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
b
Chemistry Department, Ben-Gurion University of the Negev,
Beer-Sheva 84105,Israel
*
E-mail address: tsuker@bgumail.bgu.ac.il
In this article we report evaluation of the vibronic Jahn-Teller (JT) coupling
parameters and the vibronic optical bands related to the 5 E 5T2 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
advantage from the 5 E 5T2 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
the infrared range [10] were based on the model of static JT effect for the ground 5T2
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
of optical band shape arising from the 5 E 5T2 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
2. HAMILTONIAN FOR THE IMPURITY CENTER
The ground 5 D term of a free Cr 2  ion is split by the tetrahedral crystal field in
a fairly well known zinc-blende lattice into the orbital triplet 5T2 (t22e 2 ) and orbital
doublet 5 E (t23e) , the former being the ground term. The standard cubic basis sets for
the one-electron d-functions T2 ( , , )   yz ,  xz,   xy and E (u, )


 
 
( u  3 z 2  r 2 ,   3 x 2  y 2 ) are used. The levels  5T2 and  5 E
separated by the gap 10Dq0 with Dq 0 being the cubic crystal field parameter.
are
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
embedding of Cr 2  ion in the ground state 5T2 (t22e 2 ) . 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 H e 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 R op by
R p  R p  Rop ( p is the index of the position ) we obtain
q   =
1
l
 U
p

R p ,
(2)
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
displacements R p into the dimensionless coordinates q   , l μ   (  μ  / f μ  )1/ 2 ,
 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:
H 
  μ  2q 2
μ 
μ

  2 q 2μ   .
(3)
In the case of the Td complex we are dealing with the full symmetric A1,
tetragonal E and two trigonal vibrations, T2(1) and T2( 2) . The operator H e becomes:
H e   v   r  q   .
 
The operator
v   (r ) (possessing the dimension of energy) can be expressed as:
(4)
3
v    r  = 

W ri  R p
q   
p ,i

 l  
q   0
p ,i

W ri  R p

R p
U p    v   ri  ,
R p  Rop
(5)
i
where W (ri  R p ) is the potential energy of the interaction between the i -th electron
of the chromium ion and the p -th atom of the host crystal in the position R 0p .
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:
VT22
 A1 2
T2
(6)
U Q   4 Dq 0  21B0 

Q
2 A1
2
with the energy of the minimum being lowered by  VT2 2  A1 , VT  2t  2e ,
e  e  vA e 
1
and t  t 2  v A1 t 2  are the orbital contributions to the overall
vibronic parameter VT of t 2 - and e- electrons. For the excited term 5 E one can find
the following expression for the adiabatic energy:
U E Q   6 Dq 0  21B0 
 A1
VE2
Q  Q0 2 ,

2 A1
2
(7)
 
where VE  3t  e is the vibronic parameter characterizing coupling in 5 E t 23e
term with full symmetric vibrations. The value Q0 is the shift of the equilibrium
position that accompanies the one-electron jump e  t 2 corresponding to the
 
 
transition 5T2 t 22 e 2  5 E t 23 e :
Q0  
  e
VE  VT
.
 t
 A1
 A1
(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):
V
(9)
10 Dq  10 Dq 0  T VT  VE  .
 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
Hamiltonian for 5T2  e  2t 2  JT problem is the following:

H e ( T2 )   E T2 OEu q Eu  OE q E   T12 T2  OT2 qT1  OT2 qT12  OT2 qT1

 T22  T2  OT2 qT2  OT2 qT22  OT2 qT2 
2
2

2
2

(10)
For the 5 E  a1  e  problems one finds:
 q Eu  O E v q Ev    A1 E O A1 Q .
H e E    E E O Eu
(11)
4
Table 1. Symmetry adapted vibrational coordinates for a tetrahedral complex
1
q A1
2 3
q Eu
x1  x2  x3  x4  y1  y2  y3  y4  z1  z 2  z3  z 4 
1
 x1  x2  x3  x4  y1  y2  y3  y4  2 z1  2 z2  2 z3  2 z4 
24
1
q E
2 2
q T1
2
q T1
2
q T1
2
q T2 
2
q T2
2
q T2 
2
1
2 3
1
2 3
1
2 3
1
2 3
2 3
2 3
x1  x2  x3  x4  y1  y2  y3  y4  z1  z 2  z3  z 4 
x1  x2  x3  x4  y1  y2  y3  y4  z1  z 2  z3  z 4 
x1  x2  x3  x4  y1  y2  y3  y4  z1  z 2  z3  z 4 
 x1  x2  x3  x4  y1  y2  y3  y4  z1  z 2  z3  z 4 
1
1
x1  x2  x3  x4  y1  y2  y3  y4 
x1  x2  x3  x4  y1  y2  y3  y4  z1  z 2  z3  z 4 
 x1  x2  x3  x4  y1  y2  y3  y4  z1  z 2  z3  z 4 
In Eqs. (10) and (11) the vibronic coupling constants   ( )   1 2  v Γ 
(   is the dimension of the irrep  ) are expressed in terms of the reduced matrix
element of the operators v   r  calculated with the aid of the wave-functions of
Cr 2 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


Vc   Vc ri   W ri  R op   BlmClm i ,i  ,
i
where
p ,i
(12)
i ,l ,m
Clm (  , )  4 / 2l  11 / 2 Ylm  , , Ylm  , 
are
normalized
spherical
harmonics and Blm are the parameters that depend on the geometry of the ligand
5
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 W (r  Rop ) is represented as:
nlm W r  R op  nlm  nl m W pc r  R op  nlm
2
e2
R po
 G nl nl  nlm |
p
 p nl m ,
(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 nlm with the functions  p  nl m p of the ligand p in
the reference system with Z p -axis along the ligand position vector R op , G ( nl | nl  )
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. n'' s 2 , n'' p 6 in the sum over    nl m. The overlap integrals 3d ,0 ns p ,
3d 0 np 0
p
3d ,1 np ,1
p
are assumed to mainly contribute to the metal-ligand
bond. The crystal field parameters Blm in the exchange model are found as [14,15]:
Blm  Blm ( pc)  Blm ( ec ) ,
(14)
where symbols pc and ec identify the partial contributions from point charges and
from the exchange charges correspondingly. The component Blm ( pc) is determined as
usually in the point charge crystal field theory:
 4 
Blm ( pc)  

 2l  1 
1/ 2

p
Z pe2 r l
R p 
l 1


Clm*  p ,  p ,
(15)
where Z p e is the effective charge of the ligands and Rp is the absolute value of the
position vector R p , r l
is the mean value of r l calculated with the radial wave-
functions of Cr 2  ion. The parameter Blm (ec ) is given by [15]:
Blm( ec ) 
2e 2 2l  1 Sl ( R p ) m*
Cl  p , p ,

5
Rp
p


(16)
where the following notation for the overlap integrals is used:
 
 
 
 
Sl R p  Gs Ss2 R p  Gσ Sσ2 R p  Gπ γl Sπ2 R p ,  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:
 
 
S s R p  nd ,m  0 ns , S R p  nd ,m  0 np ,m  0 ,
6
 
S R p  nd ,m  1 np ,m  1 .
(18)
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 .
Operators v   ( r ) for a tetrahedral complex formed by Cr 2  ion and its
surrounding are obtained by substitution of the crystal field potential at arbitrary R p
into eq.(5). The final expressions for the vibronic parameters are the following:
 A1 ( E ) 
27 R 6
E ( E )  
 E (T2 ) 
T(1) (T2 ) 
2
 25 Z
2 e 2 l A1

r 4  18 R 4G( S 4 ( R)  R S 4 ( R)) ,

8 e 2l E
5 Z r 4  9 Z r 2 R 2  18 R 4G S 2 ( R )  18 R 4 G S 4 ( R ) ,
6
63 R
4 2 e 2l E
6
189 R
e 2 lT(12 )
20 Z r
4

 27 Z r 2 R 2  54 R 4G S 2( R)  72 R 4G S 4 ( R) ,
8 r 100
2
378 R 6 2

r 2  81 R 2 Z
 9 R 4 G 24 3 S2 ( R)  2 R S2 ( R)   79 S4 ( R)  64 R S4 ( R)  ,
 (T2 ) 
( 2)
T2
4e 2lT(22)
189 R
6
 r  50 r
2
2


 27 R 2 Z  18 R 4 G 3S 2 ( R)  10 S 4 ( R)  .
(19)
The operators T(12) (T2 ) and T( 22 ) (T2 ) are related to two types of T2 vibrations. The
value R is the distance between the impurity ion and ligands in which the adiabatic
dS ( R )
potential U T2 Q  has minimum, S l R   l
, Ze is the effective ligand charge.
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, 3 p functions of sulfur and 4 s,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 10 5 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:
Dq  
2 (5 Ze 2 r 4  18 R 4 G S 4 ( R))
135 R 5
.
(20)
7
Table 2. Parameters of the exchange-charge model for II-VI crystals
Doped with Cr2+ ions ( a0 is the Bohr radius)
Dq
Crystal
G
R
(Å)
1
( cm )
S 2 ( R)
S 4 ( R)

S2 ( R )
1
( a0 )
S 4 ( R )
1
( a0 )
(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  E (T2 ) , T( 2) (T2 ),  E ( E ) in most
2
cases comes from the field of point charges. Meanwhile, the exchange charge field
yields a dominant contribution to the vibronic parameters T(12 ) (T 2) and  A1 ( E ) . The
data listed in Table 3 show also that for 5 E term the interaction with the E and A1
modes is approximately the same. The JT interaction with T2(1) vibrations proves to be
dominant within 5T2 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
Crystal
 E T2 
T( 21 ) T2 
T( 22 ) T2 
 A1 E 
 E 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 E JT   calculated for each kind of active vibration and the corresponding
Pekar-Huang-Rhys factors (“heat release” parameters) a   EJT    (Table 4).
One can see that the JT interaction with the trigonal modes T2(1) in 5T2 term is strong
8
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
term is estimated to be weak a  1 as well as the interaction with the full symmetric
mode.
Table 4. JT energies E JT   (in cm 1 ) and Pekar-Huang-Rhys factors
a   E JT    (in parentheses) for II-VI crystals doped with Cr2+ ion
5
Terms
5
T2
E
Active
mode
E
T21
T22 
A1
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
for the 5T2 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,
in the evaluation of the shape function for the 5 E 5T2 transitions we face a twomode JT problem e  a1   E for the 5 E state and a five-mode Jahn-Teller problem
e  t2  T2 for the 5T2 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
E T2 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 T2(1) -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:
5
5
 C

 n(k ), n , n   n (q ) n (q ) n (q )
 ( k ) 
C


  , , ;
n , n , n
nu , n
( k )
, n , n , n , nu , n
u , v;
nu , n
(k )
, nu , n
 nu (qu ) nv (qv ) 
 n (q ) n (q ) n (q ) nu (qu ) nv (qv )   ,
(21)
9
here  , (  =u,v) and   are the electronic wave functions of the excited 5E and
ground 5T2 -terms, respectively, n (q ) , n (q ) , n (q ) ,  nu ( qu ) ,  nv ( qv )
denote the harmonic oscillator wave functions. The symbols k and k enumerate the
vibronic levels and the vibronic wave functions corresponding to the terms 5E and 5T2.
The energies of the vibronic levels arising from the e  E problem can be presented
as
(22)
Enk , n , n  10 Dq   (n  n  n  3 / 2)   k ,
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
F ( ) 
here z 
1
z

k ,k 
n , n , n



 ( k ) ud  ( k ) 2exp  Enk , n , n / k BT f nk ,,kn , n () ,
 Exp[ E
k
n , n , n
(23)

/ k BT ] 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
group according to the irreducible representation T1. The symbol 
averaging over the light polarization,
f nk ,,kn , n
2
means the
() 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:
nk, k, n , n
() 
 ( Enk , n , n   k   ) 2 
1



.
exp 
22
2


(24)
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     60 cm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
E E   120 cm1 ,  E T2  120 cm1 , T(1) T2   180 cm1 and Dq  422 cm . The values
2
10
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
dimensionless parameter G the value Dq  500 cm1 . Insofar as the fitting of the
luminescence curve was obtained for Dq  422 cm1 the G value changes and
1.0
2+
CdSe: Cr , T=4K
Intencity, a.u.
0.8
0.6
0.4
0.2
3500
3750
4000
, cm
4250
4500
-1
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
for the best fit parameters: E E   120 cm1 ,  E T2  120 cm1 ,
T(12) T2   180 cm1 , 10 Dq  4220 cm1 ,     60 cm1
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:
E E   90 cm1 , E T2  120 cm1 , T(12 ) T2   272 cm1 . It is interesting to note that
the calculated constant  E E  is little smaller than the obtained from the fit, while the
calculated constant T(12 ) T2   272 cm1 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
2002409) is highly appreciated. B.S. Ts. thanks the Council for Higher
Israel for the financial support. S. K., O. R., A. P. and S. O. thank
Council on Science and Technological Development of Moldova for
support (grant No. 4-013P).
(Project No
Education of
the Supreme
the financial
11
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