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QUANTIZATION OF EXCITONIC POLARITONS IN CdTe-CdZnTe DOUBLE HETEROSTRUCTURES
Y. Merle d’Aubigné, Le Dang, A. Wasiela, N. Magnea, F. d’Albo, A. Million
To cite this version:
Y. Merle d’Aubigné, Le Dang, A. Wasiela, N. Magnea, F. d’Albo, et al.. QUANTIZATION OF EXCI-
TONIC POLARITONS IN CdTe-CdZnTe DOUBLE HETEROSTRUCTURES. Journal de Physique
Colloques, 1987, 48 (C5), pp.C5-363-C5-366. �10.1051/jphyscol:1987578�. �jpa-00226782�
Colloque C5, supplement au n0ll, Tome 48, novembre 1987
QUANTIZATION OF EXCITONIC POLARITONS IN CdTe-CdZnTe DOUBLE HETEROSTRUCTURES
Y. MERLE ~'AUBIGNE, LE SI DANG, A. WASIELA, N. MAGNEA*, F. ~ ' A L B O * and A. MILLION*"
Laboratoire d e Spectrometric Physj-que, USTMG, BP 87, F-38402.Saint-Martin-d'Heres Cedex, France
*centre d'Etudes Nucleaires, DRF/SPh, BP 85X, F-38041 Grenoble Cedex, France
* *
Centre d l E t u d e s Nucleaires, LETI/LIR, BP 85X, F-38041 Grenoble Cedex, France
RESUME-La quantification des polaritons excitoniques est 6tudi6e dans les doubles h6t6rostructures CdTe-CdZnTe.
ABSTRACT-Reflectivity, transmission and excitation spectra at energies near the 1S exciton resonance of CdTe-CdZnTe double heterostructures are reported. These spectra allow a detailed study of the quantized states of the excitonic polariton resulting from its confinement in the CdTe layer. Selection rules for the excitation of these quantized states are demonstrated.
I-INTRODUCTION-Free exciton luminescence and reflectivity spectra in semiconductors are generally affected by the contamination of the surface. Charged surface states give rise to a surface electric field and to non-radiative recombination centers which affect both the intensity and the shape of the free exciton emission line and modify the reflectivity spectrum. As demonstrated by Schultheis and Ploog [I]
for GaAs, such effects may be avoided by cladding the optically active layer between two confining GaAlAs layers. As reported in a preceeding paper [2] the luminescence spectra of CdTe-CdZnTe double heterostructures are effectively very different from those observed in bulk CdTe or unprotected epilayers [31: the free exciton emission is intense and shows a single peak. These double heterostructures have a well defined slab geometry with a thickness of the central layer which can be made arbitrarily small (100 and 50 nm in the present case).
Then the effects of the quantization of the excitonic polariton states [4,11 are clearly visible in the reflectivity, transmission and excitation spectra. In this paper, selection rules for the excitation of these quantized states are predicted and shown to be indeed observed. The parameters defining these quantized states are close to those defining the excitonic polariton dispersion in bulk CdTe.
11-EXPERIMENTAL-The samples were grown by molecular beam epitaxy on (100) Cd.96Zn.04Te substrates. The details of the growth procedure are given elsewhere [21. The CdTe with nominal thickness 1=100 nm and 50 nm for samples I and I1 respectively is stacked between two 80 nm thick Cd.gaZn.07Te confining layer's. The CdTe and ZnTe fluxes were not accurately controlled during the growth and the actual thickness may differ from these values by 15%. As a result of a lattice parameter mismatch of 0.23% with the substrate, the CdTe layer is uniformly strained [5,6] (a biaxial compressive strain).
All the measurements were made at 1.8 K in an immersion cryostat.
A tungsten lamp was used for the transmission and reflectivity measurements. The reflectivity is recorded under normal incidence.
Excitation spectroscopy is performed with a cw dye laser (Styryl 8 1 .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987578
C5-364 JOURNAL D E PHYSIQUE
111-RESULTS-For CdTe double heterostructures of thickness 50 nm the confinement energy of the electrons, 1.2 meV is small with respect to the exciton binding energy ( 10 meV). So it is the confinement of the exciton which needs to be considered, the only effect of the confinement of the carriers being a possible slight modification of the gap and of the binding energy of the exciton. The dispersion of the polariton in an infinite CdTe crystal is well known. The upper polariton branch starts at W L , the longitudinal eigenfrequency at K=O and becomes rapidly photon-like as K increases. The lower polariton branch is photon-like below W T , the transverse eigenfrequency, and becomes exciton-like above WL
.
Then the polariton dispersion is well approximated by the exciton dispersion:E = ~ W T+ ( 6 2 / 2 ~ ) ~ 2 (1)
In a thin layer of thickness L, polaritons have discrete wavevectors in the direction perpendicular to the layer plane: KI=Nfl'/L where N is a non zero integer. Because of this quantization of the exciton wavevector KJ, selection rules for the coupling to photons of given wavevector k holds. The exciton eigenstates are standing polarization waves. The wavefunctions (for the center of mass motion) are identically zero outside the CdTe layer, they may be approximated by the following expressions which respect the symmetry of the true wavefunctions :
' + '
N (z) = COS(K~Z) Q(Z/L) for N odd K I = Nn/L (2) WN (z) = sin(K~z) Q(z/L) for N even
Q(z/L) is the gate function equal to 1 for -L/2<z<L/2 and to zero outside this interval. The quantization of the excitonic states gives rise to a structure in the optical spectra, which is well resolved at energies larger than h w ~ as shown in fig. 1. Far enough from this resonance, the exciton-photon coupling leading to the excitation of a given mode can be treated in perturbation theory and the transition probability per unit time is Wt =
1
C 12 (FN (k) 12 .The electric dipole matrix element is included in C as well as the factor relating the oscillator strength of the free exciton to the envelope function of the relative motions of the electron and hole 171. FN (k) is the Fourier component of '+'N (z) at the wavevector k=2fl/J of the photon inside the layer :FN (k) = Akcos(kL/2) for odd N , Ak D( (Nfl/L)/( (Nfl/L)2
-
k2 ) ( 3 ) F N ( ~ ) = Ak sin(kL/2) for e v e n NThese expressions show the existence of selection rules for the exciton-photon coupling when ~=')(/2,>, 3>/2
...
: if kL = 2fl modulo 2fl the coupling to the exciton modes of even N is zero, if kL = fl modulo 2fl it is the coupling to the modes with odd N which vanishes. It is verified that when the width L of the double heterostructure is very small one finds the well known selection rules for the quantum well :sin(kL/2) 0 and the coupling to the modes with N even, ie to the modes with an odd wavefunction Y ) N (z), vanishes.
The reflectivity, absorption and excitation spectra of samples I and I1 are shown in fig. 1. In these strained CdTe layers the main line at 1595 meV is due to the heavy exciton, the broad line at 1609 meV in the excitation spectrum of sample I1 was assigned [51 to the light exciton. The series of lines in between are due to the excitation of the various quantized states of the heavy exciton. The excitation spectrum of fig. la is a good demonstration of the selection rules discussed above. The thickness of sample I of the order of 100 nm is nearly equal to one half the photon wavelength (232 nm if the index n=3.3) then the condition kl=fl is nearly satisfied and the excitation of the quantized exciton states with odd N should be nearly forbidden. This is what is observed in the excitation spectrum of fig. la : every alternate line is much less intense. For sample I1 with a thickness of 50 nm, kLwfl/2 and, as predicted by ( 3 ) , the intensities of the quantized exciton peaks in fig. lb decrease steadily above WL. For sample I1 the assignment of the excitation peaks to given values of K L = Nn/L is unambiguous : the best fit of the excitation spectrum to equation (1) allows the determination of
1 590 1 600 PHOTON ENERGY me^)
PHOTON ENERGY me^)
Fia.1-Excitation (of D O X ) , transmission and reflectivity spectra, dashed curves calculated spectra. a)sample I;
blsample I1,in the excitation spectrum the short period oscillations (looking like noise) are interferences due to reflexion of the back surface of the substrate.
the effective thickness L = 50 nm and of the transverse frequency %WT=
1594.8 meV. The heavy exciton mass M was taken 181 equal to 0.69 mo (mo free electron mass). For sample I the peaks are more closely spaced and two fits, corresponding to two different indexations of the peaks, are possible for L values close to 100 nm. Using the selection rules demonstrated above one can eliminate one of these fits and one gets SWT = 1594 meV and L = 102 nm. If one takes into account the existence of an exciton free "dead layer" 191 of thickness d of the order of the exciton Bohr radius, the effective L is not equal to the actual thickness 1 of the CdTe layer but to the difference 1
-
2d. Itis unfortunate that 1 is not known precisely, its comparison with the experimentally determined value of L would have allowed the determination of the dead layer thickness.
Taking into account spatial dispersion and writing the boundary conditions at the various interfaces, calculation of the reflectivity and transmission spectra presents no difficulty [Ill. In practice, the index discontinuity at the confining layer
-
substrate interface can be neglected and, because of the poor quality of the back substrate surface and of a defect of parallelism, interferences with the waves reflected at this surface may be neglected. Thus only the first three interfaces need to be considered. In agreement with expression (2), the Pekar additional boundary condition is chosen. Examples of calculated spectra are given in fig. la. A longitudinal transverse splitting TIWLT = 0.7 meV was chosen to fit the width of the reflectivity maximum and a damping r =0.2 meV to fit the minimum of transmission. This value of W L T is found equal to the one J.C. Merle et al. [lo] determined in bulk CdTe by Brillouin scattering. In the strained CdTe layers we are dealing with heavy excitons and one would have expected a reduction of the bulk W L T by a factor 3/4 which takesC5-366 JOURNAL DE PHYSIQUE
care of the sharing of the oscillator strength of the M = +1 states of the "cubic" exciton between heavy and light excitons. The damping constant
r
-0.2 meV in sample I is one order of magnitude larger than the value determined by Merle et al. [lo] in good quality bulk CdTe.This can be due to a higher donor concentration in our layers or, as suggested by the fact that it is even larger in sample I1 (T=0.5), to scattering at the interfaces.
The simulated excitation spectra shown in fig. 1 were obtained by substracting the calculated transmitted and reflected intensities from unity. The simulations shown in fig. la were calculated for %WT=
1594.2 meV, which allows a better fit of the reflectivity and transmission spectra than the value -l'fw~=1594.0 meV which gives the best fit of the excitation spectrum. All these fits could be improved by introducing a small inhomogeneous broadening and allowing the damping
r
to vary with the polariton energy Ell]. Similar calculated spectra are shown in fig. lb for sample I1 (*~~=1594.8 meV, M=0.69 mo, X w ~ r = 0 . 7 meV,. r=0.5 meV).The agreement with the experimental spectrais very good.
The transverse energy 4fw~ determined for the samples with thicknesses 50 and 100 nm are 1594.8 and 1594.2 meV respectively. The difference (0.620.4 meV) is small and comparable to the difference of the energy gaps, 1 meV, which is calculated by taking into account the confinement of the carriers. This suggests, that for these double heterostructures the binding energy of the exciton is not strongly modified by the confinement.
IV-SUMMARY-The structure in the reflectivity, transmission and excitation spectra allows a detailed study of the quantization of the excitonic polariton within the CdTe layer. For layers with well defined thicknesses L = % / 2 , 3X/2
. . . or, alternatively, L = 0,
I , . . .
()photon wavelength in CdTe) selection rules for the excitation of the quantized states are predicted. They are satisfied for sample I which nearly fullfils the first condition. Comparison of sample I and I1 with thickness 100 and 50 nm respectively is instructive since in sample I1 the double heterostructure starts to approach the quantum well limit. The parameters wr and W L T are very similar in these two samples and close to their values in bulk CdTe. The dampingr
is much larger than in bulk CdTe and increases as the width of the double heterostructure decreases. We thank R. Cox and G. Fishman for fruitful discussions and H. Tuffigo for communicating her luminescence results prior to publication.REFERENCES
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