Vibronic coupling to nearly localized modes in diamond

Solid State Communications, Printed in Creat Britain.

VIBRONIC

Vo1.6l,No.2,

COUPLING Estele

qepertamento

pp.7578,

TO NEARLY Pereira

1987.

0038-1098/87 $3.00 + .OO Pergamon Journals Ltd.

LOCALIZED

and M. Isabel

MOOES

IN DIAMOND

B. Jorge

end Centro de Ffsica [INICI. Universidede 3600 Aveiro, Portugal

[Received

16 September

1966

by R.A.

de Aveiro

Cowleyl

The presence of sharp low energy vibronic progressions in several luminescing bends of brown diamonds is interpreted es due to e nearly localized vibrational showing a week coupling to resonant lattice phonons. The effect of mode. this mode on the vibronic coupling is discussed and compared with experimental date.

1. Introduction

bend shape can be obtained. assuming linear coupling, only if in the phonon spectrum e second maximum at twice the frequency of the dominant mode is included. This is clearly the situation in the 2.145 eV band3. [section 2.11 the Sf a band4 and the 2.E 16 aV band5.

Most of the known luminescing bands in diamond can be accounted for using the linear electron-phonon coupling model for the bend shape end temperature dependence of zero phonon line [ZPLI intensity coupling approximative quadratic model and an for the ZPL energy and width dependence on temperature that takes into account the scrambling of normal modes between the two electronic states’. It assumes that the quadratic coupling parameter bij for modes i and j is proportional to the linear terms ai end ajz: bij -C1 ai aj

2. Analysis

of the 2.145 eV band

2.1. Within the standard to lattice phonons In fig

1 the

initial

model

parts

ce and luminescence excitation band are shown by the full lines.

In brown diamonds several luminescing bands have e behaviour that departures from this model. In the table the main characteristics of several of these bends are sumerized. namely the ZPL energy, the mode frequency in the two electronic states [whenever a “mirror image” is found]. its width at 10 K and the Huang-Rhys factor @I. taken from the ratio of ZPL over the whole bend intensity.

of

vibronic

of both the spectra

coupling

luminescenof

the

red

Table Yode Name

ZPLfeVI

I

freq

[meV

G.S.

E.S.

Half width smissSion1 QI fmeV1

2.424

46

44

2.5

1.5

-9 Yellow band

2.721

34

36.5

7.5

1.5

red band3

2. I45

30

23

10

S1 cLband4 2.429 -5 2.616

57

-

7

4.5

50

-

6.6

4.5

[G.S.

State

1.5

432101234 - Ground

E.S. - Excited

+

State1 Fig.

The vibrational quantum dominant in the progression has an energy [ranging from 23 to 57 maV1 where the perfect diamond lattice has a low density of states. In bends where a “mirror image can be found there is e difference relationship” in frequency of the mode between the two states. Most of the bends have large S values. The most striking feature is that in bends with a large S the

75

I - Full line: experimental corrected luminescence excitation spectra of luminescence and the 2.145 eV band: broken line: band shape by the linear electron-phonon coupling modal with phonon spectrum shown in insert: X - progression [peak intensity] due to linear + quadratic coupling to a single hermanic mode (30 meV in G.S.. 23 in E.S.1.

76

Vol. 61, No. 2

VIBRONIC COUPLING TO NEARLY LOCALIZED MODES IN DIAMOND

electron-phonon coupling model The linear can reproduce the luminescence band shape at low temperature [below 77 Kl with the phonon spectrum shown in the insert of fig 1. the theoretical band shape being shown by the dotted lines in the same figure. The plot of the logarithm of ZPL intensity also departs from the linearity expected with T* by the linear coupling theory, as shown in fig 2 [dots

-

I

510

0

I

I

100 150 Temoerorurei K)

40

Fig. 3 - ZPL

i7

3

1

4

energy shift versus temperature. X - experimental values: full line: values calculated assuming coupling to a single mode of different frequency: broken line: values calculated by the coupling to lattice modes model.

IO’

727K2,

Fig. 2 - In

lo versus T2. e- experimental full line: experimental behaviour: nearly linear dependence at line: temperatures: chain line: expected dependence with proportionality from the high temperature region.

values; broken higher I inear factor

2 are experimental points]. It is interesting to note that the seme behaviour is found in the TR12 bands. where a difference of frequency of the mode in the two states is also observed. If we assume that the higher temperature data gets closer to the usual lattice coupling model, the proportionality coefficient would be obtained from the broken line in fig 2. If this beheviour was valid over all the temperature the ZPL intensity would decrease range studied. from the lower temperature value much faster [chain line]. In figs 3 and 4 the experimental date on the ZPL energy shift [assuming a constant contribution due to lattice ex ansion as found usually in this temperature range PI and width [deconvoluted from inhomogeneous broadening taken as constant in this temperature range’1 are plotted against temperaTo account for these data within ture Ccrossesl. the usual quadratic coupling model we have to use an a value 22 times smaller for the width variation than for the energy shift of ZPL with temperature. The results are shown by the broken line in fig 3 and the chain line of fig 4. The o value taken from the energy shift data would give the behaviour shown by the broken line in fig 4. 2.2.

Considering mode

the

coupling

to

a

nearly

localized

In order to explain the experimental behaviour we interprete the coupling as due mainly to a vibrational mode characteristic of the defect. that is, a nearly localized mode. This may well explain why the mode frequency changes with the electronic state. as this was not to be expected from coupling to long wavelength lattice modes. Of course the mode cannot be considered just as a localized mode as there are lattice modes [however sparse1 isoenergetic with the mode.

0,

lo-

>

0

s-

Fig. 4 - ZPL

width versus temperature. X - experimental values: full line: values calculated assuming coupling to a narrow nearly localized mode: broken line: values calculated by the coupling to lattice modes model, same o as for the energy shift data: chain line: as the latter but with a 22 times smaller.

From the negligible widening of the vibronic peak progression in luminescence we may take the intensity to be nearly proportional to the height of the peak. Therefore we may assume a single frequency for the mode, taken as harmonic, and take into account the quadratic term that arises from the change of frequency in the two states. For the calculation we used the Manneback recursion formula7. the nth peak with

intensity

being

I, = Azone

Aon+~=[-cos~201n1’2Aon_~-sin[28]2-“2acoseA,n]/[n+

11”2

VIBRONIC COUPLING TO NEARLY LOCALIZED MODES IN DIAMOND

Vol. 61, No. 2 tan

where

2 0-

Wl -

and

r[Tl

a-fZ[S,+S11

-rl

[P J1 [PI/J,

77

[PI]1'2,

WO

with

So =

mu0

qo2

2h

and

S1 --

with P - 25 [5 [h + 11 l12. 5 the average number of the mode at temperature T intrinsic width of the mode.

mwi 902 2fl

in the where w. and 01 are the mode frequencies two electronic states. m the mass of the mode and q, the difference in equilibrium configuration of coordinate q of the mode. The results are shown in fig 1 by the crosses. the agreement being good in emission. In luminascanca excitation the experimental data are more difficult to interpret. as there are several luminescence bands with ZPL energy close to this band3. giving a base line difficult to determine. Moreover just above the energy of the gth vibrational replica a second stronger absorption band in the same centre originates. hiding the lower energy one. Therefore the agreement cannot be expected to be as good as in luminescence. From these results we may conclude that the coupling to a single mode of different frequency in the two electronic states accounts for the experimental band shape. We may regard at twice the frethe inclusion of the Znd maximum quency of the lSt in the linear coupling model as an empirical way to account for this quadratic term. Accepting the nearly localization at the defect we can explain at least the ZPL intensity data. The coupling mode would make the ZPL intensity vary rature as: IO

hw [Tl= Jo [S cschzkT

I

axp C-S coth

of the mode qualitatively to a single with tempe-

$$I

modified Bessel function. In the where Jo is the continuous lattice modes each mode coupling to i has a vanishingly small shift in equilibrium configuration and therefore a small value of Si. So the value of the Bessel function becomes close to 1. In the case where a nearly localized mode is present we may expect that the resonance with lattice phonons increases with temperature. Therefore it is in the low temperature range that the effect of the mode localization will be more important. giving a value of the Bessel function greater than 1. slowing down the decrease due to the thermal population of higher vibrational levels. Thus the experimental behaviour can be explained. If we consider now the affect of the change of frequency of the mode in the two electronic states. the ZPL energy shift AE will be given by

AE - q

[coth

RI w/ZkTl

-

11

This behaviour is shown for the 2.145 eV ZPL by the full line in fig 3. The results are close to the experimental values, and also to the values obtained by the other model. However the ZPL width dependance on temperature considering the affect of a nearly localized mode is significantly different. Assuming the mode to be sufficiently narrow to use an average thermal population term within the the ZPL width dependence in temperature mode. [rl

may be given

by8

and

quantum rl the

Assuming a constant width for the mode the result is shown by the full line of fig 4. We interpret it as a further indication that the mode is indeed a nearly localized one. In fact. and if it was a truly localized mode, its intrinsic width shouldn’t vary with temperature. However as it is in resonance with lattice phonons its lifetime will shorten upon increase increasing temperature and therefore an in width will be observed. The experimental data can be interpreted in terms of a coupling to a nearly localized mode with a five-fold increase in width from 10 K to 200 K. 3. Comments

on other

bands and conclusions

The analysis made shows that the presence of e nearly localized mode in the defect where the luminescence takes place is a likely hypothesis. as the behaviour found can be accounted for on the basis of a coupling to such a mode. In the 2.145 aV band. due to the large energy difference of the mode in the two electronic states, the quadratic term in the vibronic coupling is large. In the yellow band whose bahaviour can be accounted for in the coupling to lattice phonons modalg. the energy change of the mode is much smaller, and therefore the daparture from the linear coupling model for the band smaller. Moreover the ZPL energy shift shape is due to the different mode frequency in the two states leads to a shift towards higher energy. while the coupling to lattice phonons has the opposite effect. Indeed the energy shift is smaller than in other ZPL found in diamond. This will require a rather small ci quadratic coupling parameter. small enough to give a good fit to the width data. Unfortunately the other bands with large S don’t show absorption directly into the emitting state [their lifetimes are in the ms range. indicating that they originate in metestable statas51: therefore no direct measure of the difference of frequency of the mode in the two states can be made. However preliminary studies on the available data also show departures from the coupling to lattice phonons model, giving indication that hare also a nearly localized mode is present. Further work is currently order way on these bands. Although the softning of the force constant as cause of the low energy of the mode cannot be ruled out it seems more likely that it is due to the presence of heavier atoms at the defect. Brown diamonds are known to have as impurities atoms like Mg, Fe, Mn. Al and CrlO. Although so far no luminescence associated with these atoms has been identified in diamond. the presence of low energy nearly localized modes may be a hint for the presence of these atoms in the defects where these luminascence bands originate. Also the bands are insensitive to annealing at temperatures as high as gOO” C indicating that they are fairly stable.

identify

Further work is being carried the nature of these defects.

out

to

try

to

Vol. 61, No. 2

VIBRONIC COUPLING TO NEARLY LOCALIZED MODE6 IN DIAMOND

78

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