Exciton distribution function and secondary radiation in polar semiconductors

HAL Id: jpa-00210582

https://hal.archives-ouvertes.fr/jpa-00210582

Submitted on 1 Jan 1987

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Exciton distribution function and secondary radiation in polar semiconductors

C. Trallero Giner, O. Sotolongo Costa

To cite this version:

C. Trallero Giner, O. Sotolongo Costa. Exciton distribution function and secondary ra- diation in polar semiconductors. Journal de Physique, 1987, 48 (9), pp.1505-1512.

�10.1051/jphys:019870048090150500�. �jpa-00210582�

Exciton distribution function and secondary ^{radiation in} polar

semiconductors

C. Trallero Giner and O. Sotolongo ^Costa

Department of Theoretical Physics, ^Havana University, San Lázaro y L, Havana, ^Cuba (Reçu ^{le 10} fgvrier 1986, révisé le 9 avril 1987, accept6 le 12 mai 1987)

Résumé.

Nous calculons une fonction de distribution explicite ^hors d’équilibre pour des excitons dans l’état fondamental n = 1, dans des semi-conducteurs polaires, ^{dans le cas où on ne tient} compte que de l’interaction

avec les phonons acoustiques. ^Nous l’utilisons pour obtenir ^une expression générale pour la section efficace de rayonnement secondaire (valable pour les processus Raman ainsi que pour la luminescence chaude ^et

thermalisée). ^Nous appliquons les résultats à l’explication ^{de la} dépendance ^en température ^des demi-largeurs

des raies de luminescence 1LO et 2LO dans des monocristaux de CdS. Nous étudions la valeur relative des contributions des intensités Raman 3LO et de luminescence et la variation du spectre d’émission secondaire en

fonction de la durée de vie des excitons. Nous obtenons un accord quantitatif ^avec des résultats

expérimentaux.

Abstract.

An explicit non-equilibrium distribution function for excitons in the ground ^{state n}

^{1 in the case}

when the fundamental interaction is with acoustical phonons is calculated for polar semiconductors. Using it, ^a general expression ^{for the} secondary ^radiation cross-section (valid ^for Raman, ^hot and thermalized luminescence processes), is obtained. The results are applied ^to explain ^the temperature dependence ^{of the}

1LO and 2LO luminescence lines half-width in CdS single crystals. The relative contributions of 3LO Raman and luminescence intensities and the variation of the secondary ^emission spectrum ^{as a} function of the exciton life-time are studied. Comparison ^with experimental ^results yields quantitative agreement.

Classification

Physics ^Abstracts

71.35

78.30

78.55

1. Introduction.

The study ^of secondary ^{radiation of} light by ^conden-

sed matter is a powerful ^{method of} investigating elementary excitations within matter. There is a

number of investigations dealing ^{with this} problem

in polar semiconductor (see [1]).

It has been shown [2-6] that the excitonic model is

a successful one to explain the process of secondary

radiation in many polar semiconductors such as CdS, ZnTe, CdSe, ^etc.

In CdS the excitonic spectrum ^of secondary ^radia-

tion [2] and the relative intensities of 3LO-Raman and luminescence [3, 4] ^were investigated. ^In general

these processes of light scattering ^with participation

of excitons as intermediate states of the crystal ^can

be viewed in the following way :

a) Indirect creation of an exciton with kinetic energy Eo in the energy ^state with principal quantum

number n

1.

b) Exciton relaxation.

c) ^{Indirect exciton} annihilation with the emission

of a photon líw

^{and at least one} LO-photon ^with

energy líw Lo. On the other hand, it is known [7] ^that

the differential cross-section of the secondary ^emis-

sion radiated by ^{an excited solid is} given by ^the

formula :

where c is the velocity ^of light ^{in vacuum,} n ( w ) ^the frequency dependent refraction index, ^and W ( w S )

the emission probability ^{of a} quantum ^of secondary

radiation líw s by unit of time and of solid angle ^li,

when an incident light quantum ^{hw 1 is in the volume}

Vo ^{with a} refraction index equal ^to unity.

In terms of the exciton distribution function

(EDF), W(ws) ^is given by [8] :

where Wi(E, w,) is ^the annihilation probability per unit time and solid angle ^with photon ^{emission of}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048090150500

1506

energy hw,

and emission of LO-phonons by ^an

exciton having ^kinetic energy E, averaged through

the angle between the directions of the exciton and

photon ^{momenta, and} P(E)dE is the number of excitons within the energy interval (E, ^{E +} dE).

Although ^{the EDF for} polar semiconductors has been found when the exciton kinetic energy is greater than that of an optical phonon (i.e. ^E

hw LO) ^{[8], this} problem up ^{to now} has not been solved for E

líúJLO. ^This EDF, taking ^{into account} W, (E, co,), gives ^the possibility ^to investigate ^the light scattering processes by the above mentioned semiconductors when the incident photon ^{has an}

energy :

Here Eg is the gap and AE is the exciton binding

energy. In the following, this is the energy interval that will be considered.

If the scattering processes ^are described by ^the

relaxation time _r, and the exciton annihilation by ^T,

the stationary population in the band is conditioned

by both life-time and relaxation. If -r > TS, ^a thermal

quasi-equilibrium of excitons takes place leading ^to

the formation of thermalized luminescence. If this condition is not fulfilled, the emission line tends to become a Raman line with the participation ^{of real}

intermediate states.

In the case of thermalized luminescence, ^{it is} logical ^to expect ^{the EDF to be a} Maxwell-Bol- tzmann one, whereas for the Raman line, ^the corresponding ^{EDF must be a} strong non-equili-

brium one.

In section 2 ^we deal with the balance equation, obtaining ^an approximate expression for the EDF valid for both cases.

In section 3 an application ^{of our results to} experimental observations is made.

2. Exciton distribution function.

The determination of the EDF is a complex problem, governed by ^the specific physical conditions of the system. ^{Let us} neglect exciton-exciton interaction and consider that the exciton relaxation is con-

ditioned only by ^the interaction with acoustical

phonons (this ^{is the case in clean} enough polar

semiconductors when the exciton kinetic _energy E .-- h(o Lo at ^low temperatures). ^{In the case of an} isotropic ^{EDF the balance} equation ^{is :}

where W;n represents ^{all the} contributions per unit time to the excitonic state with energy E, ^and conversely ^for Wout. ^So, for the processes contained in Wout ^{we have :} 1) ^exciton scattering w (E, E’)

from energy E ^to energy E’ in the quantum ^state

n = 1 performed by acoustical phonons ;

2) phonon-assisted transitions to states with n > 1

(Wn); 3) ^exciton decay (Wd ) ; 4) LO-phonon ^as-

sisted exciton annihilation with the emission of a

light quantum líúJ = hw ^{1- Khw} LO (K ::. 1 ) (1/’rR) ; ^and 5) processes of non-radiative annihila- tion (11’rNR). ^{For the case} of exciton-LA phonon

interaction when the deformation potential ^{model is} used, ^{it was} shown [9] that processes 2 and 3 for E H(OLO ^are negligible. Similarly, ^{this can be} proved by taking ^the piezoelectric model into ac- count. Then we can write :

In the (5) the total lifetime of excitons 1 / T

1/TR ⁺ I/TNR ^is governed by radiative and non-

radiative annihilation processes.

In the same way ^we may represent :

where W, ,,, ^{is the rate} of indirect creation of a

Wannier-Mott exciton with energy Eo. This energy is

strongly determinated by the energy of the exciting light líúJ 1. ^{If we} suppose, for example, that exciton creation is performed by LO-phonon ^emission (indirect exciton creation by light, ^see [10]), ^then Eo is the exciton kinetic energy in the n = 1 ^state

equal ^to fiw -Eg+AE- ^{HCOLO -}

In the appendix ^a calculation of w (E, E’) ^is performed ^for piezoelectric and deformation poten- tial interaction Hamiltonians.

Using ^relations (4)-(6) ^and (A. 14) from the appen-

dix, the balance equation is transformed to :

where

Noting that the exciton energy is varied in ^a quite

small amount after acoustical phonon absorption ^or emission, ^we may expand f (E’, t ) ^{as :}

In the stationary case, and retaining ^terms up ^to

second order in (10), ^from (7) it follows that :

being :

From inspection ^of (12), (A.9) ^and (A.13) ^{it can be}

shown that for exciton kinetic energies E > mu 2

al is negative. Although the coefficients a,,(E) ^can

be calculated by ^direct integration supposing

N [ (E’ - E)IKB T] = kB TI (E’ - E) leading ^to complex expressions ^{for these} coefficients, ^{it can be}

seen that a direct solution of (11) ^is impossible.

Nevertheless, ^{it is} possible ^to evaluate al and a2 by ^the following reasoning : owing ^{to the fact} that, according ^to (A.9) ^and (A. 13), w (E, E’ ) ^{is different}

from zero in a narrow interval, ^{we evaluate the}

energy multiplicator (E’ - E ) ⁱⁿ (12) ^{as As, a}

characteristic energy of the order of 2 mu2 and remains in (12) ^the integral ^{of the} scattering proba- bility which may be identified with the inverse of the relaxation time (l/Ts(E». ^So, al and a2 may be expressed approximately ^{as :}

and for simplicity ^we consider that T s (E) ^{is a}

constant independent of the energy.

Imposing ^{that P} (E) -+ ^{0 for E oo} and that the total number of particles ^{in the} stationary ^{state is}

we obtain for the EDF the expression :

where :

-0 (x) ^{is the error} function and B (x ) the Heaviside unit step ^{function defined as :}

If ^T is great enough, ^EDF (14) ^{reduces to a Maxwell-}

Boltzmann one :

3. Comparison ^with experimental ^results.

In this section we will apply ^our theoretical results to the experimental observations reported ⁱⁿ [2-4].

3.1 HALF-WIDTH OF 1LO AND 2LO LUMINESCENCE LINES.

In [2] the half-width of 1LO and 2LO

luminescence lines in CdS was measured as a func- tion of temperature using ^an exciting radiation of

wavelength ^A

^{485.3 nm} creating excitons with a

kinetic energy E -- ftw LO.

For the 1LO luminescence line the annihilation

probability W,(E,co ) ^is given by :

where q is the phonon wavevector and M is the transition matrix element given by :

The sum in (19) ^runs for all virtual excitonic states.

H,, - 1 (H,, - p) ^{is the} exciton-light (exciton-LO phonon) interaction Hamiltonian. In [10] this matrix element was calculated. For the energy range ^we deal with, the influence of the continuous spectrum

can be proved ^{to be} negligible. ^{Then we have :}

1508

where :

ml (m2) is the effective electron (hole) ^{mass, and}

where M = M 1 + M2, EO (EOO) ^{is the static} (high frequency) dielectric constant, A is the exciton’s reduced _mass, Pcv ^{the momentum} matrix element

connecting ^{the K}

0 valence and conduction band Bloch functions and e the polarization ^{vector of the}

incident light. Using (1), (2), ^and (18) ^{we obtain :}

Formula (23) permits ^{us to} obtain the thermal in-

crease of 1LO luminescence line half-width. A detailed analysis ^{of the} experimental ^{results leads us}

to conclude that in the sample used in the experiment

T is a big one, ^{so we} may ^use the Maxwell-Boltzmann EDF (17) ⁱⁿ (23).

Concerning 2LO luminescence line, ^{it can be seen}

that two-phonon ^exciton annihilation probability ^is independent ^{of energy as was shown in} [2]. By ^this

reason the differential cross-section can be written

as :

where for this case the matrix element M is indepen-

dent of energy.

Using ^{the matrix element} calculated in [10] ^and P (E) ^from equation (17), equation (23) ^{can be}

calculated in arbitrary ^{units for} any energy. The

same applies ^to equation (24) taking ^{into account}

that M in that case is a constant. The half-widths of the curves given by (17) ^are independent ^{of the} employed ^{units, and the} comparison with exper- imental results obtained in [2] ^{can be} performed

without _any adjustment.

Fig. ^1.

Broadening ^of exciton emission lines in units of

hw Lo in ^{CdS with} temperature. Lines 1 and 2 were

obtained using ^formulae (21) ^and (22) respectively. ^The

half-widths are those at lle of the maximum intensity

value.

In figure ^{1 the} comparison of theoretical (using

formulae (23) ^and (24)) ^and experimental ^{results is}

shown. The half-widths are given in units of

hW _LO (for ^CdS !iw LO ^is equal ^{to 0.038} eV).

3.2 TEMPORAL EVOLUTION OF EXCITON SECOND- ARY EMISSION SPECTRUM. - In [3] ^the experimental study ^{of the} secondary emission in CdS single crystals, using samples with different free exciton lifetimes was reported. ^{It was seen} that the relation of 3LO luminescence integral intensity ^{with 3LO}

Raman emission line is highly dependent ^{on the}

relation between exciton lifetime and acoustical relaxation time Tg. ^The crystals ^{were excited at 2K} by ^{a laser line A}

^{476.5 nm. In this case 1LO}

assisted exciton creation with kinetic energy E

/iw LO ^is performed. ^{This means that 3LO emission}

line is obtained by 2LO assisted exciton annihilation,

whose probability, ^{as we} already ^{know, is} indepen-

dent of energy. Then, ^{3LO Raman} intensity (l3LO )

is due to the two-phonon exciton annihilation from the energy Eo, ^and we can use equation (24). ^This

3LO Raman intensity ^is proportional ^to equation (24) multiplied by ^{a narrow} frequency interval Aw,, ^{which is the} frequency interval used in the experiment. ^The intensity ^is proportional ^{to the} jump of the EDF P (E > Eo ) - ^P (E EO) = ^AP.

Then, ^from (14) ^{it can be seen that:}

The integral intensity Iz ^can be obtained by ^inte- gration ^of (24) ⁱⁿ WS. This leads to the result that

I_v ^{will be} proportional ^to We., ^T, the whole number of excitons in the band. So, ^we may obtain :

Ií dúJ s

where A = h Aw, ^{is a} constant to adjust ^with

AE

experimental data. In table I a comparison of exper-

imental values with those obtained from (24) ^is presented for different samples.

In figure ² secondary ^emission spectra ^{of CdS} samples with different lifetimes of excitons, reported

in [3] ^are presented ^and compared with results obtained from (24) ^{with EDF} given by (14). ^{As can}

be seen the EDF given by (14) satisfactorily ^reflects

the temporal evolution of secondary emission spec- trum as a function of exciton lifetime.

Fig. ^2.

^Exciton secondary ^emission spectra ^{for CdS} samples with different lifetimes. Dotted curves are theoret-

ically predicted with formula (22). ^The samples ^{are those} corresponding ^{to table I.}

The observed emission spectrum ^{in the} region

near 499 nm is due to the contribution of bound excitons which are not taken into account in our

model.

In [4] ^a comparison ^{of the} integral luminescence

intensity IA ^with I3LO ^{as a} function of ’r/T, ^was presented. This luminescence IA reveals itself as a

band of hot luminescence in the long wavelength part of the 3LO line. Its integral intensity ^is equal

to :

Table I.

Comparison of ILO ^{obtained rom 24 with} experimental . ^{data i} i I3LO ^and II ^are reported ⁱⁿ [3] for

different samples. ^The proportionality ^{constant A was} adjusted to sample N 1.

1510

Then from (26) ^{we have :}

In figure ^{3 the} comparison ^with experimental

results is shown. The theoretical curve reveals that,

as in the experiment, the increase of ^T leads to a more effective exciton thermalization.

Fig. ^3. - I AI 13LO ^{vs. s = T . The dots} represent exper-

Ts

imental observations for different samples. ^{The curve was}

obtained with formula (28).

Finally, from the obtained results it is important ^to

note that formula (1) derived from the general theory ^of secondary ^radiation [7] ^{and the EDF} given by (14) may be used ^to investigate ^Raman scattering

processes ^as well as hot and equilibrium ^lumi-

nescence for exciton kinetic energy E -- H(OLO. ^The

obtained EDF, expressed ^{as a} function of _T, TS, and temperature reflects the above showed

experimental facts that the concurrence of acoustic

relaxation and exciton lifetime determines the exci- ton population in the band.

Acknowledgments.

We are gratefully ^{indebted to Drs I. G.} Lang,

S. T. Pavlov and S. Permogorov for their helpful

discussions and comments to our work.

Appendix.

THE SCATTERING PROBABILITY w(E, E’).

^The exciton-piezoelectric phonon Hamiltonian is :

where j = 1, ² corresponds ^to electron and hole, respectively, ^r is the radius vector, bq ^{are the}

annihilation (creation) operators of the acoustic

phonon ^with wavevector q, and

e is the electron charge, 13,,,,, ,, ^{the tensor of}

piezoelectric moduli, Emn ^the dielectric tensor,

Vo the normalization volume, p o the crystal density ;

w

uq and ^e the frequency ^{and the} polarization

vector. Considering the exciton wavefunctions for the initial and final states as being ^{of the} type :

where R is the radius vector of the centre-of-mass of the exciton, and a its Bohr radius, ^and

P = Irl-r2l-

The calculation of w(E, E’ ) ^{can be} performed through Fermi’s Golden Rule, according ^{to which}

the transition probability ^{is :}

Using (A.1)-(A.4) ^{we obtain :}

where :

y ^{is defined in} [11] ^{like the mean value of:}

with respect ^{to the} angles ^formed between the direction of q ^{relative to the} crystals ^{axes. In} [11,12] ^{this value}

was found to be y

( 4£: ) 2. ^{2.68 x 10 for CdS. The} scattering probability by unit of time and energy is :

where :

In the same way w (E, E’) may be calculated using ^the deformation potential model, where the exciton-

phonon interaction Hamiltonian is :

where :

and Ei ^{is the} deformation potential for electrons or holes. In this cases :

In both models (piezoelectric ^and deformation potential) ^{it can be seen that:}

1512

References

[1] CARDONA, M., Light Scattering ^{in Solids} (Springer- Verlag) ^1975.

[2] GROSS, E., PERMOGOROV, S., RAZBIRIN, B., ^J.

Phys. Chem. Solids 27 (1960) ^1647.

[3] PERMOGOROV, S., TRAVNIKOV, V., Solid State Com-

mun. 29 (1979) ^615.

[4] PERMOGOROV, S., TRAVNIKOV, V., ^Fiz. Tverd. Tela.

22 (1980) ²⁶⁵¹ (Sov. Phys. Solid State 22 (1980))

1547.

[5] ^TRALLERO GINER, C., ^SOTOLONGO COSTA, O., PAVLOV, S. T., ^Fiz. Tverd. Tela. 26 (1984) ²⁴¹ (Sov. Phys. Solid State 26 (1984)) ^1.

[6] ^TRALLERO GINER, C., LANG, ^I. G., PAVLOV, ^S. T., Fiz. Tverd. Tela. 23 (1981) ¹²⁶⁵ (Sov. Phys.

Solid State 23 (1981)) ^743.

[7] GOLTSEV, ^A. V., LANG, I. G., PAVLOV, S. T., BRYZHINA, ^M. F., ^J. Phys. ^{C. 16} (1983) ^4221.

[8] ^TRALLERO GINER, C., LANG, ^I. G., PAVLOV, ^S. T., Phys. Status Solidi B 106 (1981) ^349.

[9] ^TRALLERO GINER, C., LANG, ^I. G., PAVLOV, ^S. T., Fiz. Tej. Poluprov ¹⁴ (1980) ²³⁵ (Sov. Phys.-

Semicond. 14 (1980)) ^138.

[10] ^TRALLERO GINER, C., ^SOTOLONGO COSTA, O., Phys. Status Solidi B 127 (1985) ^121.

[11] ^ZINOVIEV, H.-H., IVANOV, L. P., LANG, ^I. G., ^PAV-

LOV, S. T., PROKAZNIKOV, A. V., ^YARO-

SHETSKY, I. D., Zh. Eksp. Teor. Fiz. 84 (1983)

2153 (Sov. Phys.-JETP ⁵⁷ (1983)) ^1254.

[12] HUTSON, A.-P., Conference on Semiconducting

Compounds Suppl. ^{to the J.} Appl. Phys. ³²

(1961) ^2287.