Resonant Raman scattering in Ga(As, P) mediated by (localized exciton-LO phonon) complex

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Resonant Raman scattering in Ga(As, P) mediated by (localized exciton-LO phonon) complex

N. Meskini, M. Kanehisa, M. Balkanski, M. Zouaghi

To cite this version:

N. Meskini, M. Kanehisa, M. Balkanski, M. Zouaghi. Resonant Raman scattering in Ga(As, P) mediated by (localized exciton-LO phonon) complex. Journal de Physique, 1982, 43 (6), pp.973-976.

�10.1051/jphys:01982004306097300�. �jpa-00209476�

Resonant Raman scattering ⁱⁿ Ga(As, P) ^mediated by (localized exciton-LO phonon) complex

N. Meskini (*), ^M. Kanehisa, M. Balkanski and M. Zouaghi (*)

Laboratoire de Physique ^des Solides, ^associé

^{Centre National} de la Recherche Scientifique,

Université Pierre et Marie Curie, 4, place Jussieu, ⁷⁵²³⁰ Paris Cedex 05, France

(Reçu le 10 novembre 1981, ^{révisé le} 2 fevrier 1982, accepté ^le 16 fevrier 1982)

Résumé.

Dans les cristaux polaires, ^les expériences de diffusion résonnante de la lumière suggèrent que les excitons

dressés

sont les principaux états intermédiaires dans le processus de diffusion pour des énergies ^inci-

dentes 012703C9i

Eex ⁺ n012703C9LO.

Le spectre de résonance Raman (Intensité des Raies Raman

fonction de l’énergie incidente) du cristal mixte

GaAs0,2P0,8 ^est calculé dans le cadre du modèle des coordonnées de configuration. ^La comparaison

^{les résul-}

tats expérimentaux ^{de Oueslati, Hirlimann et Balkanski} (J. Physique ⁴² (1981) 1151-1156) donne les coefficients de couplage V1 ^et V2 de l’exciton localisé

les phonons LO1 ^et LO2 respectivement : V1/012703C9LO1 ^{= 0,7 ± 0,2;}

V2/012703C9LO2

^{1,4 ± 0,3, et} l’énergie de l’exciton E0

2,442 ^eV ainsi que ^sa largeur ^03B3

0,030 eV.

Abstract

Resonance light scattering experiments ⁱⁿ polar crystals ^at energies corresponding ^to Eex ⁺ n012703C9LO

have suggested that dressed excitons

the relevant intermediate states. Configuration coordinate model is

applied ^to calculate the

Raman spectrum (intensity ^of phonons ^lines

function of incident frequency)

of the mixed crystal GaAs0.2P0.8 ^and compared ^with experimental results of Oueslati, Hirlimann and Balkanski

(J. ^{Physique 42 (1981)} 1151-1156). ^This analysis gives coupling ^constants V1 ^and V2 ^of

localized exciton with the LO1 phonon ^{and the} LO2 phonon respectively : V1/012703C9LO1

^{0.7 ± 0.2;} V2/012703C9LO2 = 1.4 ± 0.3, and exciton energy E0

^{2.442 eV with its width} 03B3

0.030 eV.

Classification

Physics ^Abstracts

71.38

78.30

1. Introduction.

In the last decade, ^intensive studies, experimental ^{as well as} theoretical, ^{of the}

III-V mixed crystals have been carried out. Lattice

dynamics and electronic properties ^{have been} widely investigated [2, 3]. ^In infrared and Raman scattering experiments, ^{two different} phonon ^{behaviour were}

noted : the one-mode behaviour where the mixed components ^{behave as a virtual} crystal ^{and the two-}

mode bahaviour where each component keeps ^its individuality [2, 3]. ^The crystal Ga(As, P) exhibits the two-mode behaviour [4].

In resonant conditions, ^{the Raman} spectra ^of (GaIn)P ^and Ga(AsP) ^mixed crystals ^show repetition

of the first order spectrum ^{with a} period equal ^{to the}

LO phonon energy. This feature ^was termed the LO

phonon replica of the first order spectrum [5]. ^There

seems to be some similarity ^{with the overtones} [6, 7]

observed in polar crystals. ^Two interpretations ^{of the}

overtones have been given : the first considers a loca-

lized electron strongly coupled ^to phonons [8] ^{and the}

second is the cascade model [9, 7] where the deexcita- tion of the crystal ^takes place through ^successive

emission of LO phonons ^followed by the radiative recombination. The first model is based ^on the _assump- tion of the existence of localized electronic excitations.

Defects (impurities, vacancies) [2] ^{or the} alloy-induced

disorder in a crystal ^are of the main mechanisms of localization. In the case of Ga(As, P), ^{even if} impurities

are absent, ^a local concentration fluctuation can be

responsible ^{for the} localization of excitons [13].

Balkanski et al. [1] suggested ^a (localized ^exciton-LO phonon) complex ^as the relevant intermediate state in the scattering process involving replicas. ^Our

purpose, in this work, ^{is to} investigate ^the possibility ^of explaining replicas using the localized electron model, through ^a quantitative comparison ^with experiment.

In the present paper, ^we calculate the resonance

Raman curves of GaAso.2P 0.8 using ^the configuration

coordinate model and _compare them with experimen-

tal results [10]. ^{In section} 2, ^a brief review of the model is presented. ^{In section 3,} results of calculations are

compared ^with experiment ^{and the} electron-phonon

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004306097300

974

coupling ^{constants are} determined. In section 4, discussion of the model and some concluding ^remarks

are given.

2. ModeL - The configuration co-ordinate model

[11] ^{assumes there are} localized electrons strongly coupled ^to phonon system. ^For simplicity, ^we suppose that there are two electronic states. In the electronic

ground ^state (no ^exciton present) the Hamiltonian for the phonons ^can be written as

where bi(bi) is the destruction (creation) operator ^for the longitudinal phonons L01 (GaAs-like) ^and L02 (GaP-like) ^{for i}

^{1 and 2} respectively, ^and (OLO, and WL02 ^are their respective frequencies. ^Its eigen-

states are direct product I n, > I n2 > ^{with corres-} ponding energies Enln2

^{n 1} Ii WLOl ^{+ n2} Ii WL02’ ^where

Fig. ^1.

Raman intensities of LO, ^and L02 peaks (a)

and LO, ⁺ L02 ^{and 2} L02 peaks (b)

function of the effective incident photon energy. The relative scales of the figures (a) ^and (b)

^the

^The

the

experimental points of reference [10] and the continuous

curves are

calculated within the configuration ^coordinate

model. The

indicate the energies ^{of the} contributing

intermediate states, in the frequency range of the experi-

ment, ^{to the} scattering. ^{It is} remarkable that

choice

of LO1- ^and L02-exciton coupling ^constants Vllh(OLI, ^=0.7

and V21h(OLO, ^{= 1.4} reproduces ^overall intensity ^and (oi

dependence of all four

at the

time.

nl and _n2 are the numbers of LO1 ^and L02 phonons, respectively. ^In the excited electronic state (an ^exciton created), ^{one must take account of the} exciton-phonon

interaction and the Hamiltonian becomes :

where _so is the bare localized exciton _energy, V,

and Y2 ^the exciton-phonon coupling ^{constants for} L01 ^and L02, respectively. This Hamiltonian is that of two displaced harmonic oscillators with frequencies

WLOl and COL02- ^Its eigenstates ^{are denoted} by I nl > 17’2 > ^with corresponding energies

The Raman scattering intensity ^{of the} (11 hWLOl ⁺ 12 hWL02) Stokes line is proportional ^to [1] :

.1

where 1iWi ^{is the} incoming photon energy, Eo

Eo - Vî/1iWLOl - V/1iWL02 is ^{the energy of the}

« dressed » exciton, y is the width (inverse lifetime)

of the localized exciton and ( II n > ^is the Franck- Condon overlap matrix element given by [11] :

where Li-Z(V2) is the associated Laguerre polynomial and vi = .Vilh(OLOi(’

^{1, 2) are} dimensionless coupling

constants for L01 ^and L02 respectively. ^When h > ni, one must interchange ni and li ^and change vi by ( - vJ ⁱⁿ (4).

From (3) ^and (4), ^{we can} write the intensity for

L01 ⁺ L02, ^for example, ^{in the form}

and in table I, ^we give the contributions to the matrix elements f(nl, n2), ^{for the} L01 ⁺ L02 ^Raman line, arising from different intermediate states with ni L01 phonons ^{and n2} L02 phonons. ^{It is} important ^to

note the frequent change ^of sign of the different contributions so that _many terms can interfere with each other and it becomes essential to sum the ampli-

tude over nl and _n2 before squaring ^it.

3. Calculations and results.

Since the Raman

intensity contains contributions from non-localized

Table I.

The first three contributions f (nl, n2) of the intermediate states with 0, ^{1 and 2} phonons ^to

the L01 ⁺ L02 scattering amplitude (eq. (5)). ^Here

vi and _V2 are dimensionless coupling ^constants of ^the

localized exciton to L01 ^and L02 phonons respective- ly. ^{Note the} frequent change of sign of ^{these contri-} butions, ^so large interferences ^{are to be} expected

and it becomes essential to sum the amplitude before squaring.

1

electrons, ^{the total} intensity should be of the form

where v runs over the modes LOI, L02, L01 ⁺ L02

and 2 L02. ^{C is a} multiplicative ^{constant which is}

the same for all the four modes _v, I’(wi) ^{is the resonant}

contribution to the intensity ^{and is} given by (5)

for the case of v

L01 ⁺ LO2, N y is the contri- bution from other electronic states and is non- resonant in the frequency range of interest. We expect

to have :

which results from the form of the Raman tensor

obtained by perturbation ⁱⁿ electron-phonon ^inter-

action [12]. ^{It is} important ^{that a} unique ^{set of} para- meters Eo, V 1, V2 ^and y ^can reproduce ^{all observed}

features. The best set of parameters is listed in table II.

The uncertainties in the values arise from the fact that when the parameters ^{are out} of the indicated ranges, ^no simultaneous fit to the four experimental

curves is obtained. The value of _{y is} of the order of

Table II.

Energy Eo ^{and width} y of ^{the localized}

exciton and its coupling constants to L01 ^and L02 phonons V 1 ^and V 2. ^The relatively high ^value of y indicates the short lifetime of excitons in disordered

crystals. Localization of ^{the exciton makes the} coupling

constants V 1 ^and V 2 large ^{even in} group III V semi- conductors like Ga(As, P).

a phonon energy, this is not unreasonable because of the short lifetime of excitons in disordered crystals.

We have also obtained a value of the localized

« dressed » exciton _energy.

In table III, the values of the residual intensities N v are listed for the four modes. We first note that N L02 is relatively high ; this reflects the experimental

fact that, ^even off-resonance, this line is always ^the

strongest [10]. ^As expected, ^{the ratio} NLo2/NLo1

^5.4

is of the same order of magnitude ^as N 2L02 /NLo1 +L02

3.7. Another feature to note is the relatively small values of N LO, ^+LO2 and N 2LO2 compared ^{to NLo1}

^and NLo2, ^{this can be} _explained

Table III.

Non-resonant contributions Nv of ^the

mode v (= LOb L02, L01 ⁺ L02, ^{and 2} L02) ^{to the}

total Raman intensity compared ^with typical ^resonant

contributionsl V(w) ^at hWi

^{2.444 eV.} The intensities

are all normalized by ^the factor ^C of eq. (6). ^{Note the} relatively small values of NL01 + L02 and N2L02 com-

pared ^{to NLo1}

^{and NLo2} signifying that the residual part NV behaves like usual second order Raman ^inten- sity off-resonance. ^{In contrast to} this, ^{the resonant} contribution I V(w) for two-phonon processes

(L01 ⁺ L02 ^{and 2} L02) ^is of ^{the same order} of magnitude ^as for one-phonon processes (L01 ^and L02).

by the observation that, when the incident photon

energy is far from resonance, the replica disappears leaving behind the usual weak second order Raman

spectrum ^{and to} the fact that the first order spectrum is always ^{sizeable at on- or} off-resonance conditions.

Finally ^we compare in table IV our theoretical Raman intensity ^with experimental ^results [10] ^for

a given incident laser wavelength Ai

^{458 nm at}

T

100 K. This corresponds ^{to effective} photon

energy 1iWi

^{2.597 eV. The} agreement ^is satisfactory.

Table IV.

Comparison of experimental and theore- tical Raman intensities in logarithmic ^{scale at} 1iWi

^{2.597 eV.} Experimental points ^{are taken} from reference [10]. ^Because of ^{the enormous} dispersion of

the experimental points ^{two extreme} experimental ^values

are given for ^{each mode.}

976

4. Concluding ^remarks.

^{In this} paper ^we have shown that Raman intensity ^for ternary alloy Ga(As, P)

can be decomposed ^{into the resonant} part ^{I’ and the} residual part N v. The residual part ^is non-resonant

in the frequency range of interest and decreases

rapidly for second or higher-order ^Raman processes.

The resonant part, ^{on the other} hand, ^{is due to loca-}

lized excitons coupled strongly ^with LO, ^and L02 phonons and, at resonance, it can be equally ^intense

even for higher-order processes. It is the spatial

localization of exciton responsible ^{for the resonant}

process that yields relatively large coupling constants, V,

^0.7 nWLOl ^and V2

^1.4 nWL02 even for a semi- conductor like Ga(As, P) with small polaron coupling

constants.

Finally, ^{let us consider an} extension of the present

treatment. The localized exciton model can be _gene- ralized to include other modes than L01 ^and L02,

such as T01, T02 ^{and X} peaks ^{in the Raman} spectrum of Ga(As, P). Since these modes are non-polar, ^their

interaction with localized excitons is weak and can

be treated as a perturbation ^{to the} main Hamil- tonian given ^above. Resonances in these modes

are similarly expected. ^The explicit calculations will be described in a subsequent paper.

Acknowledgments.

^{It is a} pleasure ^{to thank} C. , Hirlimann ^for showing ^{us his} preprint prior ^to publication ^{and for} stimulating discussion. One of us’ (N.M.) would like to thank M. Jouanne and J. F. Morhange ^for initiating him in numerical _compu- tation.

References

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