Bulk polariton beatings and two-dimensional radiative decay: Analysis of time-resolved transmission through a dispersive film

Solid State Communications,

102. No. 7, pp. 505-509, 1997 Q 1997 Ekevier Science Ltd Printed in Great Britain. All rights reserved 003%1098/97 $17.OO+.OO

Pergamon

Vol.

PII: SOO38-1098(!V)OOO47-1

BULK POLARITON BEATINGS ANALYSIS OF TIME-RESOLVED Giovanna

AND TWO-DIMENSIONAL RADIATIVE DECAY: TRANSMISSION THROUGH A DISPERSIVE FILM

Panzarini”

and Lucia Claudio Andreanib

“Istituto Nazionale %tituto

Nazionale

per la Fisica della Materia- Dipartimento di Fisica, Universim di Milano, via Celoria 16, I-20133 Milano, Italy per la Fisica della Materia- Dipartimento di Fisica “A. Volta”, Universita di Pavia, via Bassi 6, I-27100 Pavia, Italy

(Received

16 January 1997; accepted 28 January 1997 by G. Bastard)

The time-resolved transmission (TRT) through a slab described by a local dielectric function with resonance behavior is studied. For films with a thickness d > X (where X is the wavelength of light in the crystal) an analytic formula which describes the recently observed polariton beatings [Frohlich, D. et al., Phys. Rev. L&t. 67, 1991,2343] is obtained. For d < h the TRT is expressed in terms of the normal modes of the slab: the radiative decay of the quasi-two-dimensional excitation is recovered when only the lowest modes contribute, as it happens for very narrow slabs, while the interference between different modes evolves into the polariton beatings on increasing the film thickness. 0 1997 Elsevier Science Ltd

When the film thickness d becomes smaller than the wavelength of light in the medium X, the concept of bulk propagation at the group velocity is not useful anymore. In the quasi-two-dimensional (2D) limit d < X no stationary polariton states exist: the excitation in the medium becomes instead radiative [9], i.e. it acquires an intrinsic lifetime. An example is the exciton in a quantum well (QW) [lo, 111, where it has been shown that the radiative lifetime can be measured by TRT [12, 131. In this paper we express the TRT at normal incidence in terms of the normal modes of the slab [ 141 and show that the radiative lifetime of the quasi-2D excitation is recovered when only the lowest modes contribute (as it happens for d << A). The present results yield a simple physical picture for the crossover from radiative decay to bulk polariton beatings when the film thickness is increased. In the linear approximation the transmitted signal is the convolution between the incident pulse and the response function G(t), which is expressed in terms of the transmission coefficient as

The recent observation of quantum beats of quadrupole polaritons in Cu20 [l] has stimulated new interest for a classic problem of electromagnetism, namely pulse propagation in a dispersive medium [2-51. A slab supporting an excitonic or phononic resonance can in fact be described by a dispersive (frequency-dependent) dielectric constant. When a light pulse whose frequency is close to resonance propagates through the medium, the response of the slab involves the mixed modes of the matter-radiation system, i.e. the polaritons [6]. When the frequency spread of the pulse is such that both lower and upper polariton modes are simultaneously excited, the interference between the two modes gives rise to beatings [ 11, which are a manifestation of coherent propagation of the two polaritons at the group velocity. Similar pulse propagation and beating effects on the four-wave mixing signal have also been investigated [7]. In this Communication we derive an analytic formula for the time-resolved transmission (TRT) of a &pulse through a slab characterized by a local dielectric function with resonance dispersion. The case of normal incidence has been considered. The formula gives a quantitative description of polariton beatings, yielding the dependence of the oscillations on the various parameters (film thickness, longitudinal-transverse (LT) splitting, etc.). Effects of nonlocality in the medium [8] are neglected.

(1) --3c

The function 505

G(t) describes

the linear response

of the

506

BULK POLARITON

system to a &pulse. If the medium is modeled by a local dielectric function with resonance dispersion E(W)=

E,

2WOWLT

1+ Wi -

w2 -

2iyw

1’

where w. and y represent the resonance energy and the non radiative broadening (HWHM) respectively, the transmission coefficient is given by

t(w) =

42&w) [Be + n(w)]‘e-

ikd- [nB - n(w>12eikd

(3)

where n(w) = m E w is the complex refractive index, ng = ,,& is the background refractive index assumed to be the same for the slab and the surrounding medium, and k = k(w) = n(w)w/c. WLT is the LT splitting. The present model can be used to describe an excitonic resonance in a slab without spatial dispersion, or a phonon resonance in a polar material. It can also be applied to a multi-quantum well (MQW) system of global thickness d (as long as the period I < X, otherwise Bragg effects occur [ 151); the effective LT splitting for a MQW is given by 2?r e2

WLT = ---

fx,

e, mow0 1’

(4)

where fryis the oscillator strength per unit area for in-plane -polarization [ 16, 171. Propagation effects in MQWs have been studied experimentally [ 181 as well as theoretically [13, 191. In the thick film limit d > X [where X = 2?rc/(nBwo)] multiple reflections at each interface of the slab, which are described by the second term in the denominator of t(w), may reasonably be disregarded. With this approximation G(t) if found to depend on the phase d(w) = k(w)d - wt. When absorption is negligible (y = 0) the phase is real and the response function can be evaluated analytically by means of the stationary phase method [3, 41. The frequencies which give a dominant contribution to G(t) at a fixed time t are those which are close to the points where the phase 4(w) is stationary: this condition corresponds to polaritons which propagate with the right group velocity vR = dwldk to arrive at a time t at the second interface of the slab. After a transient, characterized by the arrival of the first (or Sommerfeld) and second (or Brillouin) precursors [2], a polariton with a group velocity vg = d/t is present in both lower (LP) and upper (UP) polariton branches at each time t, so that interference effects between the two modes are expected to occur. Starting from the arrival of the second precursor, at times t comparable with the transit time of radiation in the slab (t 2 d/v, where v = c/ng) the response function is dominated by frequencies which satisfy the condition lw - WOI>> (wowL7/2)“2. In this extremely short temporal

BEATINGS

Vol. 102, No. 7

window the polaritonic effect is unimportant and the photon-like modes propagate with a group velocity which is close to the phase velocity. The time window d/v < t < (d/2v)(wo/wLT) (“intermediate time domain”) is the most interesting one for evidencing polariton effects in a TRT experiment. In fact in this temporal regime the response function is predominantly determined by polaritons with aLr << lw range characWOI s (WOWLT12) ‘I=, i.e. by the frequency terized by a strong interaction between the excitation and the radiation field (the light-matter coupling is in fact measured by [(wowLT/2) “‘I. Evaluation of the response function at intermediate times leads to 1 ‘@)

=

(2dwOW~T/V)“4

,,&

(t -

(5)

d/v)3’4

where

and 40 = wo

(7)

While decaying in time like tp314,the response function shows an oscillatory behaviour which is the signature of the interference between UP and LP which have the proper group velocity to arrive at time t [l]. The minima of transmission are given by the condition A+? + (7r/4) = (n + $r, or d t=

-+ V

A

~(n+

$)‘,

n= 1,2 ,....

The stationary phase method can be shown to be valid provided t >> 2vl(dwowLT): thus the value n = 0 in equation (8) is not allowed and the oscillations start from n = 1. The time spacing T between two successive minima increases with time like T = ?r[2vt/(wowLTd)]“*, since the relevant frequencies become closer to resonance for increasing time and therefore the splitting between UP and LP decreases. The period T of the beatings clearly cannot become longer than r/wLT: thus equations (5)-(g) cease to be valid when t- tr = (d/2v)/(wo/wLT). At longer times (t >> tl, “long time domain”) the response function is determined by polaritons with frequencies w. - w < WLT and w - wL < wLT (WL is the longitudinal frequency), which have a vanishingly small V~ The response function can still be evaluated with the stationary phase method, but in this case no beating is detectable because the contribution to G(t) coming from the LP branch is much larger than that from the UP branch. In the presence of absorption described by a damping

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Vol. 102, No. 7

y # 0, the response function has to be calculated using the steepest descent method [3, 5, 201. As a result, the response function turns out to be given by equation (5) multiplied by a damping factor exp[ - y(r- (d/v)]. Under these approximations the beatings are damped but the positions of the minima are unchanged;. this follows from the assumption of equal dampings for LP and UP. This assumption is a reasonable one for the intermediate time region, in which the relevant frequencies are sufficiently far from o. or wL; a frequency dependent y should be introduced for frequencies close to resonance [21], thus in the long time domain, where beatings do not occur anyway. Figure 1 shows our analytical evaluation of the transmission intensity ]G(t)12 through a slab of Cu20 of thickness d = 0.91 mm supporting a quadrupole excitonic resonance; the parameters are those of the experiment of Frohlich et al. [l], namely Ed = 6.5, liwa = 2.0329 eV, liwLr = 3.66 X 10e6 meV, 2&y = 6.5 X 10e4 meV. The intermediate time region extends from about 2dlv = 15 ps up to tl 2 2 ms, i.e. it is very extended since the LT splitting is extremely small. We have checked that the numerically computed transmission intensity is in very good agreement with the results of Fig. 1, confirming that the approximations of neglecting multiple reflections and of using the steepest descent method are justified. The results of Fig. 1 reproduce very accurately the data of [ 11. It is interesting to remark that the decrease of intensity is determined to a larger extent by the tm3’*dependence implied by equation (5) than by damping. Notice that the wavevectors of the states involved in the beatings are restricted to a very narrow region around kc = wa/v: this shows that spatial

BEATINGS

507

dispersion effects have a negligible influence on polariton beatings. The finite temporal duration of the incident pulse can also be incorporated within the stationary phase method. For a gaussian or lorentzian pulse of width At centred at wo, beatings can be observed provided At < 11~~~ (physically this corresponds to the fact that both UP and LP modes must be simultaneously excited), and the finite width of the pulse has a negligible effect for t > dwowLT(At)2/(4v). In the two experimental configurations of [l] the duration of the pulse is either 60 ps or 30 ps, so the assumption of a b-pulse is a good one for times t > 100 ps. Use of the stationary phase method in the evaluation of G(t) relies on the neglect of multiple reflections; the latter play a major role in determining the TRT through slabs of thicknesses d 5 X, so that a different approach is required for thin films. In this case a more convenient evaluation of the response function proceeds through an analysis in terms of the normal modes of the system [14], which are the solutions of Maxwell equations with the constitutive relation for the slab and are also the poles of the transmission coefficient (3). The response function may be written - by means of the theorem of residues - as the sum of the contributions stemming from each normal mode. Although a local slab of arbitrary thickness d sustains an infinite number of normal modes, only poles which fall in the neighbourhood of w. and wL give a non negligible contribution to the response function in the temporal domain of interest for a TRT experiment. The approach is most useful when thin films are considered. In this case the response

IO4 IO3 IO2

t (ps) Fig. 1. Transmitted

intensity

(analytic formula) through a Cu20 slab of thickness d = 0.91 mm. Parameters:

see text.

508

BULK POLARITON BEATINGS

Vol. 102. No. 7

function (neglecting absorption and for t > d/v) is expressed as

mode index, the response function for thin films may be obtained by keeping only a small number of poles in the sum. For d < X the m = 0 mode has the largest decay G(~) = _~e-iae(D)~e-rf rate and dominates the TRT. One can wonder if the present local model can also be + ~(_l)mrmeiRe(w,.)re-r,“f, (9) applied to an excitation whose radius is comparable to m=l the thickness of the slab, like the exciton in a QW. When the LT splitting is written in terms of the oscillator where the index m = 0,1, . . . labels the different modes strength as in equation (4), the imaginary part of equation and the complex energies close to resonance are approxi(10) is found to be given by I’ = re*f,l(nsbc), which mately given by coincides with the radiative decay rate’of the amplitude WLT cj=Re(G)-ir=wo-lof a QW exciton at k = 0 [ 10, 111. Also, for very thin 1 + W(~od)l* films (d << X) an adequate description of the transmitted d signal is given in terms of the mode m = 0 alone. This is m = 0, (IO) - ‘uO(‘JLT 2y consistent with the results of the more fundamental nonlocal theory with one excitonic resonance [ 10, 171, in which the modes with m L 1 do not appear at all, since w, =Re(w,)-iI’, = ws- $ the QW exciton cannot support an electromagnetic field with a wavelength smaller than the exciton radius. A relation similar to equation (9) is obtained also in the (11) nonlocal theory for a single and a double QW [ 121. In The mode index m represents the number of half wave- these cases, however, one and two poles respectively lengths in the slab. The present treatment of normal should be retained in equation (9), corresponding to the modes is simplified compared to that of [14], since we number of exciton states supported by a nonlocal system. Figure 2 compares the transmitted signal through are at normal incidence. The m = 0 mode has a nearly constant electric field in the slab: it corresponds to OTH, slabs of thicknesses d = 20 nm and d = 50 nm as calculated analytically by means of relation (9) (solid line) and while the modes m = 1,2,3, . . . correspond to lCL, 212, 3CL, . . . respectively (see Fig. 3 of [ 141). Since the decay numerically (dashed line). Parameters are chosen in order rates of the modes with m 2 1 decrease rapidly with the to describe a GaAs/AlAs MQW of period 1= 10 nm:

IO0 -1

10

h + J CI 2 3

IO

-2

-3

IO

-4

10 10

2

-5

3

0

5

IO

15

20

25

t (PS)

Fig. 2. Transmitted intensity through a GaAs/AlAs MQW of period I= 10 nm, with a number of periods N = 2 20 nm) and N = 5 (d = 50 nm). Solid line: analytic formula, eqns (9)-( 11). Dashed line: numerical evaluation. Parameters: see text. (d =

BULK POLARITON BEATINGS

Vol. 102, No. 7

we assume hwa = 1.58 eV, l@r = 0.46 meV and ng = 3.46. The wavelength of light in the medium is h = 226.4 nm. It can be seen that the exponential decay becomes faster but restricted to shorter intervals on increasing d. Responsible for these features is the mode m = 0 [equation (lo)], which exhibits for thin films a decay rate proportional to d [17,22,23]: in the context of molecular films or MQWs this is often called “superradiant” behavior-u. At longer times a slower, non exponential decay characterizes the attenuation of the transmitted signal through the thicker slab: this is due to interference between the mode with m = 0 and those with m 2 1. These first oscillations, which originate from the superposition between different normal modes, will then develop in the quantum beats between LP and UP branches occurring in thick slabs. It should be noticed that the position of the “ oscillation” in Fig. 2 coincides with the time t I of the n = 1 minimum calculated from equation (8); the minima in the TRT are well described by equation (8) even for films with d < h. In conclusion, we have given an analytic formula for the time-resolved transmission at normal incidence through a slab of thickness d > h, which gives a quantitative account of polariton beatings [ 11, in particular for the increase of beat period with time. For thin films with d < h, the response function has been expressed in terms of the standing modes of the slab: the mode with m = 0 corresponds to the solution found in the nonlocal theory for a QW exciton and the TRT is dominated by the radiative decay whose rate increases with the thickness. The interference between the mode with m = 0 and those with m 1 1 becomes more important with increasing thickness and yields oscillations, which eventually evolve into the polariton beatings. The predicted features may be observed with excitons (e.g. in GaAs/AlGaAs MQWs) as well as with phonons: a suitable system could be GaP, where the quasi-2D regime occurs for thicknesses smaller than 10 pm. REFERENCES 1. Frohlich, D. et al., Phys. Rev. L&t., 67, 1991,2343; Phys. Status Solidi (b), 173, 1992, 31.

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