ivchenko_bookreg

5.2 Emission Spectra of Localized Excitons

211

εmax(t) = εmaxcw + ε0	2 ln ln ω0τ0		+ X¯ − X¯
		ln ω0t

more convenient for comparison between theory and experiment on 2D systems.

According to (5.17, 5.30) the simplest theory of localized-exciton energy relaxation contains five parameters: E0, ε0, N0, ω0, τ0 and, additionally, the

¯ − ¯

sixth parameter X X. The recombination lifetime τ0 can be found by measuring the PL integral intensity J as a function of time delay and comparing the observed dependence with (5.18). A single-exponential character of the decay J(t) is of a crucial importance: linear variation of ln J(t) with time serves as the main reason for assuming τ0 to be energy-independent. The mobility edge E0 can be found from the peak position of the absorption or PLE spectrum. This allows to relate the emitted photon energy ω to the localization energy ε = E0 − ω, see (5.17). A value of εcwmax is found from the Stokes shift of the cw PL spectral peak with respect to E0. This imposes one condition on three parameters, ε0, ω0 and N0. The final choice of these parameters is made from fitting both the experimental curve εmax(t) and PL spectral shape I(ω, t).

For the purposes of illustration, Figs. 5.1 and 5.2 show the time evolution εmax(t) and cw spectrum of low-temperature PL measured at T = 2 K

˚

from a 20A-thick Zn0.8Cd0.2Se/ZnSe QW. Other experimental data including the time-resolved PL integral intensity J(t), time-resolved spectral intensity I(ω, t) for several frequencies ω and spectra I(ω, t) at di erent delay times

t are presented in [5.3]. The best fit parameters are τ0 = 170 ps, εcwmax = 13 meV, ε0 = 8 meV, ω0 = 1013 s−1, N0 = 0.053. Note that for this set of

¯ − ¯ ≈ parameters the correction X X in (5.30) equals to 0.18.

It should be noted that in the transfer term (5.14) we neglect the spatial correlation between two successive hops related to a non-Markovian nature of the hopping processes. An existence of the correlation can be explained by considering the nth and the (n + 1)th hops, respectively, from site rn−1 to rn and from site rn to rn+1 (n > 1). Since we take into account only hops to the optimal neighbors then, for the given configuration of sites, the site rn+1 certainly lies outside the sphere Vd of radius |rn − rn−1| centered at rn−1. Otherwise the exciton would have jumped directly from rn−1 to rn+1 avoiding the site rn. Thus, the straightforward theory must exclude the possibility of hopping into this sphere from the site rn, see details in [5.2]. The dotted curve in Fig. 5.2 is calculated by making allowance for the correlation between successive hops. One can see no remarkable deviation from the theory disregarding the non-Markovian correlation (dashed curve).

5.2.2 Nonmonotoneous Behavior of the Stokes Shift with Temperature

Baranovskii et al. [5.4] used the Monte-Carlo simulation procedure to model the same scenario of multi-hopping relaxations of localized excitons and, for

212 5 Photoluminescence Spectroscopy

ε

˚

Fig. 5.1. The temporal shift of the PL peak measured at T = 2 K in the 20 A Zn0.8Cd0.2Se/ZnSe QW and calculated from (5.30) with the values of parameters indicated in the text. The dashed horizontal line indicates the position of the PL peak under steady-state excitation. [5.3]

the identical set of parameters, obtained an excellent agreement between the simulation and the kinetic analytical theory represented by (5.14, 5.22). They employed a similar simulation technique for finite temperatures and confirmed the non-monotonous temperature dependence of the cw PL maximum demonstrated previously by Zimmermann et al. [5.5]. Physically, this curious phenomenon can be interpreted taking into account that, at T = 0, the PL is dominated by excitons finding themselves on accidentally isolated localization sites acting as pores. For such sites, or traps, the lifetime with respect to hopping down to the nearest lower-energy neighbor exceeds the recombination time. At low, but finite temperatures, an exciton trapped by an e ective pore has an opportunity for further energy relaxation by hopping

5.2 Emission Spectra of Localized Excitons

213

Photon energy (eV)

	2.62		2.61		2.60		2.59
	1.0
PL intensity	0.8
	0.6
Normalized	0.4
	0.2
	0.0
	0	5	10	15	20	25	30	35

Localization energy (meV)

Fig. 5.2. Comparison of the experimental time-integrated PL spectrum (solid) with the steady-state PL spectrum calculated neglecting (dashed) and taking into account (dotted) the correlation between successive hops. [5.3]

first up to the nearest higher-energy neighbor and then down to a deeperenergy site.

To illustrate and make the physics of the PL-peak temperature shift more transparent, let us consider three sites O, O and O1 with the localization energies ε, ε , ε1 (ε < ε < ε1) assuming the initial exciton generation to occur only to the site O. Then the steady-state occupancies f, f , f1 of these three sites satisfy the following set of rate equations

214 5 Photoluminescence Spectroscopy

(τ0−1 + wO O + wO1O)f − wOO f = ΓO , (τ0−1 + wO1O + wOO )f − wO Of = 0 ,

τ0−1f1 − wO1Of − wO1O f = 0 .

Here ΓO is the generation rate to the site O, wαβ is the hopping rate for the acoustic-phonon assisted transition from site β to the site α, and the upward transitions O1 → O and O1 → O are ignored. If the site O is a pore at zero temperature then wO1O < τ0−1. Let site O be the optimal at T = 0 with respect to O, which means that wO O > wO1O. Site O1 is assumed to satisfy the condition wO1O > wOO . It can be shown that, under the above assumptions and for wO O > τ0−1, the ratio f1/(f +f ) is given approximately by the product τ0wO O. At T = 0, i.e., for vanishing wO O, this ratio is equal to τ0wO1O which is small as compared with τ0wO O. Thus, the average exciton energy indeed shifts downwards (the localization energy shifts upwards) with allowance made for the e ective relaxation channel O → O → O1.

Here we generalize the kinetic theory presented in the previous subsection from zero to finite temperatures [5.6]. We use the Miller-Abrahams expression

w(ε, ε , r) = ω		exp	2r		θ(ε		ε )	ε − ε	(5.31)
			− L	−		−
	0							kB T

for the hopping transition rate between the sites ε and ε separated by the distance r. In the limit T → 0 this expression reduces to (5.6). In the light of (5.31) we characterize each localized-exciton state by three parameters ε, ε and r, where ε is the localization energy for a given site, while ε and r are the localization energy and the distance to an optimal site with the maximum value of w(ε, ε , r) in the given local configuration. The probability to occupy such a state is denoted by f (ε, ε , r). The energy distribution N (ε) which determines the photoluminescence spectral intensity, I(ω) N (E0 − ω), is related to f (ε, ε , r) by

N (ε) = g(ε) ∞ dε ∞ drPε(ε , r)f (ε, ε , r) ,

(5.32)

00

where Pε(ε , r) is the distribution of optimal neighbors. In the following we restrict the consideration to 2D systems. Then, for uncorrelated localized sites distributed in the 2D space, one can write instead of (5.15)

Pε(ε , r) = 2πrg(ε ) exp [−U (ε, ε , r)] ,

(5.33)

U (ε, ε , r) =	dε2dr2 2πr2 g(ε2) ,	(5.34)
	Ω

the integration is performed over the area Ω in the (ε2, r2) space, where

w(ε, ε2, r2) > w(ε, ε , r) .

ρ(ε)

5.2 Emission Spectra of Localized Excitons

215

If the correlation between the successive hopping processes is ignored the kinetic equation for f (ε, ε , r) under the cw photoexcitation has the form

1

f (ε, ε , r, t) + I

{

f

}

= Γ (0)

(t)

(5.35)

τ0

ε,ε ,r

similar to (5.13) but with the modified transfer term

Iε,ε ,r

f

}

= w(ε, ε , r) f (ε, ε , r)

(5.36)

{

∞

∞

g(ε )

− 0

0

dε1dr1

1

Pε1

(ε, r1) w(ε1, ε, r1) f (ε1, ε, r1) .

g(ε)

In the limit of zero temperature one has, respectively,

w(ε, ε , r) = w(r)θ(ε − ε) = ω0 exp(−2r/L) θ(ε − ε) ,

Pε(ε , r) = g(ε ) Pε(r)θ(ε − ε) , ρ(ε)

f (ε, ε , r) = f (ε, r) θ(ε − ε) ,

and (5.35) reduces to (5.20).

Figure 5.3 compares the results of calculation with the PL experimental

˚ ˚

data obtained in a cubic CdS/ZnSe 19A/19A SL in the temperature range 5–35 K. The circles (experiment) and curves (theory) show the purely kinetic contribution to the PL-peak shift

∆Emax(T ) = ωmax(T ) − ωmax(0) − δEg (T ) ,

(5.37)

where ωmax(T ) is the spectral position of the PL maximum at the temperature T, δEg (T ) is the non-kinetic shift which appears due to temperature variation of the SL band gap and is found from the micro-PL spectra, see the next subsection. The solid curve in Fig. 5.3 is the result of computer simulation, the dashed curve was obtained by numerically solving (5.35, 5.36). In order to qualitatively take into account the inhomogeneous broadening, the theoretical spectra I(ω, E0) calculated for a fixed value of the exciton mobility edge E0 were convoluted with a Gaussian

− ¯ −

J( ω) dE0 F (E0 E0)J0(E0 ω) ,

0 −	0	√π∆				−	∆
		1					¯	2
F (E	E¯ ) =				exp		E0 − E0 .

The dimensionless parameters chosen to calculate the PL spectra are as follows: ω0τ0 = 103 and N0 = g0ε0π(L/2)2 = 0.4. The inhomogeneous broadening was described by a Gaussian with ∆ = 2 ε0. At zero temperature the PL

216 5 Photoluminescence Spectroscopy

	0.00
PL peak shift	-0.04
	-0.08
Relative
	-0.12
	0	10	20	30	40
		Temperature			(K)

˚ ˚

Fig. 5.3. Temperature dependence of the PL-peak shift in a 19 A/19 A CdS/ZnSe SL. The shift is defined in accordance with (5.37) and normalized to the lowtemperature PL Stokes shift of 18 meV. Full circles, experiment; full curve, computer simulation; broken curve, kinetic theory. [5.6]

peak occurred at εmax ≡ E0 − ωmax = 3.53 ε0. The Stokes shift εcwmax was estimated from the energy di erence between the PL and PLE maxima to

be 18 meV. From the PL Stokes shift a value of (18/3.53) meV or ≈ 5 meV was obtained for ε0. One can see that the simulated PL shift ∆Emax(T ) and that calculated by using the kinetic theory are in good agreement.

5.2.3 Micro-Photoluminescence Spectroscopy

Usually optical studies of nanostructures, including those presented in Figs. 5.1–5.3, are carried out by illuminating macroscopic sample areas. This macro-PL probes large ensembles of localized sites in QWs or QWRs and

5.2 Emission Spectra of Localized Excitons

217

QDs in quantum-dot arrays. In this case narrow spectral features of individual quasi-0D excitons are hidden in PL spectral peaks inhomogeneously broadened and smoothed. At present, in PL or catodoluminescence spectroscopy, it has become possible to probe only a few localized sites or dots using a micrometer and even submicrometer spatial-resolution technique and combining it with high spectral and temporal resolution [5.7–5.10]. This can be achieved either by reducing the size of the laser spot on the sample or by reducing the area of the sample, e.g., by opening a series of apertures in an opaque metal film deposited on the surface of semiconductor nanostructure [5.11, 5.12]. Novel techniques have been developed that have led to the measurements of the Raman scattering, optically-detected nuclear- magnetic-resonance and nonlinear-optical microscopical spectra of individual 0D excitons, in addition to the micro-PL spectra.

Figure 5.4 shows three selected micro-PL spectra taken with a spatial resolution of 1 µm and recorded at di erent temperatures from the

˚ ˚

CdS/ZnSe 19A/19A SL used to measure the temperature shift of macro-PL peak (Fig. 5.3). The spectra are normalized with respect to the PL-intensity maximum and shifted vertically against each other for clarity. In addition to a broad PL background, a structure of narrow superimposed lines is observed with the full-width half-magnitude of 300 µeV. These lines show up due to the radiative recombination of strongly localized excitons forming quasi-0D states. Above 35 K, the narrow line emission is hardly observable, indicating enhanced delocalization of excitons. Within the experimental accuracy, the shift of the line positions with increasing temperature is identical for all the lines and corresponds to the temperature-induced shift of the SL band gap δEg deducted in (5.37) in order to define a purely kinetic change, ∆Emax, in the PL-peak shift. The shifts δEg (T ) for T=5 K and T=35 K di er by −1.4 meV. As one can see from Fig. 5.3, with the temperature increasing from 5 to 35 K, macro-PL spectra show a more pronounced red shift of the emission peak (−3.5 meV). This e ect, also seen for the envelope of the micro-PL spectra, is clear evidence in favor of phonon-assisted exciton multi-hopping to deeper localized states, described in the previous subsection.

In Fig. 5.5, the results of a computer simulation of micro-PL spectra are displayed. For the simulation 50 subsystems were taken each containing 1000 localized-exciton sites randomly distributed with equal probabilities in the 2D space within a square area and with weight g(ε). The exciton mobility edge in each subsystem was chosen randomly in accordance with the Gaussian distribution. The spectra were calculated for the same configuration of localized sites but for di erent temperatures, kB T = 0, 0.4ε0 and 0.8ε0. The parameters of the theory are the same as those used to calculate the macroPL spectra and the temperature dependence of ∆Emax presented in Fig. 5.3 by solid line. The simulated PL spectrum is calculated as the sum

Mn

I(ω) = fi(m) ∆(ω − ωi(m)) ,

m=1 i=1

218 5 Photoluminescence Spectroscopy

Fig. 5.4. Temperature dependent µ-PL spectra (spatial resolution ≈ 1 µm) taken

˚ ˚

from a 19 A/19 A CdS/ZnSe SL. [5.6]

where the index m enumerates the subsystems from 1 to M = 50, ωi(m) is the resonance frequency of the i-th localized-exciton state (1 ≤ i ≤ 1000) in the m-th subsystem, fi(m) is the occupation number of the states (m, i), and the function ∆(Ω) describes the homogeneous broadening of a single line. It is taken in the Lorentzian form

1 γ

∆(Ω) = π γ2 + Ω2

with γ being equal to 0.02ε0/ .

In agreement with the experiment, a smooth spectral background can be seen in Fig. 5.4, reflecting the macro-PL spectrum, with a set of narrow lines corresponding to individual contributions of localized sites. Varying the

5.2 Emission Spectra of Localized Excitons

219

Fig. 5.5. µ-PL spectra computer-simulated for three di erent temperatures. [5.6]

temperature leads to exciton redistribution over the localization sites and, therefore, to changes in the intensities of the narrow lines and to evolution of the PL background. Since the band gaps are kept constant in the simulation procedure, the energy positions of the individual narrow lines remain unchanged.

5.2.4 Excitons in Quantum Wells Containing Free Carriers

Doping of QW structures can be unintentional, homogeneous, or selective. Selectivelyor modulation-doped quantum well is a heterostructure in which a layer of donors (or acceptors) is introduced within the barrier region. The spatial separation between these donors and the 2D gas of electrons (or holes) formed in the well, strongly inhibits carrier scattering. The flexibility in the

220 5 Photoluminescence Spectroscopy

choice of the structure parameters, such as the spacer width, doping concentration, etc., allows one to obtain a 2D electron gas (2DEG) with widely varying properties. Applying a gate voltage enables one to alter the system from a high-density quasi-metallic 2DEG to a low-density noninteracting one. In another method to study the system of coexisting excitons and 2D-carrier gas, the latter is generated by photoexciting the structure above the band gap and its density is controlled by the excitation intensity. At a large electron density the excitons are suppressed due to the screening and band-filling e ects. When the electrons are removed from the well, an exciton line is gradually restored in the optical spectra. We discuss here two problems related to excitons localized in modulation-doped QWs, namely localized trions (see, e.g., [5.13,5.14]) and scattering of free carriers by localized excitons [5.15, 5.16].

Although extensive work has been carried out on the 2D trions X−, in many of these studies a recurrent and controversial question keeps arising from time to time: how localized are the trion states and how does confinement a ect the Coulomb interaction. An existence of localized trion states in modulation-doped GaAs/AlGaAs QWs was demonstrated by Finkelstein et al. [5.13] who observed a correlation between the decrease of the conductivity and the appearance of excitons and trions as the electron density is lowered. Another evidence of localized trions comes from comparison of the theoretical and experimental binding energies of negatively charged exciton X− as a function of the well width a. Although the qualitative behavior of the binding energy εtr as a function of well width a agrees with theoretical predictions, values of εtr for the narrow QWs is at least twice the predicted value [5.17]. This discrepancy between theory and experiment is likely a consequence of the localization of the trion due to QW width fluctuations. Tischler et al. [5.14] have demonstrated single-trion spectroscopy using high spatial resolution. They presented a comparative study of the fine structure of single localized excitons and trions discussed in Sect. 5.4.1.

In addition to the formation of a three-particle bound state, negatively or positively charged exciton, a free carrier and exciton can collide scattering each other. In the quantum theory of scattering, the problem of ‘electron - hydrogen atom’ scattering is next in simplicity to that concerning collisions between two elementary particles. In the physics of semiconductors, similar process is represented by scattering of a free electron or hole by WannierMott exciton. New aspects of this three-body problem were revealed in recent studies of the ‘free carrier - exciton’ interaction in 2D systems, namely in QW structures [5.18, 5.19]. Of special interest is a problem of interaction between localized excitons and free carriers photoexcited by an additional above-gap radiation at low temperatures [5.15, 5.20]. The theoretical aspects of this particular problem were studied by Golub et al. [5.16]. Following this study, we discuss the delocalization of localized excitons by free carriers in a QW structure. We use the Born approximation, take into account that the