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Multiphonon sideband intensities in rare earth ions in crystal
Journal of Luminescence 83}84 (1999) 171}175
Multiphonon sideband intensities in rare earth ions in crystal Konstantin K. Pukhov!,*, Tasoltan T. Basiev!, Johann Heber", Sergey Mirov#, Francois Auzel$ !Laser Materials and Technologies Research Center of General Physics Institute, 38 Vavilov Street, Block D, 117942, GSP-1, Moscow, Russia "Institute of Solid State Physics, Darmstadt University of Technology, Hochschulstr. 8, D-64289 Darmstadt, Germany #The University of Alabama at Birmingham, 1300 University Boulevard, Birmingham, AL 35294-1170, USA $GOTR, UPR211, CNRS, Laboratoire de Physicochimie des Materiaux, 1 Pl. A-Briand, 92195 Meudon Cedex, France
Abstract Intermultiplet transitions involving emission (or absorption) of one photon and n phonons are theoretically investigated in trivalent rare earth (RE) ions in crystals. Transitions are induced by joint action of electromagnetic "eld and high-order optical anharmonicity (M-process). Hence, this work extends Judd's study of one photon}one phonon transitions to the multiphonon case. The processes under consideration give rise to the Stokes and anti-Stokes multiphonon sidebands of light emission or absorption. Besides, these processes can signi"cantly increase the rate of intermultiplet transitions in the presence of stimulated emission. As far as we know the problem of multiphonon sidebands have not been considered previously in the theory of M-processes. Here we obtained general expression for transition probabilities. In doing this we used the crystal "eld model that takes into account both Coulomb and non-Coulomb interactions of 4f-electrons with ligands. (Previously we used this model for the calculation of multiphonon rates in RE ions.) The `electronic parta of obtained expression is presented exactly in the same form as the well-known Judd}Ofelt expression for radiative transition probability between multiplets. A simpli"ed form of general expression is proposed. Other mechanisms of photon}phonon transitions are brie#y discussed. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Multiphonon; Sideband; Rare earth ion
1. Introduction In this paper we present results of the theoretical study of intermultiplet photon}phonon transitions in RE ions. These transitions are induced by joint action of electromagnetic "eld and high-order optical anharmonicity. Hence, this work extends
* Corresponding author. Tel.: #7-95-1328376; fax: #7-0951350270. E-mail address: [email protected] (K.K. Pukhov)
Judd's study [1] of one photon}one phonon transitions to the multiphonon case. The processes under consideration give rise to the Stokes and anti-Stokes multiphonon sidebands of light emission or absorption. Besides, these processes can signi"cantly increase the rate of intermultiplet transitions in the presence of stimulated emission. Here we obtained general expression for transition probabilities. In doing this we used the crystal "eld model that takes into account both Coulomb and non-Coulomb interactions of 4f-electrons with ligands.
0022-2313/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 0 9 3 - 9
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K.K. Pukhov et al. / Journal of Luminescence 83}84 (1999) 171}175
The probability of induced absorption transition from initial state D i ) with energy E to "nal state D f ) i with energy E (E (E ) can be written as f i f 2pasP = =" exp [i(D!X)t] + ~= ]Sd (t)d (0)Te e dt, (1) q q{ q q{ where a"e2/+c is the "ne-structure constant; s"[(e#2)/3]2/Je (Je is the refractive index); P is the intensity of incident light; X is the light frequency; e are components of light polarization q vector; D"(E !E )/+. In the expression for f i correlation function Sd (t)d (0)T the symbol S2T q q{ denotes an average on lattice vibrations; d "(i D D D f ); d "( f D D D i); D"p/e, p is the q q q{ q{ 4f-electrons dipole moment; d (t)"exp (iH t/+) q L d exp (!iH t/+); H is the lattice vibrations q L L Hamiltonian. The probability of induced radiation transition (= ) is related to the probability of the %. induced absorption transition (= "=) by the !"4 equation
P
= /= "exp [+ (D!X)/k¹]. (2) %. !"4 The probability of spontaneous emission in frequency region [X, X#dX] can be written as = (X)"w (X) dX with 41 = 2aes w (X)" X3 3pc2 ~=
P
]exp [i (D!X)t]+ Sd (t)d (0)T dt. (3) q ~q q In Eqs. (1)}(3) the magnitude + (D!X) constitutes the energy which the crystal obtains or loses. When D(X (D'X) the expression (1) gives the intensity of the Stokes (anti-Stokes) wing of absorption band. The mechanism considered here is sometimes called M-process [2,3]. In order to proceed with the = and = calcu41 lation we will use an `exchange}chargea model [4] of the crystal "eld. The crystal "eld is considered in this model as H"H #H , where H is the M E M Coulomb interaction of 4f-electrons with a "eld of point charges (H1#) and that of the dipole moments due to the ligands. The non-Coulomb Hamiltonian H "+ + B (r ) + >H (r ) + > (n ) km s km a E k s a s k m
(4)
(r and n are the instantaneous radius vectors of sth s a ligand and ath 4f-electron, respectively), deals with interaction that is mainly due to the overlap of electron wave function with wave functions of ligands and includes the corrections to Coulomb interaction resulting from the spatial distribution of ligand electronic charge, the energy of exchange interaction, and contributions from the charge transfer states. Previously we used this model and correlation function method for the calculation of rates of the multiphonon transition [5}7].
2. Coulomb interaction contribution to muiltiphonon sideband Here we consider the contribution to multiphonon sideband arising from the modulation of Coulomb "eld by crystal vibrations. In this case Sd (t) d (0)T q q{ "(!1)q`q{ + + (!1)m [j] [j@] kmj k{m{j{ 1 j k 1 j@ k@ ] ) q !m!q m !q@ !m@#q@ m@
A
BA
B
Di) N (k, j)N (k@, j@) ](i D ;(j) D f ) ( f D;(j{) ~m{`q{ 4f m`q 4& (5) ]SA (t)AH (0)T. k{m{ km The notation of Eq. (5) follows Judd in Ref. [1]. Subscript 4f in matrix elements of Eq. (5) denotes that operator ;(j) connect states of 4f-con"gurap`q tion, and [j]"2j#1 (j"2, 4, 6). Expressions for A in point charge model are given by Eq. (21) in km Judd paper [1]. Using this expression we obtain SA (t)AH (0)T km k{m{ (6) "4pe2([k] [k@])~1@2 + q q G(SS{) (t), S S{ km,k{m{ SS{ where q is the e!ective charge of sth ligand. The S result of the calculation of correlation function G(SS{) (t) is given by Eqs. (A.1)}(A.3) in the appenkm,k{m{ dix. Substituting Eq. (5) in Eqs. (1)}(3) gives the expression for probabilities of vibronically induced forced electric dipole transitions in the framework of the point charge model. However, the obtained
K.K. Pukhov et al. / Journal of Luminescence 83}84 (1999) 171}175
expression is too cumbersome. By applying the Judd}Ofelt method [1,8] for the calculation of probabilities intermultiplet transitions, one "nds that the averaged probability = M of photon absorption accompanied by absorption or emission of phonons is given by = M "(2psaP/3+ [J][J@])
P
]
=
exp [i(D!X)t]S (t) dt, JJ{ ~=
(7)
where S (t), + Sd (t)d (0)T JJ{ q ~q q,MM{ " + [j](JDD;(j)DDJ@)2 kmj (8) ]SA (t)AH (0)TN2(k, j)/[k]. km km It should be noted that S (0) is the usual expresJJ{ sion for the zero phonon line (ZPL) strength of JPJ@ intermultiplet transition. Hence, Eq. (7) can be rewritten as = M "(2psaP/3+[J][J@]) + X(n)(JDD;(j)DDJ@), j n/1 where
P
=
where Z is the number of anions nearest to the RE ions, R is the equilibrium distance between RE ion and ligand, q is the e!ective charge of ligand. L Finally, in a single-frequency (u ) model of crys0 tal vibrations we obtain 2pZe4q2 (2n#2k)![j] L+ X(n)" j u [k]!n!R2(k`1) 0 k ]N2 (k, j)gnH , (13) n where H "SN#1Tn and SNTn for Stokes and n anti-Stokes absorption band, respectively (SNT" [exp (+u /k¹)!1]~1 is a mean phonon-occupa0 tion number). The quantity g"Su2T/R2 in Eq. (13) is the parameter characterizing dynamical properties of lattice [5}7]. The value of Su2T can be roughly estimated as +/2Mu , where M is a re0 duced mass of ions involved, mainly ligands. Hence, values of g are expected in the region 10~3}10~4. In Ref. [1] quantity N includes a 3j, 6j-symbols and sum +(nl D r D n@l@) (nl D rk D n@l@)/D(n@l@). This sum was calculated for a number of RE ions in Refs. [9,10]. So, the X(n) parameter can be easily estimated by j using Eq. (13).
(9)
X(n)"[j] + 4pe2q q N2(k, j) j S S{ kSS{ ]¹SS{(n)J(n) (D!X)/[k]2. (10) k SS{ In Eq. (10) ¹SS{(n) is given in the appendix, and the k spectral density is given by J(n) (u)" SS{
173
(11) e*utKn (t) dt. SS{ ~= In Eqs. (9)}(11) n is the number of phonons involved in the intermultiplet transition. Eq. (9) is precisely of the same form as Eq. (16) [1]. The expression (10) for X(n) can be simpli"ed by using j the approximation of the additive contributions of the ligands motion to the sideband intensities (when K (t)"K(t)d ): SS{ SS{ (2n#2k)![j] X(n)"Ze2q2 + L [k]!n!2nR2(k`1) j k ]N2(k, j)J(n)(D!X), (12)
3. Non-Coulomb interaction contribution to multiphonon sideband As noted above, the non-Coulomb part of crystal "eld is described by the Hamiltonian H (see E Eq. (4)). In this case [4]
A
1 j k d " + [j](!1)m`q q q !m!q m jkmj ](i D;(j) D f ) A (r ) N ( j, k ; j), m`q 4& km j where
B (14)
A (r )"([k]/4p)1@2>H (r ), km j km j
G
]N ( j, k ; j)"2+ [l ][l@ ] (!1)l`l{ ]
A
l
1 l@
0 0 0
BA
l@
)
k l
H
1 j k l
l@
l
B
0 0 0
](nl D m D n@l@)(n@l@ D B (r ) D nl)/D(n@l@), k j
(15)
174
K.K. Pukhov et al. / Journal of Luminescence 83}84 (1999) 171}175
(n@l@ D B (r )D nl)"M8pe2/r ([l ][l@])1@2NG(nl, n@l@) k j j ][S ( j, nl)S ( j, n@l@)#S ( j, nl) S (j, n@l@) s s p p #c (ll@) S ( j, nl)S ( j, n@l@)]. (16) k n n Here G(nl, n@l@) are dimensionless "tting parameters (which, in particular, determine ZPL intensity), c (ll@)"[l(l#1)#l@(l@#1)!k(k#1)]/[l(l#1)l@(l@ k #1)]1@2, and S ( j, nl) (l"s, p, n) denotes the overl lap integral of the nl-con"guration electron wave function of RE ion with the wave function of the external electron shell in jth ligand. In the important case of RE ions surrounded by oxygen or #uorine ions, these ligand wave functions are of 2pp, 2pp, and 2s type. A general expression for Sd (t)d (0)T is too cumbersome. Here we give only q q{ an expression for S (t) in the approximation of JJ{ additive contributions of ligands to multiphonon sidebands. In this case Z S (t)" + [j]S N (k, j; t) N (k, j)T, JJ{ 4p2 kmj N (k, j; t)
(17)
"exp (iH t/+) N (k, j) exp (!iH t/+). (18) L L The quantity N (k, j) in Eqs. (17) and (18) can be written as N (k, j)"+ C (kj)o (r), (19) k k k where o (r)"exp (!a r)/r. Here we take into ack k count that to good approximation the overlap integrals S (nl), S (n@l@) in Eq. (16) can be represented in s s the form S exp (!cr). Coe$cients C (kj) can be 0 k found from Eqs. (15) and (16). Hence, we have
crystal vibrations we obtain Z X(n)" + C (kj)C (kj) U (n) gn. (21) j k k{ kk{ 2pu 0 kk{ The expression for U (n) follows from Eq. (A.6) in kk{ appendix. 4. Discussion The M-process implies that lattice Hamiltonian does not change during D i )PD f ) transition. On the contrary, the D-process is essentially caused by the change (shifts of equilibrium position of ions and/or change in lattice frequencies) in lattice state under D i )PD f ) transition [11]. The theory of both processes has a common problem related to the mathematical di$culties of phonon factor calculation, namely, the spectral density J(n)(u) and Huang} Rhys parameter S. At present, it is not clear which process will dominate. To our knowledge the only experimental paper on multiphonon sidebands in RE ions is the paper [12]. Further experimental and theoretical studies are necessary for a detailed description of phonon sidebands formation. Acknowledgements We wish to thank graduate student D. Martychkine (University of Alabama at Birmingham) for assistance and Prof. B.Z. Malkin for helpful discussions. This work is partially supported by the joint grant 97-02-00143 of the DFG and the Russian Foundation for Basic Research (RFBR), by the RFBR grants 97-02-16950 and 97-02-17669, by INTAS-96-0232 grant, and by NSF International Cooperation grant ECS 9710428.
SN (k, j; t) N (k, j)T Appendix A "+ C (kj)C (kj)F (t), (20) k k{ kk{ kk{ where the expression for the correlation function F (t) is given by Eqs. (A.5) and (A.6). In the case kk{ considered, the modulation of the RE ion}ligand distance r results in the modulation of the overlap integrals, which, in turn, contribute to the multiphonon sideband. In a single-frequency model for
A.1. The correlation functions in the harmonic approximation G(SS{) "Sexp (iH t/+) (>H (r )/rk`1) km S S km,k{m{ L ]exp (!iH t/+) (>H (r )/rk{`1)T k{m{ S{ S{ L = (A.1) " + ¹(SS{) (n) Kn (t), SS{ km,k{m{ n/0
K.K. Pukhov et al. / Journal of Luminescence 83}84 (1999) 171}175
175
where
where
¹(SS{) (n)" km,k{m{ (!1)k`k{ ) M[k][k@](2n#2k)!(2n#2k@)!/(2k)!(2k@)!N1@2 Rk`1Rk{`12nn! S S{
U(SS{) (n)"o (R )o (R )p~1@2(q q )n`1@2 kk{ k S k{ S{ Sk S{k{ I/5(n@2) (2n!4p#1)P (n n ) n~2p S0 S{0 ]exp(q #q ) + Sk S{k{ C(p#1)C(n!p#3/2) p/0
A
n#k n k ]+ (!1)m`m{ !p!m p m p n#k@ n k@ ] !p!m@ p m@
A
B
]K (q )K (q ) (A.6) n~2p`1@2 Sk n~2p`1@2 S{k{ q "a R , Int(n/2) is the integer part of n/2, C(x) Sk k S is a gamma function, and K (x) is a n~2p`1@2 MacDonald function.
B
(n ) ) > (n ) ) >H n`k,p`m S0 n`k{,p`m{ S{0 and
(A.2)
K (t)"Su (t)u T/3R R . SS{ S S{ S S{
(A.3)
References
¹(SS{)"+ ¹(SS{) (n) k km,km m [k] (2n#2k)! " ) P (n n ), 4p (2k)!n!2n(R R )k`1 n`k S0 S{0 S S{ (A.4) where P (n n ) is the Legendre polynomial. n`k S0 S{0 (In Eqs. (A.1)}(A.4) r "R #u , R is equilibrium S S S S distance between RE ion and sth ligand, n " S0 R /R ). S S F(SS{)(n)"Sexp (iH t/+) [exp (!a r )/r ] kk{ L kS S ]exp (!iH t/+) [exp (!a r )/r ]T L k{ S{ S{ " + U(SS{)(n)[K (t)/2]n, kk{ SS{ n/1
(A.5)
[1] B.R. Judd, Phys. Rev. 127 (1962) 750. [2] M.P. Hehlen, A. Kuditcher, S.C. Rand, M.A. Tischler, J. Chem. Phys. 107 (1997) 4886. [3] C.M. Donega, A. Meijerlink, G. Blasse, Condens. Mat. 4 (1992) 8889. [4] B.Z. Malkin, in: A.A. Kaplyanskii, R.M. Macfarline (Eds.), Spectroscopy of Solids Containing Rare Earth Ions, Elsevier, Amsterdam, 1987, p. 13. [5] T.T. Basiev, Yu.V. Orlovskii, K.K. Pukhov et al., J. Lumin. 68 (1996) 241. [6] Yu.V. Orlovskii, R.J. Reeves, R.C. Powell, T.T. Basiev, K.K. Pukhov, Phys. Rev. B 49 (1994) 3821. [7] T.T. Basiev, Yu.V. Orlovskii, K.K. Pukhov, F. Auzel, Laser Phys. 7 (1997) 1139. [8] G.S. Ofelt, J. Chem. Phys. 37 (1962) 511. [9] R.P. Leavitt, C.A. Morrison, J. Chem. Phys. 73 (1980) 749. [10] T.R. Faulkner, F.S. Richardson, Mol. Phys. 35 (1977) 1141. [11] T. Miyakawa, D.L. Dexter, Phys. Rev. B 1 (1970) 2961 and references therein. [12] F. Auzel, Phys. Rev. B 13 (1979) 2809.
Multiphonon sideband intensities in rare earth ions in crystal Konstantin K. Pukhov!,*, Tasoltan T. Basiev!, Johann Heber", Sergey Mirov#, Francois Auzel$ !Laser Materials and Technologies Research Center of General Physics Institute, 38 Vavilov Street, Block D, 117942, GSP-1, Moscow, Russia "Institute of Solid State Physics, Darmstadt University of Technology, Hochschulstr. 8, D-64289 Darmstadt, Germany #The University of Alabama at Birmingham, 1300 University Boulevard, Birmingham, AL 35294-1170, USA $GOTR, UPR211, CNRS, Laboratoire de Physicochimie des Materiaux, 1 Pl. A-Briand, 92195 Meudon Cedex, France
Abstract Intermultiplet transitions involving emission (or absorption) of one photon and n phonons are theoretically investigated in trivalent rare earth (RE) ions in crystals. Transitions are induced by joint action of electromagnetic "eld and high-order optical anharmonicity (M-process). Hence, this work extends Judd's study of one photon}one phonon transitions to the multiphonon case. The processes under consideration give rise to the Stokes and anti-Stokes multiphonon sidebands of light emission or absorption. Besides, these processes can signi"cantly increase the rate of intermultiplet transitions in the presence of stimulated emission. As far as we know the problem of multiphonon sidebands have not been considered previously in the theory of M-processes. Here we obtained general expression for transition probabilities. In doing this we used the crystal "eld model that takes into account both Coulomb and non-Coulomb interactions of 4f-electrons with ligands. (Previously we used this model for the calculation of multiphonon rates in RE ions.) The `electronic parta of obtained expression is presented exactly in the same form as the well-known Judd}Ofelt expression for radiative transition probability between multiplets. A simpli"ed form of general expression is proposed. Other mechanisms of photon}phonon transitions are brie#y discussed. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Multiphonon; Sideband; Rare earth ion
1. Introduction In this paper we present results of the theoretical study of intermultiplet photon}phonon transitions in RE ions. These transitions are induced by joint action of electromagnetic "eld and high-order optical anharmonicity. Hence, this work extends
* Corresponding author. Tel.: #7-95-1328376; fax: #7-0951350270. E-mail address: [email protected] (K.K. Pukhov)
Judd's study [1] of one photon}one phonon transitions to the multiphonon case. The processes under consideration give rise to the Stokes and anti-Stokes multiphonon sidebands of light emission or absorption. Besides, these processes can signi"cantly increase the rate of intermultiplet transitions in the presence of stimulated emission. Here we obtained general expression for transition probabilities. In doing this we used the crystal "eld model that takes into account both Coulomb and non-Coulomb interactions of 4f-electrons with ligands.
0022-2313/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 0 9 3 - 9
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K.K. Pukhov et al. / Journal of Luminescence 83}84 (1999) 171}175
The probability of induced absorption transition from initial state D i ) with energy E to "nal state D f ) i with energy E (E (E ) can be written as f i f 2pasP = =" exp [i(D!X)t] + ~= ]Sd (t)d (0)Te e dt, (1) q q{ q q{ where a"e2/+c is the "ne-structure constant; s"[(e#2)/3]2/Je (Je is the refractive index); P is the intensity of incident light; X is the light frequency; e are components of light polarization q vector; D"(E !E )/+. In the expression for f i correlation function Sd (t)d (0)T the symbol S2T q q{ denotes an average on lattice vibrations; d "(i D D D f ); d "( f D D D i); D"p/e, p is the q q q{ q{ 4f-electrons dipole moment; d (t)"exp (iH t/+) q L d exp (!iH t/+); H is the lattice vibrations q L L Hamiltonian. The probability of induced radiation transition (= ) is related to the probability of the %. induced absorption transition (= "=) by the !"4 equation
P
= /= "exp [+ (D!X)/k¹]. (2) %. !"4 The probability of spontaneous emission in frequency region [X, X#dX] can be written as = (X)"w (X) dX with 41 = 2aes w (X)" X3 3pc2 ~=
P
]exp [i (D!X)t]+ Sd (t)d (0)T dt. (3) q ~q q In Eqs. (1)}(3) the magnitude + (D!X) constitutes the energy which the crystal obtains or loses. When D(X (D'X) the expression (1) gives the intensity of the Stokes (anti-Stokes) wing of absorption band. The mechanism considered here is sometimes called M-process [2,3]. In order to proceed with the = and = calcu41 lation we will use an `exchange}chargea model [4] of the crystal "eld. The crystal "eld is considered in this model as H"H #H , where H is the M E M Coulomb interaction of 4f-electrons with a "eld of point charges (H1#) and that of the dipole moments due to the ligands. The non-Coulomb Hamiltonian H "+ + B (r ) + >H (r ) + > (n ) km s km a E k s a s k m
(4)
(r and n are the instantaneous radius vectors of sth s a ligand and ath 4f-electron, respectively), deals with interaction that is mainly due to the overlap of electron wave function with wave functions of ligands and includes the corrections to Coulomb interaction resulting from the spatial distribution of ligand electronic charge, the energy of exchange interaction, and contributions from the charge transfer states. Previously we used this model and correlation function method for the calculation of rates of the multiphonon transition [5}7].
2. Coulomb interaction contribution to muiltiphonon sideband Here we consider the contribution to multiphonon sideband arising from the modulation of Coulomb "eld by crystal vibrations. In this case Sd (t) d (0)T q q{ "(!1)q`q{ + + (!1)m [j] [j@] kmj k{m{j{ 1 j k 1 j@ k@ ] ) q !m!q m !q@ !m@#q@ m@
A
BA
B
Di) N (k, j)N (k@, j@) ](i D ;(j) D f ) ( f D;(j{) ~m{`q{ 4f m`q 4& (5) ]SA (t)AH (0)T. k{m{ km The notation of Eq. (5) follows Judd in Ref. [1]. Subscript 4f in matrix elements of Eq. (5) denotes that operator ;(j) connect states of 4f-con"gurap`q tion, and [j]"2j#1 (j"2, 4, 6). Expressions for A in point charge model are given by Eq. (21) in km Judd paper [1]. Using this expression we obtain SA (t)AH (0)T km k{m{ (6) "4pe2([k] [k@])~1@2 + q q G(SS{) (t), S S{ km,k{m{ SS{ where q is the e!ective charge of sth ligand. The S result of the calculation of correlation function G(SS{) (t) is given by Eqs. (A.1)}(A.3) in the appenkm,k{m{ dix. Substituting Eq. (5) in Eqs. (1)}(3) gives the expression for probabilities of vibronically induced forced electric dipole transitions in the framework of the point charge model. However, the obtained
K.K. Pukhov et al. / Journal of Luminescence 83}84 (1999) 171}175
expression is too cumbersome. By applying the Judd}Ofelt method [1,8] for the calculation of probabilities intermultiplet transitions, one "nds that the averaged probability = M of photon absorption accompanied by absorption or emission of phonons is given by = M "(2psaP/3+ [J][J@])
P
]
=
exp [i(D!X)t]S (t) dt, JJ{ ~=
(7)
where S (t), + Sd (t)d (0)T JJ{ q ~q q,MM{ " + [j](JDD;(j)DDJ@)2 kmj (8) ]SA (t)AH (0)TN2(k, j)/[k]. km km It should be noted that S (0) is the usual expresJJ{ sion for the zero phonon line (ZPL) strength of JPJ@ intermultiplet transition. Hence, Eq. (7) can be rewritten as = M "(2psaP/3+[J][J@]) + X(n)(JDD;(j)DDJ@), j n/1 where
P
=
where Z is the number of anions nearest to the RE ions, R is the equilibrium distance between RE ion and ligand, q is the e!ective charge of ligand. L Finally, in a single-frequency (u ) model of crys0 tal vibrations we obtain 2pZe4q2 (2n#2k)![j] L+ X(n)" j u [k]!n!R2(k`1) 0 k ]N2 (k, j)gnH , (13) n where H "SN#1Tn and SNTn for Stokes and n anti-Stokes absorption band, respectively (SNT" [exp (+u /k¹)!1]~1 is a mean phonon-occupa0 tion number). The quantity g"Su2T/R2 in Eq. (13) is the parameter characterizing dynamical properties of lattice [5}7]. The value of Su2T can be roughly estimated as +/2Mu , where M is a re0 duced mass of ions involved, mainly ligands. Hence, values of g are expected in the region 10~3}10~4. In Ref. [1] quantity N includes a 3j, 6j-symbols and sum +(nl D r D n@l@) (nl D rk D n@l@)/D(n@l@). This sum was calculated for a number of RE ions in Refs. [9,10]. So, the X(n) parameter can be easily estimated by j using Eq. (13).
(9)
X(n)"[j] + 4pe2q q N2(k, j) j S S{ kSS{ ]¹SS{(n)J(n) (D!X)/[k]2. (10) k SS{ In Eq. (10) ¹SS{(n) is given in the appendix, and the k spectral density is given by J(n) (u)" SS{
173
(11) e*utKn (t) dt. SS{ ~= In Eqs. (9)}(11) n is the number of phonons involved in the intermultiplet transition. Eq. (9) is precisely of the same form as Eq. (16) [1]. The expression (10) for X(n) can be simpli"ed by using j the approximation of the additive contributions of the ligands motion to the sideband intensities (when K (t)"K(t)d ): SS{ SS{ (2n#2k)![j] X(n)"Ze2q2 + L [k]!n!2nR2(k`1) j k ]N2(k, j)J(n)(D!X), (12)
3. Non-Coulomb interaction contribution to multiphonon sideband As noted above, the non-Coulomb part of crystal "eld is described by the Hamiltonian H (see E Eq. (4)). In this case [4]
A
1 j k d " + [j](!1)m`q q q !m!q m jkmj ](i D;(j) D f ) A (r ) N ( j, k ; j), m`q 4& km j where
B (14)
A (r )"([k]/4p)1@2>H (r ), km j km j
G
]N ( j, k ; j)"2+ [l ][l@ ] (!1)l`l{ ]
A
l
1 l@
0 0 0
BA
l@
)
k l
H
1 j k l
l@
l
B
0 0 0
](nl D m D n@l@)(n@l@ D B (r ) D nl)/D(n@l@), k j
(15)
174
K.K. Pukhov et al. / Journal of Luminescence 83}84 (1999) 171}175
(n@l@ D B (r )D nl)"M8pe2/r ([l ][l@])1@2NG(nl, n@l@) k j j ][S ( j, nl)S ( j, n@l@)#S ( j, nl) S (j, n@l@) s s p p #c (ll@) S ( j, nl)S ( j, n@l@)]. (16) k n n Here G(nl, n@l@) are dimensionless "tting parameters (which, in particular, determine ZPL intensity), c (ll@)"[l(l#1)#l@(l@#1)!k(k#1)]/[l(l#1)l@(l@ k #1)]1@2, and S ( j, nl) (l"s, p, n) denotes the overl lap integral of the nl-con"guration electron wave function of RE ion with the wave function of the external electron shell in jth ligand. In the important case of RE ions surrounded by oxygen or #uorine ions, these ligand wave functions are of 2pp, 2pp, and 2s type. A general expression for Sd (t)d (0)T is too cumbersome. Here we give only q q{ an expression for S (t) in the approximation of JJ{ additive contributions of ligands to multiphonon sidebands. In this case Z S (t)" + [j]S N (k, j; t) N (k, j)T, JJ{ 4p2 kmj N (k, j; t)
(17)
"exp (iH t/+) N (k, j) exp (!iH t/+). (18) L L The quantity N (k, j) in Eqs. (17) and (18) can be written as N (k, j)"+ C (kj)o (r), (19) k k k where o (r)"exp (!a r)/r. Here we take into ack k count that to good approximation the overlap integrals S (nl), S (n@l@) in Eq. (16) can be represented in s s the form S exp (!cr). Coe$cients C (kj) can be 0 k found from Eqs. (15) and (16). Hence, we have
crystal vibrations we obtain Z X(n)" + C (kj)C (kj) U (n) gn. (21) j k k{ kk{ 2pu 0 kk{ The expression for U (n) follows from Eq. (A.6) in kk{ appendix. 4. Discussion The M-process implies that lattice Hamiltonian does not change during D i )PD f ) transition. On the contrary, the D-process is essentially caused by the change (shifts of equilibrium position of ions and/or change in lattice frequencies) in lattice state under D i )PD f ) transition [11]. The theory of both processes has a common problem related to the mathematical di$culties of phonon factor calculation, namely, the spectral density J(n)(u) and Huang} Rhys parameter S. At present, it is not clear which process will dominate. To our knowledge the only experimental paper on multiphonon sidebands in RE ions is the paper [12]. Further experimental and theoretical studies are necessary for a detailed description of phonon sidebands formation. Acknowledgements We wish to thank graduate student D. Martychkine (University of Alabama at Birmingham) for assistance and Prof. B.Z. Malkin for helpful discussions. This work is partially supported by the joint grant 97-02-00143 of the DFG and the Russian Foundation for Basic Research (RFBR), by the RFBR grants 97-02-16950 and 97-02-17669, by INTAS-96-0232 grant, and by NSF International Cooperation grant ECS 9710428.
SN (k, j; t) N (k, j)T Appendix A "+ C (kj)C (kj)F (t), (20) k k{ kk{ kk{ where the expression for the correlation function F (t) is given by Eqs. (A.5) and (A.6). In the case kk{ considered, the modulation of the RE ion}ligand distance r results in the modulation of the overlap integrals, which, in turn, contribute to the multiphonon sideband. In a single-frequency model for
A.1. The correlation functions in the harmonic approximation G(SS{) "Sexp (iH t/+) (>H (r )/rk`1) km S S km,k{m{ L ]exp (!iH t/+) (>H (r )/rk{`1)T k{m{ S{ S{ L = (A.1) " + ¹(SS{) (n) Kn (t), SS{ km,k{m{ n/0
K.K. Pukhov et al. / Journal of Luminescence 83}84 (1999) 171}175
175
where
where
¹(SS{) (n)" km,k{m{ (!1)k`k{ ) M[k][k@](2n#2k)!(2n#2k@)!/(2k)!(2k@)!N1@2 Rk`1Rk{`12nn! S S{
U(SS{) (n)"o (R )o (R )p~1@2(q q )n`1@2 kk{ k S k{ S{ Sk S{k{ I/5(n@2) (2n!4p#1)P (n n ) n~2p S0 S{0 ]exp(q #q ) + Sk S{k{ C(p#1)C(n!p#3/2) p/0
A
n#k n k ]+ (!1)m`m{ !p!m p m p n#k@ n k@ ] !p!m@ p m@
A
B
]K (q )K (q ) (A.6) n~2p`1@2 Sk n~2p`1@2 S{k{ q "a R , Int(n/2) is the integer part of n/2, C(x) Sk k S is a gamma function, and K (x) is a n~2p`1@2 MacDonald function.
B
(n ) ) > (n ) ) >H n`k,p`m S0 n`k{,p`m{ S{0 and
(A.2)
K (t)"Su (t)u T/3R R . SS{ S S{ S S{
(A.3)
References
¹(SS{)"+ ¹(SS{) (n) k km,km m [k] (2n#2k)! " ) P (n n ), 4p (2k)!n!2n(R R )k`1 n`k S0 S{0 S S{ (A.4) where P (n n ) is the Legendre polynomial. n`k S0 S{0 (In Eqs. (A.1)}(A.4) r "R #u , R is equilibrium S S S S distance between RE ion and sth ligand, n " S0 R /R ). S S F(SS{)(n)"Sexp (iH t/+) [exp (!a r )/r ] kk{ L kS S ]exp (!iH t/+) [exp (!a r )/r ]T L k{ S{ S{ " + U(SS{)(n)[K (t)/2]n, kk{ SS{ n/1
(A.5)
[1] B.R. Judd, Phys. Rev. 127 (1962) 750. [2] M.P. Hehlen, A. Kuditcher, S.C. Rand, M.A. Tischler, J. Chem. Phys. 107 (1997) 4886. [3] C.M. Donega, A. Meijerlink, G. Blasse, Condens. Mat. 4 (1992) 8889. [4] B.Z. Malkin, in: A.A. Kaplyanskii, R.M. Macfarline (Eds.), Spectroscopy of Solids Containing Rare Earth Ions, Elsevier, Amsterdam, 1987, p. 13. [5] T.T. Basiev, Yu.V. Orlovskii, K.K. Pukhov et al., J. Lumin. 68 (1996) 241. [6] Yu.V. Orlovskii, R.J. Reeves, R.C. Powell, T.T. Basiev, K.K. Pukhov, Phys. Rev. B 49 (1994) 3821. [7] T.T. Basiev, Yu.V. Orlovskii, K.K. Pukhov, F. Auzel, Laser Phys. 7 (1997) 1139. [8] G.S. Ofelt, J. Chem. Phys. 37 (1962) 511. [9] R.P. Leavitt, C.A. Morrison, J. Chem. Phys. 73 (1980) 749. [10] T.R. Faulkner, F.S. Richardson, Mol. Phys. 35 (1977) 1141. [11] T. Miyakawa, D.L. Dexter, Phys. Rev. B 1 (1970) 2961 and references therein. [12] F. Auzel, Phys. Rev. B 13 (1979) 2809.