Polariton time of flight measurement in a thin crystal of CdSe

State Communications, ~Solid ~ _ ~ / P r i n t e d in Great Britain.

Vol.43,No.12,

pp.879-881,

POLARITON TIME OF FLIGHT ~ A S U R E M E N T

1982.

0038-I098/82/360879-03S03.00/0 Pergamon Press Ltd.

IN A THIN CRYSTAL OF CdSe

P.H. DUONG %, T. ITOH'" and P. LAVALLARD + Groupe de Physique des Solides de I'E.N.S. Universit~ Paris 7 - 2, place Jussieu - 75005 Paris - France (Received 4 May, 1982 by M. Balkanski)

We measured the polariton time of flight in a thin CdSe crystal. From the analysis of the delay time of light as a function of frequency, we find the value of the longitudinal~transverse splitting energy, AEL_= I meV and the effective mass, m"= 0.45 m o. We compare the information given by our measurement with that given by the analysis of interferences in the reflectivity spectrum.

Introduction

orthogonal and the light propagated normally to the crystal surface. A polarizer was placed on each side of the cryostat ; it was oriented to decrease as much as possible the cross-correlation signal which corresponds to the very weakly delayed pulse du to the E//C polarization component of the light. The average intensity sent on the sample surface was kept down to a few watts per cm 2. For a higher incident intensity, non linear effects occur 7.

Time of flight measurements have been done in GaAs I, CuCI 2, CdS 3 and CdSe 4. The thickness of the CdSe sample used in (4) was rather large and it was not possible to observe the transmission of light near the exciton resonance frequency, vT. We present here results obtained with a very thin crystal. The information given by our measurements is compared with that obtained by analyzing the interferences in the transmission and in the reflection spectra 5. The sample thickness is not uniform and under a microscope, one can easily see m o n o c r y s t a ~ w i t h parallel hexagonal c-axis in the sample plane. The thickness of the thinnest part was deduced from the interference spectrum in the I-R and visible ; with the knowledge of the interference order and the wavelength measurement of the maxima, we could deduce from the known value of the refractive index 6, a precise value of the thickness, e = 0.93 +0.02 u. In order to select an homogeneous part ~f the sample, the incident light was focused on the sample. From one experiment to an other the measured delay time did not vary more than 5%. A great care was taken to mount the sample free of strain. It was just sandwiched between two sheets of paper drilled through in the middle and glued to the holder. The sample was immersed in pumped liquid helium and was excited by a dye laser (Rh 640 o DCM) synchronously pumped by a mode locked Ar T laser. The autocorrelation width of the pulse ~as equal to 9 ps ; the spectrum bandwidth was 2 A. We used the autocorrelation set-up to measure the time of flight of polaritons. The A-exciton in CdSe is coupled to light only for the polarization direction perpendicular to the c-axis. In our experiment,the two directions were kept

Experiments One must be careful when using the group velocity concept to describe the propagation of a wavepacket in a high/y dispersive and absorbing medium 4 and one has to define clearly. the delay and the frequency of the pulse. We took the delay time of the pulse as the delay at maximum U.V. signal in the cross-correlation measurement. We select the frequency corresponding to this delay in the following way. We measured the spectrum of the transmitted pulse. The absorption is strong for the light frequencies which are between the transverse and the longitudinal exciton frequencies, vT and ~L" Not too near vL' the spectrum of the transmitted light shows only one maxzmum at lower frequency than the maximum of the incident light intensity. We took the frequency value of this maximum as the mean frequency of the wavepacket which has propagated through the sample. Near the longitudinal exciton frequency, the transmitted spectrum shows a dip and two peaks. In the same conditions, ~e observed two transmitted pulses. The transmission of the light which corresponds to the propagation of a wavepacket of the upper polariton branch increases quite rapidly with frequency near VL5. We took then the upper peak frequency in the spectrum as the mean frequency of the less delayed pulse (upper polariton branch). The frequency of the other pulse is not precisely determined since the corresponding absorption is maximum between VT and ~L but does not decrease very quickly at higher frequency ; anyway, it is roughly equal to the lower peak frequency in the transmitted spectrum. Figure ! shows the experimental data we obtain in this way. The full line

On leave of absence from Institut of Physics Center for Scientific Research of Viet Nam, Hanoi, Viet Nam. :: On leave of absence from Tohoku University, Sendai, Japan. +

Laboratoire

associ~ au C.N.R.S. 879

880

POLARITON Tl.W£ OF FLIGHT MEASUREY~NT

Vol. &3, No.

12

4o

&

/

~- 30

Ar~" 15

=

i t"

~= 20

0

IO

I

I 822

Fig.

]

1824 Photon energy,

; 826 eV

5'

Delay time of the pulse. Points are experimental (see text) ; the curve is theoretical.

is a plot of the delay time determined as the ratio of the sample thickness to the group velocity calculated in a one-oscillator model, with respect to frequency. The best fitting is obtained with the following parameters : longitudinal frequency, ~L = 1"8259eV8 longitudinal-transverse splitting energy, ~ = I meV ; effective mass, .-LT m;: = 0.45 m (m is the free electron mass). The damping°fac~or,~ hF was taken to be equal to Ixl0 '~ eV but the fitting is not very sensitive to the value of this parameter. As an additionnal proof, in figure 2, we show the transmitted spectrum of a pulse in the conditions where the polarization direction of the light is not exactly orthogonal to the c-axis of the sample. One sees clearly oscillations which come from the interference of the beams corresponding to the two polarization directions of the light, parallel and perpendicular to the c-axis 9. That the fringes follow one another at asteady interval indicates that, in the LT region, the frequency is linear with the wavevector. From the frequency interval, &~, between adjacent maxima, we deduce the time of propagation of the pulse : !

&T = - Av

1825

1.826 Photon energy,

Fig. 2

eV

Transmission spectrum of a pulse in the LT region. The polarization direction of the light is slightly tilted with respect to ~he normal to the c-axis.

their percentage modulation is too weak to contribute significantly to a deformation of the pulse. In the LT frequency range, the attenuation is strong and we could not detect any additionnal pulse. We attribute the major part of the observed discrepancy to bound exciton transitions. In our experimental conditions, the corresponding absorption lines are weak ; they are stronger at high excitation level 7. An evaluation of the extra delay due to the bound exciton transitions can be obtained from the consideration of the energy velocity [0. The inverse of the velocity can be written : I

i

vE

Vp

+

-

I

-

LF e

= 37 ps.

This value is in good agreement with the direct measurement of the delay time. There is some discrepancy between the experimental data and the theoretical curve in figure I. It cannot be accounted for by the innacuraey of the zero delay measurement (done by taking the sample out of the beam) nor by the contribution of the background dielectric constant. The successive reflections of the pulse on the input and output faces of the sample produce successive out-going pulses, the amplitude ratio of which is determined by the reflection coefficient at the interface and by the attenuation in the bulk. These pulses interfere if the delay time between two successive pulses is smaller than the width of one pulse ; they appear as separated pulses, if not. For v < v T, we observed Fabry-Perot interferences in the frequency spectrum of the transmitted pulse but

c . . 1 where Vp =--n Is the..phase velocity, --L is the. absorptlon coeffzclent and i~Fe, the damplng factor. The extra delay is : e T

=

e

- -

%

vE

-- L

I Fe

By t a k i n g realistl_c values at low level, e = ~_ , mr" = O. I m e V , w e f i n d L This is the righ~ order of magnitude time difference.

excitation "r ~_2 p s . for the

Conclusion The parameter values we obtain are very close to those obtained by Kiselev et al 5 from the study of the Fabry-Perot interferences in the reflectivity spectrum of thin crystals. We get an effective mass a little larger than they do,

Vol. 43, No.

12

POLA-RITON TIME OF FLIGHT .~ASUREM~NT

0.45 m instead of 0.41 m . But they notice that "if it°were necessary to Take into account the dead layers, e.g. of 150 A, this would enlarge effectively the value of m up to 0.45 m " o We think that the time of flight measurement is a more straightforward method to obtain the polariton parameters than the study of interferences since to obtain precise locations of singularities, one has to calculate the intensity of reflection within some model. It is likely that the damping factor is greater in a time of flight experiment than in a reflectivity measurement since the instantaneous power is then much larger and that non-linear effects can occur on a picosecond time scale. This would not modify very much the accuracy of the determination of the parameters but would explain why we do not observe any structure for v > v in the trans. L mitred spectrum shown in flgure 2. In both measurements, the bound exciton (or forbidden exciton) transitions make the interpretation more complex and give some innacuracy to the results. Kiselev et al 5 have observed a very well defined structure due to the mutual interference of waves I and 2. We looked for such a direct

881

proof of the spatial dispersion in the time of flight measurement but we failed. We could get two transmitted pulses only in the frequency range where two peaks are observed in the transmission spectrum ; we had, then, no direct evidence of two wave packets propagating with the same mean frequency but different velocities. The study of interferences seems much more appropriate than the time of flight measurement to observe directly the co-existence of two waves. We call E l and Ep, the amplitudes of the two transmitted electric fields corresponding respectively to the lower and upper polariton branch. For v ~ v L, they are equal but the width of the spectrum prevents us from assessing that the same range of frequencies has been transmitted in each pulse. For u~ v , the amplitude E I is a small fraction of th~ a~plitude E 2 ; the ratio of the two pulses intensities in a time of flight experiment,(E /E~) 2, is much less than the percentage modulatiSn of the reflectivity, 4 •E./E 2. It is then, very difficult to discri! mlnate the weak pulse from the wing of the strong one even when it is possible to observe the interferences of the two waves.

REFERENCES

I.

2.

3.

4.

5.

R.G. Ulbrich and G.W. Fehrenbach, Phys. Rev. Lett. 43, 963 (1979). Y. Segawa, Y. Aoyagi and S. Namba, Solid State Commun. 32, 229 (1979). Y. Masumoto, Y. Unuma, Y. Tanaka and S. Shionoya, J. Phys. Soc. Jpn. 47, 1844 (1979). Y. Aoyagi, Y. Segawa, T. Baba and S. Namba in "Picosecond phenomena II" edited by R.M. Hochstrasser, W. Kaiser and C.V. Shank (Springer-Verlag 1980) p. 298. T. Itoh, P. Lavallard, J. Reydellet and C. Benoit ~ la Guillaume, Solid State Commun. 37, 925 (1981). V.A. Kiselev, B.S. Razbirin, and I.N. Uraltsev, Phys. Star. Sold. (b) 72, 161 (1975).

6. R.B. Parsons, W. Wardzynski and A.D. Yoffe, Proc. Roy. Soc. London A262, 120 (1961). 7. P. Lavallard, P.H. Duong, to be published. 8. For calculating the photon energies from the observed wavelengths, the refractive index of air is taken into account, i.e. l a i r . ~ = 1.2395 eV.~. 9. We could observe the interference of the two waves by rotating the polarizer placed after the sample in order to project each polarization vector along a common direction, neither parallel nor orthogonal to the c-axis of the sample. 10. R. Loudon, J. Phys. A : Gen. Phys., 1970 3.