Bose-einstein Condensation of Excitons: Promise and Disappointment

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 questioned if a classification as exciton BEC really works. Quite a good overview on experimental attempts to find exciton BEC and the ambiguities involved in their interpretation has been given by David Snoke at the NOEKS-7 conference in 2003 (Snoke, 2003). The paper title ‘When should we say we have observed Bose condensation of excitons?’ speaks on its own. A broader survey of relevant work is compiled in a special issue of Solid State Communications (Snoke, 2005), being an offspring of a workshop held in 2004 near Pittsburgh (USA). The actual strong interest in exciton BEC – often revitalizing old concepts – is surely triggered by the tremendous success to reach experimentally BoseEinstein condensation of atomic systems (Cornell and Wiemann, 2002; Ketterle, 2002). In contrast, clear-cut proofs of excitonic BEC in solid state systems are still missing. The small exciton mass is an advantage, but other facts are less promising. Several of them are listed in Table 18.1 and compared with their counterparts in atomic systems. In spite of the many cons and only few pros, I am rather confident that we will see clear evidence of excitonic BEC in semiconductor systems at some time. However, before being too optimistic on a too short time scale, we should try to learn more from the hard work which was needed to reach BEC in atomic systems. To formulate tasks for theory, we should concentrate more on the way towards exciton BEC, while the study of the condensate itself might be deferred to a later stage. A particular problem is to understand how the excitons can manage to cool down. Nonequilibrium Green’s functions and the ‘Keldysh technique’ (Keldysh, 1964) provide the proper machinery, which has been used in the present context by Haug and coworkers (Schmitt et al ., 2001). Even when assuming that quasiequilibrium has been established (as I will do in the following Sections), we face another complication: Since excitons are composite quantum particles with a sizable interaction, the formulation and solution of the many-exciton problem The ideal Bose gas in a trap 285 is more challenging than treating the nearly ideal atomic Bose gas. However, in doing so we will learn a lot which might be of use for other problems in interacting solid state systems. 18.2 The ideal Bose gas in a trap Let us start with compiling results being valid for ideal (i.e. noninteracting) bosons. The general expression for the boson number in dependence on temperature T and chemical potential μ is