EVALUATION OF THE HUANG-RHYS FACTOR AND THE HALF-WIDTH OF THE F-BAND IN KCl AND NaCl CRYSTALS

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EVALUATION OF THE HUANG-RHYS FACTOR

AND THE HALF-WIDTH OF THE F-BAND IN KCl

AND NaCl CRYSTALS

E. Mulazzi, N. Terzi

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C 4, suppliment au no 8-9 Tome 28, Aotit-Septembre 1967, page C 4-49

EVALUATION OF THE HUANGRHYS FACTOR

AND THE HALF-WIDTH OF

THE F-BAND

IN KC1 AND NaCl CRYSTALS

(*)

by E. MULAZZI and N. TERZI (**)

Istituto d i Fisica dell7Universit8 and Gruppo Nazionale d i Struttura della Materia del C . N. R., Milano, Italy

Abstract.

-

The Huang-Rhys factor S, the half-width H(T) and the peak position E of the F band are evaluated in terms of the forces that the l s -t 2 p electronic transition induces in the

surrounding lattice, and of the phonon projected density of states of the perturbed lattice. The symmetry coordinates of the F-center are explicitly considered ; only T:, T$, T'& are found to occur. H ( T ) and E are deduced from the distribution function I(w) of the absorption band by means of the method of moments. The evaluation of the perturbed projected densities is made by considering Hardy's D. D. model for the host-lattice dynamics and a grid of 4 096 points in the B. Z. The values 1 =

-

0.8 f,ff and 3. =

-

0.6

hff,

as deduced from the best fit on the F-center induced Raman spectra, were employed for the F-center in NaCl and KC1 respectively. R represents the change of the short-range effective force constant f e ~ t involved in the F-center harmonic perturbation for the ground state. The forces Fr,+, Fr,,+, Fr,,+ are evaluated making use of the experimental hydrostatic, tetragonal and trigonal stress-coefficients of the F-band. Good agreement is found between the experimental data and calculated half-width and Huang-Rhys factor in the temperature range 5 OK-305 OK.

The value of the peak position is found to be shifted by 0.66 eV and 0.32 eV, for NaCl and KC1 respectively, with respect to the energy of the pure electronic transition between 1 s and 2 p elastically relaxed electron states.

R6sum6. - Le facteur de Huang et Rhys, la demi-largeur et la position du maximum de la bande F o n t et6 exprimes en fonction des forces induites, par la transition Clectronique 1 s -+ 2 p , sur les ions autour du centre F, et en fonction de la densit6 &&tat projetk du rkseau perturbk. Les coordonnkes de symktrie de la perturbation du centre F ont 6tC considCrCes ; on a trouve que seules les representations irreductibles

rf,

r13 et

r2:

sont associks B la transition. Le calcul de

la densit6 d'6tat projetk a et6 fait en dkcrivant la dynamique du rCseau dans le modkle h ions deformables de Hardy et en employant gm rkseau de 4 096 points dans la premikre zone de Brillouin. Les valeurs 1 = - 0,8 f,fr pour NaCl et I =

-

0,6 fefr pour KCl, d6duites par mesures d'effet Raman, ont 6te adoptCes dans les calculs. 1 represente la variation de Ia constante de force effective B courte distance, &W, introduite, dans l'approximation harmonique, par le centre F. Les forces Frl + , Fr,, + et Fr,, + ont kt6 evalukes par les mesures expCrimentales des coefficients de la bande F sous contrainte hydrostatique, tktragonale et trigonale.

Les valeurs calculees de l a demi-largeur et du facteur de Huang-Rhys sont en trks bon accord avec les donnees expkrimentales dans tout l'intervalle des temperatures considkrees (5 OK-305 OK). On trouve enfin que la position du maximum de la bande F est dkplack de 0,66 eV et 0,32 eV, respectivement pour NaCl et KCl, par rapport B l'energie de la transition purement klectronique, entre les niveaux l S

-

2 p klastiquement relaxb.

I. Introduction.

-

I n this work we report a numerical estimate of some parameters of the F-absorption band in NaCl and KC1 crystals (the halfwidth H(T),

(*) This research has been sponsored in part by E. 0 . A. R. under Grant N. 65-05, with the European Office of Aerospace Research, U. S. Air Force.

(**) Now at Service de Physique des Solides D, FacultC

des Sciences, Orsay (France) for the 1966-67 academic year.

the phonon contribution E,, t o the peak position E, and the Huang-Rhys factor S(T)) and their temperature dependence between 5 OK and 305 OK. Only the halfwidths can be compared with high accuracy with the experimental data, S(T) and being quantities not directly measured from F-center bands and therefore badly known.

The aim of the present calculation is twofold. O n 4

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C 4 - 5 0 E. MULAZZI, N. TERZI

the one hand we want to show that, in the theoretical context developed in [l] and briefly exposed in the previous communication, the distribution function I(o) of the absorption band can be numerically evaluated if the experimental defect properties are well enough known to allow, as in the case of F-center, the estimate of the parameters entering I(w). On the other hand through comparison with the experimental results we wish to confirm the reliability of the theoretical framework (harmonic approximation, linear change in electron-phonon coupling, ecc.) adopted in [l].

The excellent agreement found here between theoretical and experimental halfwidths has in turn allo- wed a deeper understanding of both the F-center dynamical behavior and the nature of the transition. Furthemore it allows also a good theoretical estimate of the Huang-Rhys factor and the phonon contribution to the peak position.

Analogous calculations for F-center bands in a wider range of host crystals are now in progress [2] and we hope that a more complete and systematic description of the coupling between the electron tr2p- ped in the F-center and the phonons can be obtained. In [l] the distribution function I(o) has been deduced as a function of the perturbed lattice dynamics, through the one-phonon projected densities of states pr(02), [2], and of the linear change in the electron- phonon coupling, through the forces F, which the tran- sition switched on to the ions l surrounding the F-center. At this stage of the theory the actual evaluation of I(o) represents in general only a very hard numerical problem, the only conceptual difficulty being the estimate of the electronic ground state perturbation A and of the forces F, for these require a sufficiently good knowledge of both the electron wave function and the electron-phonon coupling. For the F-center, as well as for any other center, the value of the components of the forces F can be deduced empirically from some properties of the absorption bands, determined by stress experiments, and the ground state perturbation strength can be evaluated from other defect induced properties, say the infrared absorption or the lattice thermal conductivity. However, in this case also we are left with the problem of the evaluation of the projected density of states pr(02), which in general is not easy to solve. Here we have adopted a model for substitutional defects, recently developed by Benedek & Nardelli [3], which allows for the calculation of the one phonon perturbed densities in the Hardy D. D. model, and which was shown to be powerful in con- nection with some defect properties [3, 41.

The l s -t 2 p strongly-coupled local transition of

the trapped electron at the F-center in alkali halide host crystals gives rise to one of the less complex situations from the point of view of the vibrational structure. We remember that by strong-coupled local transition we mean that the forces F, even if intense, arise only from a change in the short range coupling between the trapped electron and the phonons. In this case, since the transition is a typical many-phonon one (the Huang-Rhys factor at T = OOK ranges approximatively from 30 to 40), each one-phonon vibrational structure has such a low intensity, that it cannot be actually detected. The resulting absorption band displays a smooth shape, with a well-defined maximum and an half-width of about 0.2 eV at low temperatures.

In the first approximation the band can be considered nearly Gaussian ; its square halfwidth H2(T) and peak position E, deduced with the method of moments, are the following :

and

if spin-orbit interaction, which indeed represents for NaCl and KC1 a very small correction [g], is neglected. The Huang-Rhys factor is defined as :

1 m

S(T) =

-

E

F,,, dw x r j j , 0

Here

r

runs over the irreducible representations (reps.) of the point group Oh of the symmetry opera- tions of the F-center. Frj is a suitable combination of the F, which transforms as the j-th basis of the rep. l?. pfJ,,(w2) is the one-phonon density of states of the perturbed crystal, projected on the JJ' bases of the

r

rep.. &Qug is the energy of the pure electronic transition, which takes place between the elastically relaxed upper and ground states. H(T), E and S(T)

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EVALUATION OF THE HUANG-RHYS FACTOR AND THE HALF-WIDTH C 4 - 5 1

The selection rules of process prescribe that the sums in (l), (2) and (3) must be performed only on the

r:,

and

r:,

reps. (or r , ,

r,

and

r,

reps. in Bethe's notation). This result is also consistent with a very general principle which states that any defor- mation in a cubic crystal, can always be decomposed into its cubic (r:-type), tetragonal (rT2-type) and trigonal (&type) symmetry components.

It follows that also only the

T:,

and &-type densities of states are associated with the transition. Therefore the rG-type infrared active p r G ( ~ 2 ) does not contribute to the band shape, in the linear coupling change assumption. However it may be shown [5] that, even if second order contributions in electron- phonons coupling are taken into account (as the change in normal mode frequencies), (2) is pratically the right expression for H2(T), except if true infrared active local modes, are present, as is not the case for F-center. On the contrary an additional term must be added in (3), which is partially responsible for the explicit temperature dependence of the peak position.

11. The defect model.

i. THE GROUND STATE PERTURBATION A.

-

The lattice dynamics has been described in the framework of the Hardy D. D. model [6]. The ground state perturbation A due to the F-center has been considered in a defect model in which the Coulomb type force constant matrix is assumed unaffected by the presence of the defect, while all the perturbation is attributed to the short-range type force constant matrix. In what follows only the change in nearest neighbors (n. n.) interaction is considered. The n. n. perturbation matrix can be diagonalized into submatrices in the symmetry coordinates frame. For a point defect of 0,-symmetry, as the F-center, the resulting A,;, A,:, and A,;, submatrices consist of one element as follows :

and

if changes in n. n. non-central forces are neglected. M + and M - are the masses of the alkali and halide ions respectively. Therefore also the corresponding perturbed projected densities of states prjj,(co2) are, for any rep.

r,

simple functions pr(w2) which, from (4), depend only on the change in force constants and not at all on the change in masses associated with

the defect. The numerical evaluation of the p,(w2) was made with the aid of a electronic computer by considering a grid of 4 096 points in the B. 2.

The above defect model, even if oversimplified, seems to represent a good representation of the actual situa- tion, provided that the parameters characterizing the interaction are suitably defined. This is done defining, in the D. D. model scheme, an n. n. effective force constant for the perfect crystal given by [3] :

X (es*/e)2(a+

+

a-)(a+

+

a -

+

30187~)-l) (S) where v = 2 r: is the unit cell volume, a' the polari- zabilities, e the electronic charge, e: the Szigeti effective charge, A and B the central and non central force constants, respectively, due to the n. n. core repulsive potential. Therefore jl is defined as ;

A = feff

-

feff

.

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where

jeff

is the change in the n. n. effective force constant.

The values

jllf,,,

= - 0.8 and - 0.6 here adopted for NaCl and KC1 respectively were obtained in reference [4] from the best-fit between the experimental data on F-center induced Raman spectra and the theoretical prediction. Furthermore the pr(w2) evaluated with the above values of A show low-frequency resonances which agree with the experimental results on low temperature thermoconducibility in NaCl and KC1 crystals containing F-center [7].

In order to clear up the effect of the polarizability in the definition off,,, the previous A-values are to be compared with the value

Alf,,,

=

-

I, appropriates for F-center short-range perturbation when the lattice dynamics is described in the rigid ion model.

ii) THE FORCES F. - Since one has assumed that the transition is very localized, only the short-range electron-phonons interaction is assumed to change during the transition. In particular only the forces switched into n. n. of the F-center are here taken into account. The values of the symmetry components F,;, F,;, and Fr; of the forces have been deduced [8, 9, 101, by fitting the theoretical with the experimental hydrostatic, tetragonal and trigonal stress coefficients of the F-band. Note that the elastic constant matrix

zij,

here used, has been modified to respect that of the perfect crystal cji,

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C 4 - 5 2 E. MULAZZ1, N. TERZI used (n. n. central forces change), only the ell compo-

nent of the elastic constant matrix has to be changed as follows :

& l = c11

+

(112 r,) ( v / $ . (7) where

v

is the volume of the perturbed zone and has been assumed to be 3.5 v (*).

Such a correction, which affects only the components of the cubic and tetragonal-type forces, may be not negligible in value, as one can see from table I and figure 1. In table I are reported the adopted values of the quantities :

( h ~ + ) - l F; = ( h ~ + ) - l C F;~

.i (8)

where the summation is performed on all the vectors F,,, transforming as a basis function of the same degenerate rep.

T.

In figure 1 the two determinations of H(T), with changed (full line) and unchanged (dashed line) elastic constants, are reported.

FIG. 1. - Theoretical (full and dotted lines, see 5 I1 ii) in the text) and experimental half width (Ref. (a) and (c) of table 11) versus temperature. The experimental uncertainly in the deter- mination of the forces F (see relation (1) and reference [9] in the text) turns out to give at least a 3.5 % and 3 % uncertainty in H(T), for NaCl and KC1 respectively

Experimental Frj were found to vary very little with temperature [10], so that room-temperature values were employed for the whole range of temperatures considered ( 5 OK-305 OK).

111. Discussion. - The agreement between theoretical and experimental halfwidth is also very good in the temperature dependence, as one can see from the full line in figure 1. There we have also reported, in the dashed lines, H(T) obtained from (1) using the Schnat- terly experimental forces Frj, reported in table I. Further improvement could be obtained if higher order corrections in electron-phonons interaction change were considered, as well as if a defect model extending over more neighbors than the n. n., were taken into account. The first quantity has been evaluated taking into account the change between the ground and the upper state charge distribution of the trapped electron [l] and it was found to be very small (10-' H ( 0 ) ) . The correction coming from a more extended defect model was not evaluated but, owing to the surprisingly good results obtained, seems to be small.

Therefore in consideration of the above results we may first confirm that the defect model here adopted gives a good description of the dynamical behavior of the F-center, and also state that the model of the strongly-coupled transition, localized to n. n., describes the l s -+ 2 p electronic transition of the F-center on an accurate way.

In the first column of table I1 is reported the value here obtained for the Huang-Rhys factor S at 5 OK, while its temperature dependence is shown in figure 2.

FIG. 2. - The Huang-Rhys factor as a function of the temperature.

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EVALUATION OF THE HUANG-RHYS FACTOR AND THE HALF-WIDTH C 4 - 5 3

N N

Adopted values of the symmetry components of the forces. Fr: and Fr:, (or Fr: and Fr:,, respectively) have been deduced from the experimental stress coefficients using modified (or not modified) elastic constant

c",

,

(or

c1 1).

S (5 OK), obtained from experimental results by means of the configurational coordinate (c. C.) model, is also reported and it is found to agree very well with the present theoretical value.

The phonon contribution to the peak energy E,, is reported in column 3. Note that E,, represents the

thermal energy associated to the elastic relaxation of the lattice, which takes place around the F-center soon after the transition, and is temperature independent in the approximation of the linear coupling change. Therefore the temperature shift of the peak energy is supported in this approximation only by the implicit temperature dependence of ha,,, the energy of the pure electronic transition. Its value at 5 OK, as deduced from (2) using the theoretical estimate of E,,, is reported in column 5 of table 11. Due to the many-phonon character of the transition, comes out to be a great percent of the peak value, as one can see from table 11. Higher order contributions in the electron-phonons interaction give term which are explicitly temperature dependent, through hyperbolic cotangent functions [5], but their evaluation is beyond the aim of the present work.

As a last remark, it is worthwile to make a compari- son with the usual analysis of H(T), done in the

NaCl KC1

framework of the c. c. model. In this model each one phonon density of states p,(02), which in general extend over the acoustical as well as the optical phonon frequencies [l], - is replaced with the same 6-like function 6(w2 - w2), centered at one single frequency

- -

w. o is in general deduced from the slope of the straight line which in the c. c. model is obtained plotting coth

{

H2(T)/H2(0)) against 1/T. The fitting with the experimental points gives a rough estimate of

G

which turns out to lie between the acoustical and optical branches. This model is not so bad in first approximation, since H(T) is an integrated quantity of the vibrational spectrum and therefore not too much sensible to its details, but cannot take into account with high accuracy the right temperature dependence. From figure 3 one can see that the higher the effective frequency the greater is the departure from the straight line behavior. Finally we note that it is not necessary to apply anharmonicity to reproduce the experimental data, as for instance to the anharmonic c. c. model, since the departure from the straight line occurs also in the harmonic approximation if the right frequency distribution is taken into account.

Acknoledgment. - The authors are deeply indepted to Prof. G. Nardelli for many helpful and stimulating

Some results obtained in this work and compared with the experimental ones (see text). 17.0 X 10'

3.6 X 10'

t h

/

1

Sth (5 OK)

1

Sexp (5 OK)

1 /

NaCl

1

41 f 3

1

0.66 1 0.04

/

2.77 (a)

1

2.11

-1

14.7 X 10' 3.1 X 10'

/ K C 1

1

3 1 f Z

/

30 (b) 0.32

i

0.02

1

2.31 (c)

1

1.99

1

(a) MARKAM (J. J.) & KONITZER (J. D.), J. Chem. Phys., 1961, 34, 1936-1942.

(b) LEMOS (A. M.) & MARKAM (J. J.), J. Phys. Chem. Solids, 1965, 26, 1837-1852. (c) MARKAM (J. J.) & KONITZER (J. D.), J. Chern. Phys., 1960, 32, 843-856.

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C 4 - 5 4 E. MULAZZI, N. TERW

his assistence in carrying out the numerical computa- tion as well as for same remarks on the results.

References

[l] MULAZZI (E.), NARDELLI (G. F.) & TERZI (N.), submitted to (( The Physical Review P for the

publication.

121 BENEDEK (G.) & MULAZZI (E.), t o be published. [3] BENEDEK (G.) & NARDELLI (G. F.), Phys. Rev.

1967, 155, 1004-1019.

[4] BENEDEK (G.) & NARDELLI (G. F.), Phys. Rev.

1967, 154, 872-876.

[5] BALDINI (G.), MULAZZI (E.) & TERZI (N.), Phys. Rev.,

1965, 140, A 2094-A 2101.

[6] HARDY (J. R.), Phil. Mug., 1961, 7, 315-336.

0002 0.004 0.W6 Q008 0.010 0912 to14 _[7]_WALKER_{(C. T.),}_{Phys. Rev.,}_{1963, 132, 1963-1975.} I / T ( 1 1 0 ~ )

[8] HENRY (C. H.), SCHNATTERLY (S. E.) & SLICHTER

FIG. 3. - Hyperbolic cotangent plot of the measured and (C. P.), Phys. Rev., 1965,137, A 583-A 602. theoretical values of the half width versus the reciprocal of the [9] SCHNATTERLY (S. E.), Phys. Rev., 1965, 140, A 1364-

temperature. A 1380.

[l01 GEBHARDT (W.) & MEIER (K.), Phys. Status Solidi,