Optical Jahn-Teller Effect in Ii-Vi Compounds

OPTICAL JAHN-TELLER EFFECT IN II-VI COMPOUNDS DOPED WITH Cr2+ ION

S.I. Klokishnera, B.S. Tsukerblatb*, O.S. Reua, A.V. Paliia, S.M. Ostrovskya a Institute of Applied Physics, Academy of Sciences of Moldova, Academy str. 5, 2028 Kishinev, Moldova bChemistry Department, Ben-Gurion University of the Negev, Beer-Sheva 84105,Israel *E-mail address: [email protected]

In this article we report evaluation of the vibronic Jahn-Teller (JT) coupling 5 5 parameters and the vibronic optical bands related to the E T2 transition in a series of II-VI crystals doped with Cr 2 ion. The parameters are estimated with the aid of the exchange charge model of the crystal field accounting for the exchange and covalence effects. Coupling to both trigonal and tetragonal vibrations proves to be important that gives rise to a five-mode optical JT problem in the orbital triplet. The profile of the optical band calculated using the results of the numerical solution of the dynamical JT problem proves to be in a good agreement with the experimental data.

1. INTRODUCTION

Chalcogenide type crystals doped with Cr 2 ions have got growing attention as the promising materials for the infrared solid state lasers. The laser operation takes 5 5 advantage from the E T2 broad vibronic optical band that provides tunability in a wide spectral range and at the same time negligible excited-state absorption. Recently demonstrated laser operation of Cr 2 ion in chalcogenide host materials like ZnSe [1], CdSe [2], ZnS [3], CdMnTe [4] gave a significant impact on the study of the optical JT problem in doped crystals [5-9]. The first attempts to interpret the optical spectra in 5 the infrared range [10] were based on the model of static JT effect for the ground T2 state [10-11]. Later on a dynamic JT model assuming coupling to both E- and T2 modes was developed [12]. Further study of II-VI crystals doped with Cr2+ ions [12] and by other transition metal ions [13] demonstrated important role of the dynamic JTE. This paper is aimed at the quantum-mechanical evaluation of the vibronic coupling constants for the series of II-VI crystals doped with Cr 2 ion and calculation 5 5 of optical band shape arising from the E T2 transition. The semiconducting systems under consideration are significantly covalent so that the point-charge crystal field model loses its accuracy. For this reason we employ the exchange charge model for the crystal field [14] that accounts for the covalence effects and provides relatively simple expressions for the crystal field and vibronic parameters keeping at the same time a reasonable level of accuracy [15, 16]. Calculation of vibronic optical band is based on the numerical solution of the five-mode dynamical JT problems for the orbital triplet and two-mode problem for orbital doublet.

2. HAMILTONIAN FOR THE IMPURITY CENTER 2

The ground 5D term of a free Cr 2 ion is split by the tetrahedral crystal field in 5 2 2 a fairly well known zinc-blende lattice into the orbital triplet T2 (t2 e ) and orbital 5 3 doublet E(t2e) , the former being the ground term. The standard cubic basis sets for the one-electron d-functions T2 (,, )   yz,  xz,  xy and E(u,) ( 2 2 2 2 ) are used. The levels 5 and 5 are separated u  3 z  r ,   3 x  y    T2    E by the gap 10Dq0 with Dq0 being the cubic crystal field parameter. The total Hamiltonian for the Cr 2 ion in crystal can be represented as

H  H e r,R0  H q H e r,q, (1) where r and q stand for the electronic and vibrational coordinates, respectively,

H e( r,R0 ) is the electronic Hamiltonian for a fixed tetrahedral configuration. This configuration ( R  R0 ) does not take into account the lattice relaxation due to the 2 5 2 2 embedding of Cr ion in the ground state T2 (t2 e ) . To emphasize this the crystal field parameter is denoted by 10Dq0 and the Racah parameter by B0 . Finally, H q is the Hamiltonian of the free lattice vibrations and He is the vibronic interaction. We will employ a quasi-molecular model that considers the impurity center as a complex formed by the central ion and the adjacent ions of the lattice. Denoting the o displacements of the ions of the impurity complex from their positions Rp by o p R p R p Rp ( is the index of the position ) we obtain

1    q   =  U p R p , (2) l p where q   are the symmetry adapted vibrational coordinates corresponding to the    irreps  , U p (  x, y, z) are the elements of the matrix for the transformation of 1/ 2 displacements R p into the dimensionless coordinates q   , lμ   ( μ  / f μ  ) ,

μ  - is the frequency of the vibration  and f μ  is the force constant, symbol  enumerates the repeating vibrational representations. Free lattice vibrations are assumed to be harmonic so that the Hamiltonian H is of the form:

2 2 2 H   2q   q   μ  μ   μ   . (3) μ  

In the case of the Td complex we are dealing with the full symmetric A1, (1) (2) tetragonal E and two trigonal vibrations, T2 and T2 . The operator He becomes:

H e   v   rq   . (4)   

The operator v  (r) (possessing the dimension of energy) can be expressed as: 3

W ri  R p  W ri  R p    l  U    v r  v    r=  p    i , (5) p,i q   p,i R p o i q   0 RpRp

where W (ri  Rp ) is the potential energy of the interaction between the i -th electron p 0 of the chromium ion and the -th atom of the host crystal in the position R p . The original Hamiltonian, can be transformed in order to take into account the full symmetric relaxation. The adiabatic potential of the ground term can be found as: 2 VT A U T2 Q  4Dq  21B  2  1 Q2 (6) 0 0 2 2 A1 V 2 2 V  2  2 with the energy of the minimum being lowered by T A1 , T t e ,   e v e and   t  v t  are the orbital contributions to the overall e A 1 t 2 A1 2 5 vibronic parameter VT of t2 - and e- electrons. For the excited term E one can find the following expression for the adiabatic energy:

2  E VE A1 2 U Q  6Dq0  21B0   Q  Q0  , (7) 2 2 A1 5 3 where VE  3t  e is the vibronic parameter characterizing coupling in Et2e term with full symmetric vibrations. The value Q0 is the shift of the equilibrium position that accompanies the one-electron jump e  t2 corresponding to the 5 2 2 5 3 transition T2 t2 e  Et2e:

VE VT t e Q0     . (8)   A1 A1 Due to this shift the Dq parameter proves to be redefined and can be evaluated as the energy of the Franck-Condon transition providing that the ionic configuration is self-consistent with the ground term (Q=0):

VT 10Dq 10Dq0  VT VE . (9)  A1 This selfconsistent value of the Dq is determined by a new (relaxed) equilibrium configuration..Rp.. for the impurity cluster in the ground state. In the following calculation we will use the interatomic distances for the host lattice. The vibronic 5 Hamiltonian for T2 e  2t2  JT problem is the following: 1 1 1 1 H e (T2 ) E T2 OEu qEu  OE qE T T2 OT  q  OT qT   OT  q  2 2 T2 2 2 2 T2 (10) 2 2 2 2 T T2 OT  q  OT qT   OT  q  2 2 T2 2 2 2 T2

5 For the E  a1  e problems one finds:

H E  E O q  O q  E O Q e   E   Eu Eu Ev Ev  A1   A1 . (11) 4

Table 1. Symmetry adapted vibrational coordinates for a tetrahedral complex

q 1 A1 x1  x2  x3  x4  y1  y2  y3  y4  z1  z2  z3  z4  2 3 q 1 Eu  x  x  x  x  y  y  y  y  2 z  2 z  2 z  2 z  24 1 2 3 4 1 2 3 4 1 2 3 4

qE 1 x1  x2  x3  x4  y1  y2  y3  y4  2 2 q1 1 T2 x1  x2  x3  x4  y1  y2  y3  y4  z1  z2  z3  z4  2 3 q1 1 T2 x1  x2  x3  x4  y1  y2  y3  y4  z1  z2  z3  z4  2 3 q1 1 T2 x1  x2  x3  x4  y1  y2  y3  y4  z1  z2  z3  z4  2 3 q2 1 T2  x1  x2  x3  x4  y1  y2  y3  y4  z1  z2  z3  z4  2 3 q2 1 T2 x1  x2  x3  x4  y1  y2  y3  y4  z1  z2  z3  z4  2 3 q2 1 T2  x1  x2  x3  x4  y1  y2  y3  y4  z1  z2  z3  z4  2 3

1 2 In Eqs. (10) and (11) the vibronic coupling constants  ( )     v Γ  ( is the dimension of the irrep  ) are expressed in terms of the reduced matrix element 2 of the operators v   r calculated with the aid of the wave-functions of Cr ions, symbol  is omitted. The symmetry adopted coordinates q [5] are given in Table

1. The matrices O and O  are given in [7, 8].

3. VIBRONIC INTERACTION IN THE EXCHANGE CHARGE MODEL

The crystal field potential acting on the electronic shell of the Cr 2 ion looks as follows

o m m Vc  Vc ri   W ri  Rp   Bl Cl i ,i  , (12) i p,i i,l ,m

m 1/ 2 where Cl (, )  4 / 2l 1 Ylm ,, Ylm , are normalized spherical m harmonics and Bl are the parameters that depend on the geometry of the ligand surrounding. For the calculation of the vibronic coupling constants we employ the exchange charge model of the crystal field developed in ref. [14]. In this model the o matrix element of the one-electron operator W (r  Rp ) is represented as:

o pc o nlmW r  R p  nlm  nlmW r  R p  nlm e2 2 G nl nl nlm | nlm  o      p  p , (13) Rp  where the first term is the matrix element of the operator of interaction of the valent electron of the impurity with the point charges. The second term comes from the   nlm p overlap of the functions nlm with the functions p p of the ligand in Z Ro G ( nl | nl ) the reference system with p -axis along the ligand position vector p ,  are the phenomenological parameters. This term includes the effects of exchange, covalence and non-orthogonality of the metal and ligand wave-functions. We restrict ourselves to electronic states of external closed shells of the ligands, '' 2 '' 6 3d ,0 ns , e.g. n s ,n p in the sum over    nlm. The overlap integrals p 3d0 np0 3d ,1 np,1 p p are assumed to mainly contribute to the metal-ligand m bond. The crystal field parameters Bl in the exchange model are found as [14,15]:

m m( pc) m(ec) Bl  Bl  Bl , (14) where symbols pc and ec identify the partial contributions from point charges and m( pc) from the exchange charges correspondingly. The component Bl is determined as usually in the point charge crystal field theory:

1/ 2 Z e2 rl m( pc)  4  p m* Bl     Cl p , p , (15)  2l 1 l 1 p Rp 

where Z pe is the effective charge of the ligands and Rp is the absolute value of the

l l position vector Rp , r is the mean value of r calculated with the radial wave-

2 m(ec) functions of Cr ion. The parameter Bl is given by [15]:

2 m( ec ) 2e 2l 1 Sl ( Rp ) m* Bl   Cl p , p , (16) 5 p Rp where the following notation for the overlap integrals is used:

2 2 2 Sl Rp  Gs Ss Rp  Gσ Sσ Rp  Gπ γl Sπ Rp ,  2 1, 4  4 / 3, (17)

The overlap integrals for the 3d wave functions of Cr 2 and n''s, n'' p functions of the ligands are introduced as follows:

Ss Rp  nd ,m  0 ns , S Rp  nd ,m  0 np,m  0 ,

S Rp  nd ,m  1 np,m  1 . (18) 6

The values Gs ,G ,G are the dimensionless parameters. We employ the simplest version of the exchange charge model [14] with the only phenomenological parameter

G  Gs  G  G which can be found from the value of 10 Dq . 2 Operators v   ( r ) for a tetrahedral complex formed by Cr ion and its surrounding are obtained by substitution of the crystal field potential at arbitrary Rp into eq.(5). The final expressions for the vibronic parameters are the following:

2e2l A1 4 4  A (E)   25Z r 18R G(S4 (R)  R S4 (R)), 1 27 R6 2 8 e lE 4 2 2 4 4 E (E)   5Z r  9Z r R 18R G S2 (R) 18R G S4 (R), 63R6 2 4 2 e lE 4 2 2 4 4 E (T2 )  20Z r  27Z r R  54R G S 2(R)  72R G S4(R), 189R6 e2 l (1) (1) T2 2 2 2 T (T2 )  8 r 100 r  81R Z 2 378R6 2 4  9 R G 24 3S2 (R)  2 R S2(R) 79 S4 (R)  64 R S4(R), 4e2l (2) (2) T2 2 2 2 4 T (T2 )   r  50 r  27 R Z 18R G 3S2 (R) 10S4 (R). (19) 2 189 R6

(1) (2) The operators  (T ) and  (T ) are related to two types of T2 vibrations. The T2 2 T2 2 value R is the distance between the impurity ion and ligands in which the adiabatic dS (R) potential U T2 Q has minimum, SR  l , Ze is the effective ligand charge. l dR

4. NUMERICAL ESTIMATES FOR THE PARAMETERS

The combinations Sl (R) of the overlap integrals, their derivatives SlR and the values r l have been computed using the radial atomic “double zeta” 3d wave functions of chromium, 3s, 3p functions of sulfur and 4s,4 p functions of selenium given in ref. [17]. The values R for ZnS, ZnSe,CdS and CdSe crystals were taken from refs. [18], while the effective charge Z of the ligands was put equal to 2. The mean value of f was taken approximately 2105 dyn / cm for all vibrations. The frequency  is taken the same for all vibrations and identified with that for TA phonons [13] which have been found active as low-frequency JT modes in previous studies of Cr 2 and other transition metal ions in II-VI compounds [19]. The parameters Dq were estimated in [12] from the analysis of the experimental data. The parameter G was calculated with the aid of the relation:

2(5Ze 2 r 4 18R 4G S (R)) Dq   4 . (20) 135R5 Table 2. Parameters of the exchange-charge model for II-VI crystals 2+ Doped with Cr ions ( a0 is the Bohr radius) 7

S (R) Dq G R S 2 (R) 4 S2( R ) S4( R )  Crystal -1 ( cm 1 ) (Å) 1 1 (cm ) ( a0 ) ( a0 )

ZnS -480 1.6 2.34 0.0250 0.0138 -0.0296 -0.0103 90

ZnSe -460 2.1 2.45 0.0216 0.0116 -0.0238 -0.0074 70

CdS -500 3.0 2.52 0.0163 0.0102 -0.0212 -0.0102 80

CdSe -500 3.7 2.62 0.0149 0.0091 -0.0180 -0.008 60

The evaluated overlap integrals and their derivatives and the parameters used for calculations of the vibronic coupling constants are collected in Table 2. The calculated vibronic coupling constants for all active modes are given in Table 3. The (2) main contribution to the vibronic coupling constants E (T2 ) , (T2 ),E (E) in most T2 cases comes from the field of point charges. Meanwhile, the exchange charge field  (1) (T )  (E) yields a dominant contribution to the vibronic parameters T2 2 and A1 . The 5 data listed in Table 3 show also that for E term the interaction with the E and A1 (1) modes is approximately the same. The JT interaction with T2 vibrations proves to be 5 dominant within T2 term. Although the interaction of this term with the E  vibrations is smaller it is appreciable. At the same time the interaction with the second vibration of T2 symmetry is negligible.

Table 3. Vibronic coupling constants ( in cm1 ) for II-VI crystals doped with Cr2+ ion

(1) ( 2 ) E T2   T   T   E  E Crystal T2 2 T2 2 A1 E  

ZnS 202 398 38 -140 164

ZnSe 160 319 26 -106 127

CdS 164 387 15 -156 120

CdSe 132 315 7 -122 94

More distinct insight on the role of different JT vibrations provide the JT energies EJT   calculated for each kind of active vibration and the corresponding

Pekar-Huang-Rhys factors (“heat release” parameters) a   EJT    (Table 4). (1) 5 One can see that the JT interaction with the trigonal modes T2 in T2 term is strong a  7  9. At the same time the interaction with E  vibrations can be considered as intermediate a  2.5. On the contrary, the interaction with the E  vibrations in 5 E 8

term is estimated to be weak a 1 as well as the interaction with the full symmetric mode.

1 Table 4. JT energies EJT   (in cm ) and Pekar-Huang-Rhys factors 2+ a  EJT    (in parentheses) for II-VI crystals doped with Cr ion

5 5 Terms T2 E Active T 1 T 2 A mode E 2 2 1 E ZnS 226 (2.51) 586 (6.51) 5 (0.06) 110 (1.22 ) 74 (0.82)

ZnSe 181 (2.59) 485 (6.93) 3.5 (0.05) 80 (1.14) 57 (0.81)

CdS 168 (2.10) 622 (7.78) 0.8 (0.01) 152 (1.90) 45 (0.56)

CdSe 145 (2.42) 556 (9.19) 0.24 (0.004) 124 (2.06) 37 (0.61)

The results obtained do not confirm the assumption that the interaction of the ground state or the interaction of both the ground and excited states with the tetragonal vibrations is dominant. The calculations show that the trigonal vibrations 5 for the T2 state cannot be neglected and also play a significant role in the formation of the optical bands of Cr 2 doped II-VI compounds in the infrared range. In general, 5 5 in the evaluation of the shape function for the E T2 transitions we face a two- 5 mode JT problem e  a1  E for the E state and a five-mode Jahn-Teller problem 5 e  t2 T2 for the T2 state.

5. DYNAMICAL JAHN-TELLER PROBLEM. EVALUATION OF THE SHAPE-FUNCTION FOF THE VIBRONIC BAND

In this section we will calculate the luminescence band arising on the transition 5 5 2+ E T2 of the Cr ion in the CdSe crystal [20]. For the first step we will take into account the coupling of the E-state with the tetragonal E-mode, while, basing on the calculations of the vibronic constants above performed, for the T2 –term we will include into consideration the interaction of this term with the tetragonal E-mode and (1) the trigonal T2 -mode. The hybrid vibronic states corresponding to the dynamical

pseudo Jahn-Teller problems for the e  E and e  t2 T2 cases can be expressed as the expansion over the unperturbed electronic and vibrational states:

 (k )   (q ) (q ) (q ) C (k )  (q ) (q )  n ,n ,n n  n  n    ,nu ,n nu u nv v  u,v; nu ,n  (k)  C (k)  (q ) (q ) (q ) (q ) (q )     ,n ,n ,n ,nu ,n n  n  n  nu u nv v   ,, ; , (21) n ,n ,n nu ,n here  , ( =u,v) and   are the electronic wave functions of the excited 5E and 5  (q )  (q )  (q )  (q )  (q ) ground T2 -terms, respectively, n  , n  , n  , nu u , nv v denote the harmonic oscillator wave functions. The symbols k and k enumerate the 9

5 5 vibronic levels and the vibronic wave functions corresponding to the terms E and T2. The energies of the vibronic levels arising from the e  E problem can be presented as E k  10Dq  (n  n  n  3/ 2)   k n ,n ,n    , (22) where the values  k will be determined by diagonalization of the vibronic Hamiltonian for the E-term. By  k we denote the energies of vibronic levels arising from the e  t2 T2 problem. For numerical calculations of the eigenvalues and eigenvectors of the dynamic vibronic problem e  E the number N of the oscillation states satisfied the condition

(nu  nv  N) , where in the calculation was taken N=12. For the e  t2 T2 problem the same type inequality (n  n  n  nu  nv  12) was hold. The general dimension of the vibronic matrix for the E-term was 182182. For the T2 -term the dimension of the matrix was 1856418564. The diagonalization was performed with the use of the Lanczos method [21-23]. The shape of the luminescence band was calculated with the aid of the formula  1 (k)  (k) 2 k k,k F()   ud  exp En ,n ,n / kBT  f ()     n ,n ,n z k,k , (23) n ,n ,n z  Exp[E k / k T]  here  n ,n ,n B is the partition function, u is the light polarization k,n ,n ,n    vector and d - dipole moment vector whose component transform in the Td point 2 group according to the irreducible representation T1. The symbol  means the f k,k () averaging over the light polarization, n ,n ,n is the shape-function of the individual line related to the transition between hybrid vibronic states k and k . The individual lines are assumed to be of Gaussian shape:

 (E k  k  )2  k,k 1 n ,n ,n n ,n ,n ()  exp   . (24)    22 2   The phonon dispersion in the crystal results in the structureless broad band. In order to smooth quantum discrete structure of the calculated band the second central moment of the individual lines  should be comparable with the  value. We put here     60cm1 (see Table 2). For numerical calculation of the luminescence band at T=4K the transitions between the two vibronic levels arising from the E-state and 9000 vibronic levels arising from the T2 –state were taken into account so that the full number of the transitions that are taken into account was 18000. The luminescence band for the CdSe:Cr2+ crystal at T=4K is shown in Fig.1. A quite a good agreement with the experimental data [20] is observed for the vibronic coupling constants 1 1  (1) T  180cm1 1 E E  120 cm , E T2  120cm , T2 2 and Dq  422cm . The values of the vibronic parameters obtained from fitting differ from those listed in Table 3 for CdSe. The reason for this constants are the following. While calculating the vibronic coupling constants given in Table 3 we used for the determination of the 10 dimensionless parameter G the value Dq  500cm1 . Insofar as the fitting of the luminescence curve was obtained for Dq  422cm1 the G value changes and

1.0 CdSe: Cr2+, T=4K

0.8 . u . a

0.6 , y t i c n e t

n 0.4 I

3500 3750 4000 4250 4500

-1 , cm

Fig.1 Comparison of the calculated emission curve (solid line) with the experimental data [20] (circles) the for the CdSe: Cr2+ crystal at T=4K 1 1 for the best fit parameters: E E  120 cm , E T2  120cm ,  (1) T   180cm1 1 1 T2 2 , 10Dq  4220cm ,     60cm become equal to 2.967. Correspondingly, the vibronic coupling constants calculated with the aid of the methods developed in previous sections take on the values: 1 1  (1) T   272cm1 E E  90 cm , E T2  120cm , T2 2 . It is interesting to note that the calculated constant E E is little smaller than the obtained from the fit, while the  (1) T   272cm1 calculated constant T2 2 is higher than the obtained from simulation. At the same time the E T2  constant holds its value. In summary, it should be mentioned that developed model reproduces well the observed luminescence band in CdSe:Cr2+ crystal and , in particular, the pronounced shoulder at the left side of the band that undoubtedly has JT origin.

ACKNOWLEDGMENT Financial support from USA-Israel Binational Science Foundation (Project No 2002409) is highly appreciated. B.S. Ts. thanks the Council for Higher Education of Israel for the financial support. S. K., O. R., A. P. and S. O. thank the Supreme Council on Science and Technological Development of Moldova for the financial support (grant No. 4-013P). References 1. M. Mond, D. Albrecht, H.M. Kretschmann, E. Heumann, G. Huber, S. Kuck, V.I.Levchenko, V.N.Yakimovich, V.G.Shcherbitsky, V.E.Kisel, N.V.Kuleshov, 11

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