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Time-resolved studies of excitonic polaritons
57
Journal of Luminescence 48&49 (1991) 57-66 North-Holland
Time-resolved studies of excitonic polaritons J. Aaviksoo Institute of Physics, Estonian Acad. Sci., 142 Riia SIr., 202400 Tartu, Estonia, USSR
The kinetic properties of the secondary emission of excitonic polaritons are reviewed, based on experiments on molecular (anthracene) and semiconductor (CdS, GaAs, InP, etc.) crystals at low temperatures. First, coherent propagation effects are considered, including group velocity renormalization, the appearance of optical precursors and photon echo type responses. Second, the problem of the radiative lifetime of polaritons is addressed, elucidating the appearance of gigantic oscillator strengths and the light (polariton) trapping phenomenon. Third, the kinetic characteristics of the luminescence spectrum are discussed with the emphasis on polariton relaxation, propagation, and the distinction between Raman scattering and luminescence. Fourth, time-resolved reflection experiments are explained in the polariton framework.
1. Introduction 1.1. The polariton concept
The interaction of light (photons) with material resonances (excitons) brings about the formation of new self-consistent excitations called polaritons [1]. In classical electrodynamic theory, light in a homogeneous dielectric medium is described by the dielectric constant e, which can be calculated from microscopic theory. In the case of excitons ~ is very well modelled by a set of Lorentz oscillators and one has 2
coo—co —icoF
(1)
the new normal waves in the medium; the polaritons. Quantum theory regards the polariton as a superposition of a photon state and an exciton state with common wave vector k, which forms an eigenstate of the coupled exciton-photon Hamiltonian. The general shape of the polariton dispersion curve is given in fig. 1. The splitting of the dispersion curves at exact resonance k = k0 =
~f~w0/c is given by /4
2V=
\
1/2
( _2~~~)coo,
(3)
“ E0
and reflects the coupling strength of excitons and photons. Another important parameter is the LTsplitting (the width of the stop band) 4’Traw0
where ~ is the dielectric constant due to higher lying resonances, a the polarizability, coo the frequency, and F the phenomenological damping rate of the material resonance [2]. If coupling of the local oscillators (spatial dispersion) is important w0 is replaced by 2m~
(2)
~LT
~
=
‘
which is proportional to the polarizability a of the resonance. The oscillator strength F and the dipole moment D, which are often used to characterize the resonance, are related to the polarizability by the following relations 2 a— 2e F —
w
0 my
where m~is the effective mass of excitons, and ~ is, wave dependent wave ~ = ~(w,equation k) [3]. By hence, solving thevector homogeneous e(w, k) = c2k2/w2 one gets the dispersion curve of 0022-2313/91/$03.50
© 1991
—
2 2, 3j~ v D where v is the volume of the unit cell. =
Elsevier Science Publishers B.V. (North-Holland)
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58
J. Aaviksoo
/
Time-resolved studies of excilonic polaritons
~I
/~ / /
/ ,“~
>.
c.o
a
I i i I i i
i
/
~
_.
j~LT —
—
~
—
/
n TPh=~l(k~wIvPhlk~w)l(1)P~
//
(6)
where V~his the polariton(exciton)-phonon coup-
(
ling strength, n the phonon occupation number, T’ph varies a very broad timescale depends and p over the density of final polaritonand states. strongly on polariton frequency through the density of states function p. At low temperatures, when processes with phonon absorption are sup-
iii III IL
and therefore the initial polariton state 1k, co) decays into new polariton states 1k’, co’). In the weak coupling limit this decay rate can be calculated from the Fermi Golden Rule formula 2ir
~
ii,!
15 x 10— 15 s in the case of exciton—photon oscillations in anthracene). In real crystals, one has to take into account at least two additional processes, which bring about the decay of the polariton states. First, polaritons (strictly speaking their exciton components) are scattered by phonons (impurities, carriers, etc.),
pressed, the lifetime
k 0
WAVEVECTOR Fig. 1.The polariton dispersioncurve w region,
w(k)inthe resonance
T
=
F~’is in the range
10_52_
10_s s. The decay rate F, after the summation over all possible scattering channels, broadens the initial polariton state and when multiplied by the corresponding group velocity ~5= aco(k)/ak determines also the mean free path of the polariton, i.e. its absorption length.
The polariton description is useful for both Wannier-Mott and Frenkel excitons as well as for (homogeneous) crystals with resonant impurities, For some common crystals, where polariton effects have been observed 4’rrcs 3of the lowest1.5x102 exciton (GaAs), resonance equals 1.3x10 (CdS) and 7.6x 10_2 (anthracene). 1.2. The temporal characteristics of polaritons Polaritons in the above sense are stationary eigenstates of an infinite dielectric medium and therefore they exhibit no temporal characteristics per se. However, if a nonstationary initial state is created in the medium (e.g. a bare exciton with k = k0) it evolves in time in an oscillatory way. The period T of these oscillations is determined by the corresponding coupling strength and may be extremely short (T = 1/ V = 2(4’rraco~/~) 1/2 =
The second inevitable decay channel is the transformation of polaritons into external photons on the crystal surface. The rate of this process is determined by v FR(1—R)~, (7) where R is the reflection coefficient of polaritons at the surface and L the length of the crystal. FR strongly depends on the polariton state under consideration. For a number of polariton states with energies w~, co 0, R = 1 due to total internal reflection (TIR), and FR~ 0. These polariton modes are termed nonradiative or waveguide 5-i07 cm/s around w modes. For R < 1, c ~ V8 = i0 0, and even for comparatively thin samples (L= 10 p~m)radiative lifetimes of the order of iOns are observed. On the other hand, for some polariton states, especially in ultra-thin (quasi two-dimensional)
J. Aaviksoo / Time-resolved studies of excitonic polaritons
crystals, the radiative lifetime can be less than 1O12 s. It is important to note that all the aforementioned parameters (except the exciton—phonon coupling strength) which determine the polariton decay rates can be calculated from the dielectric function E(co, k). In order to give an overview of the range of these relevant parameters: the group velocity Vg, the density of states p(co) dco ~x k2v,’ dII dco, and the index of refraction n = (r(co, k))’~2are plotted for the anthracene crystal in the vicinity of the lowest exciton resonance in fig. 2. As one readily sees, v 5 and p vary over several orders of magnitude in a narrow frequency range below the resonance. Hence a corresponding range of decay rates may be expected. The present article reviews some recent results of time-resolved studies of excitonic polaritons. In section 2 the coherent propagation of polaritons is studied yielding direct information on the polariton dispersion curve. Further, the existence of optical precursors [4,5] is demonstrated experimentally, as is the appearance of multiple photon echoes in the linear response of photochemically bleached media. Section 3 elucidates the phenomenon of gigantic oscillator strengths, deduced from the viewpoint. measured Section radiative lifetimes, from the polariton 4 deals with luminescence kinetics and their dependence on
5
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1
0
toil
:
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_—‘~
~ ,-.i
~
—
2. Coherent propagation experiments 2.1. Time-of-flight measurements
The most direct proof of the polariton model is the measurement of its dispersion curve. Besides other methods [6], the time-of-flight method has been used, where the frequency dependence of the group velocity v8 is measured. For a number of materials (GaAs [7], CuCI [8], CdSe [9], anthracene [10,11], etc.), good agreement between theory and experiment has been found. The experimental data for anthracene crystal are presented in fig. 3 [11]. As one can see, a considerable reduction of the light speed takes place. At a large detuning co0= coo— V (V= 2000 cm~’ in anthracene) V5 = ~c/V~, but immediately below t)the group velocity the resonance (coo—co0 ~ 2 cm~ of polaritons (light) in anthracene becomes smaller
0
~k2! 4~J2Ô0
.j~iZ
12ó0
600
0 200
600
D
___,_~__-“
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detection frequency and location with respect to excitation. The different kinetic properties of Raman and luminescence processes are discussed. Section 5 presents some results of time-resolved reflection experiments, relating them to polariton decay and propagation. The final section draws the main conclusions.
•io~
II ~
-i0~,~
59
I
1111
—50
0
BETUNING (cm’) Fig. 2. Density of states p, refraction index n, and group velocity Vg of polaritons in anthracene below the exciton resonance at = 25097 cm~.The following parameters have been used: e0=2.5, F0.24, and m5 = 100 m~.n0=../~ and the background values of n and v5 are shown,
TIME (ps) Fig. 3. The time-of-flight curves of polaritons in anthracene. The base lines indicate the pulse frequency with respect to the dispersion curve. The pulses at zero delay are due to another polarization and serve as a reference. T= 1.8K, L= 18 urn.
60
J. Aaviksoo
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Time-resolved studies of excitonic polaritons
than the velocity of sound (2 x i05 cm/s [10]). The group velocity concept is directly applicable only in the region of weak absorption and for sufficiently long (spectrally narrow) light pulses. Spectrally broader pulses suffer from group velocity dispersion that results in a broadened and asymmetric shape of the transmitted pulse (see fig. 3). In the strong absorption region pulses are severely distorted and the group velocity does not describe the propagation of the pulse as a whole. In these cases the derivative of the real part of the dispersion curve may become larger than c, or even negative. As shown in ref. [12], this “abnormal” velocity describes the propagation ofthe maximum of the Gaussian pulse in the limit of an infinitely thin sample. Of course, in this case the energy velocity VE v 5 still obeys the requirement VE < c. In conclusion: light velocities in a crystal vary over several orders of magnitude and coherent 5 times slower than in propagation of light i0 vacuum has been observed. 2.2. Optical precursor detection As a special case of light propagation in resonant media, the propagation of a truncated sinusoidal monochromatic wave has been considered [4,5,13,14]. It has been predicted that the steep front of the truncated wave should travel at the limiting speed of light (Sommerfeld precursor), whereas the main wave arrives at the appropriate group delay for the given frequency. Recently, the observability of optical precursors was analysed [15], showing that they can be expected in the propagation of an exponential pulse with a steep (<1 ps) rising edge near exciton resonances. Experiments performed on a 0.2 p.m thick GaAs crystal near the lowest exciton resonance at 1.515 eV [16] have confirmed the theoretical predictions; the fast rising front of the exponential pulse appears, independent of detuning of the pulse, at zero delay, whereas the rest of the pulse is delayed depending on the detuning of the carrier frequency co~ with respect to the resonance frequency (see fig. 4). The appearance of optical precursors can be easily understood in the polariton framework. The rising edge of the incident exponential pulse contains frequency
I—
~
A 9.2
\ \
‘,.L
~
14 ~
,1 \
~ 2 5 ~
\ ~-11
.
..1.
.1
D~ AY IPS Fig. 4. The cross-correlation functions of the transmitted pulses dependent on detuning i (in cm~)with respect to the exciton resonance in GaAs (w~=12221 cm~’).T= 1.8 K. The uppermost curve depicts the incident exponential pulse, the spike at zero delay corresponds to the Sommerfeld precursor.
components which are far away from the resonance, and travel undisturbed through the crystal at y = c. Most of the pulse energy, however, is concentrated at the carrier frequency w~near the resonance and travels at the corresponding group velocity v 5~ffc. It has been shown theoretically[17], that all discontinuities of the incident pulse (or its derivatives) should propagate at the limiting velocity of the medium. 2.3. Photon echoes from a linear medium Coherent pulse propagation in a linear dispersive medium, characterized by 6 = 6(0)), can result in very complicated transmitted pulse shapes. Making use of the photochemical hole-burning effect [18], one can modify F by bleaching the medium at given frequencies and hence influence the polariton dispersion curve in a specific way. It can be easily shown, that a sinusoidal modulation of the transmission spectrum results in the appearance of a delayed satellite to the transmitted pulse
J. Aaviksoo / Time-resolved studies of excitonic polaritons
[19]; a photon echo. In general, arbitrary responses to the incident short pulse can be generated at the output, subject only to the causality requirement [20]. As an example, we have observed two satellites, at 30 and 400 ps delays, of the transmitted 75 fsmm pulse passing a plate 66 fs pulse 0.7 thickbypolystyrene doped through with dyea molecules and subjected to a photochemical holeburning process [21]. This technique of combined hole-burning and recall of the burnt-in responses has been developed into a method of time-andspace domain holography [22].
3. Radiative lifetime of polaritons 3.1. Gigantic oscillator strengths Polaritons in a finite crystal decay into external photons at the crystal boundary. In the case of a homogeneous polariton wave inbyaeq. 3D(7). crystal the rate of this process is governed A series of experiments in thin high-quality GaAs samples [23] have shown that extremely short radiative lifetimes can be observed, compared to the lifetimes determined from the oscillator strength of the underlying resonance according to the formula [24] 3 T= m0c 2e2co2f~ (8) For GaAs (f= i0~) one gets r=40 p.s [23], whereas the experimentally observed radiative decay time 3.3 ns yields f=’ 1. These giant oscillator strengths are even more pronounced for excitons in quantum well structures [25,26], where a radiative lifetime of 330 ps has3-fold beenincrease observedof [27], to a 6 of x f Thecorresponding radiative coupling 10 (quasi) 2D-excitons to the external photon field was first analysed in ref. [28] and later treatments were given in refs. [29,30]. It was shown that in a 2D array of dipoles the total oscillator strength of the N x N dipoles is concentrated in those k-states whose wave vector is less than that of light of the same energy in vacuum: i.e. k <2 times co0/c. less The than number the of those states of is about (k0/ kB) states, where k total number 2D polariton 8=
61
‘Tv/a is the Brillouin zone boundary wave vector. Therefore, the effective oscillator strength of these excitons may be very large 2
f2D/f~(kB/kO) 2. (9) = (A/a) For 2D surface excitons on anthracene crystals [31] this factor is i05, which means that the corresponding radiative lifetime of these states T2D < 1 ps. This result has been proved experimentally in ref. [32]. For Wannier-Mott excitons, a in (9) must be replaced by the exciton Bohr radius a~, and for GaAs (a = 12 the predicted x 3-fold shortening0 of thenm) radiative lifetime is7 in 10 good agreement with the experimental observation. This strong (superradiant [29]) coupling of the 2D exciton states with k < w 0/c to the 3D photon field can be understood as a coherent emissionThe of 2 dipoles. alatter phased array of N~ N = (A/a) possesses a joint oscillator strength, which exceeds by N2 times that of an isolated dipole. As a simple example one may consider a dimer molecule, which has a symmetric excited state with a radiative lifetime T 2 = T/2. In conclusion: in thin (quasi-2D) crystals some of the exciton (polariton) states a very radiative which in the have limiting case short of a 2D crystal lifetime, may be (A/a)2 times shorter than the lifetime of a localized exciton with the same oscillator strength f 3.2. “Blind” states Alongside polariton states with gigantic radiative decay rates, polariton states exist in a finite crystal which transformed into undergo external photons at all.cannot Thesebenonradiant modes total internal reflection (TIR) at crystal boundaries and cannot be excited by external light. In the case of a slab dielectric all polaritons with k~ 1>co0c/n, where k~is the wave vector component parallel to the slab plane, are nonradiative. It is easy to see that for the limiting case of a 2D crystal these nonradiativestates statesof form of N the excitonic the the 2D majority slab; Nnr/ 2~’=l—10~~. I —(a/A)
62
J. Aaviksoo
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Time-resolved studies of excitonic polaritons
In an isotropic 3D slab crystal, polaritons within the solid angle 12 =4’rr(l —cos OTJR), where °TJR= arcsin(1/n), are nonradiative. As an example, 12 = 0.13 4’rr for n = 2, and 12 = 5 x iO3 4ir for n = 10 (values up to n = 20 have been observed experimentally in anthracene [33]). One sees that in both cases most of the polaritons at the given frequency are trapped in the crystal. These modes are made visible through crystal imperfections (impurities, phonons, etc.), which couple those nonradiative modes to radiative modes. It is important that the waveguide modes may have rather large group velocities and hence transfer energy efficiently over considerable distances compared to pure excitonic transport.
4. The kinetics of polariton luminescence Until now the temporal properties of a specific polariton state or their coherent superpositions (polariton wave packets) have been considered, which are manifested in time-resolved translinear wave-mixing experiments. In time-resolved mission, reflection (see below) and different nonluminescence experiments, on the other hand, the evolution of a (quasi-stationary) distribution (in energy, wave vector, space, etc.) of polaritons is followed. Hence the individual properties of the underlying states are averaged and the problem of polariton kinetics is replaced by that of the energy relaxation and transport. Out of a great number of effects the dependence of the luminescence decay kinetics location, and temperature, on detection andwavelength, the Raman spatial versus luminescence distinction problem are considered here.
3Ops after the excitation a narrow (—=5 cm1) distribution of polaritons is formed just above the bottleneck region at coo. Due to low group velocities and high refractive indexes these polaritons cannot escape the crystal and therefore no resonant emission line is observed in the spectrum (fig. 5). These polaritons can emit an acoustic phonon (co 5~=1O-3Ocm~) or one of the high frequency optical phonons (49, 394, 1403 cm~, etc.). The rates of these processes obey (6) and determine the lifetime of this narrow distribution (TNDE 1 ns), which is manifested by the decay times of the corresponding narrow emission lines in the spectrum (E in fig. 5). Polaritons, which are further scattered by acoustic phonons, form a broad distribution below W~ (A, B, C, D in fig. 5) and can leave the crystal due to enhanced radiative decay. However, the lifetime TBD 2 ns> TND due ID
E
ANTHRACE T 2K
NE A 25i00
25050 WAVENUMBER
(cm~)
25000
—
F
_____________________________ z ID w _______________________ ZC U)
/
4.1. The dependence on detection wavelength
B
The polariton luminescence spectrum is formed in the course of relaxation in the bottleneck region around the resonant exciton frequency w~[34]. The time-resolved luminescence experiments, carried out on CdS [35] and anthracene [36,37], have revealed a strong dependence of the decay kinetics on detection frequency. Let us illustrate this by anthracene luminescence at low temperatures. In
A
]
0
500
TIME
1000
1500
(ps)
Fig. 5. The luminescence spectrum of anthracene in the bottleneck region (upper curve) and the corresponding kinetic curves at frequencies A to G as indicated above.
J. Aaviksoo
/ Time-resolved studies
to the strongly reduced relaxation rate on acoustic phonons in this region. v5= 5 x 10~cm/s and the polaritons can propagate already a distance of about 100 p.m, i.e. away from the excitation spot. Further scattering on high frequency phonons brings polaritons into the spectral region, where radiative decay dominates and they escape easily from the crystal. A more detailed study of the frequency dependence allows one to determine the underlying scattering rates and coupling strengths. So, the decay times of polariton luminescence are influenced by both the polariton-phonon scattering rate and radiative decay rate of polaritons both being strongly frequency dependent. The emission maximum lies at the frequency where these decay rates become equal (F~= F0h) and the corresponding time constant can be used as an estimate of the integral polariton luminescence decay time. 4.2. The dependence on spatial location
Concurrent with energy relaxation polaritons propagate through the crystal and may travel across considerable distances due to TIR effects in a thin crystal slab. To study this effect we have measured the luminescence kinetics at a different location than the excitation spot. Figure 6 represents the corresponding kinetic curve of the 46 cm’ phonon line (E in fig. 5) D = 0.17 cm away from the excitation spot. The observed decay consists of two maxima, one at zero delay and with a decay time r = 2 ns, and the other at a delay ~ T = 8.4 ns. The first maximum appears due to the direct excitation of excitons at the location of detection by the fastThe incident lightsignal and therefore is observed. delayed is due to no thedelay propagation of the 46 cm~ polaritons, created at the excitation spot as waveguide modes across the sample. The slightly larger delay as the direct group delay at the given frequency D/ v 5 = 6.4 ns can be accounted for by averaging over all possible path lengths of the waveguide modes. At the given frequency n = 4.65 41’r, and i.e. the97% corresponding TIR of the polaritons solid angle is frequency 0.97 created at this are nonradiant and travel as waveguide modes across the crystal over several mm. They are made visible by defect or low
of excitonic polaritons
D/v
63
o
=6.4 ~
I—I —IF’— b
*
C
L. 3.2 flS
t2flS LU
I
I
0
5
10
TIME
Ins)
Fig. 6. The kinetics of the 46 cm~ luminescence line of anthracene at a distance D = 0.17cm from the excitation spot. The calculated group delay is indicated. The inset shows the actual geometry, T =2 K.
frequency acoustic phonon scattering. The frequency dependence of this effect is very well demonstrated in fig. 7, where the spatially resolved luminescence kinetics is given for different detection wavelengths. A detailed analysis of these data will be published elsewhere [38], but let us make here some general remarks in this connection. First, the lifetime of polariton emission in the whole crystal can exceed that of the emission in the excitation spot up to a factor of ten. Second, polaritons travel as waveguide modes over macroscopic distances groupmeasuring velocity luminesVg(Wo 3c.Tat hird,a when 46cm’) = l0~ cence decay at the excitation spot the propagation of polaritons away from the detection region must be accounted for alongside with other decay channels. It is interesting to note that the observed maximum delays ~2O ns are close to the luminescence lifetime in anthracene at room temperature [39]. 4.3. Temperature dependence of the decay kinetics
A very illuminating series of experiments has been carried out on 2D excitons in GaAs-GaA1As
J. Aaviksoo
64
10
20
~
10
20
Time-resolved studies of excitonic polariions
0
2
0
/
2
0
10
20
20
0
10 20 TIME Ins)
ANTHRACENE
0
—46 —100 ENERGY ~ (cm1) Fig. 7. The kinetics of the luminescence spectrum of anthracene at a distance D
quantum wells [27], where a direct dependence of the luminescence lifetime was established on the homogeneous line width (resp. temperature). This somewhat surprising result can be understood in the polariton framework by recalling the existence of a relatively small number of (super)radiative states and a large reservoir of nonradiant blind states. Scattering of the former by phonons (or impurities, etc.) couples them effectively to nonradiant states and the larger the share of the blind states in the broadened polariton state, the longer the radiative lifetime. In the high temperature limit, when all possible exciton states are effectively mixed up, the luminescence decay time should approach that of a localized exciton in correspondence with (8).
0
=
0.24cm from the excitation spot.
and we show that time-resolution of the line can provide additional insight into the above distinction problem. In a pre-resonant (coo—wL<~iLT) Raman scattering experiment on anthracene the kinetics of the 81 cm1 line was measured. The latter appeared as a well pronounced peak in the emission spectrum shifting together with the excitation frequency ([40], fig. 8). However, the kinetic curve shows two distinct components, a fast and
~
~ DETUNINC
(dIrt)
>-
4.4. Raman scattering versus luminescence
I-
T~i0K uJ
In the polariton picture the secondary emission spectrum is formed in the course of multiple scattering of polaritons by phonons and a subsequent transformation of the polariton into an external photon. Hence a clear-cut difference between scattering and luminescence is missing. Nevertheless, in several cases some spectral features can be unambiguously related to a certain Raman process
T~2K ‘—I
_____________________
0 200
TIME
600
(ps)
Fig. 8. The spectrum of the preresonant scattering of anthracene (upper curve, arrow indicates the excitation frequency, *: the
81 cm~ line). Lower: the kinetic curves of the 81 cm ‘line at two temperatures.
I Aaviksoo
/
Time-resolved studies of excitonic polariions
a slow one. The interpretation of the data is as follows. The fast component has the lifetime of the initial polariton wave packet, propagating perpendicular to the slab plane, and creating a Raman polariton by emitting the 81 cm~optical phonon. At the same time the initial polariton undergoes (quasielastic) scattering by acoustic phonons producing a distribution of polaritons, most of which are waveguide modes and hence have a much longer lifetime than the initial polariton. Subsequent scattering of these polaritons by the 81 cm~ phonon ends up with polaritons of the same energy as in the straightforward Raman process, but their kinetics reflects the lifetime of the secondary polariton distribution, i.e. they give us the slow component of the Raman-like line. Increasing the temperature enhances the slow component due to an increased acoustic phonon scattering rate. In conclusion: temporal resolution of polariton luminescence/scattering lines can provide additional insight into the hidden dynamics of multiple scattering by phonons.
5. Time resolved reflection experiments The reflection spectrum of crystals exhibits distinct structures near excitonic resonances. This structure results in a corresponding transient behaviour of resonantly reflected light pulses. The effect of transient reflection has been theoretically analysed in refs. [41,42,15] and was experimentally demonstrated on GaAs [43] and InP [44]. Pulsed excitation has been shown to induce resonant polarization in the reflecting surface layer of the crystal, which in turn re-emits light in the form of a reflected pulse. Time-resolved studies thus provide a direct means to follow the polarization dynamics in the surface layer. In the case of InP crystals transient reflection near the lowest exciton resonance (fig. 9) can be well described by the straightforward linear response model calculating the corresponding reflection coefficient r(co) proceeding from (1). Three processes contribute to the reflection kinetics [43,45]: dephasing due to scattering by phonons, polariton propagation into the crystal due to spatial dispersion, and radiative decay due to reflection itself. The last two have
65
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-
tO” ~
Z II
~o-~
i
5
0
5
I
I
I
I
10
15
20
20
DELAY
(ps)
Fig. 9. Logarithmic plot ofexperimental cross-correlation functions of pulses (FWHM = 3 ps) reflected from an lnP crystal at the frequency of the exciton resonance (dotted) and theoretical fits to the data (solid line) at two excitation densities. The fit yielded ni~= 0.2m~and F = 1.5 cm’ and 6.1 cm~as adjustable parameters, T= 1.8 K.
essentially nonexponential kinetics. It is noteworthy in the present treatment that the radiative decay rate of the polariton wave packet in the reflection experiment is approximately equal to the LT-splitting siLT, which yields picosecond lifetimes for GaAs and InP crystals. Hence, transient reflection provides a means to study polariton dynamics in the surface layer and yields the material parameters, which determine the dielectric function F.
6. Conclusions As is evident from the above treatment, polaritons in a real crystal show a large variety of transient features. They can be crudely divided into three categories. First, temporal properties of a coherent polariton state, which are determined by F and the phenomenological damping constant F. Second, temporal properties of a distribution of polaritons, which is largely determined by the interrelation of polariton relaxation (on phonons), propagation, and radiative decay. Third, in specific polariton experiments complicated temporal
66
J. Aaviksoo
/
Time-resolved studies of excitonic polaritons
behaviour may be observed as in the case of photon echo type responses and transient reflection measurements. The appropriate timescales of the transients vary from about 100 fs in the case of a 2D surface polariton on anthracene to about 20 ns for a nonradiant 3D polariton mode in the same crystal.
[16] J. Aaviksoo, J. KuhI and K. Ploog, Phys. Rev. Lett., to be published. [17] L.A. Vainshtein, Soy. Phys. Usp. 19 (1976) 189. [18] W.E. Moerner ed., Persistent Spectral Hole-Burning. Science and Applications (Springer, Berlin, 1988). [19] A. Rebane ci al., Opt. Common. 47 (1983) 173. [20] H. Sönajalg et al., Opt. Common. 71(1989) 377. [21] A. Rebane, J. Aaviksoo and J. KuhI, AppI. Phys. Lett. 54 (1989) 93.
[22] P. Saari, R. Kaarli and A. Rebane, J. Opt. Soc. Am. B 3 (1986) 527.
Acknowledgements
[23] G.W. ‘t Hooft et al., Phys. Rev. B 35 (1987) 8281.
[24] DL. Dexter, Solid State Physics Vol. 6, eds. F. Seitz and
Fruitful cooperation and discussions with many colleagues are gratefully acknowledged, e.g. with V. Agranovitch, J. Kuhl, A. Rebane, K. Rebane, I. Reimand, T. Reinot, P. Saari and I. Tartakovskii.
References [1] U. Fano, Phys. Rev. 103 (1956) 1202. J.J. Hopfield, Phys. Rev. 112 (1958) 1555. V.M. Agranovitch, Soy. Phys. JETP 10 (1960) 307. [2] R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973). [3] V.M. Agranovitch and V.L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer, Berlin, 1984). [4] A. Sommerfeld, Ann. Phys. 44(1914)177. [5] L. Brillouin, Ann. Phys. 44 (1914) 203. [6] B. Hönerlage et al., Phys. Rep. 124 (1985) 161. [7] R.G. Ulbrich and G.W. Fehrenbach, Phys. Rev. Lett. 13 (1979) 963. [8] Y. Segawa, Y. Aoyagi and S. Namba, Solid State Common. 32 (1979) 229. [9] T. Itoh, P. Lavallard, J. Reydellet and C. Benoit a Ia Guillaume, Solid State Common. 37 (1981) 925. [10] NA. Vidmont, A.A. Maksimov and 1.1. Tartakovskii, Pis’ma Zh. Eksp. Teor. Fiz. 37 (1983) 578. [ii] J. Aaviksoo, J. Lippmaa, A. Freiberg and A. Anijalg, Solid State Common. 49 (1984) 115. [121 S. Chu and S. Wong, Phys. Rev. Lett. 48 (1982) 738.
[13] L. Brillooin, Wave Propagation and Group Velocity (Academic, New York, 1960). [14] M. Frankel and J.L. Birman, Phys. Rev. A 15(1977)2000. [15] J. Aaviksoo, J. Lippmaa and J. Kohl, J. Phys. Soc. Am. B 5(1988)1631.
D. Turnbull (Academic, New York, 1958) 353. [25] and K. Ploog, Phys.Vol. Rev.15,B ed. 29 (1984) 7058. [26] L. R. Schultheis Dingle, Festkörperprobleme, H.J. Quisser (Pergamon, Oxford, 1975) 25. [27] J. Feldmann et al., Phys. Rev. Lett. 59 (1987) 2337. [28] V.M. Agranovitch and O.A. Dubovskii, Pis’ma Zh. Eksp. Teor. Fiz. 3 (1966) 345. [29] M.R. Philpott and P.G. Sherman, Phys. Rev. B 12 (1975) 5381. [30] M. Orrit, C. Aslangul and P. Kottis, Phys. Rev. B 25(1982) 7263. [31] M. Orrit and P. Kottis, Adv. Chem. Phys. Vol. 74, eds. 1. Prigogine and S. Rice (Wiley, New York, 1988). [32] J. Aaviksoo, J. Lippmaa and T. Reinot, Opt. Spectrosc. 62 (1987) 706. [33] M. Orrit, J. Bernard, J.M. Turlet and P. Kottis, J. Chem. Phys. 78 (1983) 2847. [34] J. Toyozawa, Prog. Theor. Phys. Suppl. 12 (1959) ill. [35] P. Wiesner and U. Heim, Phys. Rev. B 11(1975) 3071. [36] M.D. Galanin, E.N. Myasnikov and Sh.D. KhanMagometova, Mol. Cryst. Liquid Cryst. 57 (1980) 119. [37] J. Aaviksoo, A. Freiberg, J. Lippmaa and T. Reinot, J. Lumin. 37 (1987) 313. [38] J. Aaviksoo and T. Reinot, to be published. [39] Y. Nozue, J. Phys. Soc. Jpn. 57 (1988) 3204. [40] J. Aaviksoo, A. Freiberg, T. Reinot and S. Savikhin, J. Lumin. 35(1986) 267. [41] D. Elert, Ann. Phys. Ser. 5 7 (1930) 65. [42] G.P. Agrawal, J.L. Birman, D.N. Pattanayak and A. Pun, Phys. Rev. B 25 (1982) 2715. [43] J. Aaviksoo and J. Kohl, IEEE J. Quantum Electron. QE-25 (1989) 2523. [44] J. Aaviksoo, J. Kohl and I. Reimand, Solid State Common. 72 (1989) 49. [45] J. Aaviksoo and I. Reimand, Opt. Common., to be published.
Journal of Luminescence 48&49 (1991) 57-66 North-Holland
Time-resolved studies of excitonic polaritons J. Aaviksoo Institute of Physics, Estonian Acad. Sci., 142 Riia SIr., 202400 Tartu, Estonia, USSR
The kinetic properties of the secondary emission of excitonic polaritons are reviewed, based on experiments on molecular (anthracene) and semiconductor (CdS, GaAs, InP, etc.) crystals at low temperatures. First, coherent propagation effects are considered, including group velocity renormalization, the appearance of optical precursors and photon echo type responses. Second, the problem of the radiative lifetime of polaritons is addressed, elucidating the appearance of gigantic oscillator strengths and the light (polariton) trapping phenomenon. Third, the kinetic characteristics of the luminescence spectrum are discussed with the emphasis on polariton relaxation, propagation, and the distinction between Raman scattering and luminescence. Fourth, time-resolved reflection experiments are explained in the polariton framework.
1. Introduction 1.1. The polariton concept
The interaction of light (photons) with material resonances (excitons) brings about the formation of new self-consistent excitations called polaritons [1]. In classical electrodynamic theory, light in a homogeneous dielectric medium is described by the dielectric constant e, which can be calculated from microscopic theory. In the case of excitons ~ is very well modelled by a set of Lorentz oscillators and one has 2
coo—co —icoF
(1)
the new normal waves in the medium; the polaritons. Quantum theory regards the polariton as a superposition of a photon state and an exciton state with common wave vector k, which forms an eigenstate of the coupled exciton-photon Hamiltonian. The general shape of the polariton dispersion curve is given in fig. 1. The splitting of the dispersion curves at exact resonance k = k0 =
~f~w0/c is given by /4
2V=
\
1/2
( _2~~~)coo,
(3)
“ E0
and reflects the coupling strength of excitons and photons. Another important parameter is the LTsplitting (the width of the stop band) 4’Traw0
where ~ is the dielectric constant due to higher lying resonances, a the polarizability, coo the frequency, and F the phenomenological damping rate of the material resonance [2]. If coupling of the local oscillators (spatial dispersion) is important w0 is replaced by 2m~
(2)
~LT
~
=
‘
which is proportional to the polarizability a of the resonance. The oscillator strength F and the dipole moment D, which are often used to characterize the resonance, are related to the polarizability by the following relations 2 a— 2e F —
w
0 my
where m~is the effective mass of excitons, and ~ is, wave dependent wave ~ = ~(w,equation k) [3]. By hence, solving thevector homogeneous e(w, k) = c2k2/w2 one gets the dispersion curve of 0022-2313/91/$03.50
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—
2 2, 3j~ v D where v is the volume of the unit cell. =
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58
J. Aaviksoo
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Time-resolved studies of excilonic polaritons
~I
/~ / /
/ ,“~
>.
c.o
a
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i
/
~
_.
j~LT —
—
~
—
/
n TPh=~l(k~wIvPhlk~w)l(1)P~
//
(6)
where V~his the polariton(exciton)-phonon coup-
(
ling strength, n the phonon occupation number, T’ph varies a very broad timescale depends and p over the density of final polaritonand states. strongly on polariton frequency through the density of states function p. At low temperatures, when processes with phonon absorption are sup-
iii III IL
and therefore the initial polariton state 1k, co) decays into new polariton states 1k’, co’). In the weak coupling limit this decay rate can be calculated from the Fermi Golden Rule formula 2ir
~
ii,!
15 x 10— 15 s in the case of exciton—photon oscillations in anthracene). In real crystals, one has to take into account at least two additional processes, which bring about the decay of the polariton states. First, polaritons (strictly speaking their exciton components) are scattered by phonons (impurities, carriers, etc.),
pressed, the lifetime
k 0
WAVEVECTOR Fig. 1.The polariton dispersioncurve w region,
w(k)inthe resonance
T
=
F~’is in the range
10_52_
10_s s. The decay rate F, after the summation over all possible scattering channels, broadens the initial polariton state and when multiplied by the corresponding group velocity ~5= aco(k)/ak determines also the mean free path of the polariton, i.e. its absorption length.
The polariton description is useful for both Wannier-Mott and Frenkel excitons as well as for (homogeneous) crystals with resonant impurities, For some common crystals, where polariton effects have been observed 4’rrcs 3of the lowest1.5x102 exciton (GaAs), resonance equals 1.3x10 (CdS) and 7.6x 10_2 (anthracene). 1.2. The temporal characteristics of polaritons Polaritons in the above sense are stationary eigenstates of an infinite dielectric medium and therefore they exhibit no temporal characteristics per se. However, if a nonstationary initial state is created in the medium (e.g. a bare exciton with k = k0) it evolves in time in an oscillatory way. The period T of these oscillations is determined by the corresponding coupling strength and may be extremely short (T = 1/ V = 2(4’rraco~/~) 1/2 =
The second inevitable decay channel is the transformation of polaritons into external photons on the crystal surface. The rate of this process is determined by v FR(1—R)~, (7) where R is the reflection coefficient of polaritons at the surface and L the length of the crystal. FR strongly depends on the polariton state under consideration. For a number of polariton states with energies w~, co 0, R = 1 due to total internal reflection (TIR), and FR~ 0. These polariton modes are termed nonradiative or waveguide 5-i07 cm/s around w modes. For R < 1, c ~ V8 = i0 0, and even for comparatively thin samples (L= 10 p~m)radiative lifetimes of the order of iOns are observed. On the other hand, for some polariton states, especially in ultra-thin (quasi two-dimensional)
J. Aaviksoo / Time-resolved studies of excitonic polaritons
crystals, the radiative lifetime can be less than 1O12 s. It is important to note that all the aforementioned parameters (except the exciton—phonon coupling strength) which determine the polariton decay rates can be calculated from the dielectric function E(co, k). In order to give an overview of the range of these relevant parameters: the group velocity Vg, the density of states p(co) dco ~x k2v,’ dII dco, and the index of refraction n = (r(co, k))’~2are plotted for the anthracene crystal in the vicinity of the lowest exciton resonance in fig. 2. As one readily sees, v 5 and p vary over several orders of magnitude in a narrow frequency range below the resonance. Hence a corresponding range of decay rates may be expected. The present article reviews some recent results of time-resolved studies of excitonic polaritons. In section 2 the coherent propagation of polaritons is studied yielding direct information on the polariton dispersion curve. Further, the existence of optical precursors [4,5] is demonstrated experimentally, as is the appearance of multiple photon echoes in the linear response of photochemically bleached media. Section 3 elucidates the phenomenon of gigantic oscillator strengths, deduced from the viewpoint. measured Section radiative lifetimes, from the polariton 4 deals with luminescence kinetics and their dependence on
5
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0
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2. Coherent propagation experiments 2.1. Time-of-flight measurements
The most direct proof of the polariton model is the measurement of its dispersion curve. Besides other methods [6], the time-of-flight method has been used, where the frequency dependence of the group velocity v8 is measured. For a number of materials (GaAs [7], CuCI [8], CdSe [9], anthracene [10,11], etc.), good agreement between theory and experiment has been found. The experimental data for anthracene crystal are presented in fig. 3 [11]. As one can see, a considerable reduction of the light speed takes place. At a large detuning co0= coo— V (V= 2000 cm~’ in anthracene) V5 = ~c/V~, but immediately below t)the group velocity the resonance (coo—co0 ~ 2 cm~ of polaritons (light) in anthracene becomes smaller
0
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600
0 200
600
D
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detection frequency and location with respect to excitation. The different kinetic properties of Raman and luminescence processes are discussed. Section 5 presents some results of time-resolved reflection experiments, relating them to polariton decay and propagation. The final section draws the main conclusions.
•io~
II ~
-i0~,~
59
I
1111
—50
0
BETUNING (cm’) Fig. 2. Density of states p, refraction index n, and group velocity Vg of polaritons in anthracene below the exciton resonance at = 25097 cm~.The following parameters have been used: e0=2.5, F0.24, and m5 = 100 m~.n0=../~ and the background values of n and v5 are shown,
TIME (ps) Fig. 3. The time-of-flight curves of polaritons in anthracene. The base lines indicate the pulse frequency with respect to the dispersion curve. The pulses at zero delay are due to another polarization and serve as a reference. T= 1.8K, L= 18 urn.
60
J. Aaviksoo
/
Time-resolved studies of excitonic polaritons
than the velocity of sound (2 x i05 cm/s [10]). The group velocity concept is directly applicable only in the region of weak absorption and for sufficiently long (spectrally narrow) light pulses. Spectrally broader pulses suffer from group velocity dispersion that results in a broadened and asymmetric shape of the transmitted pulse (see fig. 3). In the strong absorption region pulses are severely distorted and the group velocity does not describe the propagation of the pulse as a whole. In these cases the derivative of the real part of the dispersion curve may become larger than c, or even negative. As shown in ref. [12], this “abnormal” velocity describes the propagation ofthe maximum of the Gaussian pulse in the limit of an infinitely thin sample. Of course, in this case the energy velocity VE v 5 still obeys the requirement VE < c. In conclusion: light velocities in a crystal vary over several orders of magnitude and coherent 5 times slower than in propagation of light i0 vacuum has been observed. 2.2. Optical precursor detection As a special case of light propagation in resonant media, the propagation of a truncated sinusoidal monochromatic wave has been considered [4,5,13,14]. It has been predicted that the steep front of the truncated wave should travel at the limiting speed of light (Sommerfeld precursor), whereas the main wave arrives at the appropriate group delay for the given frequency. Recently, the observability of optical precursors was analysed [15], showing that they can be expected in the propagation of an exponential pulse with a steep (<1 ps) rising edge near exciton resonances. Experiments performed on a 0.2 p.m thick GaAs crystal near the lowest exciton resonance at 1.515 eV [16] have confirmed the theoretical predictions; the fast rising front of the exponential pulse appears, independent of detuning of the pulse, at zero delay, whereas the rest of the pulse is delayed depending on the detuning of the carrier frequency co~ with respect to the resonance frequency (see fig. 4). The appearance of optical precursors can be easily understood in the polariton framework. The rising edge of the incident exponential pulse contains frequency
I—
~
A 9.2
\ \
‘,.L
~
14 ~
,1 \
~ 2 5 ~
\ ~-11
.
..1.
.1
D~ AY IPS Fig. 4. The cross-correlation functions of the transmitted pulses dependent on detuning i (in cm~)with respect to the exciton resonance in GaAs (w~=12221 cm~’).T= 1.8 K. The uppermost curve depicts the incident exponential pulse, the spike at zero delay corresponds to the Sommerfeld precursor.
components which are far away from the resonance, and travel undisturbed through the crystal at y = c. Most of the pulse energy, however, is concentrated at the carrier frequency w~near the resonance and travels at the corresponding group velocity v 5~ffc. It has been shown theoretically[17], that all discontinuities of the incident pulse (or its derivatives) should propagate at the limiting velocity of the medium. 2.3. Photon echoes from a linear medium Coherent pulse propagation in a linear dispersive medium, characterized by 6 = 6(0)), can result in very complicated transmitted pulse shapes. Making use of the photochemical hole-burning effect [18], one can modify F by bleaching the medium at given frequencies and hence influence the polariton dispersion curve in a specific way. It can be easily shown, that a sinusoidal modulation of the transmission spectrum results in the appearance of a delayed satellite to the transmitted pulse
J. Aaviksoo / Time-resolved studies of excitonic polaritons
[19]; a photon echo. In general, arbitrary responses to the incident short pulse can be generated at the output, subject only to the causality requirement [20]. As an example, we have observed two satellites, at 30 and 400 ps delays, of the transmitted 75 fsmm pulse passing a plate 66 fs pulse 0.7 thickbypolystyrene doped through with dyea molecules and subjected to a photochemical holeburning process [21]. This technique of combined hole-burning and recall of the burnt-in responses has been developed into a method of time-andspace domain holography [22].
3. Radiative lifetime of polaritons 3.1. Gigantic oscillator strengths Polaritons in a finite crystal decay into external photons at the crystal boundary. In the case of a homogeneous polariton wave inbyaeq. 3D(7). crystal the rate of this process is governed A series of experiments in thin high-quality GaAs samples [23] have shown that extremely short radiative lifetimes can be observed, compared to the lifetimes determined from the oscillator strength of the underlying resonance according to the formula [24] 3 T= m0c 2e2co2f~ (8) For GaAs (f= i0~) one gets r=40 p.s [23], whereas the experimentally observed radiative decay time 3.3 ns yields f=’ 1. These giant oscillator strengths are even more pronounced for excitons in quantum well structures [25,26], where a radiative lifetime of 330 ps has3-fold beenincrease observedof [27], to a 6 of x f Thecorresponding radiative coupling 10 (quasi) 2D-excitons to the external photon field was first analysed in ref. [28] and later treatments were given in refs. [29,30]. It was shown that in a 2D array of dipoles the total oscillator strength of the N x N dipoles is concentrated in those k-states whose wave vector is less than that of light of the same energy in vacuum: i.e. k <2 times co0/c. less The than number the of those states of is about (k0/ kB) states, where k total number 2D polariton 8=
61
‘Tv/a is the Brillouin zone boundary wave vector. Therefore, the effective oscillator strength of these excitons may be very large 2
f2D/f~(kB/kO) 2. (9) = (A/a) For 2D surface excitons on anthracene crystals [31] this factor is i05, which means that the corresponding radiative lifetime of these states T2D < 1 ps. This result has been proved experimentally in ref. [32]. For Wannier-Mott excitons, a in (9) must be replaced by the exciton Bohr radius a~, and for GaAs (a = 12 the predicted x 3-fold shortening0 of thenm) radiative lifetime is7 in 10 good agreement with the experimental observation. This strong (superradiant [29]) coupling of the 2D exciton states with k < w 0/c to the 3D photon field can be understood as a coherent emissionThe of 2 dipoles. alatter phased array of N~ N = (A/a) possesses a joint oscillator strength, which exceeds by N2 times that of an isolated dipole. As a simple example one may consider a dimer molecule, which has a symmetric excited state with a radiative lifetime T 2 = T/2. In conclusion: in thin (quasi-2D) crystals some of the exciton (polariton) states a very radiative which in the have limiting case short of a 2D crystal lifetime, may be (A/a)2 times shorter than the lifetime of a localized exciton with the same oscillator strength f 3.2. “Blind” states Alongside polariton states with gigantic radiative decay rates, polariton states exist in a finite crystal which transformed into undergo external photons at all.cannot Thesebenonradiant modes total internal reflection (TIR) at crystal boundaries and cannot be excited by external light. In the case of a slab dielectric all polaritons with k~ 1>co0c/n, where k~is the wave vector component parallel to the slab plane, are nonradiative. It is easy to see that for the limiting case of a 2D crystal these nonradiativestates statesof form of N the excitonic the the 2D majority slab; Nnr/ 2~’=l—10~~. I —(a/A)
62
J. Aaviksoo
/
Time-resolved studies of excitonic polaritons
In an isotropic 3D slab crystal, polaritons within the solid angle 12 =4’rr(l —cos OTJR), where °TJR= arcsin(1/n), are nonradiative. As an example, 12 = 0.13 4’rr for n = 2, and 12 = 5 x iO3 4ir for n = 10 (values up to n = 20 have been observed experimentally in anthracene [33]). One sees that in both cases most of the polaritons at the given frequency are trapped in the crystal. These modes are made visible through crystal imperfections (impurities, phonons, etc.), which couple those nonradiative modes to radiative modes. It is important that the waveguide modes may have rather large group velocities and hence transfer energy efficiently over considerable distances compared to pure excitonic transport.
4. The kinetics of polariton luminescence Until now the temporal properties of a specific polariton state or their coherent superpositions (polariton wave packets) have been considered, which are manifested in time-resolved translinear wave-mixing experiments. In time-resolved mission, reflection (see below) and different nonluminescence experiments, on the other hand, the evolution of a (quasi-stationary) distribution (in energy, wave vector, space, etc.) of polaritons is followed. Hence the individual properties of the underlying states are averaged and the problem of polariton kinetics is replaced by that of the energy relaxation and transport. Out of a great number of effects the dependence of the luminescence decay kinetics location, and temperature, on detection andwavelength, the Raman spatial versus luminescence distinction problem are considered here.
3Ops after the excitation a narrow (—=5 cm1) distribution of polaritons is formed just above the bottleneck region at coo. Due to low group velocities and high refractive indexes these polaritons cannot escape the crystal and therefore no resonant emission line is observed in the spectrum (fig. 5). These polaritons can emit an acoustic phonon (co 5~=1O-3Ocm~) or one of the high frequency optical phonons (49, 394, 1403 cm~, etc.). The rates of these processes obey (6) and determine the lifetime of this narrow distribution (TNDE 1 ns), which is manifested by the decay times of the corresponding narrow emission lines in the spectrum (E in fig. 5). Polaritons, which are further scattered by acoustic phonons, form a broad distribution below W~ (A, B, C, D in fig. 5) and can leave the crystal due to enhanced radiative decay. However, the lifetime TBD 2 ns> TND due ID
E
ANTHRACE T 2K
NE A 25i00
25050 WAVENUMBER
(cm~)
25000
—
F
_____________________________ z ID w _______________________ ZC U)
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4.1. The dependence on detection wavelength
B
The polariton luminescence spectrum is formed in the course of relaxation in the bottleneck region around the resonant exciton frequency w~[34]. The time-resolved luminescence experiments, carried out on CdS [35] and anthracene [36,37], have revealed a strong dependence of the decay kinetics on detection frequency. Let us illustrate this by anthracene luminescence at low temperatures. In
A
]
0
500
TIME
1000
1500
(ps)
Fig. 5. The luminescence spectrum of anthracene in the bottleneck region (upper curve) and the corresponding kinetic curves at frequencies A to G as indicated above.
J. Aaviksoo
/ Time-resolved studies
to the strongly reduced relaxation rate on acoustic phonons in this region. v5= 5 x 10~cm/s and the polaritons can propagate already a distance of about 100 p.m, i.e. away from the excitation spot. Further scattering on high frequency phonons brings polaritons into the spectral region, where radiative decay dominates and they escape easily from the crystal. A more detailed study of the frequency dependence allows one to determine the underlying scattering rates and coupling strengths. So, the decay times of polariton luminescence are influenced by both the polariton-phonon scattering rate and radiative decay rate of polaritons both being strongly frequency dependent. The emission maximum lies at the frequency where these decay rates become equal (F~= F0h) and the corresponding time constant can be used as an estimate of the integral polariton luminescence decay time. 4.2. The dependence on spatial location
Concurrent with energy relaxation polaritons propagate through the crystal and may travel across considerable distances due to TIR effects in a thin crystal slab. To study this effect we have measured the luminescence kinetics at a different location than the excitation spot. Figure 6 represents the corresponding kinetic curve of the 46 cm’ phonon line (E in fig. 5) D = 0.17 cm away from the excitation spot. The observed decay consists of two maxima, one at zero delay and with a decay time r = 2 ns, and the other at a delay ~ T = 8.4 ns. The first maximum appears due to the direct excitation of excitons at the location of detection by the fastThe incident lightsignal and therefore is observed. delayed is due to no thedelay propagation of the 46 cm~ polaritons, created at the excitation spot as waveguide modes across the sample. The slightly larger delay as the direct group delay at the given frequency D/ v 5 = 6.4 ns can be accounted for by averaging over all possible path lengths of the waveguide modes. At the given frequency n = 4.65 41’r, and i.e. the97% corresponding TIR of the polaritons solid angle is frequency 0.97 created at this are nonradiant and travel as waveguide modes across the crystal over several mm. They are made visible by defect or low
of excitonic polaritons
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63
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=6.4 ~
I—I —IF’— b
*
C
L. 3.2 flS
t2flS LU
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I
0
5
10
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Fig. 6. The kinetics of the 46 cm~ luminescence line of anthracene at a distance D = 0.17cm from the excitation spot. The calculated group delay is indicated. The inset shows the actual geometry, T =2 K.
frequency acoustic phonon scattering. The frequency dependence of this effect is very well demonstrated in fig. 7, where the spatially resolved luminescence kinetics is given for different detection wavelengths. A detailed analysis of these data will be published elsewhere [38], but let us make here some general remarks in this connection. First, the lifetime of polariton emission in the whole crystal can exceed that of the emission in the excitation spot up to a factor of ten. Second, polaritons travel as waveguide modes over macroscopic distances groupmeasuring velocity luminesVg(Wo 3c.Tat hird,a when 46cm’) = l0~ cence decay at the excitation spot the propagation of polaritons away from the detection region must be accounted for alongside with other decay channels. It is interesting to note that the observed maximum delays ~2O ns are close to the luminescence lifetime in anthracene at room temperature [39]. 4.3. Temperature dependence of the decay kinetics
A very illuminating series of experiments has been carried out on 2D excitons in GaAs-GaA1As
J. Aaviksoo
64
10
20
~
10
20
Time-resolved studies of excitonic polariions
0
2
0
/
2
0
10
20
20
0
10 20 TIME Ins)
ANTHRACENE
0
—46 —100 ENERGY ~ (cm1) Fig. 7. The kinetics of the luminescence spectrum of anthracene at a distance D
quantum wells [27], where a direct dependence of the luminescence lifetime was established on the homogeneous line width (resp. temperature). This somewhat surprising result can be understood in the polariton framework by recalling the existence of a relatively small number of (super)radiative states and a large reservoir of nonradiant blind states. Scattering of the former by phonons (or impurities, etc.) couples them effectively to nonradiant states and the larger the share of the blind states in the broadened polariton state, the longer the radiative lifetime. In the high temperature limit, when all possible exciton states are effectively mixed up, the luminescence decay time should approach that of a localized exciton in correspondence with (8).
0
=
0.24cm from the excitation spot.
and we show that time-resolution of the line can provide additional insight into the above distinction problem. In a pre-resonant (coo—wL<~iLT) Raman scattering experiment on anthracene the kinetics of the 81 cm1 line was measured. The latter appeared as a well pronounced peak in the emission spectrum shifting together with the excitation frequency ([40], fig. 8). However, the kinetic curve shows two distinct components, a fast and
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4.4. Raman scattering versus luminescence
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In the polariton picture the secondary emission spectrum is formed in the course of multiple scattering of polaritons by phonons and a subsequent transformation of the polariton into an external photon. Hence a clear-cut difference between scattering and luminescence is missing. Nevertheless, in several cases some spectral features can be unambiguously related to a certain Raman process
T~2K ‘—I
_____________________
0 200
TIME
600
(ps)
Fig. 8. The spectrum of the preresonant scattering of anthracene (upper curve, arrow indicates the excitation frequency, *: the
81 cm~ line). Lower: the kinetic curves of the 81 cm ‘line at two temperatures.
I Aaviksoo
/
Time-resolved studies of excitonic polariions
a slow one. The interpretation of the data is as follows. The fast component has the lifetime of the initial polariton wave packet, propagating perpendicular to the slab plane, and creating a Raman polariton by emitting the 81 cm~optical phonon. At the same time the initial polariton undergoes (quasielastic) scattering by acoustic phonons producing a distribution of polaritons, most of which are waveguide modes and hence have a much longer lifetime than the initial polariton. Subsequent scattering of these polaritons by the 81 cm~ phonon ends up with polaritons of the same energy as in the straightforward Raman process, but their kinetics reflects the lifetime of the secondary polariton distribution, i.e. they give us the slow component of the Raman-like line. Increasing the temperature enhances the slow component due to an increased acoustic phonon scattering rate. In conclusion: temporal resolution of polariton luminescence/scattering lines can provide additional insight into the hidden dynamics of multiple scattering by phonons.
5. Time resolved reflection experiments The reflection spectrum of crystals exhibits distinct structures near excitonic resonances. This structure results in a corresponding transient behaviour of resonantly reflected light pulses. The effect of transient reflection has been theoretically analysed in refs. [41,42,15] and was experimentally demonstrated on GaAs [43] and InP [44]. Pulsed excitation has been shown to induce resonant polarization in the reflecting surface layer of the crystal, which in turn re-emits light in the form of a reflected pulse. Time-resolved studies thus provide a direct means to follow the polarization dynamics in the surface layer. In the case of InP crystals transient reflection near the lowest exciton resonance (fig. 9) can be well described by the straightforward linear response model calculating the corresponding reflection coefficient r(co) proceeding from (1). Three processes contribute to the reflection kinetics [43,45]: dephasing due to scattering by phonons, polariton propagation into the crystal due to spatial dispersion, and radiative decay due to reflection itself. The last two have
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N
-
tO” ~
Z II
~o-~
i
5
0
5
I
I
I
I
10
15
20
20
DELAY
(ps)
Fig. 9. Logarithmic plot ofexperimental cross-correlation functions of pulses (FWHM = 3 ps) reflected from an lnP crystal at the frequency of the exciton resonance (dotted) and theoretical fits to the data (solid line) at two excitation densities. The fit yielded ni~= 0.2m~and F = 1.5 cm’ and 6.1 cm~as adjustable parameters, T= 1.8 K.
essentially nonexponential kinetics. It is noteworthy in the present treatment that the radiative decay rate of the polariton wave packet in the reflection experiment is approximately equal to the LT-splitting siLT, which yields picosecond lifetimes for GaAs and InP crystals. Hence, transient reflection provides a means to study polariton dynamics in the surface layer and yields the material parameters, which determine the dielectric function F.
6. Conclusions As is evident from the above treatment, polaritons in a real crystal show a large variety of transient features. They can be crudely divided into three categories. First, temporal properties of a coherent polariton state, which are determined by F and the phenomenological damping constant F. Second, temporal properties of a distribution of polaritons, which is largely determined by the interrelation of polariton relaxation (on phonons), propagation, and radiative decay. Third, in specific polariton experiments complicated temporal
66
J. Aaviksoo
/
Time-resolved studies of excitonic polaritons
behaviour may be observed as in the case of photon echo type responses and transient reflection measurements. The appropriate timescales of the transients vary from about 100 fs in the case of a 2D surface polariton on anthracene to about 20 ns for a nonradiant 3D polariton mode in the same crystal.
[16] J. Aaviksoo, J. KuhI and K. Ploog, Phys. Rev. Lett., to be published. [17] L.A. Vainshtein, Soy. Phys. Usp. 19 (1976) 189. [18] W.E. Moerner ed., Persistent Spectral Hole-Burning. Science and Applications (Springer, Berlin, 1988). [19] A. Rebane ci al., Opt. Common. 47 (1983) 173. [20] H. Sönajalg et al., Opt. Common. 71(1989) 377. [21] A. Rebane, J. Aaviksoo and J. KuhI, AppI. Phys. Lett. 54 (1989) 93.
[22] P. Saari, R. Kaarli and A. Rebane, J. Opt. Soc. Am. B 3 (1986) 527.
Acknowledgements
[23] G.W. ‘t Hooft et al., Phys. Rev. B 35 (1987) 8281.
[24] DL. Dexter, Solid State Physics Vol. 6, eds. F. Seitz and
Fruitful cooperation and discussions with many colleagues are gratefully acknowledged, e.g. with V. Agranovitch, J. Kuhl, A. Rebane, K. Rebane, I. Reimand, T. Reinot, P. Saari and I. Tartakovskii.
References [1] U. Fano, Phys. Rev. 103 (1956) 1202. J.J. Hopfield, Phys. Rev. 112 (1958) 1555. V.M. Agranovitch, Soy. Phys. JETP 10 (1960) 307. [2] R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973). [3] V.M. Agranovitch and V.L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer, Berlin, 1984). [4] A. Sommerfeld, Ann. Phys. 44(1914)177. [5] L. Brillouin, Ann. Phys. 44 (1914) 203. [6] B. Hönerlage et al., Phys. Rep. 124 (1985) 161. [7] R.G. Ulbrich and G.W. Fehrenbach, Phys. Rev. Lett. 13 (1979) 963. [8] Y. Segawa, Y. Aoyagi and S. Namba, Solid State Common. 32 (1979) 229. [9] T. Itoh, P. Lavallard, J. Reydellet and C. Benoit a Ia Guillaume, Solid State Common. 37 (1981) 925. [10] NA. Vidmont, A.A. Maksimov and 1.1. Tartakovskii, Pis’ma Zh. Eksp. Teor. Fiz. 37 (1983) 578. [ii] J. Aaviksoo, J. Lippmaa, A. Freiberg and A. Anijalg, Solid State Common. 49 (1984) 115. [121 S. Chu and S. Wong, Phys. Rev. Lett. 48 (1982) 738.
[13] L. Brillooin, Wave Propagation and Group Velocity (Academic, New York, 1960). [14] M. Frankel and J.L. Birman, Phys. Rev. A 15(1977)2000. [15] J. Aaviksoo, J. Lippmaa and J. Kohl, J. Phys. Soc. Am. B 5(1988)1631.
D. Turnbull (Academic, New York, 1958) 353. [25] and K. Ploog, Phys.Vol. Rev.15,B ed. 29 (1984) 7058. [26] L. R. Schultheis Dingle, Festkörperprobleme, H.J. Quisser (Pergamon, Oxford, 1975) 25. [27] J. Feldmann et al., Phys. Rev. Lett. 59 (1987) 2337. [28] V.M. Agranovitch and O.A. Dubovskii, Pis’ma Zh. Eksp. Teor. Fiz. 3 (1966) 345. [29] M.R. Philpott and P.G. Sherman, Phys. Rev. B 12 (1975) 5381. [30] M. Orrit, C. Aslangul and P. Kottis, Phys. Rev. B 25(1982) 7263. [31] M. Orrit and P. Kottis, Adv. Chem. Phys. Vol. 74, eds. 1. Prigogine and S. Rice (Wiley, New York, 1988). [32] J. Aaviksoo, J. Lippmaa and T. Reinot, Opt. Spectrosc. 62 (1987) 706. [33] M. Orrit, J. Bernard, J.M. Turlet and P. Kottis, J. Chem. Phys. 78 (1983) 2847. [34] J. Toyozawa, Prog. Theor. Phys. Suppl. 12 (1959) ill. [35] P. Wiesner and U. Heim, Phys. Rev. B 11(1975) 3071. [36] M.D. Galanin, E.N. Myasnikov and Sh.D. KhanMagometova, Mol. Cryst. Liquid Cryst. 57 (1980) 119. [37] J. Aaviksoo, A. Freiberg, J. Lippmaa and T. Reinot, J. Lumin. 37 (1987) 313. [38] J. Aaviksoo and T. Reinot, to be published. [39] Y. Nozue, J. Phys. Soc. Jpn. 57 (1988) 3204. [40] J. Aaviksoo, A. Freiberg, T. Reinot and S. Savikhin, J. Lumin. 35(1986) 267. [41] D. Elert, Ann. Phys. Ser. 5 7 (1930) 65. [42] G.P. Agrawal, J.L. Birman, D.N. Pattanayak and A. Pun, Phys. Rev. B 25 (1982) 2715. [43] J. Aaviksoo and J. Kohl, IEEE J. Quantum Electron. QE-25 (1989) 2523. [44] J. Aaviksoo, J. Kohl and I. Reimand, Solid State Common. 72 (1989) 49. [45] J. Aaviksoo and I. Reimand, Opt. Common., to be published.