Query: Bose–Einstein condensation of excitons in bilayer electron systems

Candidate papers:

abstract: An exciton is the particle like entity that forms when an electron is bound to a positively charged 'hole' An ordered electronic state in which excitons condense into a single quantum state was proposed as a theoretical possibility many years ago. We review recent studies of semiconductor bilayer systems that provide clear evidence for this phenomenon and explain why exciton condensation in the quantum Hall regime, where these experiments were performed, is as likely to occur in electron electron bilayers as in electron hole bilayers. In current quantum Hall excitonic condensates, disorder induces mobile vortices that flow in response to a supercurrent and limit the extremely large bilayer counterflow conductivity.
author_names: James Philip Eisenstein

Allan H MacDonald
title: Bose Einstein condensation of excitons in bilayer electron systems
venue: Nature
year: 2004

author_names: James Philip Eisenstein

Allan H MacDonald
title: Bose Einstein Condensation of Excitons in Bilayer Electron Systems
year: 2005

abstract: Bilayer electron hole systems, where carriers in one layer are electrons and carriers in the other are holes, have been actively investigated in recent years with the focus on Bose Einstein condensation of excitons. This condensation is expected to occur when the layer separation $d$ is much smaller than the interparticle distance {r}_{s}{a}_{B} within each layer. In this Brief Report, we argue for the existence of a state, Wigner supersolid, in which excitons are phase coherent but form a Wigner crystal due to dipolar repulsion. We present the qualitative phase diagram of a bilayer system and discuss properties and possible signatures of the Wigner supersolid phase.
author_names: Yogesh N Joglekar

Alexander V Balatsky

Sankar Das Sarma
title: Wigner supersolid of excitons in electron hole bilayers
year: 2006

abstract: We report on small cluster exact diagonalization calculations which prove the formation of electron hole pairs (excitons) as prerequisite for spontaneous interlayer phase coherence in bilayer systems described by the extended Falicov Kimball model. Evaluating the anomalous Green's function and momentum distribution function of the pairs, and thereby analyzing the dependence of the exciton binding energy, condensation amplitude, and coherence length on the Coulomb interaction strength, we demonstrate a crossover between a BCS like electron hole pairing transition and a Bose Einstein condensation of tightly bound preformed excitons. We furthermore show that a mass imbalance between electrons and holes tends to suppress the condensation of excitons.
author_names: Tatsuya Kaneko

Satoshi Ejima

Holger Fehske

Yukinori Ohta
title: Exact diagonalization study of exciton condensation in electron bilayers
year: 2013

abstract: At a total filling factor of one two closely spaced two dimensional electron systems (2DES) can be viewed as a Bose Einstein condensate of excitons, where electrons from one 2DES couple to vacant states from the other 2DES. The pairing of electrons and holes (i.e. excitons) is mediated by Coulomb interactions. In this thesis this excitonic state was investigated by interlayer tunneling and Coulomb drag experiments. The Coulomb drag experiments on Corbino ring structures have shown a minimum in the conductivity, implying an energy gap at the Fermi energy due to exciton condensation. The tunneling experiments revealed a critical behavior and an I/V curve reminiscent of the Josephson effect of superconductivity.
author_names: Lars Tiemann
title: Phase coherence in the regime of bilayer exciton condensation
year: 2008

abstract: Macroscopic quantum states such as superconductors, Bose Einstein condensates and superfluids are some of the most unusual states in nature. In this project, we proposed to design a semiconductor system with a 2D layer of electrons separated from a 2D layer of holes by a narrow (but high) barrier. Under certain conditions, the electrons would pair with the nearby holes and form excitons. At low temperature, these excitons could condense to a macroscopic quantum state either through a Bose Einstein condensation (for weak exciton interactions) or a BCS transition to a superconductor (for strong exciton interactions) While the theoretical predictions have been around since the 1960's, experimental realization of electron hole bilayer systems has been extremely difficult due to technical challenges. We identified four characteristics that if successfully incorporated into a device would give the best chances for excitonic condensation to be observed. These characteristics are closely spaced layers, low disorder, low density, and independent contacts to allow transport measurements. We demonstrated each of these characteristics separately, and then incorporated all of them into a single electron hole bilayer device. The key to the sample design is using undoped GaAs/AlGaAs heterostructures processed in a field effect transistor geometry. In such samples, the density of singlemore 2D layers of electrons could be varied from an extremely low value of 2 x 10{sup 9} cm{sup 2} to high values of 3 x 10{sup 11} cm{sup 2} The extreme low values of density that we achieved in single layer 2D electrons allowed us to make important contributions to the problem of the metal insulator transition in two dimensions, while at the same time provided a critical base for understanding low density 2D systems to be used in the electron hole bilayer experiments. In this report, we describe the processing advances to fabricate single and double layer undoped samples, the low density results on single layers, and evidence for gateable undoped bilayers. less
author_names: Sungkwun Kenneth Lyo

Roberto G Dunn

Michael P Lilly

D R Tibbetts-Russell

Larry L Stephenson

J A Seamons

John Louis Reno

Edward S Bielejec

Wes Edmund Baca

Jerry A Simmons
title: LDRD final report on engineered superconductivity in electron hole bilayers.
year: 2005

abstract: Abstract The aim of this project is to explore the strongly correlated states, more specifically Bose Einstein condensation of indirectly bound excitons, in bilayer graphene. During this final reporting period, we have continued our work on the extension of the many body calculations to an atomistic model, examined the electron phonon coupling, studied the possibility of finding the same type of condensation in a 3D topological insulator, and started our work on a time dependent theory to understand the behavior of the time dependent many body systems. In this final progress report, we will outline our most important results and draw some final conclusions from our 3 year project.
author_names: Matthew J Gilbert
title: Progress Report for Grant Number W911NF 09 1 0347 (University of Illinois Urbana)
year: 2012

abstract: The bilayer systems exhibit the Bose Einstein condensation of excitons that emerge due to Coulomb pairing of electrons belonging to one layer with the holes belonging to the other layer. Here we present the microscopic derivation of the dynamic equation for the condensate wave function at a low density of electron hole $e h$ pairs in a strong magnetic field perpendicular to the layers and an electric field directed along the layers. From this equation we obtain the dispersion law for collective excitations of the condensate and calculate the electric charge of the vortex in the exciton condensate. The critical interlayer spacing, the excess of which leads to a collapse of the superfluid state, is estimated. In bilayer systems with curved conducting layers, the effective mass of the $e h$ pair becomes the function of the $e h$ pair coordinates, the regions arise, where the energy of the $e h$ pair is lowered (exciton traps) and lastly $e h$ pairs can gain the polarization in the basal plane. This polarization leads to the appearance of quantized vortices even at zero temperature.
author_names: A I Bezuglyj

Sergej I Shevchenko
title: Dynamic equation for quantum hall bilayers with spontaneous interlayer coherence The low density limit
year: 2007

abstract: The Bose Einstein condensation of excitons has a long history with seminal contributions from Leonid V. Keldysh. Despite numerous efforts, however, a compelling experimental evidence is still missing. A brief survey of attempts to realize exciton condensation in different semiconductor systems is given. Specific problems compared with atomic Bose condensation are highlighted. More details are given on coupled quantum wells as a possible candidate in the search for exciton condensation. While here extremely long radiative lifetimes of indirect excitons can be achieved, their strong dipole dipole repulsion leads to a genuine non ideal behavior. Theoretical results from a dynamical T matrix theory are presented which allow to explain blue shift and line broadening under strong excitation which have been seen in recent high density photoluminescence experiments using a lateral trap. 18.1 A short history of exciton condensation My first encounter with exciton physics was not reading one of the famous textbooks like the 'Knox' (Knox, 1963) but trying to understand the 'Excitonic Insulator' from a review written by Halperin and Rice (1968) Soon we realized that this branch of solid state theory was shaped and brought forward by (at that time) Soviet physicists, with cornerstone papers by Keldysh and coworkers (Keldysh and Kopaev, 1964; Keldysh and Kozlov, 1968) The basic idea was to look at a small gap semiconductor whose energy gap could be tuned to values below the exciton binding energy. Then, the ground state must become unstable against formation of excitonic correlations. The theoretical description is very close to the famous BCS (Bardeen Cooper Schrieffer) theory of superconductivity, only that here the attraction between electron and hole is the standard Coulomb force, while in the BCS the dynamical screening by acoustic phonons provides the necessary attraction. More important, however, are two other facts. In the exciton case, anomalous propagators have a simple physical meaning they are the optically driven interband polarization. Therefore, what needs sophisticated experiments using Josephson junctions in superconductors, can be 282 Bose Einstein condensation of excitons by R. Zimmermann easily done in the excitonic case via interband optics. Secondly, outside the excitonic ground state (or condensate) we find excitons as normal bound states, being thermally excited. There is a long standing claim that the intermediate valent compound TmSeTe fits to the original idea of the excitonic insulator (Bucher et al 1991; Wachter, 2005) When driving the band gap through zero by applying hydrostatic pressure, the material exhibits a phase transition with a critical temperature as high as 250K, which is deduced from measuring thermal properties. Further specific experiments would be needed to clarify if excitons are the main players in this game ruling out competitive mechanisms like charge density wave or lattice instability. A new turn came in when the excitonic insulator was identified as a possible phase transition in wide gap semiconductors, too. If optical recombination is slow enough, excited electrons and holes may reach quasi equilibrium (with common temperature, but different chemical potentials) Although more speculative than the small gap case with strict equilibrium, a lot of interesting physics was to be expected. However, the pressing question 'Does it work, even in principle? was always lurking behind this scenario. This feeling of uncertainty was often seen already in the paper titles, as, e.g. 'Possibility of the excitonic phase in insulators' (which was my first scientific publication, Zimmermann, 1970) The story with the excitonic insulator was coming to rest for some time when a new phase transition from dilute exciton gas to the electron hole liquid was shown to dominate the physics in highly excited semiconductors. Again, Leonid V. Keldysh was among the pioneers of this new 'electron hole droplet physics' (Keldysh, 1968) which is mainly electron gas theory extended to two species (electrons and holes) In fact, the ground state energy minimum for multi valley semiconductors (Si or Ge) is rather deep and lies at relatively high densities. Therefore, the excitonic character of the dilute gas around the droplets was of not much importance, and a random phase like approximation for the twocomponent electron hole plasma did rather well. Indeed, at these high densities, strong screening prevents the formation of excitons as bound states (Mott transition) Things were expected to be different for single valley semiconductors (GaAs, CdS) Theoretical work revealed that in this case even the high density side of the first order phase diagram bears excitonic signatures (Zimmermann, 1976) However, these materials have a direct band gap with dipole allowed optical transitions, and the much shorter radiative lifetime prevents the build up of quasi equilibrium under high excitation. Still it was general belief that here a quite interesting sequence could be expected with rising density: Formation of excitons, their Bose Einstein condensation (BEC) strong nonideal effects due to the underlying fermionic structure (excitonic insulator) and a high density plasma which looses the excitonic correlations gradually (Zimmermann, 1988) When including polar optical phonons into the electron hole plasma theory, Leonid V. Keldysh was once more paving the way (Keldysh and Silin, 1975) which gave me the first chance to come into personal contact with him, and to A short history of exciton condensation 283 learn quite a lot (which continued to be the case at all further meetings) How to overcome the exciton lifetime problem? One way was to use wide gap semiconductors with dipole forbidden optical transitions. The paradigm material is here cuprous oxide which has indeed a long history on the search for exciton BEC. Following earlier claims on biexciton condensation in CuCl (Nagasawa et al 1975) pioneering work on Cu2O was done by Wolfe (Lin and Wolfe, 1993) and Mysyrowicz (Mysyrowicz et al 1996) However, several findings were not really conclusive, for instance to read off directly the bosonic distribution function from the (phonon assisted) photoluminescence. Other BEC claims found a different and much less spectacular explanation, as, e.g. exciton super transport being driven by phonon wind (Bulatov and Tikhodeev, 1992; Tikhodeev, 1997) Quite recently, new findings on Cu2O give new hopes, as the exploration of 1s 2p transitions using infrared femtosecond spectroscopy (Kuwata Gonokami, 2005) Another way to achieve extremely long radiative lifetimes of excitons is using coupled quantum wells. Application of a static electric field in the growth direction allows to tilt the confinement potentials such that the lowest state has electrons and holes residing in different quantum wells. A spatially indirect exciton is formed which acquires microsecond lifetime since the overlap between electron and hole is exponentially small. Already in 1976, Lozovik made a first theoretical prediction for the excitonic insulator in such a system (Lozovik and Yudson, 1976) The first experimental hint on a possible BEC was reported quite early, too (Fuzukawa et al 1990) But here again, spectacular findings as the nearly millimeter sized ring emission has been considered first as BEC evidence (Butov, 2002; Snoke et al 2002) but are now discussed in terms of a dynamic p n junction. Still, the quite regular fragmentation of this ring emission is waiting for a conclusive explanation. Another interesting issue is to take the fluctuations (noise) of the photoluminescence as indicator of BEC (Butov, 2003) In Sections 18.3 and 18.4 we give more details and own theoretical results for this very promising system, focussing on the emission lineshape. It is quite common knowledge that at a given density, the BEC critical temperature is larger if the Bose particles have a lighter mass (a specific example is given in eqn (18.3) below) Usually, the exciton mass is dictated by the underlying semiconductor material. However, if exciton polaritons are formed within a microcavity, the dispersion is dominated by the cavity mode, and the relevant polariton mass can be orders of magnitude smaller than the exciton mass. However, there is always a price to be paid! Here, the system has to be pumped hard in order to get reasonable polariton densities. Furthermore, the cooling is slowed down since only the exciton part in the polariton is able to transfer energy to the lattice (Doan et al 2005) Both effects hinder to establish quasi equilibrium, which we consider to be one condition for classification as BEC. At least bosonic stimulation has been seen (Deng et al 2003) and interesting features like parametric scattering under resonant pumping have been reported, too (Baumberg et al 2000) For the latter, a description as 'driven condensate' would be probably right. In a genuine BEC, we hope to see a condensate whose phase coherence 284 Bose Einstein condensation of excitons by R. Zimmermann Table 18.1. Bose Einstein condensation of excitons is rather improbable if not impossible since: Excitons are: due to: While for atoms, instable radiative decay stability is given hard to equilibrate slow cooling sophisticated within lifetime cooling in use composite bosons electron hole pairs, electron ion Mott transition! plasma is far away strongly non ideal dipolar repulsion repulsion is weak (hinders condensation) builds up spontaneously, and is not triggered from outside. A more exotic version of exciton condensation is discussed for the quantum Hall effect in electron bilayer systems. At half filling, a description in terms of electrons and holes within the Landau level can be applied. Using the exciton terminology provides a new look on this system (Eisenstein and MacDonald, 2004) but it may be que
author_names: R Zimmermann
title: BOSE EINSTEIN CONDENSATION OF EXCITONS: PROMISE AND DISAPPOINTMENT
year: 2008

abstract: The bosonic atoms used in present day experiments on Bose Einstein condensation are made up of fermionic electrons and nucleons. In this Letter we demonstrate how the Pauli exclusion principle for these constituents puts an upper limit on the Bose Einstein condensed fraction. Detailed numerical results are presented for hydrogen atoms in a cubic volume and for excitons in semiconductors and semiconductor bilayer systems. The resulting condensate depletion scales differently from what one expects for bosons with a repulsive hard core interaction. At high densities, Pauli exclusion results in significantly more condensate depletion. These results also shed a new light on the low condensed fraction in liquid helium II.
author_names: Stefan M A Rombouts

Kris Van Houcke
title: Consequences of the Pauli exclusion principle for the Bose Einstein condensation of atoms and excitons
year: 2005