Gamma (4mG+Q^2{\hslash }^2)}.$$

$$R_{\text{NTT}}(\alpha )$$

$$R_{\text{NTB}}(\alpha )$$

$$R_{\text{ATT}}(\alpha )$$

$$R_{\text{ATB}}(\alpha )$$

$$R_{\text{NTB}}$$

$$R_{\text{ATT}}$$

$$R_{\text{NTT}}$$

$$\frac{\partial I}{\partial V}$$

$$\displaystyle\hskip 42.67912pt+R_{\text{AT}\beta}(eV,\alpha)\Big{]}\cos\alpha d\alpha\,.$$

$$\frac{dI}{dV}=G_0\left[1-8{\delta }^2\frac{(eV{)}^2{\Gamma }^2}{{\Delta }^2({\Gamma }^2-(eV{)}^2)}\right].$$

$${\psi }_{\text{SM}}$$

$$+r_{\text{ATT}}{\text{e}}^{\text{i}{\kappa }_{-}x}\left(\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right)+r_{\text{ATB}}{\text{e}}^{-\text{i}{\kappa }_{+}x}\left(\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right)\},$$

$${\kappa }_{\pm }=\sqrt{\frac{2m}{{\hslash }^2}(G\pm \mathcal{E})-Q^2}.$$

$${\psi }_{\text{SSM}}$$

$$+t_{\text{QHB}}{\text{e}}^{\text{i}{k}_{+}x}\left(\begin{array}{c}0\\ u_{\text{B}}\\ 0\\ v_{\text{B}}{\text{e}}^{\text{i}\varphi }\end{array}\right)+t_{\text{QEB}}{\text{e}}^{-\text{i}{k}_{-}x}\left(\begin{array}{c}0\\ v_{\text{B}}\\ 0\\ u_{\text{B}}{\text{e}}^{\text{i}\varphi }\end{array}\right)].$$

$$k_{\pm }=\sqrt{\frac{2m}{{\hslash }^2}(G\pm \sqrt{{\mathcal{E}}^2-{\Delta }^2})-Q^2}.$$

$$u_{\text{T/B}}$$

$$v_{\text{T/B}}$$

$${\psi }_{\text{EC}}(x)$$

$$+\left(a_2{\text{e}}^{\text{i}{q}_{+}x}+c_2{\text{e}}^{-\text{i}{q}_{+}x}\right)\left(\begin{array}{c}\\ 0\\ 0\\ v_h\\ u_h{\text{e}}^{-\text{i}\chi }\end{array}\right)+\left(b_2{\text{e}}^{-\text{i}{q}_{-}x}+d_2{\text{e}}^{\text{i}{q}_{-}x}\right)\left(\begin{array}{c}\\ 0\\ 0\\ u_h\\ v_h{\text{e}}^{-\text{i}\chi }\end{array}\right)\}.$$

$$q_{\pm }=\sqrt{\frac{2m}{{\hslash }^2}(G\pm \sqrt{{\mathcal{E}}^2-{\Gamma }^2})-Q^2}.$$

$$u_{e/h}$$

$$v_{e/h}$$

$${\psi }_{\text{SM}}(0^{-})$$

$${\psi }_{\text{EC}}(L^{-})$$

$${\psi }_{\text{EC}}^{\prime }(0^{+})-{\psi }_{\text{SM}}^{\prime }(0^{-})=$$

$${\psi }_{\text{SSM}}^{\prime }(L^{+})-{\psi }_{\text{EC}}^{\prime }(L^{-})=$$

$$S_z={\mathbb{I}}_2\otimes {\sigma }_z=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right).$$

$$k_{\beta ,x}$$

$$k_{\beta ,y}$$

$${\kappa }_{\text{TL}}$$

$${\kappa }_{\text{BL}}$$

$${\alpha }_{\frac{\text{ATT}}{\text{NTB}}}=\arcsin \left[\frac{\hslash \mathcal{Q}}{\sqrt{2m(G-\mathcal{E})}}\right],$$

$${\alpha }_{\text{c}}=\arcsin \left[\sqrt{\frac{|G-\mathcal{E}|}{G+\mathcal{E}}}\right].$$

Full text

Introduction.— Semimetals (SMs) could undergo at sufficiently low temperatures, a phase transition into an insulating state described by electron-hole bound pairs. These pairs form a so-called exciton condensate (EC), as theoretically predicted a long time ago,  Keldysh and Kopaev (1965); Jérome et al. (1967); Combescot and Shiau (2016) and one refers to the system being in an excitonic insulating phase. The ground state of such a phase can be described with the help of a BCS-like theory, in analogy to the superconducting phase. However, the coupling strength in an excitonic insulator is expected to be even weaker than in a superconductor (S). Furthermore, electron-hole recombination can be quite fast, thereby preventing the formation of the condensate. For these reasons the EC remains an elusive phase of matter Keldysh and Kopaev (1965); Jérome et al. (1967); Halperin and Rice (1968). Possible SM candidates suggested to undergo a transition to the excitonic insulating phase with an EC are transition-metal dichalcogenide TiSe2 Monney et al. (2010); Rossnagel (2011) and HgTe QWs with a thickness of 20 nm Minkov et al. (2013); Knap et al. (2014); Kononov et al. (2016). However, there is no conclusive evidence for an EC in such systems. So far the most successful attempt to obtain an EC is based on the condensation of excitons coupled to light confined within CdTe/CdMgTe micro-cavities — the so-called exciton-polaritons Kasprzak et al. (2006).

Besides bulk semi-metals, there have been several proposals to create an EC in systems with spatially separated electron and hole gases in order to reduce the electron-hole recombination rate. The exciton formation in such electron-hole bilayers can be detected by Coulomb drag measurements Croxall et al. (2008); Seamons et al. (2009); Li et al. (2016); Lee et al. (2016); Fogler et al. (2014); Calman et al. (2016). According to the theory, if the excitons form a condensate one expects a discontinuity in the drag at the critical temperature and a divergence when $T\rightarrow 0$ Vignale and Macdonald (1996). Although certain anomalies of the Coulomb drag as a function of temperature have been observed Croxall et al. (2008); Seamons et al. (2009), it is hard to attribute them to the formation of an exciton condensate.

The primary motivation of this Letter is to propose and explore an additional type of measurements to validate the existence of an EC in electron-hole bilayers. Instead of measuring the Coulomb drag, we suggest to perform a differential conductance measurement using normal and superconducting electrodes (see Fig. 1), so to directly unveil the presence of the excitonic gap. We show that the transport at low voltages is determined by the competition between the intra-layer Andreev reflection and the inter-layer normal reflection between the SM and the EC parts of the system. The latter process is analogous to the one introduced by Rontani and Sham for direct SM/EC interfaces Rontani and Sham (2005a, b); Wang et al. (2005). We discuss the competition between these two type of reflections as a function of the length of the EC region and show that this competition leads to a minimum of the full differential conductance at a voltage $V_{0}$ of the order of the EC order parameter.

Model and Formalism.— We consider an ideal two-dimensional electron-hole bilayer, characterized by two parallel two-dimensional electron gases with opposite particle filling, as illustrated in Fig. 1(a) — the top layer (TL) is “electron-doped” whereas the bottom layer (BL) is “hole-doped”. By gating the two layers independently it is possible to modulate the band overlap $G$ between the bands of the TL/BL Zhu et al. (1995). We assume a spatial modulation along the $x$-axes of the Coulomb interaction in the electron-hole bilayer. In the parts of the bilayer covered by the metallic electrodes, the charge screening allows neglecting the Coulomb interaction. Thus the left and right parts of the system, are described by a SM consisting of two bands — one from each layer — crossing at the Fermi level [see Fig. 1(a)]. Due to the proximity effect, we assume an induced superconducting gap $\Delta$ on the part of the layers below the S contacts [see Fig. 1(c)]. In the region between the contacts the reduced screening results in an indirect EC described by the order parameter $\Gamma$ Zhu et al. (1995) [see Fig. 1(b)].

To be precise, the structure shown in Fig. 1(a) is modelled as a SM, i.e., $\Gamma=\Delta=0$, in contact with a central region of length $L$ with a finite EC coupling ($\Gamma\neq 0$ and $\Delta=0$). In the region $x>L$ we assume $\Delta\neq 0$ due to the proximity from the S electrodes Beenakker (2006) and $\Gamma=0$. We denote this region as SSM. This junction is described by the following Hamiltonian written in an extended Nambu space Dolcini et al. (2010):

[TABLE]

where

[TABLE]

and $\Theta(x)$ is the Heaviside step function. The Hamiltonian (1) is written in the basis defined by the bi-spinor $\Psi=~{}(\psi_{\text{TL},\sigma},\psi_{\text{BL},\sigma},\psi^{\dagger}_{\text{TL},-\sigma},\psi^{\dagger}_{\text{BL},-\sigma})$, where $\sigma$ is the spin Dolcini et al. (2010); Peotta et al. (2011). We account for possible elastic reflection at the SM/EC and EC/SSM interfaces by introducing delta-barriers in the system Hamiltonian Blonder et al. (1982), $\mathcal{H}_{\text{int}}=H_{\text{SM/EC}}\delta(x)+H_{\text{EC/SSM}}\delta(x-L)$. This reflection can be ascribed, for example, to the mismatch of the Fermi wave vector in the different regions Imp .

We analyse the scattering properties of this hybrid SM/EC/SSM junction by matching the scattering states at each interface separating these three regions SuM (a). In the SM region, there is no coupling between the TL and BL nor between electron/hole of the same layer. Thus, we can consider an electron (hole) in the TL (BL) as an initial scattering state. In the middle, EC, region, there is a finite coupling between electrons of TL and BL proportional to $\Gamma$. And in the SSM region the superconducting pairing $\Delta$ couples electrons and holes within the same layer.

We summarize all possible processes that an incoming electron from the TL of the SM at energies smaller than $\Gamma$ and $\Delta$, may experience. If we look at the SM/EC interface the electron can be either normal reflected in the same layer with amplitude — r_{\text{N,T\toT}}\equiv r_{\text{NTT}}, or in the opposite layer — r_{\text{N,T\toB}}\equiv r_{\text{NTB}}. The latter process resembles the Andreev reflection at the S/N interface and was studied by Rontani and Sham in Ref. Rontani and Sham (2005b, a); Wang et al. (2005). It results in an exciton in the EC region. Whereas at the EC/SSM interface the electron can be Andreev-reflected into a hole either in the same layer — r_{\text{A,T\toT}}\equiv r_{\text{ATT}}, or into the other layer — r_{\text{A,T\toB}}\equiv r_{\text{ATB}}. It is interesting to notice that the reflection amplitude $r_{\text{ATB}}$ is the analog to the Andreev specular-reflection in Dirac-material/superconductor hybrid junctions Beenakker (2006).

For energies higher than the EC gap, $\mathcal{E}>|\Gamma|$, the particles travel through the EC region as propagating waves. In contrast, for $\mathcal{E}<|\Gamma|$, the quasi-bound states are characterized by complex momenta describing evanescent modes. The characteristic decay length of these modes $\xi_{\Gamma}$, is energy dependent and given by:

[TABLE]

where $\chi_{Q}=Q^{2}-\frac{2mG}{\hbar^{2}}$, here $Q$ is the momentum parallel to the interfaces that is conserved in the scattering process SuM (a). When the band overlap $G$ is the dominant energy scale, i.e., $G\gg\max[\Gamma,\Delta,\mathcal{E}]$, the EC characteristic length scale is approximated by

[TABLE]

In what follows, we focus on the subgap transport, i.e. we consider the injection of electrons from the TL with energies smaller than the SSM gap, $\mathcal{E}<|\Delta|$. Furthermore, we also assume that $\Gamma<\Delta<G$. The probability for the four possible reflection channels in the SM electrode are then given by SuM (a):

[TABLE]

where $\alpha$ is the injection angle. For angles larger than a critical $\alpha_{\text{c}}=\arcsin[\pm\sqrt{{\mid G-\mathcal{E}\mid}/{\mathcal{E}+G}}]$, $R_{\text{NTB}}=R_{\text{ATT}}=0$ because of the lack of propagating states on the TL/BL. The actual form of the reflection and transmissions amplitudes are obtained by solving the scattering problem at the two interfaces SuM (a).

Results.— We now analyze the dependence of the probabilities (5) on the length of the EC region. In the limiting case $L=0$ the system consists of a simple SM/SSM junction. In this case, the absence of the EC leads to $r_{\text{NTB}}=0$. Moreover, $r_{\text{ATB}}$ also vanishes and the results for a clean interface coincide with the standard N/S junction case Blonder et al. (1982); SuM (a). Here we stress the analogy between the interlayer Andreev reflection and the specular Andreev reflection typical of SM with Dirac spectrum again Beenakker (2006); Efetov et al. (2015); Chen et al. (2013). If there is no EC then electrons from the TL and BL are decoupled, and hence all reflections occur within the same layer only.

We now focus on the more interesting case of a finite EC region, $L\neq 0$. In Fig. 2 we present the reflections $R_{\text{NTB}}$ and $R_{\text{ATT}}$ as a function of the injection energy and length of the EC region for a normal injection angle ($\alpha=0$). For a length smaller than $\xi_{\Gamma}$, the probability for an injected electron in the TL to reach the EC/SSM interface is large. The electron is then Andreev reflected as a hole of the same layer. This explains that for small $L$, the Andreev reflection, $R_{\text{ATT}}$, dominates over the interlayer one, $R_{\text{NTB}}$, at all energies.

By increasing the length of the EC region, the probability to reach the EC/SSM interface, for an injected electron with energy $\mathcal{E}<\Gamma$ decreases and hence the $R_{\text{ATT}}$ processes are suppressed, whereas the $R_{\text{NTB}}$ ones are enhanced. The crossover between these two behaviors depends on the injection energy. In the limiting case $L\gg\xi_{\Gamma}$ the probability for the injected TL electron of reaching the SSM electrode is exponentially small and the system behaves as a SM/EC junction with the reflection probability $R_{\text{NTB}}=1$ Rontani and Sham (2005a, b). For injection energies larger than the EC gap $|\Gamma|$ (but smaller than $\Delta$), all electrons propagates towards the EC/SSM interface and hence Andreev processes dominate, whereas $R_{\text{NTB}}$ decreases to zero by increasing the injection energy. The oscillatory behavior we observe for this energy range is a feature of quasi-bound states in the EC region.

Analytically we can calculate the reflection probabilities (5b) and (5c) in the short-junction limit, $\delta=L/\xi_{\Gamma}\ll 1$ within the Andreev approximation, i.e. $G\gg\text{Max}[\mathcal{E},\Delta,\Gamma]$. In this limits and for zero injection angle $\alpha=0$, we obtain

[TABLE]

These expressions are in agreement with the numerical results shown in Fig. 2 within the small $L$ region. Notice that for the large value of $G$ chosen, $G=100\Gamma$, the probability for the reflection $R_{\text{ATB}}$ is negligibly small. However, if one choose a smaller $G$ and a larger injection angle $\alpha$, $R_{\text{ATB}}$ is finite SuM (a). In this latter case both types of Andreev reflections, $R_{\text{ATB}}$ and $R_{\text{ATT}}$, may take place simultaneously. This differs from the case of retro- and specular-Andreev reflections in SM with Dirac spectrum, where one or the other is finite by crossing the charge neutrality point Beenakker (2006); Efetov et al. (2015); Chen et al. (2013).

To make a connection with possible transport experiments, we now turn the focus upon the differential conductance (DC), which is a quantity accessible in conventional transport experiments. For this purpose, we assume that the two left normal contacts are placed at the same potential $V$ and that the two right superconducting contacts are grounded. The DC can be expressed in terms of the reflection probabilities (5), by using a generalized expression based on the Blonder-Tinkham-Klapwijk formula Blonder et al. (1982); Mortensen et al. (1999),

[TABLE]

Here $G_{0}(eV)$ is the differential conductance of the normal state.

Figure 3(a) shows the DC as a function of the injection energy, which is proportional to the applied voltage, and for different values of the length $L$. Notice first that the DC is not equal to $2G_{0}$ for $L=0$. This is a consequence of considering a finite chemical potential Beenakker (2006). For a finite length EC region the DC decreases faster as a function of the voltage, it reaches a minimum and then increases up to a voltage of the order of $\Delta$. It is also worth to notice that the DC of the normal state in the presence of a finite EC is also smaller compared to the case without it.

The decrease of the DC is due to a normal reflection channel that accounts for electrons injected from the TL and reflected into the BL. Thus, the presence of a finite length EC region entirely accounts for the minimum in the DC. If the length $L$ exceeds the decay length $\xi_{\Gamma}$, injected electrons with $E<\Gamma$ rarely reach the superconducting electrode (c.f. Fig. 2) and therefore the DC remains small up to voltages of the order $|\Gamma|$ Rontani and Sham (2005a).

We can obtain an analytical expression for the low bias behavior of the DC in the short-junction limit, using Eqs. (6), and within the Andreev approximation:

[TABLE]

According to this expression the low voltage peak in the DC is of the order of $\Gamma$. This suggests that transport measurement using superconducting electrodes could be used to directly estimate the size of the order parameter $\Gamma$.

Our results for the differential conductance may help to understand measurements of the differential resistance of a superconductor/HgTe-quantum well junction. The quantum well has a width of 20 nm Kononov et al. (2016). One of the most striking findings in this experiment is a zero-bias peak in the measured differential resistance Kononov et al. (2016) that can be seen as the manifestation of an EC gap. Indeed, in the light of our model the size of the zero-bias peak corresponds to the EC gap. This comparison is shown in Fig. 3(b) where we plot the differential resistance by choosing parameters consistent with the experiments on HgTe quantum wells Kvon et al. (2011); Kononov et al. (2016). Although the agreement between theory and experiment looks promising we would like to be cautious at this point because our approach assumes a full ballistic system, whereas the samples measured in Ref. Kononov et al. (2016) have a size of several $\mu$m, and disorder might play a role SuM (b). Further experiments and analysis are needed to draw definite conclusions.

Conclusions.— We have studied the electronic transport through an electron-hole bilayer in contact with normal and superconducting electrodes. We have assumed that the electron-hole bilayer hosts an exciton condensate. The transport properties of this junctions are determined by the competition of different coherent reflection processes occurring at the interfaces with the normal and superconducting electrodes. As a consequence of this competition, the differential conductance has a minimum at voltages of the order $\Gamma/e$, where $\Gamma$ is the EC order parameter. The observation of this minimum in an electron-hole bilayer system represents a unique hallmark of the presence of the EC. Good candidates for our proposal are double bilayer-graphene systems separated by hexagonal boron nitrate Li et al. (2016); Lee et al. (2016). In bilayer-graphene, superconductivity induced by proximity effect has been already observed Efetov et al. (2015). Moreover, from our model one might interpret the zero bias-peak observed in the differential resistance of a HgTe quantum well/superconductor junction as a manifestation of an EC phase.

Acknowledgments.— Discussions with B. Bujnowski, E. Deviatov, F. Konschelle, V. Golovach, T. T. Heikkilä, P. Lucignano, M. Rontani and T. van den Berg are acknowledged. The work of DB and FSB is supported by Spanish Ministerio de Economía y Competitividad (MINECO) through the project FIS2014-55987-P and by the Transnational Common Laboratory QuantumChemPhys. TMK acknowledges the financial support from the European Research Council Advanced grant no. 339306 (METIQUM), and the Ministry of Education and Science of the Russian Federation, contract N 14.B25.31.0007 of 26 June 2013.

Appendix A The scattering states

In this section we briefly introduce the scattering states in the three regions of the electron-hole bilayer system we have considered in the main text, this is constituted by a SM region on the left, an EC region of length L in the center and a superconducting region on the right. The scattering states in the SM region read:

[TABLE]

here $Q$ is the component of the momentum along the $y$ direction. Being that we consider a translational invariant interface between the three regions, the component of the momentum parallel to this interface is a conserved quantity in the overall scattering process. At a fixed injection energy $\mathcal{E}$, the momenta $\kappa_{\pm}$ are expressed by the following relations [c.f. Fig. 1(b) main text]:

[TABLE]

In the SSM region, the superconducting pairing couples electrons and holes of the same layer, but there is no direct coupling between electrons of the two layers. The incoming electron from the SM region can be transmitted as quasi-electron or quasi-hole either in the top later (TL) ($t_{\text{QET}}$ and $t_{\text{QHT}}$) or bottom layer (BL) ($t_{\text{QEB}}$ and $t_{\text{QHB}}$). The transmitted wave function reads:

[TABLE]

The momenta in the SSM [c.f. Fig. 1(d) main text] region are

[TABLE]

The coherence factor are the $u_{\text{T/B}}$ and $v_{\text{T/B}}$ are the standard one of a $s$-wave superconductor, here T/B is for TL and BL, respectively. Note that there is change of sign between two layers, this is due to their opposite curvature of the energy dispersion:

[TABLE]

In the middle EC region, there is a finite Coulomb coupling in the electron-hole bilayer. Here the wave function is given by counter-propagating states for the four possible states: quasi-electron/hole in TL/BL:

[TABLE]

The momenta in EC region [c.f. Fig. 1(c) main text] are defined as:

[TABLE]

For this case the the coherent function are very similar to the superconducting one, they are here given by: Rontani and Sham (2005a)

[TABLE]

The density plot of the four reflection channels are presented in S-Fig. 4 for the case of large and small band overlap $G$. It can be observed that for small injection angle $\varphi$ the $R_{\text{NTB}}$ and the $R_{\text{ATT}}$ reflection are the dominant processes, on the other side at large angles the $R_{\text{NTT}}$ and $R_{\text{ATB}}$ reflection are finite, but smaller than $R_{\text{NTB}}$ and $R_{\text{ATT}}$.

Appendix B The boundary conditions

The scattering amplitudes of the various scattering processes can be obtained by imposing the continuity of the wave function in $x=0$ and $x=L$:

[TABLE]

the continuity of the first derivative in the same points. When considering a possible mismatch of the Fermi velocity at the interface between the various region, we can model this via delta-barriers of strength $H_{\text{SM/EC}}$ in $x=0$ and $H_{\text{EC/SSM}}$ in $x=L$. Blonder et al. (1982) In this case the first derivative has a jump that is opposite for the TL and BL Rontani and Sham (2005a):

[TABLE]

where we have introduced the boundary matrix $S_{z}$ defined as:

[TABLE]

Appendix C The critical angle $\alpha_{\text{c}}$

Using simple kinematic consideration we can determine the expression for the critical angle $\alpha_{\text{c}}$ Eq. (4) of the main text. We need only to account for the conservation of the momentum parallel to the interface $k_{y}=\mathcal{Q}$. For each of the layer involved in the transport mechanism, momenta in polar coordinate read (see S-Fig. 5):

[TABLE]

The normal reflection inside the same layer has a propagation direction opposite to the incoming one, the same is true for electrons that are converted into electrons of the BL. For electrons injected from the TL and converted into holes of the TL or converted into holes of the BL, the reflection angle is given by:

[TABLE]

for injection energies $\mathcal{E}$ exceeding the band overlap $G$, the angle is becoming complex and the corresponding mode becomes evanescent. By imposing $2m/\hbar^{2}(G-\mathcal{E})-\mathcal{Q}^{2}\leq 0$ we can determine the critical injection angle $\alpha_{\text{c}}$:

[TABLE]

Appendix D Case of a EC region of zero length

Here we will presents the results in the case the length of the EC region is zero. This corresponds to a lack of a region where carriers of the electron-hole bilayer are coupled. Thus, the hybrid junction reduced to a SM place in contact with a proximized SM. This corresponds to solve the matching problem with wave function (A) and (A). In the case of the scattering state (A), we are considering an initial incoming electron in the TL. Using this initial incoming state, the amplitudes for reflection process read:

[TABLE]

whereas for transmissions read:

[TABLE]

here $t_{\text{QET/QHT}}$ is the transmission as the quasi-electron/quasi-hole in the TL, whereas $t_{\text{QEB/QHB}}$ is the transmission as the quasi-electron/quasi-hole in the BL. For both sets of reflection and transmission we have introduced the factor:

[TABLE]

Here $Z=2mH_{\text{int}}/\hbar^{2}$ is the dimensionless strength of the delta barrier at the SM/S interface. Importantly, all the interlayer crossing processes have zero probability amplitudes. In the limit of $Z\to 0$ we recover the standard results for S/N junctions. Blonder et al. (1982) Starting with a different initial scattering state, e.g., an hole in the BL, we get expressions similar to Eqs. (23)-(24) but still without a direct process between TL and BL.

Appendix E Analytical results

In this appendix we show how to get the analytical results (6a)-(6b) of the main text. Both results are obtained assuming that the band overlap is the dominant energy scale $G\gg\max[\mathcal{E},\Gamma,\Delta]$ — as the standard Andreev-like approximation in superconducting junctions. Blonder et al. (1982) We further assume the superconducting gap is larger that the excitonic one $\Gamma<\Delta$ and injection energy is smaller than superconducting gap : $\mathcal{E}<\Delta$. For zero incidence angle, in these limit we can strongly approximate the values of the momenta in the three regions:

[TABLE]

For this choice of the energy scale, the coherent function of the superconductor lead $u_{\text{T/B}}$ and $v_{\text{T/B}}$ are complex conjugates, thus we need to carry information of only one of them $v_{\text{T/B}}=u_{\text{T/B}}^{*}$ where only $u_{\text{T}}$ is relevant. Whereas in the EC region we need to keep track of all of them but we can simplify the notation into $u_{e}=u_{h}=u$ and $v_{e}=v_{h}=v$. We find the coefficients for the components of EC wave function (A) by solving the boundary problem as first at $x=0$ and at $x=L$, we than impose the equality of the coefficients obtained by at the two different boundaries. By combining all of these we obtain the following sets of equations for the reflection and transmission amplitudes:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We can solve these eight equations with respect to the 4 reflections and transmissions amplitudes we obtain:

[TABLE]

where in Eqs. (26a) and (26b) we have introduced $a=\sqrt{\Gamma^{2}-\mathcal{E}^{2}}/\mathcal{E}$ and $b=\sqrt{\Delta^{2}-\mathcal{E}^{2}}/\mathcal{E}$, the constant $\delta=L\xi_{\Gamma}$ has been introduced in the main text. The other two reflection amplitudes $r_{\text{NTT}}$ and $r_{\text{ATB}}$ are smaller than the two in Eqs. (26) by an order $1/\sqrt{2G}$ and thus negligible, further, for energies smaller than the superconducting gap $\Delta$ the transmission probabilities are zero.

If we consider the limit of a long junction $\delta\gg 1$ we find that $r_{\text{ATT}}\to 0$ and $r_{\text{NTB}}\to 1$, exactly as we have sound for the full numerical problem in the main text (c.f. Fig. 2 main text).

Appendix F Case of HgTe quantum wells

The experiment by Kononov et al., Kononov et al. (2016) explored the transport properties of a hybrid junction consisting of a HgTe quantum well (QW) with a width of circa 20 nm. From previous studies, the authors know that the system is a indirect SM, with a small bands overlap. Kvon et al. (2011) This makes HgTe QWs candidates for observing a direct EC phase. Specifically in the experiment of Ref. [Kononov et al., 2016], the QW is sandwiched between a normal and a superconductor contact. The transport measurements showed a zero bias anomaly in the differential resistance with an energy size comparable with the estimated EC gap of the system. Kononov et al. (2016)

We can use the results presented in the main text in order to explain the zero bias anomaly of the hybrid SM/EC/S junction. Assuming that the momentum displacement between the conduction and the valence band is small enough to be negligible, we can evaluate the differential resistance so as explained in the main text. The actual values of the system parameters can be extracted by Refs. [Kvon et al., 2011; Kononov et al., 2016] and have: $G=3.0$ meV, $\Gamma=0.05$ meV and $\Delta=0.6$ meV, which should correspond to a (Nb/FeNb)/HgTe structure explored in Ref. [Kononov et al., 2016], we have further set a finite transparency of the two interface barriers $H_{\text{L}}=H_{\text{R}}=0.5$. Kononov et al. (2016) For these parameters value we have an EC coherence length of $\xi_{\Gamma}=955$ nm.

The results for the differential resistance are shown in S-Fig. 6. The black solid line describes a SM/SSM junction ($L=0$), and coincides with the well-known results of the BTK theory after integration over injection angle (main text). Mortensen et al. (1999) It is important to note that $\Delta/G=0.50$ and hence the differential resistance at zero voltage is not exactly equal to $0.5G_{0}^{-1}$, which would be the result obtain in the leading order when $\Delta/G\ll 1$. By increasing the length of the EC region we obtain peak in the differential resistance at energies smaller than $\Delta$. This result resemble the experimental observations of Ref. [Kononov et al., 2016] that has been attributed as a manifestation of the EC phase.

The experiment of Ref. [Kononov et al., 2016] also shown features in the superconducting sub-gap conductance that resemble the multiple-Andreev reflections (MARS) processes in voltage biased Josephson junctions. Averin and Bardas (1995). In principle such sub-gap features are unexpected, since in the experiment by Kononov et al. Kononov et al. (2016) there is only one superconducting electrode. Our model is also done for an unique S-electrode and in a quasi-equilibrium situation and therefore cannot describe MARS-like processes.