Biexcitons and Exciton-Exciton Interactions in Semiconductors

The concept of the excitonic molecule was introduced independently by Moskalenko [1] and Lampert [2]; later, the excitonic molecule was called the biexciton [3]. The biexciton represents the bound state of four Fermi quasiparticles, namely two electrons and two holes. More simply, it can be regarded as a bound state of two excitons.

Besides biexcitons, one can in general talk about “trions” and “quaternions,” which are bound states of any combination of three or four electrons and holes, respectively. A trion always carries charge; a quaternion may or may not have charge, biexcitons being one kind of quaternion. Several different types of trions and quaternions are expected to be stable in semiconductors in both bulk and two-dimensional (2D) structures; later in this chapter we discuss a proposal [4] for superconductivity based on charged quatemionic bound states of electrons and holes.

Since biexcitons are also integer-spin bosons, they are expected to obey the same Bose statistics as excitons, including Bose narrowing of the energy distribution, discussed in Chapter 1. This has been seen in the semiconductor CuCl. We review experiments on Bose effects of biexcitons in CuCl at the end of this chapter.

We have already briefly discussed the exciton-exciton interaction in Subsection 2.1.3 in terms of the estimate of the s-wave-scattering length that goes into models of the weakly interacting Bose gas. As we mentioned there, this interaction is essentially a van der Waals force, but with the additional complications of comparable electron and hole masses and electron-hole exchange. In this section, we treat this interaction in greater detail, since it determines whether excitonic molecules and/or electron-hole liquids (EHLs) can form.

Introduction

As discussed briefly in Chapter 1, the general theory of phase transitions in the electronhole system predicts that in a system with a simple band structure, when annihilation and polariton effects are neglected, there will always be a region of phase space in which BEC of electron-hole pairs occurs. In this chapter we examine some of the more complicated features of the electron-hole phase diagram.

The many-body system of electrons and holes appears deceptively simple. If one neglects the possibility for electron-hole annihilation, one can consider the following apparently simple system: two types of Fermi particles with equal mass and opposite charge, interacting only by means of Coulomb interaction. As we will see in this chapter, however, the phase diagram for this system is extraordinarily complex, and the full theory for the phase diagram is not yet complete. We emphasize that most of these complexities arise solely from the above simple model and not from the other complicated features of solids such as band structure, band-to-band recombination, etc. Therefore much of this theory applies equally well to the case of electron-positron plasma or electron-ion plasma. The similarity of the e-h system to the electron-proton system, for example, suggested the existence of excitonic molecules [1,2] and, moreover, the possibility of a gas-liquid phase transition of exciton and biexciton gas into a Fermi electron-hole liquid (EHL). The EHL is the bound state of a macroscopically large number of electrons and holes [3-6], similar to the many-electron-proton system.

As stressed by Keldysh, however, this similarity is not complete. In a comprehensive and instructive review [7], he pointed out the peculiar properties distinguishing the e-h system from any other, which we summarize here.

Condensate Formation in the Bose Gas

An important question in connection with the experiments aimed at achieving BEC of a weakly interacting Bose gas with a finite lifetime is the time needed for the formation of the condensate. It is not at all evident that the condensation can always take place within the lifetime of the system. For the case of liquid He, the assumption of equilibrium seems satisfied because 4He atoms, which do not decay, undergo roughly 1011 interparticle-scattering events per second, assuming classical hard-sphere scattering. In other boson systems, however, nonequilibrium effects may have a significant effect. For excitons in a semiconductor, the lifetime of the particle may be only a few hundred scattering times. Spin-polarized hydrogen also has a finite lifetime to nonpolarized states [1], and alkali atoms can recombine into molecules [2]. Naively judging from classical behavior, one might assume that hundreds of scattering times may seem more than adequate time to establish equilibrium. The case of Bose condensation is unique, however, since a macroscopic number of particles must enter a single quantum state that by random-scattering processes alone is highly improbable. Therefore we need not assume that the time for Bose condensation is short.

Early experiments with orthoexcitons in CU2O seemed to indicate that the excitons might not have enough time to condense within their lifetime [3]. As discussed in Subsection 1.4.3, the orthoexciton density increased to approach the BEC boundary given in Eq. (1.18) and highly degenerate Bose-Einstein momentum distributions were observed, but a significant condensate fraction did not appear. Instead the gas increased its temperature to accommodate extra particles, reaching temperatures well above the lattice temperature.

Introduction

In this chapter we generalize the theory of the weakly nonideal Bose gas to the case of the interaction of excitons with acoustic or optical phonons, i.e., in the case of interaction with another Bose subsystem, the quasiparticle number of which is not conserved. To simplify the question, we assume that the exciton-phonon interaction leads to only intraband exciton scattering and does not change the total number of excitons in the band. In other words, we consider an isolated exciton band or suppose that the processes of interband scattering are not important because of the symmetries of the exciton and the phonon states and the values of the interaction constants.

The theory of the interaction of condensed excitons with phonons has primary importance in the question of whether excitons will be superfluid. In Chapter 2, we saw that the fact that the excitons are particle-antiparticle pairs does not prevent them from becoming superfluid. At first glance, however, it is not easy to draw an analogy to other Bose condensates such as liquid He. In the case of liquid He, interactions with the external world are limited to the surfaces of the container, while for excitons, the phonon field interpenetrates the same region of space as the exciton gas. The situation is more analogous to the case of a superconductor. Unlike a BCS superconductor, however, in an exciton condensate the lattice phonons have nothing to do with the pairing. We shall see that nevertheless the excitonic condensate remains superfluid in the presence of a phonon field. At the end of this chapter we review recent experiments that may be evidence of excitonic superfluidity.

The Bogoliubov Model of the Weakly Nonideal Bose Gas

In this chapter we review the basic theory of Bose-Einstein condensation (BEC) of excitons. To start, this means we must review the basic theory of BEC of any kind of particle, the theory known as the Bogoliubov model, after the foundational contributions made by N. N. Bogoliubov. This model introduces all the strange things associated with BEC: spontaneous symmetry breaking, off-diagonal long-range order (ODLRO), macroscopic occupation of a single quantum state, etc.

It is often said that physicists who spend years studying quantum mechanics eventually warp their intuition so much that things like ODLRO seem normal. It is well worth stepping back every now and then to think about just how strange a Bose condensate is. First, consider the idea of spontaneous symmetry breaking. Many systems exist in which the underlying physics, expressed in the Hamiltonian, do not favor one state over another. The a priori probability of occupation of the different states by a particle is equal, i.e., symmetric. Nevertheless, in some cases, thermodynamics requires that a macroscopic number of particles must somehow “choose” one of the states preferentially. If they did not, the system would not be in equilibrium, i.e, could not have a definable temperature. If the particles preferentially choose one state, then the underlying symmetry has been broken.

In the context of magnets, the breaking of the symmetry of the magnet (to have all its constituent dipoles lined up one way and not another) is energetically favored, since there are terms in the Hamiltonian that give lower energy for aligned dipoles, e.g., μS1 · S2.

Turbulence and chaos in a semiconductor crystal? These are just some of the intriguing predictions of nonlinear optics with coherent excitons. In this chapter we discuss some aspects of nonlinear coherent optics with the participation of the excitons, photons, and biexcitons. As in many nonlinear systems, these effects arise because of feedback mechanisms, i.e., the light impinging on the system affects the electronic states, which in turn affect the dielectric constant of the medium through which the light passes. These effects are closely related to the phenomena of self-organization [1-8] and optical bistability [7-10] and self-pulsations and the appearance of chaos [2-6]. The cooperative nonlinear coherent processes in optical systems have recently attracted much attention [7-20]. These investigations have stimulated the development of new mathematical methods [21-31] and have opened up possibilities for many new applications. The theoretical and the experimental investigations in these directions give rise to the theory of solitons [32∧2] and lead to the new developments in the optics of ultrashort pulses [43-58].

Ultrafast phenomena such as self-induced transparency and nutation are related to the propagation of solitons and pulse trains, whereas the transient stages of the ultrashort lightpulse penetration in the media are described by area theorems. These coherent ultrafast phenomena typically take place during intervals of time less than the relaxation time, i.e., femtoseconds to picoseconds in typical solids. Collisions and incoherent scattering processes destroy the coherent evolution of the excited particles. But if coherent macroscopic states of the excitons, photons, and biexcitons have been formed because of spontaneous or induced BEC, in this case one can consider their time evolution over intervals of time much greater than the relaxation time.

Introduction. Polaritons and Semiconductor Microcavities

As we have mentioned previously, BEC of excitons or biexcitons and lasing can be seen as two limits of the same theory. Lasing occurs in the case of strong electron-photon coupling (recombination rate fast compared with interparticle-scattering rate) while excitonic BEC occurs in the case of weak electron-photon coupling (recombination rate slow compared with the interparticle-scattering rate.) A laser can be seen as a Bose condensate in which the long-range phase coherence exists in the photon states [1], while in the exciton condensate the coherence exists in the electronic states. This is one of the reasons for some of the confusing debates in the early history of excitonic Bose condensation – in many systems there is not a sharp distinction between an excitonic condensate and supperradiance, i.e., luminescence with enhanced intensity due to stimulated emission [2-4].

In the previous chapters, we have so far considered a laser light source only as a source of excitons by means of quantum transitions. In Chapter 6, the laser radiation was taken into account as a given external factor. This chapter is dedicated to phenomena related to a strong and noticeable exciton-photon interaction, when the light significantly influences the energy spectrum of the high-density excitons [5-9]. As we will see, this leads to many fascinating effects related to instabilities.

A strong exciton-photon interaction implies a strong polariton effect, in which photon and exciton states are mixed (as in Fig. 1.3). When this mixing is strong, it is not obvious how to define the ground state of the system.

Because in many of the experiments discussed in this volume the semiconductor CU2O was used, and because this material is less well known than other semiconductors such as Si and GaAs, we review here some of the basic properties of CU2O.

CU2O is actually one of the earliest semiconductors studied [1], and the proof of the existence of excitons in this crystal [2, 3], which confirmed the theories of Frenkel [4], Wannier [5], and Mott [6], led to an exciting period of expansion in the field of excitonic physics in the late 1950s and early 1960s, during which time many of the optical properties of this intriguing crystal were established [7-19]. (For general reviews, see Refs. 20 and 21.) The same property that makes it an excellent material for nonlinear optical effects, namely, a very large excitonic binding energy (150 meV), makes it a poor electrical conductor even at room temperature. Therefore CU2O has not been studied or fabricated extensively by the electronics industry, and high-quality samples are still not widely available.

Band Structure. Most of the complexities of the band structure of CU2O stem from the d orbitals of the Cu atoms in the valence band. CU2O forms into a cubic lattice with inversion symmetry, which is the Oh symmetry group [22]. In the Oh symmetry, the five 3d orbitals are split by means of the crystal field into a higher threefold degenerate Γ+5 level and a lower twofold degenerate Γ+3level. The lowest conduction band, on the other hand, is formed from the 4s orbitals that have Γ+1 symmetry in the Oh group.

What is an Exciton?

Many people seem to have trouble with the concept of an exciton. Is it “real” in the same sense that a photon or an atom is? Does the motion of an exciton correspond to the transport of anything real in a solid?

Simply put, an exciton is an electron and a hole held together by Coulomb attraction. Of course, for some people the idea of a “hole” is a difficult concept, so this may not help much. Nevertheless, a hole is a “real” particle and so is an exciton.a Modern solid-state theory [1,2] gives equal footing to both free electrons and holes as charge carriers in a solid, exactly analogous to the way that electrons and positrons are both “real” particles, even though a positron can be seen as the absence of an electron in the negative-energy Dirac sea, i.e., a backwards-in-time-moving electron.

All excitons are spatially compact. The strong Coulomb attraction between the negatively charged electron and the positively-charged hole keeps them close together in real space, unlike Cooper pairs, which can have very long correlation lengths because of the weak phonon coupling between them. The sizes of excitons vary from the size of a single atom, e.g., approximately an angstrom up to several hundred angstroms, extending across thousands of lattice sites. Excitons are roughly divided into two categories based on their size. An exciton that is localized to a single lattice site is called a “Frenkel” exciton, after the pioneering work of Frenkel [3] on excitons in molecular crystals. Frenkel excitons appear most commonly in molecular crystals, polymers, and biological molecules, in which they are extremely important for understanding energy transfer.

Nonequilibrium Theory of the Optical Stark Effect in the Excitonic Range of the Spectrum

One of the most important applications of the theory of Bose condensation of excitons is the optical Stark effect, or “AC Stark effect, ” first demonstrated for excitons in Cu2O [1] and since seen in several semiconductors and semiconductor heterostructures (e.g., Refs. 2-5). This effect is a promising tool for the field of optical communications. One of the major interests in this field is the development of all-optical-switching methods, by which light signals are switched on or off directly by other light signals, just as in electronics, electrical signals switch other electrical signals. At the present, most optical communications systems use electrical signals (e.g., electro-optic or acousto-optic devices) to switch the light signals, which means that the limiting bandwidth of the system is controlled by that of the electrical signals, not the optical bandwidth. The optical Stark effect offers this possibility. When one laser beam impinges on a sample, it can drastically alter the excitonic absorption line shape. Therefore a second beam can see either a transparent or an absorbing medium, depending on the presence of the control beam. This is the optical equivalent of a transistor.

The basic effect is shown in Fig. 6.1. When an intense laser is tuned to a photon energy that is just below the excitonic ground state, the exciton ground state is shifted in frequency and the oscillator strength is altered. This leads to a strong reduction in the optical absorption at the original wavelength of the exciton ground state.

Abstract

We review the theory of coherent pairing of excitons and biexcitons in two-dimensional and three-dimensional semiconductor structures.

Introduction

The coherent pairing of bosons, formally analogous to Cooper pairing of electrons in superconductors, and the coexistence of particle and pair Bose–Einstein condensation (BEC) was first studied in Refs. [1–4]. BEC of excitons in semiconductors was investigated using the approximation of coherent pairing of the electrons and holes [5]. Later on, the idea of coherent pairing of bosons was applied to excitons [6–8]. In Ref. [6], it was assumed that the boson pairs consist of two electron–hole pairs instead of two excitons, and it was suggested that the new collective state is a BEC even though the two-pair bound state does not exist. It was proved independently [7, 8] that in a system of bosons and excitons with an attractive pair interaction sufficient for biexciton formation, the coherent pairing of excitons with momenta k and −k coincides with the BEC of biexcitons with zero translational momentum.

The possibility of particle and pair BEC in a system of four species of excitons with different values of spin projections of the electron and hole was studied in Ref. [9]. It is known [10, 11] that the interaction of an exciton of either type with other exciton species is repulsive on the average, but that pairs of excitons with antiparallel spins of electrons and holes interact attractively, and biexcitons can be formed.