ELECTRONIC STATES OF MIXED CRYSTALS

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ELECTRONIC STATES OF MIXED CRYSTALS

D. Craig

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C 3, supplkment nu no 5-6, Tome 28, mai-juitz 1967, page C 3-129

ELECTRONIC STATES

OF

MIXED CRYSTALS

by D. P. CRAIG

William Ramsay and Ralph Forster Laboratories, University College London

Resume. - L'auteur Ctudie deux problkmes. Tout d'abord il constate qu'en incluant les cor- rections de retard electromagnetique dans l'energie potentielle intermoleculaire entre l'impurete et les moltcules h8tes d'un cristal a pieges profonds, il apparait un nouveau terme qui depend de I'inverse du carre de la distance qui Ies separe. Les calculs montrent que la divergence correspon- dante de I'energie de couplage est extremement lente, et sans importance dans les cristaux de dimensions de I'ordre des centimetres.

I1 examine ensuite les modifications du spectre d'absorption de I'h8te par les impuretes a pitge profond, et montre que les changements de la largeur d'absorption et de la dkcomposition de Davidov (pour autant qu'ils dependent des effets de rCseau rigide) sont lies li la distribution des niveaux dans le spectre d'excitons du cristal pur. Ces changements sont les plus faibles pour les zones d'bnergie oh ces niveaux ont la repartition la plus dense.

Abstract. - Two problems are discussed. On inclusion of electromagnetic retardation correc- tions in the intermolecular potential energy between impurity and host molecules in a deep-trap mixed crystal a new term appears depending on the inverse square of the distance of separation. Calculations are reported to show that the corresponding divergence in the coupling energy is extremely slow and unimportant in crystals of up to a few centimetres in size.

Changes produced by deep-trap impurities in the host absorption spectrum are discussed. The changes in absorption width and Davydov splitting (so far as they depend on rigid lattice effects) are shown to be related to the distribution of levels in the pure crystal exciton spectrum, and are least for energy regions where the levels are most densely packed.

Introduction.

-

The profound effect on solid- state fluorescence of small amounts of impurities is well known, being shown, for example, in the exclu- sive green (tetracene) emission from anthracene crystals with as little as one tetracene present per lo4 anthra-

cenes. Many such cases are known where excitation first absorbed by the host crystal is captured or sca- venged highly efficiently by a small number of impurity molecules (traps) and emitted as trap fluorescence. In the absorption spectrum the influence of low-concentration impurities in the region of pure host absorption is not very great, while in the guest absorption region one gets very weakly the absorption spectrum of the impurity according to its concentration in the host. Thus there is typically a striking difference between fluorescence and absorption, so far as the effect of an impurity on a host crystal is concerned. The changes in the absorption spectra, even though not so marked, have up t o the present been found easier to measure, and they also are easier to calculate for the reason that in absorption the processes of energy transfer t o the lattice modes and other many-

particle motions do not enter, and one is mainly dealing with an intermolecular electronic problem.

Deep (chemicaI) traps.

-

I t is useful t o classify impurities acting as traps by referring to the trap- depth 6, defined as the difference between the lowest transition energies of host and guest. The shallow trap limit applies where 6 is much less than the band- width of the pure host crystal excitation; a typical example is that of partially deuterated naphthalene in perdeuteronaphthalene host crystal, where 6 is a few tens of wave numbers in energy. The deep trap limit applies where 6 is greater than the pure host band- width, as in many cases of chemical (as opposed to isotopic) impurities, for example, tetracene or carbazole in anthracene crystal.

In this case the measurement that has most been .used to characterise the interaction between the impurity molecules and surrounding host matrix is the polarisation ratio, namely the ratio of the absorbed intensity along two host crystal directions. A quantity

that at first seems more useful, the displacement AE

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C 3

-

130 D. P. CRAIG of the energy of the guest transition by interaction

with the host is usually too small to be unambiguously interpreted. In a perturbation approximation [I] it is given by the expression (1)

where is the interaction matrix element between molecules at two sites in the crystal. The right-hand side may be summed to include other host transitions where necessary.

For the first transition of tetracene in anthracene host the energy depression given by (1) is about 200 cm-I compared with the vapour, but this is much smaller than the correction to the transition energy assignable to host-guest dispersion forces, and cannot be separated from it.

If there were no coupling between guest and host transitions the polarisation ratio would have the oriented-gas value found from the crystal structure constants of the host lattice, insofar as it can be taken that the orientation of the axes of the guest are the same as those of the host. For tetracene in anthracene the PR of the first (short-axis) system would be bla = 7.611. The most recent measurements give the much lower values in Table 1.

The reduction from the oriented gas values are caused by a degree of mixing of long-axis polarised transitions of the host molecules with the short-axis guest system. According to perturbation theory [I] the guest transition moment is given by expression (2), where only a single host level is included,

where Mg and Mh are the guest and host free-molecule transition moments.

Absorption Polarisation ratios of tetracene in anthracene crystal

(Akon and Craig, Trans. Far. Soc., 1967, 63, 58)

(*) Extinction coefficient ratio.

(**) Oscillator strength ratio.

(+) P. D. Dacre, unpublished. Wave-

length

( m ~ )

The results for tetracene-anthracene are included in table 1, where one sees that they are of the right order. The calculations [I] were made with the static potential (3) for the coupling of transition dipole moment components

M i

in the two molecules before the problem

of the retarded potential had been worked out. The tensor is that for the interaction of permanent dipoles in classical theory. There seemed to be no particular difficulty, since the result in (I) depends on the sum of squares of intermolecular matrix components of the potential (3) and there is a very rapid convergence inside a radius small compared with the wavelength of light. In expression (2) the contribution to the oscillator strength by any one host molecule is also seen to depend on the squared matrix component. However, the whole question has to be reconsidered in the light of the correction to the potential (3) to allow for retardation [2]. In the form appropriate to the mixed crystal problem the potential, now used in second order, contains a term which becomes dominant at distances R y A (A = A/2n, where 1 is the wavelength of the guest transition) varying with dipole orientation and distance as in

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The bracketed quantities are the tensors for the interaction of the purely transverse components of the transition moments. One sees that the dependence on distance is an inverse square times a quantity that is never negative ; thus in a summation in a three- dimensional molecular array the potential will diverge linearly in the distance, giving a result that is dependent both on the size and shape of the crystalline sample. The interaction will be cut off at the distance travelled by light within the radiative lifetime, but this is too great (-- 100 cm) to remove the difficulty. It remains to test the magnitude of the divergent term.

&b/&a (*)

New calculations of the intermolecular coupling in anthracene-tetracene mixed crystals have now been made by Mr. P. D. Dacre. The results will be more fully reported elsewhere. They show that the lattice sums of retarded interaction are indistinguishable from those of the static interaction within summation zones of up to a few thousands of angstrom units. At such distances the wave-zone contribution by the term given, apart from natural constants, in expression (4) is infinitesimally small. The static interaction dominates, and its convergence limit has already been attained to 1 in lo3 or lo4 at a distance of 200

A.

The

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ELECTRONIC STATES OF MIXED CRYSTALS C 3

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131 scale of the divergent contribution is so small that one

must go to macroscopic distances before it becomes commensurable with the static contribution. Dacre shows that the divergence in A E given in (I) is

a(AE)/aR =

-

2.3 wave numbers per centimetre (5) and thus insignificant compared with the contribution by the static interaction based on (1) in crystals even of a few centimetres in size. The calculation of polarisation ratio including retardation (Table 1) gives results identical with the older ones. It may still seem puzzling and unsatisfactory to find a divergence of any kind in this problem, but from a practical point of view it is not important.

In-band states in the presence of deep traps. -

Although impurities do not affect crystal spectra in absorption so much as in fluorescence, one expects to find some changes in, for example, the Davydov splitting and host polarisation ratio in the presence of impurities. Not much attention has been given to this problem in the past by deliberate experimentation, although discrepancies in measured results may well have arisen from it. Matsui [3] has recently reported experiments on the Davydov splitting of anthracene host crystal doped with known amounts of tetracene impurity. The splitting is greatest in pure crystals

(- 230 cm-') and is reduced as the added tetracene goes to 1 in lo5 (splitting

-

200 cm-') to 1 in lo4

(- 170 cm-I). A full calculation of these changes cannot yet be made because insufficient is known of the details of the exciton band structure of pure anthracene in its wave vector dependence in the

3 800 system, but some general comments can be made based on an approximate exciton band structure in the similar system of crystalline naphthalene, and deriving from the theory of mixed crystal levels.

We suppose the exciton energy spectrum to be known, denoted by e(k). Following the presentation given by Craig and Philpott [4] we write the states of the impure crystal in the two forms

where Qr(k) defined in (7) are delocalised exciton wave functions built from localised functions Qsi,

The functions are defined in a crystal of N molecules, with ground state and excited state wave functions cp,

and 9: a t the q-th site. The allowed exciton energies in the mixed crystal, with one impurity molecule (trap-depth 6) in the block of N molecules, are given by solutions of the equation (8)

and the amplitude of the pure crystal state Qsr(k) in the mixed crystal wave function is given by (9) [4],

where a, is the amplitude of the Iocalised excitation function at the site of the impurity molecule, as given in the last term of expression (6).

We can now consider how the spectrum of the host crystal is affected by impurities. In the pure crystal allowed transitions in the visible and ultraviolet are confined, approximately, to states of k = 0. I n the mixed crystal the wave vector k ceases to be a good quantum number, and the new eigenstates are linear combinations, expression (6). The intensity of transitions to the levels depends upon the amplitude A(0) with which the unique, spectrally active, wave function appears in the linear combination concerned. Substituting in (9) for a,, we find for A(0) the value given in (lo),

Thus the transition moment to a mixed crystal level at energy E is given by the following,

where T is the transition moment for the transitiori in the pure crystal. One sees that, on account of the nor- malisation properties of the A's, the total oscillator strength of the transitions will remain unaltered, but will be differently distributed over the mixed crystal levels according to the trap depth and concentration. The argument here sketched applies to crystals with one molecule in the unit cell, in which in the pure crystal there is a single allowed k = 0 transition. For two crystallographically equivalent molecules in the unit cell, the energy equation (8) has the same form [4],

with the summation over k values of both exciton branches ; the transition moment ( 1 I ) then has contri-

butions from both polarisation directions, but tjlib complication will not be discussed here.

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C 3

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132 D. P. CRAIG us make the approximation of neglecting all levels e(k)

except the pair k, k' immediately above and immediately below a chosen impurity state at E. We then have the simpler expression (12) for the transition moment,

The total oscillator strength, which is proportional to the square of the transition moment, is of course conserved. The form of expression (12) permits some interesting conclusions on the character of this re- distribution through the band :

1) The more densely packed the levels in the energy spectrum the narrower is the spectral spread caused by impurities.

2) If the level spacing in the energy spectrum is not the same above and below the energy e(O), the apparent intensity maximum will be moved by impurities in the direction of the lower level density.

In the simple case that the levels are midway between the e(k), and if one defines an average level spacing A over a small part of the band near e(O), one has the intensity variation I(E)/I

-- A2/8 ( E

-

e(0)j2, varying as the square of the average level spacing. The idea of an average in this situation is very crude, but the quadratic dependence of intensity on spacing is pro- bably not far from correct in real examples. A similar

result follows from the assumption, appropriate to extremely low impurity concentrations, that each level lies very close to one of the pure crystal levels.

Detailed knowledge of the band structure of the

anthracene 3 800

A

crystal system is lacking. Calcu- lations by M. R. Philpott [ 5 ] on the corresponding system in the naphthalene spectrum at 2 750

A

do, however, give an interesting clue. Philpott finds a level density near the b-polarised pure crystal level more than twice the density near the a level. The sen- sitivity of the a-polzirised absorption maximum to impurities is thus expected to be very much greater than that of the b, a result that agrees with the fact that, experimentally, it has proved much harder to locate the a maximum in anthracene crystal spectrum than the b. It is quite uncertain whether the band structure of naphthalene is a reliable guide to anthracene but the crystal structures and intermolecular couplings are certainly similar. One can perhaps feel some confidence in the explanation that the disagree- ments in the literature over the absorption frequency of the a-polarised anthracene transition are accounted for by the low-level density as compared with the b transition, but the remarkably large changes in splitting found by Matsui in slightly impure crystals cannot be explained by analogy with the naphthalene levels, which do not show sufficient variations in density between the high and low frequency neighbourhoods of the k = 0 states. Other factors, such as broadening of the absorptions by interaction with lattice motions, may also play some part.

References

[I] CRAIG (D. P.) and THIRUNAMACHANDRAN (T.), PYOC.

Roy, Soc., 1963, A 271, 207.

[2] GOMBEROFF (L.), MCLONE (R. R.) and POWER (E. A.),

J. Chem. Phys., 1966, 44, 4148.

131 MATSUI (A.), J. Phys. Soc. Japan, 1966, 21, 2212. [4] CRALG (D. P.) and PHILPOTT (M. R.), PYOC. ROY. SOC.,

1966, A 290, 583.

[5] THESIS (Ph. D.), University of London, 1964. Niveaux _{des niveaux}Largeur Transitions Rbgles de stlection

I

I I

!

Cristal non excitBexciton

I

-I

Cristaux inorganiques Bandes Clectro- niques. Bandes excitoni- ques. Cristaux organiques Larges en fonction de K. Ou mains !ar- ges en fonctlon de K. Bandes excitoni- ques. De bande a bande. Cristal non excite-enciton

et vice-versa. Elargissement en foncti01-1 de K. AK = 0, K quel- conque. AK = 0, K = 0 ou petit.

et vice-versa. Cette transition ressemb]e des transitions entre niveaux moltculaires.

I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ .

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ELECTRONIC STATES OF MIXED CRYSTALS C 3 - 133

DISCUSSION

S. NIKITINE.

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Je ne puis ttre d'accord avec le Professeur Craig sur le tableau pessimiste qu'il a donnC au dCbut de la confirence et dans lequel il fait res- sortir les diffkrences qui existent entre les spectres de cristaux organiques et inorganiques. Les ressemblances ne se limitent pas

B

des abus de langage. Je me permets de donner un tableau analogue B celui que le Profes- seur Craig nous a donnt, mais dont il ressort que pres- que sous tous les rapports, il y a des similitudes entre ces deux sortes de spectres.

I1 rtsulte de cette comparaison que les spectres des

cristaux inorganiques sont plus compliquCs

B

cause des transitions de bande A bande, mais que pour le reste il y a une correspondance frappante.

Je voudrais en plus noter que si les spectres de cristaux inorganiques sont plus complexes, ils apportent aussi plus de renseignements sur I'ttat solide. Nous avons vu que leur Ctude donne une clef pour 1'Clabo- ration d'une structure de bandes.