MOTION OF THE SELF-TRAPPED EXCITON IN ALKALI HALIDES

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MOTION OF THE SELF-TRAPPED EXCITON IN ALKALI HALIDES

K. Song

To cite this version:

K. Song. MOTION OF THE SELF-TRAPPED EXCITON IN ALKALI HALIDES. Journal de

Physique Colloques, 1973, 34 (C9), pp.C9-495-C9-498. �10.1051/jphyscol:1973982�. �jpa-00215458�

JOURNAL DE P H Y S I Q U E

CoIloque C9, supplement au no 11-12, Tome 34, Nouembre-DPcembre 1973, page C9-495

MOTION OF THE SELF-TRAPPED EXCITON IN AL'KALI HALIDES

K. S. SONG

Department of Physics, University of Ottawa, Ottawa, Canada

R6surn6. - Une etude theorique du rnouvernent de saut de l'exciton auto-capti dans les halo- ginures alcalins est presentee. L'expression de I'energie d'activation thermique du rnouvernent de I'exciton auto-captC est obtenue. Cette quantitt depend tres fortement de l'itat interne de I'exciton. Limites superieure et infkrieure de l'energie d'activation sont evaluies avec un rnodele simple.

Abstract.

-

Hopping motion of self-trapped exciton in alkali halides is studied theoretically.

Expressions of the thermal activation energy for self-trapped exciton jump motion are derived.

It is shown that this quantity depends much on the internal state of exciton. Upper and lower limits of the activation energy are estimated employing a simplified model.

1. Introduction.

-

It is now well established that excitons in alkali Iialides (AH) created optically or by electron irradiation, undergo relaxation (self- trapping) and many of the subsequent behaviours can be adequately understood only through the self-trapped aspect. This seIf-trapped exciton in AH can most easily be regarded as a n excited halogen molecule ion X2-- o r as (V, + e*), where e:!: is the excited electron bound to the self-trapped hole.

The migration of exciton in a n ionic crystals is consi- dered to play an important role in many problems of energy transport, such as the production of colour centers.

In this paper we report the results of a preliminary study in which we derived the expression and eva- luated the activation energy for hopping motion of self-trapped excitons in A H . We follow here the theory of small polaron motion [ l ] which we previously applied success fully to the problem of V,-center reorientation [2]. Employing the continuum model we will show that the hopping motion activation energy depends critically on the internal state of the self-trapped exciton. We will discuss the results of a numerical evaluation performed on a very simple model.

2. Hopping motion of self-trapped excitons.

-

We closely follow our previous studies on V,-center migration in AH. The basic assumptions made in deriving the hopping probability are as follows

:

1) The lattice of the perfect crystal is harmonic and any distortion can be described by a set of normal coordinates.

2) The electron-phonon coupling is linear. There is n o change in the force constants introduced by the presence of a defect, only the equilibrium position of atoms changes.

3) Condon approximation is made which allows

the factorization of electron and lattice parts in the calculation of jump probability.

Under these assumptions one obtains the following expression of hopping probability W,,,, in the high temperature limit

:

This expression is identical t o the ones previously employed for V,-centers [2], [3] except that J and E:, are defined for an exciton. J is the transfer energy for a self-trapped exciton between two adjacent sites R and R'. The activation energy E, is defined as

:

with

AQ~,~(AQ;,,) is the shift of the normal coordinate Q,,, due to the electron (hole) when the exciton is located a t site R. In a truly consistent calculation AQ,(R) will be calculated as one single quantity, instead of as a difference of two independent quantities as above. It is, however, more convenient to define AQiv and AQ:" separately as we will see shortly.

An alternative expression of the activation energy is more convenient sometimes. The summation over W in eq. (2) is replaced by an equivalent one in the real space

:

E,

=

1

^{@ i j}

AXi AXj/2 (3)

where

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973982

C9-496

K. S. SONG AX,(R) is the distortion of the lattice point i (i also includes the three cartesian components) when the exciton is located at R.

^{Q i j}

is the force constant.

When the strain field around a defect is available from a reliable calculation eq. (3) is more convenient than eq. (2). Noting that E, represents relaxation energy of lattice corresponding to a strain field given by [AXi(R) - AXi(R1)]/2, the eq. (3) can further be rewritten as follows

:

where

The first term of eq. (4) is the relaxation energy of the self-trapped exciton at its equilibrium position, while the second term might be called the saddle- point (or transition point) relaxation energy. Norgett and Stoneham [3] have prefered an expression similar to eq. (4) in their calculation on V,-center in alkaline earth halides.

A consistent theory should determine the wave function of exciton together with the lattice distortion simultaneously. Such a calculation is considerably complex. We are exploring the possibility of such a study presently. Here we evaluate instead an upper and lower limit that the activation energy E., may have in some of the alkali halides. Before we proceed to this evaluation we present tlie continuum model which allows some interesting general analysis of the problem. Expressions of interesting parameters were derived in a study [4], in which optical absorption by self-trapped exciton was discussed. In the present continuum model we assume that phonon is dis- persion-less and the electron-phonon coupling is due to the interaction between the electron and the longi- tudinal polarization wave of the lattice. A treatment along the line of Pekar's polaron theory [5], in which the total hamiltonian of the system is minimized variationaly with respect to tlie wave function of the system yields the relaxation energy of the following form

:

where D , is the Fourier transform of D(r, $,,) defined as

:

D(r, $,,), which represents the strain field produced by exciton of wave function I),,, has several noteworthy properties.

1) If the effective masses of the two particles are equal, there will be no net interaction between the exciton and the polarization field of lattice, because D above will be zero everywhere.

2) In the opposite extreme where one of the two particles is localized (self-trapped for example), we can separate the motion of the two particles (neglecting the correlation between the two particles' motion) and obtain the following expression :

This last expression is very convenient to study the self-trapped exciton. As an illustration we present expression of D obtained for the following type of hole and electron wave functions

:

Substitution of eq. (8) in eq. (6') yields

where

More elaborate wave functions can be proposed in which the spatial extension, as well as the symmetry of the exciton state are considered.

The relaxation energy defined by eq. ( 5 ) can be rewritten in the real space

:

The expression of activation energy for a jump R

-+

R' of self-trapped exciton can be obtained from eq. (5') by replacing D(r, $,,) by

From what we presented above following conclusions are drawn

:

I ) At large distance,

r

> Ila, D c x becomes small

very quickly. At intermediate distance, r 5 ]/a, there

is considerable net strain produced by the self-trapped

exciton. The more extended (diffuse) the electron

wave function is, the larger the net strain is around

the defect. This leads to larger activation energy

for the jump. It is expected that for a given hole

(V,-like state) state the net strain around the sclf-

trapped exciton will depend very much on the syninie-

MOTION

OF THE SELF-TRAPPED EXCITON I N

ALKALI

HALIDES

C9-497

try and principal quantum number of the electron orbitals (exciton envelope functions) such as Isa,, 2po,, 2p7ru, etc ...

2) Close examination of D,(r, 1)~3 - DR,(r, $,J indicates that this quantity is approximately propor- tional t o the jump distance I R

-

R' I. The activation energy E, is seen t o be approximately proportional to the square of the jump distance. This is the same approximate result obtained previously by Flynn [6]

for Vk jumps.

In the following we present tlie results of a very simple evaluation of thermal activation energy of self-trapped exciton, based on the above two remarks.

We evaluated tlie upper and lower limits of E, for 600 and 900 jumps of self-trapped exciton in alkali fluorides and alkali chlorides. They are obtained by the method of [2b] in which the distortion of the two central halogen ions and the nearest six alkali ions as calculated by Jette rt al. [7] for Vk-center are used.

The upper limit (E,) max is the one calculated pre- viously for V,-center itself and corresponds to the exciton in its continuum state. We are interested here only in those self-trapped exciton states with the hole in state (npa,), which is the stable state of V,-center. The lower limit (E,) min is obtained by assuming that the electron is so tightly bound to the hole that it cancels tlie strain field produced by the hole everywhere, except for the two central halogen ions which maintain tlie same distance as in a V,- center. The obtained numerical values of E , are listed in table I. The lower limit is situated between

+ ^and + of the value obtained for the V,-centers.

The activation energy calculated from eq. (2)-(3) is always larger than tlie experimentally determined activation energies, as was discussed in [2] and [3].

The reason being that the calculated E,, is the rrc.tiva- tion energy in the high telnperature limit, while tlie actual experiments are performed a t

^{1 0 ~ 1}

ternperaticres.

The only available data on the diffusion of self- trapped excitons is from tlie sputtering rate of KI under optical or electron irradiation [8]. The consi- derable distance of energy transport (20 n n ~ ) to produce the surface effect is attributed to the migration of self-trapped exciton [8]. If this analysis is correct, their work seems to indicate that the activation energy of self-trapped exciton in Kl is somewhat smaller than that of V,-center in the same substance (assuming that the pre-exponential factor is the same for both).

Upper and lower limits for. activation energj, of seljltrapped exciton dlflusion in alkali fluorides and chlorides (in eV).

60° 900

(Ea) max (E,) min (E,) max (E,) min

- - - -

Li F 0.93 0.33 1.40 0.49

N a F 1.34 0.53 2.06 0.80

K F R b F LiCl NaCl KC1 RbCl

There is one problem which seems t o be of consi- derable interest for future experiment. When the excited electron wave function is very diffuse tlie radiative recombination probability will be very small, leading t o long lifetime (transition is assumed to be allowed). At the same time, as we have found, we expect the activation energy E;, large (approaching that of the V,-center). Conversely, self-trapped exciton with s l ~ o r t lifetime for luminescence may have smaller thermal barrier to overcome in migration. The very large difference in the lifetimes of a and n-polarized (V, - e":) recombination emission (respectively of lo-' s and lo-' s) is explained presently along two different mechanisms ('). According to Kabler [9]

the long lifetime of n-polarized emission is due to tlie forbidden character of 3C,,

-+

'Ig, whicli becomes weakly observable due to the mixing (very small) of 3C,, and '17,, via spin-orbit coupling. Earlier, Wood [lo] has proposed that this was due to a very diffuse wave function of LI,, in the allowed 'u,,

^{-t 'C,}

transition. If the first mechanism is tlie right one, tlie activation energy E, will be about the same for tlie two exciton states ; 'c, and

^{3 ~ u .}

A noticeable difference in the activation energy is expected in the second case. An experimental measurement of jump frequencies for the two self-trapped exciton states would certainly be of great interest to understand this problem.

(1) This problem is now settled by the latest

EPR

experiment of the self-trapped exciton in

KBr

by Merle dlAubigne et 01. [I I ] .

References

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