Cooperative two-photon luminescence

Journal of Luminescence 46 (1990) 209—215 North-Holland

209

COOPERATIVE TWO-PHOTON LUMINESCENCE K.C. MISHRA a J.K. BERKOWITZ

a

E.A. DALE

a

T.P. DAS

b

and K.H. JOHNSON

C

GTE Electrical Products, 100 Endicott Street, Dancers, MA 01923, USA b Department of Physics, SUNY at Albany, Albany, NY 12222, USA C Department of Materials Science, MIT, Cambridge, MA 02139, USA a

Received 8 September 1989 Revised 6 December 1989 Accepted 6 January 1990

Using perturbation theory, Dexter proposed a theoretical model for obtaining a quantum efficiency greater than one for a luminescent material. The model is based on the simultaneous transfer of excitation energy from a sensitizer to two activators. It will be shown in this paper that the nonzero transition probability obtained by Dexter is an artifact of the nonorthogonal basis set used for the perturbative expansion. The cooperative transfer of energy from the sensitizer to the activators can at best be a second or higher order process.

I. Introduction The theoretical model of a two photon phosphor proposed by Dexter [1] has stimulated considerable experimental effort in luminescent research. Several cases of two photon emission have been reported in the literature [2—4]but none of them operate on the basis of the “cooperative emission” or “photon splitting” process of ref. [1]. In this paper, it will be demonstrated that the probability of such a transition as a first order process exactly vanishes. It will also be shown that the non-zero transition probability obtained in ref. [1] is an artifact of this choice of many-electron states to describe the two photon process. cooperative process of two photon emission or the photon-splitting process can be described as follows. A sensitizer S in the ground state is excited by an ultraviolet photon such that in the excited state it can emit a photon of energy, E = 2hv. This energy is then transferred equally to two activator ions before emission can occur from the sensitizer. The two activators finally relax to the ground state by emitting two photons of energy hi’. For simplicity we have assumed: (i) an exact condition of resonance; (ii) no broadening of the energy levels; and (iii) no phonon relaxation 0022-2313/90/$03.50

~

1990

—

process. Inclusion of such effects will not affect the conclusions arrived at in this paper. Our aim in this paper is to demonstrate that such a process, which involves a change of states of three electrons simultaneously through a radiationless process, cannot occur as a first order effect. It will, however, be shown that it can be a second order process. The experimental observation of a weak inverse process supports such a possibility [5]. Two photon emission may also occur through a modified cooperative process (to be discussed later), or an interband Auger process. Both these processes are first order processes and have the appearance of a cooperative luminescence process. Such processes, in spite of this similarity, do not provide any experimental support for the first-order cooperative process of ref. [1].

2. Theory 2.1. General proof based on standard many-body perturbation theory It is obvious that the cooperative process involves three electrons, two activators, and one

Elsevier Science Publishers B.V. (North-Holland)

210

KG. Mishra et a!.

/ Cooperative two-photon

The first term within the bracket refers to the

S’ A~________ ~

charges, and the last term corresponds to electron—electron interaction. The one electron func-

A

tions, 0, are solutions of the Hartree Fock equation,12 [—2V~ +Veff(i~j)]O=O, (3) where J’~ff(r,) represents the effective Hartree

hv

S

X

________

2

A~

2 hv

hv A

~netic the energy of the energy electrons, represents potential duethetosecond the nuclear

X

I

A

S

A

Fig. 1. Schematic diagram of cooperative luminescence. The symbols “.“ and “X” represent the electronic configurations described by the many electron function ‘I’s and “F’ respec1’F electrons in a transitively. Note the change of states of three tion from ~ to ‘

sensitizer in a host lattice (fig. 1). We can describe the initial state, ‘~P 1 as a many electron state in which the sensitizer is in the excited state, (0~) 0A2 and and the two activators are in the ground state, 0A3~ For convenience, we will drop 0 from our notations representing the one electron states. This state, (‘I’~),can approximately be represented by a Slater determinant I S ‘(1), A 2 (2), A3 (3)1 (except for the normalization factor (6) 1/2), Similarly, the final state, “F can be described by I 5(1), A2~(2),A3’(3) I where the prime indicates an excited state. The “vacuum” or the ground state of the system ‘~PG is described by I S(1), A2(2), A3(3) In a similar manner, other many electron states such as: .

=

I S(1),

A2(2), A2~(3)I’

(la)

= I s(1), A3~(2),A3(3) I’ (ib) can be defined. The states described by ~I’~and ~‘2 are those in which one of the activator ions is effectively reduced and excited, while the other is oxidized. These Slater determinants can be defined in terms of one electron functions in the Hartree Fock sense. The one electron functions can be obtained by solving the many electron Schrodinger equation variationally,

~

~

~2

—

~

V(s)

+

~-

luminescence

~‘

‘>1

~-1~ =

‘

j

EP,

(2) where ~ represents a many electron wave function.

Fock one electron potential. expand 0 in terms atomic functions localized at For many centerof problems, it is convenient to different atomic centers. This is known in the literature as the linear combination of atomic orbitals molecular orbital procedure (LCAO—MO) [6]. The LCAO MO approach has many advantages. It includes overlap effect among atomic orbitals in a any rigorous manner, while the the one electron eigenfunctions, the molecular orbitals, are orthogonal to each other. One can identify the molecular orbitals as primarily being localized at one site or another by examining the coefficients of the atomic functions in their LCAO expansions. Within the framework of the molecular orbital approach one can discuss excitation or de-excitation processes in terms of different atomic centers, while the one electron states properly include any overlap effect among the atomic orbitals. The Slater determinants constructed from these one electron functions are orthogonal to each other due to the orthogonality of the one electron functions. They can be used as a complete set of basis functions for solving eq. (2). However, they are not eigenfunctions of the many electron Hamiltonian, H in eq. (2), because of the electron—electron interaction [7]. Therefore, the states, cP.,~corresponding to the stationary state solutions of H can be expanded in terms of the Slater determinants, ~l’k: =

~

(4)

C~’I’~.

The coefficients ck’ in eq. (4) can be obtained through a perturbative approach, the perturbative Hamiltonian, H’ being given by: H’ =

~

~‘

i>j

-~— —

i_I

~

J”~ff(t)

—

~

V( i

).

(5)

K C. Mishra et al.

/

Cooperative two-photon luminescence

The perturbative Hamiltonian in eq. (5) does not include any external perturbation. The application of the perturbative method in the present case can only be understood as a procedure used to obtain the correct stationary states of H, whose eigenfunction cannot be obtained by directly solving eq. (2). An excited quantum system described by a many electron state, may return to the ground state in many different ways. For example, if is predominantly composed of photons ~ and “F’ probability of getting one or two outthe from the system due to radiative transition to the ground state will be proportional to I 2 or I cF I 2 respectively. If cI~,is mostly composed of ‘I~i.e., ~zI 1, I cF 2 can be identified with the probability of transition from ‘I’j to “Frn The transition probability for the Auger process was calculated in this manner by Wentzel [8] using the time independent perturbation theory. One can also utilize the time dependent theory as in the case of the interband Auger process in solids [9]. TeWordt [9] assumed that at t = 0 the many electron state can be described by ‘I’s. Then, using the time dependent perturbation theory, he calculated the transition probability from ‘.1’~ to some final state ~,

211

solving the equation, H’ = ~ + E~ H 0~ —

H’ +

—

H’

H0 E0

+ (higher order terms). (8) Using a complete basis set, one can expand ~ as in eq. (4). Then the probability of transition from ~ to ~‘F will be proportional to the square of (‘I’F I 1). However, if the Slater determinants used for the expansion are not orthogonal other, 1’i~I to I each 2 with the it is not correct to identify I (‘. transition probability ~seeappendix). Since the many electron states are orthogonal in the present formulation, a transition from ‘I’~to “F as a first order process can occur only if H’ ‘I’~ in eq. (8) includes a term involving ~‘F~ It is therefore essential to determine whether or not H’ can scatter I ‘~) to I “F) or (~‘FI H’ ‘I’~)~ 0. This can be easily done by expressing the perturbation Hamiltonian, H’ in a second quantized form:

H’ =

-~-

~ Ku I r

—

~

(1

1 I kl>a~a~a/ak 32

i,j.k,1

I

—

V

I I> a~a 1.

~‘F which corresponds to a state with an additional electron—hole pair. In the present case, we will utilize time independent perturbation theory as Dexter did in his original formulation and calculate the probability of transition from ‘~t’~ to “F’The many electron wave function cls can be obtained using the linked cluster expansion [10], oo (L) ~

(~,

H’ H

“

0)

‘

where n refers to the order of the perturbation theory and the superscript L implies that the summation is over the linked diagrams. H0 and are the zero order Hamiltoman and eigenvalue, respectively, the latter corresponding to ~ In this case, ~° is the same as ‘,‘~, =

‘Pr.

(7) 1’F

The amount of

‘

(9)

The first term in eq. (9) leads to change of state for two electrons while the second term changes the state for one electron. In a first order process, none of the terms in eq. (9) can lead to a change in t’F differ the states of three electrons, theand matrix states in of three electrons. Since in fact ‘~I’~ ‘element (F I H’ II> will vanish. This is simply a re-statement of a very well known rule that the matrix element of a two electron operator with

(6)

~°

—

—

in ~ can be determined by

respect to many electron states expressed in the form of Slater determinants differing by more than two states is identically zero [7]. In fig. 2(a), a typical Brueckner Goldstone diagram [7,10] is presented describing the effect of the first term in eq. (9) on the state ‘.I’1. The two electron operator scatters the electron in the state O~’to O~while exciting the second electron from a state, say 02 to 02’. The excitation of the two activators can only occur in two steps (fig. 2(b)). First, thestate, second excited to an to intermediate 02” electron which isis then scattered 02’

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KG. Mishra et a!.

/ Cooperative two-photon luminescence

Kx31x2’) = 0(3, 2’),

(a)


(lOf) The symbol “0” stands for the overlap integral. The one electron functions are solutions of a zero order Hamiltonian corresponding to isolated ions.

(b)

~

~

—

0(2, 3’).

Therefore, it was among reasonable to assume nonvanishing overlap the one electrona funcby:

c

02(~.

(lOe)

~

H’=~ tions. 1The perturbative r1 Hamiltonian is then given (11) I J which inI,] the‘J second quantized form can be expressed as: H’ ~ Ku I r321 k1)a,~a~alak ~,,

Fig. 2. Diagrammatic representation of various emission processes discussed in the text. (a) Single photon emission through radiationless energy transfer from an electron in the state &~‘ to another one in the state ~2 (b) Second order two photon processes in which energy transfer from the electron in 9s’ to the electrons in states 93 and 93 occurs in two the state steps through an electron electron interaction. (c) First order two photon process representing partial energy transfer from Os’ to ~2 and subsequent emission of photons.

while the electron in 03 is scattered to 03’. The final state after these two steps will be ‘~~‘Fand thus the transition probability from ‘.1’~ to ~‘F will be of second order. Therefore, the cooperative process cannot compete with the radiative transition from O~’ to O~ through the emission of a single photon when a dipole transition among these states is allowed. 2.2. Critical analysis of Dexter’s formalism

~

—

~/

—

~

Ki I ~

)—

I J) a

~a 1

(Xs

(Xx

I X2) I Xi’)

(X2 I Xl)

=

I = 0, (Xs I X3) —0,

(lOb) (lOc)

(X2 X2’) =

0(2,

3)(x2

I Xl’)

=

0(2, 3’),

(lOd)

~~J’1.

(12)

In eqs. (10) and (12), lower and upper case subscripts denote electronic and nuclear coordinates, respectively. Using the basis functions described by eq. (10), one can express ‘I’~and “F in the forms: ‘.1’~= 6 1 2 [i 02(2 3)] 1 X IXs’(l) X2(2) X3(3)

2

I

=N

Ix~’(1)X2(2) X3(3) I~ 6 1 2[1 O2(2~,3’)] 1 2 1

=

(13)

—

I xsii~ X2~(2)X3’(3) I —NFIXS(1) x2’(2) X3’(3) I~ (14) where N1 and NF represent the normalization factors, 6 1 2[1 02(2, 3)] 1 2 and 6 1/2(1 02(2~, 3’)) 1 2, respectively. It can easily be shown that “F and ‘I’, are orthogonal to each other. Using ‘I’s and “F in this form, the matrix element (F I H’ I I) can be shown to be: —

The question arises, why was a non-zero value for (F I H’ II) obtained in ref. [1]? The initial and final state ‘I’~and “F were described in terms of normalized but non-orthogonal one electron states, Xs’ Xs” X2’ X2” or X3 and X3’ satisfying the following relations: (Xs IXs’) = KX2 1X2’> = = 0, (ba)

+

I,]

(F I H’

II)

=

—

0(2, 3’)[l O2(2~,3’)] 1 2 x[i 02(2, 3)] 1 2 1 X3(3) X2’ (3)) >< [(~S~ (2) Xs(2) r23 + (x~~(2) Xs(2) r 1 X3’(3) X2(3))1 23 (15) —

—

—

K C. Mishra et at

/ Cooperative two-photon luminescence

The matrix element term arising from the one electron term in H’ vanishes. Equation (15) can be simplified further to the “dipole form” obtained in ref. [1] by applying the Taylor series expansion to (r23) The dipole form of the transition matrix element is not crucial to our discussions and can be obtained from ref. [1]. Equations (1O)—(15) describe how ref. [1] arrived at a nonzero transition probability for cooperative transfer of excitation energy from the sensitizer to two activators. Underlying these equations are the following implicit assumptions: (1) In the perturbative expansion, the Slater determinants constructed from the one electron functions x, have been utilized as a basis set, although the choice of basis set is not explicitly mentioned in ref. [1]. (2) Since I
213

of magnitude as in eq. (15), one obtains: (Fl H’ I~)= NINF <

{(Ix~(1)x2’(2) x3’(3) II a~a~’a2as’ I

~.

x I xs’(l) X2(2) X3(3) I) X (Xs’ (2) Xs (2) +

X3’ (3) X2 (3)) a~~ a3 a3~

~ I xs(1) X2’(2) X3’ (3)

x I x~’(1)X2(2) X3(3) I) X(~s’(2) Xs(2) r23 X3(3) X2’(3))}.

(16)

Equation (16) can be easily reduced to the form in eq. (15) by using the values of N1 and NF from eqs. (13) and (14). The first term in eq. (16) can be represented diagrammatically by fig. 2(a), if we replace 2’ by 3’. The operator { a~a a 2a5~)sends the state “i to the determinantal state { I xx~x~”I which is “2 in terms of the basis functions chosen by Dexter. In the representation adopted by Dexter, “F is not orthogonal to “2 and therefore, (Fl H’ II) is not zero. In other words, Dexter’s nonzero transition probability results from a deliberate choice of nonorthogonal atomic basis functions to describe the many electron states in the form of Slater determinants. This leads to nonzero values of the integrals like xs(1)x2’(2)x3’(3) Ix~(1)X3’(2)X3(3)) =

I

<~‘F

Since the overlap integral is not a real interaction, the non-zero value of (F I H’ II) merely reflects an artifact of the representation chosen by Dexter, not a true transition. It is interesting to ask if there exists any first order process which can lead to emission of two photons. In fig. 2c, we have presented diagrammatically such a possibility. Consider a sensitizer with an intermediate state O~~’ and an activator with an excited state °2’ such that the energy difference between O~,and O~~’ is the same as the energy required to excite an electron from °2 to ~2’~ This diagram represents a situation in which the electron in 02 is excited to ~2’ as the electron in state O~relaxes to the state 8~”.This process

214

K. C. Mishra et a!.

/ Cooperative two-photon luminescence

will have non-zero transition probability since it involves the excitation of two electrons only. While it is necessary that a dipole transition should be allowed between the levels 02 and 02’, this is not necessary for states O~and Os”. This is a case of sensitized energy transfer before the emission of photons and has indeed been observed [2,3,11].

3. Conclusion It has been shown that the mechanism proposed in ref. [1] for cooperative excitation of two luminescent centers can occur only as a second order process. Since the luminescent centers in ionic crystals lead to discrete energy levels broadened by solid state effects, it is very unlikely that a second order process will be a dominant mode through the participation of intermediate states. Whenever such a process is considered, the transition probability has to be determined within the framework of second order perturbation theory. This has indeed subsequently been considered by Dexter and detailed formulation of the transition probability can be obtained from ref. [12]. There are alternative processes such as the one represented in fig. 2(c) or interband Auger processes [13] which may lead to two photon emission. Such processes are not included in ref. [1]. One may observe two photon emission by ions other than the sensitizer. Such an observation should not be considered as a verification of Dexter’s mechanism. Origin of such effects should be looked for in some other electronic processes. Sometimes a second order process may become the dominant modeofofan radiative transition through the participation intermediate state. For example, the sensitizer may relax to a metastable state from which the system cannot return to the ground state by emitting a single photon. In such a situation, the second order processes may become predominant, Finally, we have shown the problems associated with using a nonorthogonal basis set. In the appendix, it is shown that one cannot use the results of standard perturbation theory (such as attributing I (F I H’ II)) 2 to a probability of transition)

when the basis set does not involve a complete set of orthogonal functions. There is nothing sacrosanct about an orthogonal basis set. The same results can be obtained using both type of basis sets. But when using a nonorthogonal basis set, one has to be very careful regarding the interpretation of terms arising due to the nonorthogonality among the basis functions. The orbital overlap is a consequence of a choice of basis set. The overlap does not correspond to any physical interaction.

Ap~endtx Let us assume that a given set of basis functions { ~ } is not orthogonal, i.e., S,1

(“~I‘I’~)# 0,

(A.1)

Let us assume further that there exists an operator, H such that ~ = H’ I = I ‘P). (A.2) ‘.~‘

Equation (A.2) implies that H’ scatters ‘P to If we expand 1 in terms of the basis functions { ‘P’ } as is normally done in the case of an orthogonal basis set, we will have ~Ji=

~ (*~I~) I k

—

~-,

~‘k)

=

~

I

(~‘~H’

I

~‘)

I

~‘k)

k

~,

5k

L..~

1

I

A3

k)’

where eq. been 1’k I H’ I ~I’)(A.2) in thehas form (~1’kutilized I ~I’). to express (‘ Thus, .1i contains a term in ‘Irk with a coefficient Ski. Obviously eqs. (A.3) and (A.2) are not identical. The non-vanishing coefficients of ‘~k, i.e., (~1’kI H’ I ~P) are merely a consequence of the nonorthogonality of basis functions. Thus, the use of I ‘I’k)(’.l’k I as a projection operator for I ‘I’k) is not mathematically correct in the case of a non-orthogonal basis set, { “k }, Therefore, the square of the matrix element (~~I’k I H’ I ‘~) cannot be equated to the probability of transition in this case.

K C. Mishra et at

/ Cooperative two-photon luminescence

Acknowledgements It is a pleasure to acknowledge helpful discussions with Dr. B.G. DeBoer. This work was partially supported by a grant from the Lawrence Berkeley Laboratory, Subcontract No. 4553410.

References [1] D.L. Dexter, Phys. Rev. 108 (1957) 630. [2] F. Varsanyi and G.H. Dieke, Phys. Rev. Lett. 7 (1961) 442. [3] G.E. Peterson and P.M. Bridenbaugh, J. Opt. Soc. Am. 53 (1963) 301, 1129; 54 (1963) 644. J.F. Porter and H.W. Moos, Phys. Rev. 152 (1966) 300. [4] J.L. Sommerdijk, A. Brill and A.W. de Jager, J. Lumin. 8 (1974) 341.

W.W.

215

Piper, J.A. DeLuca and F.S. Ham. J. Lumin. 8 (1974) 344. [5] 5. Ostermayer and L.G. Van Uitert, Phys. Rev. B1 (1970) 4208 [6] C.C.J. Rothaan, Rev. Mod. Phys. 23 (1951) 69. [7] I. Lindgren and J. Morrison, in: Atomic Many-Body Theory (Springer, New York, 1985). [8] G. Wentzel, Z. Phys. 43 (1927) 524. [9] L. TeWordt, Z. Phys. 138 (1954) 499. [10] J. Goldstone, Proc. Roy. Soc. A239 (1957) 267; for applications to atomic problems, see for example, H.P. Kelley, Phys. Rev. 136 (1964) B896. E.S. Chang, R.T. Pu and T.P. Das, Phys. Rev. 174 (1968) 1. [11] J.P. van der Ziel and L.G. Van Uitert, Phys. Rev. Lett. 21 (1968) 1334. [12] D.L. Dexter, Phys. Rev. B 126 (1962) 1962. [13] ER. Ilmas and T.I. Savikhina, J. Lumin. 1&2 (1976) 703.