Superconductivity with excitons and polaritons
Abstract
A Bose–Einstein condensate of exciton polaritons coexisting with a Fermi gas of electrons has been recently proposed as a promising system for realisation of roomtemperature superconductivity [Phys. Rev. Lett., 104, 106402 (2010)]. In order to find the optimum conditions for exciton and excitonpolariton mediated superconductivity, we study the attractive mechanism between electrons of a Cooper pair mediated by the exciton and excitonpolariton condensate and analyze the gap equation that follows. We specifically address microcavities with embedded doped quantum wells as well as coupled quantum wells hosting a condensate of spatially indirect excitons, put in contact with a twodimensional electron gas. We show that engineering of the interaction in these peculiar BoseFermi mixtures is complex and sometimes counterintuitive, but leaves much freedom for optimization, making promising the realization of hightemperature superconductivity in multilayer semiconductor structures.
pacs:
71.35.Gg, 71.36.+c, 71.55.Eq, 74.78.w, 74.90.+nI Introduction
The electron gas undergoes, in some conditions, a phase transition to bound pairs of electrons (the socalled Cooper pairs), that replace electrons as the fundamental agent of the electronic properties. The Cooper pairs are, from the point of view of their electric charge, objects qualitatively identical to the underlying electrons. From the point of view of their spin, on the other hand, they become integerspin particles, that is, from the spinstatistics theorem, bosons rather than fermions. This shift of statistical paradigm of the carriers, from Fermi to Bosestatistics, results in the outstanding behaviour of superconductivity, that is, conduction of electric charge by a macroscopic coherent wavefunction (akin to a BoseEinstein condensate). It has taken some time to capture the fundamental and universal features of this phenomenon and set them apart from particularities proper to certain cases only. The gap of excitations, responsible for zero resistivity, for instance, results from the longrange nature of the Coulomb interaction, but gapless superconductivity is also possible. One of the central, fundamental concepts of superconductivity is that of a coherent quantum state of charged bosons. Although superconductivity was discovered empirically, and its theoretical construction consisted in assembling a puzzle, it is now possible to envisage engineering superconducting phases in other systems, based on this understanding of condensation of charged bosons. If superconducting phases can be identified in other systems, progresses will be quick for the understanding of cuprate superconductivity, which still eludes compelling theoretical explanation of its intrinsic mechanism.
A system that is making rapid and impressive progress in terms of creating and controlling macroscopic quantum states is that of microcavity excitonpolaritons Kavokin et al. (2007a) (see Deng et al. (2010) for a review). These quasiparticles that combine properties of light (cavity photons) and matter (quantum well excitons) have been noted for their predisposition to accumulate in macroscopic number in a single or few quantum states Ĭmamoḡlu et al. (1996); Baumberg et al. (2000). They have many advantages from a practical point of view, such as their 2D geometry, which allows straightforward manipulation by lasers impinging at an angle, and their short lifetime, which allows continuous monitoring of the system, reconstructing its internal dynamics also by angleresolved spectroscopy Skolnick et al. (1998). The pumping can be either coherent (driving states in parametric scattering configurations) Savvidis et al. (2000); Stevenson et al. (2000); Ciuti et al. (2001); Ciuti and Carusotto (2005); Shelykh et al. (2006) or incoherent (with a constant flow of unrelated particles relaxing into the ground state ) Kavokin et al. (2003); Laussy et al. (2004); Szymanska et al. (2006); Kasprzak et al. (2006); Balili et al. (2007); del Valle et al. (2009). In nitride systems, the formidable claim has been made of room temperature BoseEinstein condensation Christopoulos et al. (2007); Baumberg et al. (2008). Recently, there has been great interest in propagation of polariton fluids Amo et al. (2009a); Sanvitto et al. (2010a) and their superfluid properties Amo et al. (2009b), with reports of quantized vorticity Lagoudakis et al. (2008, 2009) and persistent currents Sanvitto et al. (2010b).
These rising stars of macrosopic coherence have also been proposed to service another much sought after quantum phase at high temperature: superconductivity. Polariton condensates cannot conduct electric current themselves, being neutral particles. One of the proposed implementations involves “quatrons” (or quadrions) rather than polaritons Kavokin et al. (2007b). Quatrons are bound states of two electrons with a polariton. They remain bosons from spinaddition rules but carry an electric charge. A Bose condensate of quatrons would, through its superfluid propagation, exhibit superconductivity. To date, however, the existence of the quatron, predicted theoretically Kavokin et al. (2007b), has not been found experimentally. Recently, we have approached the problem from another, and more conventional, angle, that of the BCS mechanism Laussy et al. (2010) with an important new feature: replacement of phonons by Bosecondensed excitonpolaritons in the role of a binding agent between electrons. In the present paper, we generalise the model proposed in Laussy et al. (2010) to describe a wide range of hybrid BoseFermi semiconductor systems where a BoseEinstein condensate of neutral quasiparticles (excitons or excitonpolaritons) coexists with a Fermi sea of electrons. We show that indeed such systems are promising for observation of (hightemperature) superconductivity. Moreover, as the superconducting gap and critical temperature appear to be very sensitive to the concentration of bosons in the system and the latter may be controlled by direct optical excitation of exciton, light induced superconductivity in semiconductor heterostructures appears to be possible. In this work we closely follow the BCS approach, generalised and adapted to the case of superconductivity mediated by a BoseEinstein condensate. Solving the gap equation in this case turns out to be a nontrivial problem, requiring careful analysis.
i.1 BCS with a BoseEinstein condensate as a binding agent
BCS is a pillar of superconductivity theory, which relies on three main tenets:

Instability of the Fermi sea,

Existence of an attractive interaction,

Condensation of charged bosons.
These are the three insights that were mainly contributed by Cooper, Bardeen and Schrieffer, respectively, and that they could assemble into the BCS theoretical edifice and exploit to reproduce strikingly or predict successfully most of the superconductivity phenomenology.
The first point follows from Cooper’s observation Cooper (1956) that an arbitrary small attractive interaction between two electrons on top of the Fermi sea leads to a bound state (the Cooper pair), thanks to the truncation of the momentum space for states with wavevector (Fermi wavevector). This is a general result, that follows from BetheGoldstone equation for the twoelectron problem.
The second point is the identification of an effective attraction between electrons that normally experience bare Coulomb repulsion. This attraction is attributed for conventional superconductors to an interaction through phonons, the BardeenPines potential Bardeen and Pines (1955), that consists, vividly, of one electron wobbling the lattice at a first time, which affects another electron at a second time (and at a larger timescale, since the lattice dynamics is much slower than that of electrons) Bardeen (1951). If the frequency of the lattice vibration is smaller than that of the propagating electron, the net effect results in an effective attraction [Leggett offers an insightful toy model of coupled oscillators to capture the essence of the interaction character (attractive or repulsive) Leggett (2006a)].
The last point is the socalled BCS state, which is a coherent superposition of paired bound states and which brings the twoparticle Cooper effect to a collective behaviour of all electrons in the system Bardeen et al. (1957a).
With these three ingredients put together, the BCS theory is complete Bardeen et al. (1957b). For our purposes, points 1 and 3 will be regarded as fundamental and well established features of (BCS type of) superconductivity. Point 2, that might appear a mere desiderata for point 1 to apply, leaves us room for identifying and designing new types of attractive potentials, optimizing the range of applicability and strength so as to obtain robust superconductivity (e.g., holding at high temperatures or high magnetic fields) or its manifestation in a new class of systems (in microcavities). Bardeen himself, with coworkers Allender et al. (1973), investigated possibilities to engineer a more robust BCS in a bilayer structure where excitons replace phonons as mediators of the interaction. The idea of substituting phonons by excitons was pioneered by Little Little (1964) and developed by Ginzburg Ginzburg (1970), who coined the term and theorized the possibility of hightemperature superconductivity, much before it came to fruition with cuprates Bednorz and Müller (1986). Those, however, exploit another (still unknown) mechanism different to BCS Leggett (2006b).
In the following, we revisit the Ginzburg mechanism, based on BCS, with emphasis on maximizing the strength of interaction between electrons, so as to maintain their binding, and therefore superconductivity, to higher temperatures. The text is organized as follows: in Section II, we give a short overview of the BCS mechanism and the exciton (Ginzburg) mechanism, outlining the points of special interest in our case. In particular, we introduce the gap equation. In Section III, we introduce our hybrid BoseFermi system configuration, its Hamiltonian and microscopic interactions, and the effective electronelectron Hamiltonian that results from a mean field approximation for the condensate and the usual Frölich transformation. We obtain the shape of the effective electronelectron interaction , that we find to be quite different in character to the Cooper (square well) potential. We also consider possible variations of our scheme, namely, a microcavity with a condensate of excitonpolaritons and a condensate of indirect excitons in coupled quantum wells Butov et al. (2002). The system of coupled quantum wells explored by several groups Butov et al. (2004); Rapaport et al. (2004) might be easier to realize and study (it does not need a cavity) and presents some interesting differences as compared to the polariton system. On the other hand, polaritons condense at much higher temperatures Baumberg et al. (2008). In Section IV, we study the gap equation for a Bose condensate mediated effective interaction. Because the potential is not positivedefinite, the problem is not wellposed numerically. We propose a simplified potential and an approximate solution of the gap equation, that we motivate by studying its validity on wellestablished approximations. We obtain the critical temperature in this case. In Section V, we give our conclusions and perspective on this new application of excitons and polaritons and discuss how to measure the effect experimentally.
Ii BCS and Ginzburg mechanisms
Superconductivity is a fundamental property of solids. At low enough temperatures, most metals superconduct. As the mechanism is rooted in quantum mechanics, temperatures are expected to be very low and indeed this is the case for all metals. After considerable theoretical efforts from various groups, a compelling theoretical model was assembled by Bardeen, Cooper and Schrieffer, the socalled BCS theory Bardeen (1951). The model provides the critical temperature:
(1) 
where, in conventional superconductivity, is the Debye energy, and with the density of electrons at the Fermi energy and the electronphonon coupling strength. The Debye energy is, in good approximation, the maximum energy that can be carried by a phonon. The mechanism therefore relies heavily on phonons, as was realized empirically before the advent of BCS (good conductors, for instance, are bad superconductors, since is small, or through the isotope effect, which correlates critical temperature with weight of the crystal atoms, and thus with resonance frequency). The exponential form and the presence of show that the effect is a collective one involving all electrons, which have formed a new phase of matter that cannot be approached perturbatively by the independent electron pictures (since has no Taylor expansion around zero). Based on the theory, and accumulated experience, it was widely accepted that critical temperatures would not exceed a few tens of Kelvins, since the Debye energy, which can be quite large in some systems (hundreds of Kelvins), is exponentially reduced. In all conventional superconductors (the socalled weakcoupling regime).
Ginzburg made the obvious but daring assumption that to achieve higher critical temperatures—crucial for technical applications which can easily be understood to be momentous—it is enough to find a system where and/or are increased. Replacing phonons by excitons, for instance, Ginzburg found that values , K as well as , are obtained, yielding temperatures of several hundred Kelvin Ginzburg and Kirzhnits (1982). High critical temperatures have been later reported in cuprates Bednorz and Müller (1986) and nowadays, temperatures as high as 125K are reported in systems such as TlBaCuoxide. Just as superconductivity in metals was discovered ahead of theory, cuprate superconductivity does not appear to follow the BCS pattern Monthoux et al. (2007) even with substitution of the mediating field as proposed by Ginsburg. All attempts to date to realize the exciton mechanism or a variation of it have remained fruitless, but it is this mechanism, rooted in BCS, which we consider in this text. Before we turn to the mechanism we propose, which consists in substituting the phonons of conventional BCS, or the excitons of Ginzburg scheme, by a BoseEinstein condensate (BEC) of excitons or excitonpolaritons, we recall that the starting point of a microscopic derivation of superconductivity is the gapequation:
(2) 
where is the effective interaction between electrons with wavevectors and and energy . The gap can be identified as the macroscopic wavefunction of a Cooper pair, which is also the order parameter for the superconducting phase. If it is nonzero, the system is in the superconducting state. A realistic microscopic treatment of is very complicated. A simplified version is provided by the Jellium model, which is a toy model of a metal that gives predominance to electronelectron interactions, that is, in particular, the underlying crystal is approximated as a uniform (structureless) background (like a “jelly”) in which the interacting electron gas evolves under its own selfinteractions and the overall charge cancellation of the background. A popular effective interaction is given by the BardeenPines potential Leggett (2006a) which is derived from the microscopic form of the electronlattice interaction:
(3a)  
(3b)  
with . In Eq. (3a), is the ThomasFermi screening parameter and is the phonon dispersion. The first term between the brackets is bare Coulomb repulsion, and the second term, which is frequency dependent, follows from the perturbative coupling to the lattice, tracing out the phonons.  
The physical meaning of Eq. (3a) is at the heart of the phononmediated mechanism. The interaction is dependent, which means, in Fourier transform, time dependent. This reflects the famous retardation effect in superconductivity. This effect is based on the strong difference between the electron Fermi velocity in metals and the sound velocity. Roughly speaking, a fast electron from the Fermi surface creates a slow phonon and goes away. After some time, another fast electron arrives and absorbs the slow phonon. The average distance between these two electrons remains of the order of 100nm, the distance at which the screened Coulomb repulsion can be safely neglected. Due to the retardation effect, a weak phononmediated attraction of electrons wins over their Coulomb repulsion and provides formation of Cooper pairs at low enough temperatures. 
(4) 
with and the polariton and 2DEG dispersions for the inplane wavevector , respectively. In the exciton case (without the microcavity), it suffices to replace with in the above. is the electronpolariton interaction, the polaritonpolariton interaction and the electronelectron repulsion. We now consider these terms in turn. Parameters assumed are listed in table 1.
Parameter  Meaning  Value 

Permittivity  
Electron reduced mass  
Hole reduced mass  
Distance between wells  \unit5\nano\meter  
Coulomb screening length  \unit1.2×10^9\meter  
Exciton mass  
Photon mass  
Rabi splitting  [0]  
Hopfield coefficient (exciton weight)  [1]  
Fermi wavevector  
Exciton Bohr radius  
Exciton Rydberg  
Dipole moment 
iii.1 Electronelectron interaction
In the original BCS mechanism, electronelectron repulsion is either neglected altogether or overcome by the attractive mechanism and not manifested outside of the attractive window. We take it into account here since it is a detrimental factor for binding and most of our concern for experimental realization is to optimize this value. The full form of the potential is given by the Yukawa potential:
(5) 
with screening constant . We get rid of the momentum dependence by averaging the potential over the Fermi surface (FS), where:
(6) 
and are the momenta of the two interacting electrons on the FS (so that, in particular, ), as shown in Fig. 2. In 3D, the FS is the surface of a sphere, while in 2D, it is a circle (we shall speak of surface in both cases). The vector difference is therefore joining the two endpoints on the surface. If the potential has spherical symmetry (), the average of all two vectors on the FS reduces to that where is pinned at one point of the surface (the south pole in Fig. 2) and runs overs the FS. This average, in the particular choice of Fig. 2, is the usual polar integration with describing the surface as (and in 3D) are varied, with:
(7) 
from Al Kashi’s theorem, so that the average potential reads, in 3D:
(8) 
where is the normalization, i.e., the same integral where is replaced by unity. This gives, in 3D:
(9)  
(10) 
where we integrate over since this is a natural variable in Eq. (7), and, in 2D:
(11) 
This is plotted in Fig. 3. Since we are trying to maximize attraction, that is, minimize repulsion, systems with small screening length and large wavevectors should be favored (but these parameters play critically on other aspects of the mechanism and the optimum is not compulsorily ).
iii.2 Electronexciton interaction
The electronexciton or excitonpolariton interaction is one of the most important ingredients of the mechanism, as it ultimately determines the shape of the effective potential. In the microcavity, an electron (from the 2DEG) interacts with a polariton (from the condensate) through its excitonic component, so this is really the electronexciton interaction that is to be computed, weighted by the Hopfield coefficient (the excitonic fraction) . Let us consider, therefore, the scattering of an electron in one of the parallel QW, separated by a distance from the QW with excitons. The matrix element of the direct interaction between excitons and electrons reads:
(14) 
where correspond to the 2D coordinates of the 2DEG electron, the exciton electron and the exciton hole respectively. The 2DEG electron is described by a plane wave while the electron/hole in the condensate are assumed to be in the 1s bound state with plane wave centerofmass motion:
(15a)  
(15b)  
where are inplane coordinates of the center of mass of the exciton and relative coordinate of electron and hole in the exciton, and are normal to the QW plane electron and hole envelope functions, respectively. We also consider the existence of a dipole moment for the exciton, which can be intrinsic to the structure, because of spatial separation of electrons and holes in coupled QWs, or (in the case of microcavities) be induced by an internal piezoelectric field, or result from an externally applied electric field. To account for all these possibilities, one can consider the layers of electrons and holes in the exciton shifted in the direction with respect to the position of the center of mass by a distance . The matrix element of the interaction is then computed to be: 
(16a)  
(16b)  
where Eq. (16a) is the direct
electronexciton interaction that exists even in the absence of a
dipole moment of the exciton, and Eq. (16b)
is the dipolar interaction. The direct interaction vanishes at small
exchanged momenta, while the dipolarinduced one assumes its maximum
value here of . Overall, the dipolar interaction is
naturally much larger than the direct one, since the exciton is
electrically neutral. 
(17) 
where describes the dispersion of the elementary excitations (bogolons) of the interacting Bose gas, which is very close to a parabolic exciton dispersion at large :
(18) 
where and with the renormalized bogolonelectron interaction strength:
(19) 
The last term of Eq. (17) coincides with the Fröhlich electronphonon interaction Hamiltonian, which allows us to write an effective Hamiltonian for the bogolonmediated electronelectron interaction. This results in an effective interaction between electrons, of the type . The effective interaction strength reads , with:
(20) 
Equation (20) recovers the bosonmediated interaction potential obtained for a BoseFermi mixture of cold atomic gases Bijlsma et al. (2000), in the limit of vanishing exchanged wavevectors. It describes the BEC induced attraction between electrons. Remarkably, it increases linearly with the condensate density . This represents an important advantage of this mechanism of superconductivity with respect to the earlier proposals of excitonmediated superconductivity Little (1964); Ginzburg (2009); Allender et al. (1973), as the strength of Cooper coupling can be directly controlled by optical pumping of the excitonpolariton condensate. The attractive potential is displayed for various exchanged energies in Fig. 5, as a function of the exchanged momentum expressed directly through the angle defined in the Fermi circle (cf. Fig. 2). As commented earlier, the negative part corresponds to attraction, and the potential alternates between repulsive and attractive character, obtained at different exchanged momenta. We do not want to keep track of such complicated wavevector dependence, and therefore will average the interaction over the Fermi sea. A notable feature is, for most values of , the presence of a pole , where . As is seen in the figure, separates the attractive part from the repulsive part. The average will bring the additional convenience of cancelling such divergencies. We note as well that the spectrum of excitations of the exciton BEC may be changed in the presence of the electron gas, so that their eventual dispersion may be different Shelykh et al. (2010). This has no effect on the Cooper pairing of electrons which we discuss here.
We therefore wish to perform the average
(21) 
where , as seen previously. Since is symmetric around we perform the integral only. The integral would be easily computed numerically if there were no pole. There are dedicated numerical methods to compute principal values numerically Thompson (1998); Noble (2000), but in our case, since the pole is first order, it is enough to isolate it analytically by defining
(22) 
so that then
(23) 
where and the first integral is regular (it is shown in Fig. 5 in dashed purple). The integration (average) is then straightforward and produces the results shown in Fig. (6).
If we take instead of a quadratic dispersion for the excitation
(24) 
and assume allexcitonic interactions, , we can consider the same effect in the absence of a microcavity, relying on a purely excitonic (rather than polaritonic) BEC. In this case, the same procedure as detailed above leads to an effective potential as shown in Fig. 7. We kept all parameters the same for comparison except for the dipole moment, which we have taken three times as large (nm). This corresponds to the spatial separation of electrons and holes in the system of indirect excitons studied by Butov et al Butov et al. (2002). The potential we obtain in the exciton case is very different in character from the polariton case, and is more closely related to the Cooper (conventional) shape of a square well, or the Bogoliubov potential including repulsive elbows.
From these potentials, one can proceed to solve the gap equation.
Iv Gap equation
iv.1 Cooper potential
The BCS gap equation (2) is easier to tackle as a continuous equation:
(25) 
where we have also introduced a finite temperature from the FermiDirac distribution of elementary excitations de Gennes (1999). With the BCS approximation of a step potential, the gap equation at zero temperature simplifies to . If , the dependence in the integral boundaries can be neglected (or, coming back to Eq. (2), one sees that in the initial gap equation, is exactly constant if and is zero otherwise). Thus, we can assume the gap to be of the form:
(26) 
In this case, simplifying on both side of Eq. (25), we obtain:
(27) 
Equation (27) is better known as its approximation when , in which case it takes the form of the famous BCS gap expression, .
Solving exactly the gap equation calls for some numerical method. Equation (25) is a nonlinear integral equation, of the type studied by Hammerstein, i.e.,
(28) 
where in our case and
. There are strong conditions of existence of
nontrivial (nonzero) solutions when
Kitamura (1963); Vansevenant (1985), however the case when the
kernel is not positive definite, that is, in presence of
repulsion,
iv.2 Bogoliubov potential
The potential is not always attractive when, for instance, some overall repulsion, such as direct Coulomb interaction, is superimposed on the attractive Cooper potential, as shown on Fig. 9. The Coulomb interaction is time independent and should therefore extend to all but here also a cutoff is introduced to avoid divergencies. This results in an attractive, Cooperlike potential, flanked by two repulsive windows. Such a potential is known as the Bogoliubov potential Ketterson and Song (1999); Zheng and Walmsley (2005).
This approximation has been used to show extremely counterintuitive behaviour of the gap equation and justify aposteriori another heavily criticized approximation of BCS, neglecting Coulomb repulsion: the BCS mechanism indeed assumes only attraction between electrons, which can be dimmed by Coulomb repulsion, but which never explicitly appears as such (like in the Bogoliubov scenario). The great result of Cooper was that binding occurs at arbitrarily small attraction. An important result of the Bogoliubov potential is to show that the detrimental effects of Coulomb repulsion are greatly reduced in the gap Zheng and Walmsley (2005).
We now present a linearization of the gap equation that allows one to obtain an approximate solution for the critical temperature Ketterson and Song (1999). By assuming the gap equation to be a two step valued function , the gap equation becomes with
(29) 
where
(30a)  
(30b)  
appear invariably in the columns of . In Eq. (30a), the BCS approximation has been applied while in Eq. (30b), the fact that has been used to neglect in the denominator and the temperature in the numerator. The parametrization of the matrix in terms of the potential depends on the particular configuration (for example the relative widths of the various layers of the structure). Here we have adopted the original parametrization of Bogoliubov, which assumes narrow repulsive elbows surrounding a large attractive central region. Solving the linear equation for and then for the critical temperature, we find: 
(31) 
In this form, one can see how Coulomb repulsion indeed introduces a small correction to the original BCS formula. The expression also seems to indicate that could be repulsive () and still lead to a gap as long as the denominator in Eq. (31) remains positive.
In the following, we compare these predictions with numerical solutions of the gap equation, keeping in mind that the iterative procedure is not assured, mathematically, to converge. We have observed that indeed, it sometimes encounters problems and exhibit strong instabilities, with bifurcations of solutions, for example.
In Fig. 10, we show the evolution of the gap function as is increased, from zero (BCS) to a point where the overall potential is essentially repulsive. With onset of the repulsion, the gap acquires two negative sides and becomes a highly distorted function for large . Note that the approximation of constant gap over the various regions is at least as good as for the case of BCS.
In Fig. 11, we now show the case where is held constant as the strength of the repulsion is varied independently. In foresight of what is to come later, we also allow the elbows to be negative, that is, to contribute an additional attraction to the conventional mechanism (for now we do not consider physical justification of this). Another unexpected result is obtained: the repulsive potential is in this case favouring a larger gap, as can be seen by comparing (a), where the gap function is highly oscillatory, to (d) where it recovers the BCS bellshape. In (h), the distorted but overall attractive potential still results in a BCS type gap, but wider and larger. Note how the repulsion, by “squeezing” the gap, allows it to achieve much higher values than for the case of smaller or no repulsion.
These unexpected results are confirmed phenomenologically by the Bogoliubov approximation Eq. (31), which we plot as a dashed line in Fig. 11, compared to computed numerically. Here we should emphasize that the two quantities are not meant to be compared quantitatively (as we did when comparing the BCS formula with the numerical solution), since one, (computed numerically) is the gap at zero temperature while the other, is the temperature at which the gap vanishes. There is a monotonous relationship between the two, that is, increasing implies increasing , so one can appreciate the consistency of the results by observing similar trends. The obstacle to conducting an extensive numerical comparison is that it is an intensive task numerically to compute , since this requires solution of the gap equation for various temperatures until the curve is obtained and its intersect with zero is found. A critical slowing down phenomenon makes the iterative process slower as the critical temperature is approached. In addition, numerical instabilities are stronger at nonzero . Therefore, although it is relatively straightforward to compute numerically, it is not convenient to use this method to obtain . On the other hand, the Bogoliubov approximation gives a fair estimate of , but is not able to provide the gap at zero temperature, since at the core of its method, there is an assumption of vanishing . Therefore, we have two complementary methods, each suited to provide a relevant aspect of the problem. We note that the gap at zero temperature is an important quantity which can be measured independently from by Andreev reflection in conductivity experiments, which is why we discuss it here in a great detail.
We can see, indeed, that the qualitative agreement is good and that the counterintuitive features of the gap equation are reproduced by the analytical formula.
iv.3 Polariton potential
In the case of the polariton problem, we have seen that, even when neglecting Coulomb repulsion (as in the original BCS formulation), the potential departs strongly from the Cooper potential and features two large attractive regions far from small energies, immediately followed by two strong repulsive windows. We extend the Bogoliubov method to a threestep approximation of this potential, such as displayed in Fig. 13, with, in reference to previous potentials, notations , and for the boundaries of the central, shallow attractive region, narrow, deep attractive region and repulsive region, respectively. This is a notation only and is not mean to be understood as referring to Debye, Coulomb or Bogoliubov in any strict sense. Following the same premises, we approximate the gap equation by a threestep valued function . This approximation turns out to be an exceedingly good one in certain cases, such as the one displayed in Fig. 13. Here there is even more room to choose a parametrization of the matrix. We now give general guidelines on how to build this matrix. The simplest method is to fix on the lhs of Eq. (25) at the center of each region and, in the corresponding row, take for each column the potential that is sampled more by the difference . Refinements are possible, such as weighting elements of by coefficients which reflect how much time the variables and spend in the regions that determine the matrix equation. This problem has the following mathematical expression: how is the random variable distributed when (resp. ) is uniformly distributed in an interval (resp. ). The solution is easily obtained as proportional (normalize to unity) to:
(32) 
This is easily obtained geometrically (the square root comes from Pythagoras’ theorem) and the problem results in finding the intersect of a line with the grid. There are two configurations. Working out the cases shows that Eq. (32) reduces to a triangular or a tophead truncated triangular distribution. The coefficients entering can then be taken as the potentials weighted by the area intersecting their corresponding region. The soundness of such an approach can be appreciated only if cases are checked numerically to compare quantitatively various parametrizations. For simplicity, we shall here consider cases where only the dominant potential is considered. An example of such a gap equation is defined with:
(33) 
where
(34) 
and is integrated on the respective steps, as defined by the integral boundary conditions, i.e., .
As before, we estimate the critical temperature by the condition that gives vanishing values of the gap on the whole interval. The first integral is evaluted in the same approximation as for the usual BCS (or Bogoliubov) potential:
(35) 
and are also essentially logarithmic in their energy range and we take:
(36) 
We solve the gap equation by setting its determinant to zero and solving for , giving
(37) 
We see that the impact of the polariton potential shape on the gap is rather intricate, here as well, with some unexpected behaviour, produced both numerically and from this formula. In Fig. 14, we show the effect of widening the central, attractive region of the potential. Naively one would expect this to increase the gap (or critical temperature), since increases linearly with . However, Eq. (37) predicts a decrease of the critical temperature, as shown in Fig. 15. The reason why is understood by computing the gap at zero temperature, as seen in Fig. 14, where the repulsive barrier is shown to behave as a trap for the gap function, which results, as the width is increased, in a loosening of the gap strength, akin to the quantum mechanical situation of a bound state in a square potential. The numerical and linearized models display a good qualitative agreement, as seen in Fig. 15. Let us repeat that we are not considering here quantities that can be directly compared, since should be used for this purpose.
Another example is shown in Fig. 16, where both the attractive and repulsive parts of the potential are increased together. This results in a rapid increase of the gap. The result is also confirmed by numerical simulations, as shown in Fig. 17, and a good agreement is obtained. Unexpectedly, if the repulsive part grows twice as fast as the attractive one, not only does this also result in an increase of the gap, but even a faster one. This, too, is confirmed by the numerics. Note that in this case, we reach a region where our numerical procedure jumps to other solutions (usually of a highly oscillatory character, such as those reported in Zheng and Walmsley (2005)), but behaves as expected until then. In this regard, the value of having an approximate analytical solution is obvious.
This analysis is finally applied to the case of the gap equation with the numerical results of the critical temperature obtained in both the polariton and exciton cases for the cases of Figs. 6 and 7. The parameters are gathered in Table 1 and the results plotted in Fig. 18. Both cases show a strong variation of the critical temperature with moderate variations of the condensate density (one order of magnitude). The polariton case is steeper and roughly linear while the exciton case increases less quickly but begins earlier. In both cases, temperatures are very high, and other effects will surely break the mechanism, for instance the loss of the condensate. This shows, however, the robustness of the mechanism in the conditions where it should apply.
V Conclusions
We have studied possible mechanisms of superconductivity in semiconductor heterostructure systems where a BoseEinstein condensate mediating the effective electronelectron interaction leads to enhancement of the coupling, yielding very high critical temperatures. We have considered the case of an exciton BEC, consisting of a sandwich of an doped QW containing the superconducting electrons, in contact with coupled QWs where the BEC of indirect excitons is formed, for instance by optical excitation. We have also considered the case of a polariton BEC, where doped QWs and undoped QWs hosting excitons are embedded in a microcavity. We have computed in these two cases the effective electronelectron interaction as a function of the exchanged energy , showing how the retardation effect acquires a peculiar character in the polariton case, namely, yielding a weakly attractive potential at long times, followed by a succession of strongly attractive and strongly repulsive windows. The gap equation in this case exhibits strong differences as opposed to the case of the Cooper potential (of conventional BCS and excitonBEC mechanism). To understand the physical mechanism leading to large gaps and bypass numerical instabilities, we studied in detail the gap equation in the cases of simplified stepwise potentials, and offered an analytical method to obtain the critical temperature, which is in qualitative agreement with the numerical results. Our results suggest record breaking critical temperatures in these systems. We stress, however, that the path towards achievement of exciton mediated superconductivity at high temperatures may be long and full of obstacles. Of the two experimentally relevant systems that we have considered, one (coupled QWs) only shows the BEC of excitons at very low temperatures (less than 1K Butov et al. (2002)). Consequently, one cannot expect high temperature superconductivity in this system, while at low temperatures the superconducting gap may be very large. In microcavities, polariton BEC or polariton lasing have indeed been demonstrated at room temperature Christopoulos et al. (2007); Baumberg et al. (2008). However, embedding a highquality doped QW inside the cavity, the proper choice of the experimental geometry in order to minimise optical absorption in the doped QW, and especially fabrication of quantum contacts for selective injection of carriers in the QW of interest, may pose technological difficulties. It is possible that hybrid metalsemiconductor or semimetal semiconductor systems may appear more suitable for the observation of the predicted effects. A conventional superconductor put in contact with a semiconductor also seems promising. Collective quantum phenomena in BoseFermi mixtures are extremely complicated and we foresee breakthroughs in their study in multilayer structures combining a Fermi gas of electrons and an exciton BEC.
Acknowledgements.
We are grateful to H. Ouerdanne and P. Lagoudakis for useful discussions. Support from EPSRC and EU FP7 ITN project CLERMONT4 are acknowledged. I.A.S. thanks the support from Rannis “Center of excellence in polaritonics” and FP7 IRSES POLAPHEN project.Footnotes
 Note at this point that there is an error in Ref Laussy et al. (2010) where only the direct exciton interaction has been taken into account with an incorrect power in the parenthesis, which led to an expression similar to the dipolar interaction. The direct interaction by itself turns out to be too small to evidence superconductivity with the parameters chosen in Ref. Laussy et al. (2010), therefore a dipole moment should be induced in this case, say by applying an external electric field, to restore the effect.
 as well as attraction, since the case of only repulsion admits only as a solution.
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