Optical Jahn-Teller Effect in Ii-Vi Compounds

1

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:

In this article we report evaluation of the vibronic Jahn-Teller (JT) coupling parameters and the vibronic optical bands related to the transition in a series of II-VI crystals doped with 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 ions have got growing attention as the promising materials for the infrared solid state lasers. The laser operation takes advantage from the 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 ion in chalcogenide host materials like [1], [2], [3], [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 the infrared range [10] were based on the model of static JT effect for the ground 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 ion and calculation of optical band shape arising from the 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

The ground term of a free ion is split by the tetrahedral crystal field in a fairly well known zinc-blende lattice into the orbital triplet and orbital doublet , the former being the ground term. The standard cubic basis sets for the one-electron d-functions and () are used. The levels and are separated by the gap with being the cubic crystal field parameter.

The total Hamiltonian for the ion in crystal can be represented as

, (1)

where and stand for the electronic and vibrational coordinates, respectively, is the electronic Hamiltonian for a fixed tetrahedral configuration. This configuration () does not take into account the lattice relaxation due to the embedding of ion in the ground state . To emphasize this the crystal field parameter is denoted by and the Racah parameter by . Finally, is the Hamiltonian of the free lattice vibrations and 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 displacements of the ions of the impurity complex from their positions by ( is the index of the position ) we obtain

=, (2)

where are the symmetry adapted vibrational coordinates corresponding to the irreps , ( are the elements of the matrix for the transformation of displacements into the dimensionless coordinates , , - is the frequency of the vibration and is the force constant, symbol enumerates the repeating vibrational representations. Free lattice vibrations are assumed to be harmonic so that the Hamiltonian is of the form:

. (3)

In the case of the Td complex we are dealing with the full symmetric tetragonal and two trigonal vibrations, and . The operator becomes:

. (4)

The operator (possessing the dimension of energy) can be expressed as:

=, (5)

where is the potential energy of the interaction between the -th electron of the chromium ion and the -th atom of the host crystal in the position . 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:

(6)

with the energy of the minimum being lowered by , , and are the orbital contributions to the overall vibronic parameter of - and e- electrons. For the excited term one can find the following expression for the adiabatic energy:

, (7)

where is the vibronic parameter characterizing coupling in term with full symmetric vibrations. The value is the shift of the equilibrium position that accompanies the one-electron jump corresponding to the transition :

. (8)

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):

. (9)

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 Hamiltonian for JT problem is the following:

(10)

For the problems one finds:

. (11)

Table 1. Symmetry adapted vibrational coordinates for a tetrahedral complex

In Eqs. (10) and (11) the vibronic coupling constants (is the dimension of the irrep) are expressed in terms of the reduced matrix element of the operators calculated with the aid of the wave-functions of ions, symbol is omitted. The symmetry adopted coordinates [5] are given in Table 1. The matrices and are given in [7, 8].

3.  VIBRONIC INTERACTION IN THE EXCHANGE CHARGE MODEL

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

, (12)

where are normalized spherical harmonics and 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 matrix element of the one-electron operator is represented as:

, (13)

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 overlap of the functions with the functions of the ligand in the reference system with -axis along the ligand position vector , 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, e.g. in the sum over . The overlap integrals are assumed to mainly contribute to the metal-ligand bond. The crystal field parameters in the exchange model are found as [14,15]:

, (14)

where symbols pc and ec identify the partial contributions from point charges and from the exchange charges correspondingly. The component is determined as usually in the point charge crystal field theory:

, (15)

where is the effective charge of the ligands and is the absolute value of the position vector , is the mean value of calculated with the radial wave-functions of ion. The parameter is given by [15]:

, (16)

where the following notation for the overlap integrals is used:

, (17)

The overlap integrals for the 3d wave functions of and functions of the ligands are introduced as follows:

,

. (18)

The values are the dimensionless parameters. We employ the simplest version of the exchange charge model [14] with the only phenomenological parameter which can be found from the value of .

Operators for a tetrahedral complex formed by ion and its surrounding are obtained by substitution of the crystal field potential at arbitrary into eq.(5). The final expressions for the vibronic parameters are the following:

,

,

,

,

. (19)

The operators and are related to two types of T2 vibrations. The value is the distance between the impurity ion and ligands in which the adiabatic potential has minimum, , Ze is the effective ligand charge.

4.  NUMERICAL ESTIMATES FOR THE PARAMETERS

The combinations of the overlap integrals, their derivatives and the values have been computed using the radial atomic “double zeta” 3d wave functions of chromium, functions of sulfur and functions of selenium given in ref. [17]. The values for and crystals were taken from refs. [18], while the effective charge of the ligands was put equal to 2. The mean value of was taken approximately 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 and other transition metal ions in II-VI compounds [19]. The parameters were estimated in [12] from the analysis of the experimental data. The parameter was calculated with the aid of the relation:

. (20)

Table 2. Parameters of the exchange-charge model for II-VI crystals

Doped with Cr2+ ions ( is the Bohr radius)

Crystal /
() / /
(Å) / / /
( ) /
( ) /
(cm-1)
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 main contribution to the vibronic coupling constants in most cases comes from the field of point charges. Meanwhile, the exchange charge field yields a dominant contribution to the vibronic parameters and . The data listed in Table 3 show also that for term the interaction with the and modes is approximately the same. The JT interaction with vibrations proves to be dominant within term. Although the interaction of this term with the vibrations is smaller it is appreciable. At the same time the interaction with the second vibration of symmetry is negligible.

Table 3. Vibronic coupling constants ( in ) for

II-VI crystals doped with Cr2+ ion

Crystal / / / / /
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 calculated for each kind of active vibration and the corresponding Pekar-Huang-Rhys factors (“heat release” parameters) (Table 4). One can see that the JT interaction with the trigonal modes in term is strong . At the same time the interaction with vibrations can be considered as intermediate . On the contrary, the interaction with the vibrations in term is estimated to be weak as well as the interaction with the full symmetric mode.

Table 4. JT energies (in ) and Pekar-Huang-Rhys factors (in parentheses) for II-VI crystals doped with Cr2+ ion

Terms / /
Active mode / / / / /
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 for the state cannot be neglected and also play a significant role in the formation of the optical bands of doped II-VI compounds in the infrared range. In general, in the evaluation of the shape function for the transitions we face a two-mode JT problem for the state and a five-mode Jahn-Teller problem for the 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 of the Cr2+ 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 the trigonal -mode. The hybrid vibronic states corresponding to the dynamical pseudo Jahn-Teller problems for the and cases can be expressed as the expansion over the unperturbed electronic and vibrational states: