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We review several recent theoretical and experimental results in the study of exciton condensates. This includes the present experimental advances in the study of exciton condensates both using layers and coupled bilayers. We will shortly illustrate the different phases of exciton condensates. We focus especially on the Bardeen-Cooper-Schrieffer-like phase and illustrate the similarities to superconductors. Afterwards, we want to illustrate several recent advances and proposals for measuring the different phases of superconductors. In the remainder of this short review, we will provide an outlook for the possibilities and complications for future technical applications of exciton condensates.

Excitons represent one of the most interesting composite particles in modern condensed matter physics. In its simplest form, it is a bound state of an electron and a hole and thus has bosonic properties [

Such bosonic systems have shown a huge variety of different phases including a Bose-Einstein condensate [

In this short review, we want to provide an overview of the recent advances in the generation of exciton condensates and their phase structure. We will especially focus on the superconductor-like phase [

We will shortly review the recent experimental advances and results in Section 2 and provide an overview of the different phases of an exciton condensate in Section 3. Section 4 will concentrate on techniques and theoretical proposals for measuring the different phases of an exciton condensate and Section 5 will discuss possible technical applications of exciton condensates in nanoelectronics. We will conclude in Section 6.

An exciton is bound pair of an electron and a vacancy (hole) and thus may occur in various circumstances that allow the close proximity of many electrons and holes with some mechanism to forbid immediate recombination [

where n is the electronic density, T is the temperature and m is the electron mass.

The BEC phase emerges at low densities and low enough temperatures, where the key obstacle is to cool down the excitons to a temperature below the critical temperature [

The requirement for BCS condensation being high density and a relatively long lifetime is challenging since typically electron-hole pairs have a very short lifetime [

First, one may use materials for which the direct (dipole) transition for the annihilation of the electron-hole pair is forbidden. The most promising candidate in this respect has been CuO_{2}. In such a setup it is possible to create both direct (electron and vacancy in the same layer) and indirect excitons (electron and vacancy in different layers). However, in spite of several promising results [

Second, one may use physically separated coupled bilayers which are illuminated by laser at a given frequency. In this case the excitons generated in the separate layers (indirect excitons) have a very long lifetime due to the spatial separation of the layers but have to be generated externally by a laser field. The massively increased lifetime of indirect excitons in layered semiconductor structures and the possible confinement of the electrons in those structures [

The third candidate are again coupled bilayers which are insulated from each other (e.g. Boroncarbide-hete- rostructure) but still close enough so that the electron and hole can feel their mutual Coulomb interaction. In this case an exciton condensate may emerge without the presence of an applied laser field due to the presence of electrons and holes in both layers very similar to the appearance of superconductivity.

Recently, systems of two graphene layers separated by boron nitride have been realized [

These experimental realizations are compatible with the prediction of the possibility of room-temperature superfluidity in layered graphene systems [

First results for counterflow transport measurements in a range where condensation is expected have been performed in typical drag configurations, where two Corbino disks are confined in a GaAs/AlGaAs double quantum well structure [

As the first-order coherence function has been shown to be accessible experimentally and shows the typical behavior of a BEC condensate one of the primary theoretical research directions has been to find ways how to

investigate the behavior on the BCS side in a similar way.

Typically we expect to observe a Bose-Einstein condensate (BEC) for low densities whereas for high densities exciton BCS condensates are expected. However, two key characteristics of exciton condensates allow for a richer phase structure.

The first characteristic is the presence of parallel electric dipole moments of indirect excitons due to the separation and different charges of the electron and the vacancy. These become important at intermediate densities. In this case, the electrons can be excited to form excitons but the excitons also feel the repulsion due to the dipole-interaction

We therefore observe a cubic dependence on the inter-exciton distance r.

This inter-exciton exchange leads to the fact that excitons may form Wigner crystals [

If the excitons can be observed on a timescale small enough in order to assume that they are fixed in space we should expect the formation of an emergent long-range order of Wigner-crystal type [

The second characteristic is the BCS phase itself. If we discuss one-dimensional systems two topological phases are possible either with all pairs bound or with two particles at the ends being present unpaired [

Whereas the BEC phase of excitons has been detected the BCS and intermediary phases still remain to be explored experimentally. The more we move away from the BEC phase the less adequate it will become to investigate excitons only in space but we will need to move to transport measurements.

First, transport measurements of excitons did not seem to be an adequate mean to explore excitons since these are neutral particles and therefore will not move once localized in space.

However, as discovered by Su and MacDonald [

Theoretically all important features in this BCS-condensed situation are captured by applying an effectively one-dimensional model for an exciton condensate extending from

where

The first and most intriguing prediction from this model is that for voltages below the gap of the exciton condensate the top-current

The theory described in Equation (3) has also been checked by comparison to the experimental data in [

The investigation of the non-BCS phases becomes less trivial due to the special nature both of the Kitaev phase and the crystalline phase.

The crystalline phase emerges as a consequence of the dipole interaction of excitons. It can be detected in three ways: either by modern spectroscopic techniques measuring the distribution function of generated excitons or the pair correlation function or by a transport measurement using the photoelectric effect.

We will first discuss the two possibilities to measure the phase transition by modern spectroscopic techniques. In this case one focusses on physically separated bilayers and extremely short laser pulses and experimental of τ < 0.1 ns so that the movement of the excitons can be neglected [

where

The Hamiltonian in Equation (4) can be numerically simulated for a finite system of size L. In this case, the correlation function clearly shows the transition from a liquid to a crystalline phase. For weak interactions (blue curve in

The same behavior can also be extracted by exploring the distribution function or respectively the corresponding moments of the number of excitons. The number of excitons drops by increasing interaction but the variance first increases with increasing interaction whereas for the interaction being larger than a critical value it sharply drops. This change can be associated to the formation of crystalline order [

However, the crystalline phase cannot just be observed by spectroscopic measurements, but also by exploring the photoelectric effect in an exciton bilayer.

If we assume no interaction the Hamiltonian in Equation (4) becomes exactly solvable and the number density of excitons is given by

with

where

where

Indeed Equation (7) allows always for the trivial solution (

If we move to crystalline order the excitons are fixed in space due to their mutual interaction and therefore a photoelectric effect will disappear.

If we therefore find a solution for which the excitons show a photoelectric effect and vary the interaction in order to suppress it, we have clearly shown the transition to the crystalline phase via a transport measurement.

Finally, we want to investigate the Kitaev phase. Typically the Kitaev model can be easily realized in a system involving spin-orbit coupled nanowires coupled to superconductors in a magnetic field or ferromagnetic nanowires coupled to superconductors [

In the case of exciton condensates, we may also mimick the Kitaev model via coupling a 2D exciton condensate to two p-wave superconductors as shown in

In this case, the simplest Hamiltonian describing the exciton condensate in the presence of the p-wave super- conductors is given by [

where the states a four-spinors

and top-bottom space respectively.

The spectrum can be revaled by squaring Equation (8) twice [

Consequently, we will observe a level-crossing at

The appearance of this phase can be easily observed as the Majorana fermions at the ends of the exciton condensate will open perfectly conducting transmitting channels either through the top or bottom part of the exciton condensate. As the top and bottom part will therefore be perfectly conducting transconductance between the layers has to vanish due to unitarity [

After this investigation of the different phases of exciton condensates, we want to investigate possible technological applications.

Studying technical applicatios of exciton condensates two properties are of high relevance: the existence of superconducting counterflow and the possibility to excite excitons with light.

The counterflow as such may be used for transformation of current on the nanoscale. In order to do so, one layer of the exciton condensate should not be connected to an outer voltage source but to a nanoelectronical system itself. This means that e.g. the lower layer may directly couple to systems such as quantum point contacts or quantum dots coupled to phonons. Due to the coupling to an additional system the voltage characteristics of the lower layer are changed and the system may be used to transform voltage on the nanoscale [

Furthermore, the availability of a photoelectric effect allows the usage of exciton condensates as solar cells (see

Finally, the existence of a Kitaev phase and the possible existence of Majorana fermions allow to use exciton condensates as future building blocks of topological quantum computers as discussed in [

To conclude, we have reviewed several of the recent experimental and theoretical results for exciton condensates. After the theoretical and experimental exploration of the BEC phase of exciton condensates, the BCS and crystalline phases have attained increasing attention in the last years.

Especially, the possibility of superconducting counterflow in the top and bottom layer starts to be verified experimentally and will in the future be enhanced by the study of the high-order cumulants and cross-correla- tions. Likewise, the study of exciton condensates illuminated by laser light allows to observe crystalline order as in a Wigner crystal and the photoelectric effect in contacted exciton condensates.

Finally, the existence that a rich phase structure allows for several future technological applications of exciton condensates includes the usage in nanotransformers, solar cells and topological quantum computers, which provides a promising future path to explore for exciton condensates.

The author would like to thank A. Komnik, S. Maier and D. Breyel for numerous discussions.

HenningSoller, (2015) Exciton Condensates and Superconductors-Technical Differences and Physical Similarities. Journal of Applied Mathematics and Physics,03,1218-1225. doi: 10.4236/jamp.2015.39149