11. Exciton polariton and its role in the interband magneto-optics of semiconductors

To study the exciton polariton (EP), the integrated absorption K of the exciton absorption line can be chosen as the most

useful and informative response function. It is defined as

K(γ ) = −∞∞ dω α(ω − ω0, γ ), where γ = / is the EP non-radiative damping parameter. For bulk-EP waves

propagating in a quasi-homogeneous medium with some spatial dispersion, the integrated absorption is expressed by

an analytical dependence derived and studied in paper [45]. As was shown, when parameter γ increases from γ = 0, the integrated absorption also increases in accordance with the

of the dependence inflexion corresponds to the equality of radiative and non-radiative dampings. Here, ω0 and ωLT are the resonance frequency of the transition into exciton ground state and longitudinal-transversal (LT) splitting, correspondingly,

Figure 29. Observation of the Coulomb-well oscillatory levels in the (In,Ga)As/GaAs heterosystem. n = 2 . . . 10 are numbers of the oscillatory levels.

Figure 30. Calculated ‘Coulomb well’ and CW oscillatory levels.

ε0 is the background permittivity and M is the exciton translational mass.

This dependence is fundamental in analysis of both the exciton absorption and exciton-polariton one. At γ /γ c > 1, it provides a possibility of determining the LT splitting and transition oscillator strengths. When the non-radiative damping is below the critical value, the EP integrated absorption is its linear function and decreases to zero together with it, irrelatively to the damping physical nature. Among the factors of the non-radiative damping, there may be impurities, defects, phonons, electric fields, free carriers, other excitons, etc.

Observations of the fine structure of the exciton-polariton absorption in the broadened quantum wells and barriers (1999– 2003) were conducted using samples with multiple quantum

Figure 31. Schematic band structure and cross-section of the sample for upper-barrier exciton localization. The structure consists of

10.5 periods, each including a 13 nm GaAs layer surrounded by 10 In0.1Ga0.9As/ GaAs supperlattices with layer thicknesses 3.4 and 6.5 nm, correspondingly. Computed electron and hole envelopes of the waive functions for above-barrier states are shown at the top and bottom of the band scheme.

Figure 32. Fan diagram formed by the main SL exciton state (SLE) and an above-barrier exciton (ABE) in the coordinates x =ω0(l+1/2). The diagram was constructed according to the energies of experimental peaks adjusted by the computed binding energies of excitons (see inset). The slope of these lines gives the reduced cyclotron mass as a function of the transition energy.

wells. In those experiments, the barrier layer was relatively wide, so that it did not fit the definition of ‘quantum’ one. Nevertheless, in some ‘above-quantum’ range of thicknesses, such a layer can be considered as quantizing the exciton as a whole. In the InGaAs/GaAs system, we observed a fine line

structure belonging to the exciton polariton from the GaAs barrier between quantum wells InGaAs. When the physical barrier thickness differed from the exciton one by the doubled radius of the light-and-heavy-hole exciton [46] (see figure 33), this structure demonstrated ‘exciton interference’.

In the case of phonons, one can introduce the critical temperature Tc determined by the condition γ (Tc) = γ c, provided that the function γ (T) is monotonous. The study of temperature dependence of the integrated absorption of various crystals and multiple quantum wells allowed us to observe the typical behaviour of this quantity (see figure 34), which was described above. We have been also able to determine such parameters as oscillator strength and the longitudinaltransverse splitting [46–49] (see figure 35). We partly use these results below to verify the EP nature of the diamagnetic exciton. A series of new observations of the effects of the EP integrated absorption was carried out in 1991 to 2003. These

included the first observation of EP in semiconductor solid solutions AlGaAs [49] (see figures 36 and 37; 2003) as well as the recent study of the EP integrated absorption in multiple quantum wells [50] (2006–2011).

12. Polariton nature of the diamagnetic exciton

At present, it can be regarded as fully established that the properties of the quasi-particle named ‘diamagnetic exciton’ determine practically any interband optical phenomenon in semiconductors placed in a high magnetic field [9, 11, 36]. The DE concept has passed examinations in more than ten and half as much again of semiconductor crystals. However, detailed analysis of the DE quantitative data shows a more complicated nature of the diamagnetic exciton when one takes into account the results on the EP integrated absorption. It can

Figure 35. Temperature dependence of the exciton-polariton integrated absorption in a variety of crystals and in AlGaAs/GaAs MQW.

be precisely described only in a framework of another concept, namely the concept of diamagnetic-exciton polariton (DEP).

The DEP problem was in a shadow since the qualitative analysis of the spectral features was quite sufficient to establish the nature and magnitude of the crystal band parameters in earlier magneto-optical studies. In fact, quantitative measurements could be performed in some rare cases only. Those cases required special actions, for example, simultaneous registration of transmission and reflection spectra. Such spectra have been measured, e.g. in paper [16], where some new facts were found out. Foregoing figure 8 presented a quantitative magneto-absorption spectrum [16]. It reveals rather unexpected behaviour of the spectral peak at least for the most long-wave state characterized by the electron Landau number l = 0. The peak amplitude αmax decreases with increasing magnetic field down to some value αmax and only then increases (see figure 38). At the same time, in a general consideration, both the transition oscillator strength f and amplitude of the magneto-absorption could only grow with the magnetic field B (in proportion to B2) because of the field compression of the exciton wavefunction in the plane perpendicular to the field. The oscillator strength versus magnetic field dependence is f (B) = f(0)(1 + dn B2), where dn is the dipole moment of the exciton with principal quantum number n, ground state magnetic susceptibility d0 =

ε aexc3/μc2.

Some conclusions can be drawn from available data. At low temperature (T = 1.7 K) and B = 0, the exciton ground state will have the integral equal to 8 eV cm−1 for GaAs and 16 eV cm−1 for AlxGa1-xAs where x = 0.15, saturation level Ksat being 69 and 87 eV cm−1 for GaAs and Al0.15Ga0.85As, respectively. Figure 39 presents the spectrum of the absorption edge of GaAs at two different temperatures, typical for the EP. Having measured line half-width at liquid-helium temperature and provided that the line shape is Lorentzian, one can find that the exciton amplitude appears to be significantly below

(a)

(b)

(c)

Figure 36. Extraction of absorption spectra from the optical transmission and reflection spectra (at different temperatures increasing from 5 K to 300 K in the direction to the coordinate origin).

the level of the states’ continuum. The integral absorption changes non-monotonously with magnetic field, namely first decreases and then increases (see figure 40). Thus, the structure of the exciton absorption edge is rather typical for the exciton polariton when the ground state does not reach the level of the continuum. The levels are aligned with increasing temperature, i.e. with a significant increase in the damping factor (see figure 37).

The initial part of the K(B) dependence is another interesting peculiarity. This dependence turns out to be sinking despite the expected increase in oscillator strength. For both GaAs and AlGaAs, the dependence of K (B) has the shape of the curve with a minimum (see figure 40), which is usably approximated by the expression: K(B) = K(0) (1+dnB2)(1 − pB1/2). When magnetic field increases up to the values of an order of Bmin = 2–2.5 T only, the increase of oscillator strength in the magnetic field begins to dominate, and there is overall growth of the integral absorption. The decrease can only be explained using the EP concept. On the basis of the dependence shown in the inset to figure 37,

Figure 38. The amplitude of the ‘bottom’ (the most short wave) DE state with increasing magnetic field (left to right).

this decrease corresponds to reduction of the damping with increasing magnetic field.

To explain the decreasing part of the dependence, one can involve the magnetic ‘freezing out’ of the impurity, which has been studied in papers [49–50]. Magnetic-field increase results in an increase in the binding energy of a donor ED. The latter

affects the concentration of charged donors, i.e. leads to its reduction. ND+ = ND [1 + 2exp{(EF + ED − ωc/2)/k0T}]−1, where EF is the Fermi level energy, ωcc is the electron cyclotron

energy. Variational calculation of ED has been made in paper [51, 52] making use of the test Pokatilov–Rusanov functions [53]. It has given the theoretical dependence ED(B), which agrees with the results of Hall measurements at the low field (high-field Hall-effect measurements are distorted by the field ‘freezing out’ of electrons due to the growth of ED(B)).

Considering the change in the exciton damping to be (B) = σ ND+ where σ is the cross section of the exciton scattering on a charged impurity centre, using the data of the experimental dependence K(B), and taking into account both the impurity compensation rate and the formation of near-surface volume charge, one can obtain satisfactory agreement for GaAs.

The behaviour of the integrated absorption in a magnetic field is a bit more complicated in solid solutions as well as in heavily doped semiconductors [54, 55]. Taking into consideration the potential fluctuations, the calculated magnetic-field variation of the concentration of charged donors ND+ turns out to be much more efficient. This explains the fact that the fall of integrated absorption K with the magnetic-field increase is deeper in AlGaAs than in GaAs.

So, the analysis of integrated absorption K as a function of the magnetic field for two kinds of semiconductor crystals (compound GaAs and quasi-binary solid solution Al0,15Ga0,85As) clearly shows (by an example of the lowest state related to the zero Landau level) that magneto-excitons (excitons modified by the magnetic field) are indeed not just DE but a quasi-particle of more complicated nature, namely diamagnetic-exciton polariton (DEP).

A question arises: can one extend these conclusions to the other DE states, e.g. ones with the Landau number l > 0? It is worth noting that, to observe polariton effects, the ‘subcritical’ damping γ < γ c is necessary. Besides, one can ascertain that switching on magnetic-field keeps polariton properties of the first DE series at least. So, looking at the behaviour of half-widths of lines related to higher states with greater Landau numbers lc > 1, we can suppose that the γ < γ c rate is observed up to energies exceeding Eg significantly (see, for example paper [16]). Furthermore, the oscillator-strength increase together with a possibility of simultaneous decrease of damping lead to moving up (towards higher damping) the border line of the observation of the EP effects.

There is no doubt that the DEP will become a basis of interband magneto-optical phenomena in other crystals studied usually using the most perfect samples which show the ‘exciton’ behaviour without a magnetic field already.

13. Modelling the low-dimensional state of matter in the experiments with excitons in high and ultra-high magnetic fields

Due to specific properties of low-dimensional systems, the latter will be the main ‘constructional base’ for microand opto-electronics of this century. However, the practical materialization of these properties straightforwardly depends on both the technological progress and improvement in the synthesis of appropriate nanostructures. In the case of ‘quantum-well’-like structures (quasi-two-dimensional systems), the modern growth techniques, such as molecularbeam epitaxy (MBE), allow one to control the layer-by-layer growth and to ensure nearly ideal structure quality. Still, such a success has not been achieved in the cases of either ‘quantum- wire’-like structures (one-dimensional) or ‘quantum-dot’-like ones (zero-dimensional). Such structures require the quantum confinement in more than one direction, and they cannot thus be grown in a wholly controllable process without the use of the nanolithograph technique. Besides, the processes related to the self-organization of nanostructures are the only solution in the latter case. Indeed, these processes turn out to be very effectual for the ‘quantum-dot’ synthesis, although not ensuring either ideally identical sizes or a perfect order arrangement of the ‘dots’. For ‘quantum wires’, there is not even a possibility of such a kind. Consequently, the case of creation of a 1D structure through applying either magnetic or

Figure 40. AlGaAs integrated-absorption dependence on the magnetic field.

electric restraining fields is especially important. In this case, the cyclotron orbits guarantee the confinement within the plane perpendicular to the applied magnetic field, free movement along the field axis determining the 1D state. This offers a chance to study processes that are inaccessible in practice when dealing with real technological structures. Using an MBEgrown quantum well as an initial object, one can produce an artificial zero-dimensional structure of a controllable size by means of the application of a magnetic field (see figure 41).

Thus, the simulation of low-dimensional processes with the ultrahigh magnetic field allows one to investigate