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Exciton trapping with coherent emission of phonons at impurities in molecular crystals
Journal of Luminescence 38 (1987) 317 319 North-Holland. Amsterdam
317
EXCITON TRAPPING WITH COHERENT EMISSION OF PHONONS AT IMPURITIES IN MOLECULAR CRYSTALS Jai SINGH Taco/tv of Science, tn, vers itt’ College of the Northern Territory, GPO Boy 1341, Darnin, ‘s 1’ 5794, Au5tral,a
A. THILAGAM Department of Phi’2ic3, \ ational Unitcr~iii’ of Singapore, Kent Ridge, Singapore 0 11 The process of exciton trapping at impurities with deep trap depths is studied in molecular crystals assuming that the excess exciton energy can be emitted in the form of coherent phonons. We represent the state of emitted phonons by coherent states and derive rates of exciton trapping for isotopic impurities. The calculation is done with acoustic as well as optical phonons. In naphthalene crystals with impurities oftrap depth 350cm rates are calculated in terms ofthe coherent state parameter a. For I < a~<1 8 it is found that the rate of acoustic phonon emission is 10°’s and of optical phonon emission is 10 s ‘. Such high rates suggest that a laser-like new source of intense and coherent phonon energy can be designed. 1.
Introduction
Processes oftrapping charge carriers and excitation energy in both crystalline and amorphous solids are of importance because of their industrial applications. The theuretical study of such processes is extremely complicated not only in amorphous but also in doped and mixed crystalline solids because the substitution of impurities destroys the translational symmetry of a crystal lattice. However, it may be assumed that the substitution of isotopic impurities retains the spatial symmetry ofa crystal, and then one can derive [1,21 an exciton phonon interaction operator for a crystal doped with isotopic impurities which act as traps. Such an operator of exciton—phonon—impurity interaction can be used as a perturbation for calculating the rates of exciton trapping in doped crystals. In this paper we study the trapping of excitons at deep traps emitting many phonons in molecular crystals. In a process of multiple-phonon emission, phonons may be emitted with a spread of frequencies including the possibility that all emitted phonons can also have identical frequencies. With the latter possibility it can be assumed that the exciton energy is given to exicte lattice vibrations ofa particuar mode usually called the interaction mode [3]. Considering that the lattice relaxation takes place by emitting phonons through a cascade-type process, one can easily see that an emitted phonon frequency will correspond to that of the interacting phonon mode. The cascade process as considered here describes the emission of a phonon of a particular frequency first, then the second ofthe same frequency, then the third and so on. With this picture in mind the state of phonons thus emitted can be represented by coherent states [4]. The theory is worked out for crystals at a very low temperature considering the emissin of both acoustic and optical phonons. Impur0022-231 3/87/$03.50 © Elsevier Science Publishers By.
(North-Holland Physics Publishing Division)
ity impurity interaction is neglected here and therefore the results are valid for crystals essentially with low impurity concentrations. Rates of exeiton trapping as a function oftrap depth, unperturbed exciton band width and coherent state parameter a are derived.
2. Theory It is assumed that an exciton is initially created in a pure crystal region where the exciton—impurity interaction is neghginble. The diagonalized Hamiltonian of an exciton, interacting with phonons and created through0—0 phonon transition, represents the energy operator of a composite exciton—phonon state [51 and is given by [J~,—
~
,t
0(k)A~A~.
(I)
where )~0(k)is an energy eigenvalue of a composite cxciton—phonon given by: i.(k) —E(k)
EE;(E(k)
2 (E( —
E0) + B
2k) E0)]
I
(It”)
I
+
(2) c’ being the velocity of sound in the crystal, 2B the unperturbed exciton band width and E0 the energy of the centre of the unperturbed exciton band. A~is the creation operator of a composite exciton—phonon [5,6 1. The rate oftransition from an initial state to a final state of one trapped exciton at an impurity and phonons in coherent states is calculated. The exciton phonon impurity interaction operator R,,,,. used as a perturbation responsible for the transition, is already known. The transition matrix element
8
.1 S’,n~h. 1 / liilu’ani /
1K/i
—
Si/Ion
itiJpJYitii,’
(3)
f~ o~
nit/i iithei (‘lii (‘/1/1/ /00 iii
R”
and
~) ]
+~ ~
32ii C’s, I a I J~(%1~Qdi ) e
P~cc —B,f
0.
n+ccq
(4)
as eigenvectors of initial and final states respectisel\. Here
I ii represents the phonon occupation ector before the ers stal is excited and 0, the composite cxciton phonon ‘acuum state. B,~is the creation operator of an the product impurit’~ p ectors and of 0. 0 + exciton i~ 1o localized n I cc,, at is the of site eigens the exciton sacuum state 0 . the initial phonon occupation state In and additional emitted phonons in colierent states Icc,, . The subscript q signifies the mode of the emitted phionons. A ielation between the creation op erator 1, and exeiton creation operator Bq is alreads known [5.6]. The coherent state vector as used here is gisen b~
cc,,
r[I
~ihotions
—
~ exp(
~
a’ m,,~
/flq
( )
.
X
3h(hr’i
[
0)’
i
.~
>< exp L2I(clh
(8) iou]
For optical phonon emission we used1’Einstein’s model of as:
the densit~of phonon states to get R” R~ J~Q’c[( exp[ a~
(9)
87~h a
Here J~is the trap depth, Q the olume of a unit cell, ci the mass densits of a crystal. ()j) the Debse cut-off frequcnc\. t1 1, J~,lI.and I and I are mass coefficients of Impurit\ and hosi molecules respectisels
3. Results and discussion
where cc is the coherent state parameter. The expression of the transition matrix 7,,,,, 1K: n thus obtained using eq. (3), element (4) and p: a,,11 (5)is s cry complicated and should be simplified. At lou temperature we ma\ assume that the lattice in initialls frozen before the cr\stal is excited, and therefore we can
\\e hase calculated both rates R ‘ (8) and R ‘~“ (9) for naphthalene-doped cm . B— IOU cm crystals ho using trap depth j1 0— 90 em ‘. (~, 10 and ‘if1, 1.06. which is the mass ratio of C5D~to C H,. For 1.1 cc < 1.8 we have obtained R ‘‘ tU1~ s and R ~ s . i.e. the rate of trapping with emission of
set without an~loss of gcneralit\. the initial thermalized
optical coherent phonons is about two orders of magni tude higher than that with acoutic coherent phonons. Both rates are 8 to 10 orders of magnitude higher than that of single phonon emission [7.9]. It is clear form the densits of multiple phonon states p”(hei(q)) that the phonon emission of a cascade tspe is assumed here. That is, first
aserage population of phonons to be zero (ii,, 0). It is to be noted that ii,, does not refer to the phonon population of coherent states I a,, ~. We calculate the transition rate using the transition matrix element thus obtained in R’
‘~
2mC’~,~
I
p: cc K; n
‘p’ (hw(q))
.
(6)
I) where C’, is the impuritr concentration.
,ti corresponds to phonons emitted after the trapping and p ‘(hro( q)) is the density of in phonon states calculated from the product of in single-phonon density of states as
f)
(firo(
-
it’’
q))
one phonon of ro(q)
IS
spontaneousl~emitted and
it
is
then followed by the emission of another phonon of the same frequenc~and so on. Trapping of more excitons at
impurities will enable more induced phonon emissions of the same frequency pros ided a set of reflectors are used to accumualte the emitted phonon energy and to let it build up coherently to a desired value. The mechanism of phonon emission in this situation therefore becomes quite elearl\ identical to that in lasers w’here the presence of
p( 6w (q))---p(hw, (q))
radiations ofa frequency stimulates the emission of more radiations of the same frequency. It can therefore be sug-
-
>< d( infcco — 6w~ /1w ~--- )dhw
dhw,
.
(7)
It is essential here to explain the method of calculation of the transition rate using eqs. (3) (7): eq. t 5) is an infinite series summing over the number of phonons and therefore the matrix element and its square. I p: cv,, I 1’,,,,P I K~n I . calculated using eq. (5) would also be obtained as infinite series. Using 1is calculated the in” term andofthen the latter the total series R isinobtained eq. (6) from a rateRR°’~ ,~, R’”. Using the Debye model for the density of a single acoustic phonon state we get the rate R” as:
gested on the basis of present results that from a suitable choice of host and impurity materials, a source of intense and coherent phonon energy may be designed. The subJect of stimulated emission of phonons was brought up recently by Bron [10]. The coherent state parameter is usually a complex quantity and it is difficult to assign it a particular alue in 1.1 the < ccl case < 1.8 of chosen exciton here trapping. for IccHoweser I is mainlythe because range for of I a~< I the rates tend to increase sers rapidly to a high s aluc and for Id >2 they become sery low. It is stated abose that the application ofcoherent states
J. Singh. .4 Thilagain/Evciton trapping i,,’ith coherent emis ion ofphonons
is based on the assumption that all emitted phonons have the same frequency. As the calculated rates with this assumption are so high the assumption apparently seems to be quite justified. It may be expected that if the process of emitting all phonons of the same frequency is not the most efficient it is at least as efficient as that for emission of another combination of frequencies. It is to be noted that the rates of coherent optical phonon emission are unrealistically high which may be attributed to the use of the Einstein model for the phonon density of states. The same calculation using the Debye model gives rates of coherent acoustic phonon emission less by 2—3 orders of magnitude. However, rates of many-phonon emission in discrete phonon states (not coherent states) calculated from the Debeye model are negligibly low [11] as expected.
319
References [1].1. Singh. Phys. Rev. B30 (1984) 7267. [2]J. Singh, Phys. Rev. B33 (1986) 2602. [3]H. Sumi and Y. Toyozawa, J. Phys. Soc. Japan 31 (1971) 342. [4] R. Klauder and E.C.G. Sudarshan, Fundamentals ofQuantum Optics (Benjamin, New York, 1968). [5] J. Singh, mt. J. Quant. Chem. 30 (1986) 7. [6] D.P. Craig and J. Singh, Chem. Phys. Levi. 82 (1981) 405; J. Singh, in: Solid State Physics, Vol. 38. eds. D. Turnbull and H. Eherenreich (Academic Press, New York, 1984) p. 295. [7] J. Singh, Phys. Rev. B33 (1986) 2602. [8] J. Singh, J. de Phys. C7 (1985) 61. [9] D.P. Craig. L.A. Dissado and S.H. Walmsley. Chem. Phys. Lett.46 (1977) 191. [10] W.E. Bron, in: Proc. 2nd mt. School on Condensed Matter Physics, Varna, 1982 (World Scientific, Singapore. 1983)
p. 143. [I1] J. Singh and A. Thilagam, in: Proc. ICL’87, J. Lumin. 40&41 (1987) 457.
317
EXCITON TRAPPING WITH COHERENT EMISSION OF PHONONS AT IMPURITIES IN MOLECULAR CRYSTALS Jai SINGH Taco/tv of Science, tn, vers itt’ College of the Northern Territory, GPO Boy 1341, Darnin, ‘s 1’ 5794, Au5tral,a
A. THILAGAM Department of Phi’2ic3, \ ational Unitcr~iii’ of Singapore, Kent Ridge, Singapore 0 11 The process of exciton trapping at impurities with deep trap depths is studied in molecular crystals assuming that the excess exciton energy can be emitted in the form of coherent phonons. We represent the state of emitted phonons by coherent states and derive rates of exciton trapping for isotopic impurities. The calculation is done with acoustic as well as optical phonons. In naphthalene crystals with impurities oftrap depth 350cm rates are calculated in terms ofthe coherent state parameter a. For I < a~<1 8 it is found that the rate of acoustic phonon emission is 10°’s and of optical phonon emission is 10 s ‘. Such high rates suggest that a laser-like new source of intense and coherent phonon energy can be designed. 1.
Introduction
Processes oftrapping charge carriers and excitation energy in both crystalline and amorphous solids are of importance because of their industrial applications. The theuretical study of such processes is extremely complicated not only in amorphous but also in doped and mixed crystalline solids because the substitution of impurities destroys the translational symmetry of a crystal lattice. However, it may be assumed that the substitution of isotopic impurities retains the spatial symmetry ofa crystal, and then one can derive [1,21 an exciton phonon interaction operator for a crystal doped with isotopic impurities which act as traps. Such an operator of exciton—phonon—impurity interaction can be used as a perturbation for calculating the rates of exciton trapping in doped crystals. In this paper we study the trapping of excitons at deep traps emitting many phonons in molecular crystals. In a process of multiple-phonon emission, phonons may be emitted with a spread of frequencies including the possibility that all emitted phonons can also have identical frequencies. With the latter possibility it can be assumed that the exciton energy is given to exicte lattice vibrations ofa particuar mode usually called the interaction mode [3]. Considering that the lattice relaxation takes place by emitting phonons through a cascade-type process, one can easily see that an emitted phonon frequency will correspond to that of the interacting phonon mode. The cascade process as considered here describes the emission of a phonon of a particular frequency first, then the second ofthe same frequency, then the third and so on. With this picture in mind the state of phonons thus emitted can be represented by coherent states [4]. The theory is worked out for crystals at a very low temperature considering the emissin of both acoustic and optical phonons. Impur0022-231 3/87/$03.50 © Elsevier Science Publishers By.
(North-Holland Physics Publishing Division)
ity impurity interaction is neglected here and therefore the results are valid for crystals essentially with low impurity concentrations. Rates of exeiton trapping as a function oftrap depth, unperturbed exciton band width and coherent state parameter a are derived.
2. Theory It is assumed that an exciton is initially created in a pure crystal region where the exciton—impurity interaction is neghginble. The diagonalized Hamiltonian of an exciton, interacting with phonons and created through0—0 phonon transition, represents the energy operator of a composite exciton—phonon state [51 and is given by [J~,—
~
,t
0(k)A~A~.
(I)
where )~0(k)is an energy eigenvalue of a composite cxciton—phonon given by: i.(k) —E(k)
EE;(E(k)
2 (E( —
E0) + B
2k) E0)]
I
(It”)
I
+
(2) c’ being the velocity of sound in the crystal, 2B the unperturbed exciton band width and E0 the energy of the centre of the unperturbed exciton band. A~is the creation operator of a composite exciton—phonon [5,6 1. The rate oftransition from an initial state to a final state of one trapped exciton at an impurity and phonons in coherent states is calculated. The exciton phonon impurity interaction operator R,,,,. used as a perturbation responsible for the transition, is already known. The transition matrix element
8
.1 S’,n~h. 1 / liilu’ani /
1K/i
—
Si/Ion
itiJpJYitii,’
(3)
f~ o~
nit/i iithei (‘lii (‘/1/1/ /00 iii
R”
and
~) ]
+~ ~
32ii C’s, I a I J~(%1~Qdi ) e
P~cc —B,f
0.
n+ccq
(4)
as eigenvectors of initial and final states respectisel\. Here
I ii represents the phonon occupation ector before the ers stal is excited and 0, the composite cxciton phonon ‘acuum state. B,~is the creation operator of an the product impurit’~ p ectors and of 0. 0 + exciton i~ 1o localized n I cc,, at is the of site eigens the exciton sacuum state 0 . the initial phonon occupation state In and additional emitted phonons in colierent states Icc,, . The subscript q signifies the mode of the emitted phionons. A ielation between the creation op erator 1, and exeiton creation operator Bq is alreads known [5.6]. The coherent state vector as used here is gisen b~
cc,,
r[I
~ihotions
—
~ exp(
~
a’ m,,~
/flq
( )
.
X
3h(hr’i
[
0)’
i
.~
>< exp L2I(clh
(8) iou]
For optical phonon emission we used1’Einstein’s model of as:
the densit~of phonon states to get R” R~ J~Q’c[( exp[ a~
(9)
87~h a
Here J~is the trap depth, Q the olume of a unit cell, ci the mass densits of a crystal. ()j) the Debse cut-off frequcnc\. t1 1, J~,lI.and I and I are mass coefficients of Impurit\ and hosi molecules respectisels
3. Results and discussion
where cc is the coherent state parameter. The expression of the transition matrix 7,,,,, 1K: n thus obtained using eq. (3), element (4) and p: a,,11 (5)is s cry complicated and should be simplified. At lou temperature we ma\ assume that the lattice in initialls frozen before the cr\stal is excited, and therefore we can
\\e hase calculated both rates R ‘ (8) and R ‘~“ (9) for naphthalene-doped cm . B— IOU cm crystals ho using trap depth j1 0— 90 em ‘. (~, 10 and ‘if1, 1.06. which is the mass ratio of C5D~to C H,. For 1.1 cc < 1.8 we have obtained R ‘‘ tU1~ s and R ~ s . i.e. the rate of trapping with emission of
set without an~loss of gcneralit\. the initial thermalized
optical coherent phonons is about two orders of magni tude higher than that with acoutic coherent phonons. Both rates are 8 to 10 orders of magnitude higher than that of single phonon emission [7.9]. It is clear form the densits of multiple phonon states p”(hei(q)) that the phonon emission of a cascade tspe is assumed here. That is, first
aserage population of phonons to be zero (ii,, 0). It is to be noted that ii,, does not refer to the phonon population of coherent states I a,, ~. We calculate the transition rate using the transition matrix element thus obtained in R’
‘~
2mC’~,~
I
p: cc K; n
‘p’ (hw(q))
.
(6)
I) where C’, is the impuritr concentration.
,ti corresponds to phonons emitted after the trapping and p ‘(hro( q)) is the density of in phonon states calculated from the product of in single-phonon density of states as
f)
(firo(
-
it’’
q))
one phonon of ro(q)
IS
spontaneousl~emitted and
it
is
then followed by the emission of another phonon of the same frequenc~and so on. Trapping of more excitons at
impurities will enable more induced phonon emissions of the same frequency pros ided a set of reflectors are used to accumualte the emitted phonon energy and to let it build up coherently to a desired value. The mechanism of phonon emission in this situation therefore becomes quite elearl\ identical to that in lasers w’here the presence of
p( 6w (q))---p(hw, (q))
radiations ofa frequency stimulates the emission of more radiations of the same frequency. It can therefore be sug-
-
>< d( infcco — 6w~ /1w ~--- )dhw
dhw,
.
(7)
It is essential here to explain the method of calculation of the transition rate using eqs. (3) (7): eq. t 5) is an infinite series summing over the number of phonons and therefore the matrix element and its square. I p: cv,, I 1’,,,,P I K~n I . calculated using eq. (5) would also be obtained as infinite series. Using 1is calculated the in” term andofthen the latter the total series R isinobtained eq. (6) from a rateRR°’~ ,~, R’”. Using the Debye model for the density of a single acoustic phonon state we get the rate R” as:
gested on the basis of present results that from a suitable choice of host and impurity materials, a source of intense and coherent phonon energy may be designed. The subJect of stimulated emission of phonons was brought up recently by Bron [10]. The coherent state parameter is usually a complex quantity and it is difficult to assign it a particular alue in 1.1 the < ccl case < 1.8 of chosen exciton here trapping. for IccHoweser I is mainlythe because range for of I a~< I the rates tend to increase sers rapidly to a high s aluc and for Id >2 they become sery low. It is stated abose that the application ofcoherent states
J. Singh. .4 Thilagain/Evciton trapping i,,’ith coherent emis ion ofphonons
is based on the assumption that all emitted phonons have the same frequency. As the calculated rates with this assumption are so high the assumption apparently seems to be quite justified. It may be expected that if the process of emitting all phonons of the same frequency is not the most efficient it is at least as efficient as that for emission of another combination of frequencies. It is to be noted that the rates of coherent optical phonon emission are unrealistically high which may be attributed to the use of the Einstein model for the phonon density of states. The same calculation using the Debye model gives rates of coherent acoustic phonon emission less by 2—3 orders of magnitude. However, rates of many-phonon emission in discrete phonon states (not coherent states) calculated from the Debeye model are negligibly low [11] as expected.
319
References [1].1. Singh. Phys. Rev. B30 (1984) 7267. [2]J. Singh, Phys. Rev. B33 (1986) 2602. [3]H. Sumi and Y. Toyozawa, J. Phys. Soc. Japan 31 (1971) 342. [4] R. Klauder and E.C.G. Sudarshan, Fundamentals ofQuantum Optics (Benjamin, New York, 1968). [5] J. Singh, mt. J. Quant. Chem. 30 (1986) 7. [6] D.P. Craig and J. Singh, Chem. Phys. Levi. 82 (1981) 405; J. Singh, in: Solid State Physics, Vol. 38. eds. D. Turnbull and H. Eherenreich (Academic Press, New York, 1984) p. 295. [7] J. Singh, Phys. Rev. B33 (1986) 2602. [8] J. Singh, J. de Phys. C7 (1985) 61. [9] D.P. Craig. L.A. Dissado and S.H. Walmsley. Chem. Phys. Lett.46 (1977) 191. [10] W.E. Bron, in: Proc. 2nd mt. School on Condensed Matter Physics, Varna, 1982 (World Scientific, Singapore. 1983)
p. 143. [I1] J. Singh and A. Thilagam, in: Proc. ICL’87, J. Lumin. 40&41 (1987) 457.