fulltext

An analysis of the T 7 (e 1e) two-mode Jahn-Teller system

with strong coupling

K. MEKNASSIand S. SAYOURI

Laboratoire de Physique des Solides, Département de Physique, Faculté des Sciences B.P. 1796 Fès, Atlas, Morocco

(ricevuto l’1 Luglio 1996; revisionato il 25 Gennaio 1997; approvato il 3 Febbraio 1997)

Summary. — Details of an approach for strongly coupled orbital triplet Jahn-Teller (JT) systems using a unitary transformation and energy minimization procedure are applied to the two-mode T 7 (e 1e) JT system in the case when both linear and quadratic coupling are included. The inclusion of quadratic JT coupling leads to a partial lift of degeneracy of the triplet ground states and the excited ones. The effect of static strains in strongly coupled T 7 (e 1e) JT system is also investigated analytically. We have shown that tetragonal trigonal and orthorhombic strains modify the structure of the system.

PACS 71.70 – Level splitting and interactions.

1. – Introduction

The Jahn-Teller (JT) effect in a solid describes the coupling of an atom or ion to its surroundings. In solids, this interaction is with lattice vibrations. Such interaction may have importance when determining the energy levels of impurities with orbitally degenerate states as observed, for example, in systems of deep level impurities in semiconductors.

Theoretical investigations taking account of vibronic effects lead to the prediction and the interpretation of numerous effects observed experimentally. The calculation techniques have been considerably generalized and applied to describe the experimental spectra of impurity centers in many systems [1-14]. Vibronic effects also play a role in other phenomena as phase transitions and ferroelectricity [4, 15, 16].

Systems having triplet ground states and strongly coupled to their surroundings are known as T 7 e , T 7 t and T 7 (e 1t) JT problems. Whereas the first problem is solvable analytically, the two others need approximations to be solved [17, 18].

In real crystal the orbital degeneracy state is coupled to many modes of each symmetry, which are called multimode (JT) systems

(

for example: T 7 (e 1e1R t1 797

t 1R)

)

. In general, the exact study of multimode (JT) systems including both linear and quadratic coupling is not easy due to the difficulty of finding the fundamental energy minima. For the tetragonal multimode (JT) systems T 7 (e 1e1R) the situation is less difficult and the problem can be solved exactly because of the diagonal form of the transformed Hamiltonian.

In this paper, we have investigated the exact solution for the two-mode T 7 (e 1e) (JT) system including linear coupling and quadratic one (the latter has an influence on the line shape of the optical transition [19]), using a unitary transformation and minimization procedure [20]. We have also calculated a first-order perturbation correction to the energy of the ground, first and second states.

We have extended our investigations to real systems, which differ from the ideal case of ideal systems, by considering random strains (the term strain is used to include all random internal strains and electric field due to vacancies, dislocation and other impurities. This may include trigonal or tetragonal strains due to nearest-neighbor vacancies, which are often modeled as separate crystal-field-type terms in effective Hamiltonians), generated by inherent defects as vacancies or impurities, added as a perturbation.

Transient spectroscopy of ultrashort pulses offers qualitatively new possibilities in this field from multimode aspects of vibronic interaction in optical spectra of impurity crystals.

2. – Background theory

In standard analytical models for triplet JT systems, a Hamiltonian H is written down to describe the vibrational and interaction energies of the JT center, which is a function of phonon coordinates Qiand moment Pi.

In these semiclassical theories, diagonalization of H, adopting the adiabatic approx-imation (in which the Piterms are neglected), leads to eigenstates of energy E=E(Qi).

Minimization of E with respect to the Pi(treated as dynamic variables) produces many

different solutions defining energy wells or saddle points [6, 18, 21-23].

Orbital states, associated with each well, are multiplied by simple harmonic-oscillator function centered on the well to obtain vibronic JT states.

Recently, a quantum-mechanical treatment, using a unitary transformation U which is a function of free parameters ai, permitted to separate the transformed Hamiltonian

into two parts HA1and HA2[17, 20]. HA1depends on electronic operators only whereas HA2

involves coupling to excited phonon states. Hence, in the limit of very strong coupling only HA1has to be considered.

Diagonalization of HA1followed by a minimization relatively to aiof the total energy

E=E(ai) generates wells, which correspond directly to those produced by the

standard methods. However, the states associated with them are automatically vibronic, whereas in the previous theories oscillations were added somewhat arbitrarily. In this paper, exact solution for strongly T 7 (e 1e) two-mode JT system (in which the two modes considered are the e1-type mode Qu1 and Qe1and the e2-type mode Qu2

and Qe2of Tdsymmetry) will be given including both linear and quadratic coupling, and

3. – The Hamiltonian

The basis Hamiltonian for the T 7 (e 1e) two-mode JT systems including linear and quadratic coupling can be written in the form

H 4Hvib1 Hint1 HQ, (1) where

.

`

/

`

´

Hvib4

!

i 4u e

!

n 41 2

g

mv2n 2 Q 2 in1 1 2 mPin 2

h

_, Hint4

!

n 41 2 Vn(QunTu1 QenTe) , HQ4

!

n 41 2 2 Wn[ (Qen22 Qun2) Tu1 2 QunQenTe] , (2)

Vn and Wn are the linear and the quadratic ion-lattice couplings, respectively, Ti

(i4u, e) are the components of the orbital T (L41) operators defined in second-quantized form [24] as

.

`

/

`

´

Tu4 1 2(C † 1C11 C2†C22 2 C3†C3) , Te4 k3 2 (C † 1C12 C2†C2) . (3)

The Qinand Pinare given by

Qin4 2

o

ˇ 2 mvn (bin1 bin†); Pin4 j

o

ˇ_mvn 2 (bin2 b † in) , (4) b†

inand binare the phonon creation and annihilation operators, respectively.

We write the states, as an orbital part Xo, followed by phonon contribution Xp, in

the form NXo; Xpb. Orbital creation operators C1†, C2†and C3†are defined in terms of the

orbital vacuum state N0ob such that

.

/

´

C1†N0o, Xpb 4Nx; Xpb , C2†N0o, Xpb 4Ny; Xpb , C† 3N0o, Xpb 4Nz; Xpb , (5)

where x denotes the orbital state of x symmetry etc.

Before using the unitary transformation, we have made a rotation in the space of the normal coordinates Qi1and Qi2to pass to new coordinates qi1and qi2given by

qi14 1 V(w1v1Qi11 w2v2Qi2); qi24 1 V(2w2v1Qi11 w1v2Qi2) , (6)

and the corresponding conjugate momenta pinare pi14 1 V

g

w1 v1 Pi11 w2 v2 Pi2

h

; pi24 1 V

g

w1 v2 Pi22 w2 v1 Pi1

h

, (7)

where wnare the effective constants of vibronic coupling

wn4

Vn

vn

and _{V 4}

k

w211 w22.

(8)

In terms of qinand pin the Hamiltonian can take the following form:

.

`

/

`

´

Hvib4

!

i

k

1 2 m

(

V 2 1p2i11 V22p2i21 2 ]2(pi1pi2) 1 m 2(q 2 i11 qi22)

l

, Hint4 V(qu1Tu1 qe1Te) , HQ4

!

n 41 2 Wn[ (qen22 qun2 ) Tu1 2 qunqenTe] 2 2W3[ (qe1qe22 qu1qu2) Tu1 (qu1qe21 qu2qe1) Te] , (9) where V2 14 w21v211 w22v22 V2 , V 2 24 w21v221 w22v21 V2 , ] 2 4 w1w2 V2 (v 2 22 v21) (10)

and the Wnwill be written as a function of the constants Kn, so

.

`

/

`

´

b14 W1 m 4 8

g

t1j1 1 1 1g2 1 t2j2 g2 1 1g2

h

, b24 W2 m 4 8 l

g

t1j2 l2 1 1g2 1 t2j1 g2 ( 1 1g2)

h

, b34 W3 m 4 16

g

t1j1 lg 1 1g2 2 t2j2 g l( 1 1g2₎

h

, (11) where tn4 Wn V2 n , jn4 Kn2 ˇ_vn , g 4 V2 V1 and _{l 4} v2 v1 , (12)

Knare the constant defined [20] as Kn4 2 (Vnkˇ) O(2k2 mvn).

The displacements qin and momenta pin are written in terms of phonon creation

and annihilation operators as

qin4 2

o

ˇ_Vn 2 m (bin1 bin †_{); p} in4 j

o

ˇ_m 2 Vn (bin2 bin†) , (13)

Hamiltonian H in terms of phonon creation and annihilation operators is given in the appendix A.

The advantage of this rotation in the space of the normal coordination is to reduce the number of linear interactions terms to one, and enables us obtaining manageable expressions.

4. – Unitary transformation

When Hintis supposed to be strong, it is impossible to separate the electronic motion

from the nuclear vibration. The potential function then contains minima that deepen as the magnitude of constant coupling increases. At low temperatures and in the absence of perturbation such as spin-orbit coupling, the system will be frozen into one of the lowest energy wells in the potential energy surface, so we can perform the unitary transformation U 4exp

k

j

!

i , n Rinpin

l

(14) with Ri14 1 V(V1ai11 V2ai2); Ri24 1 V(w1v2ai22 w2v1ai1) , (15)

where ain are the free parameters corresponding to Qin coordinates. The

transforma-tion displaces the origin of each of the effective displacements qin by 2ˇRin[2].

Transformed Hamiltonian HA_{4 U}21_{HU may be written as H}A

4 HA11 HA2, where

.

`

`

`

/

`

`

`

´

HA14 mˇ2 2 (R 2 u11 Ru221 Re121 Re22) I 2ˇV(Ru1Tu1 Re1Te) 1 12

!

_nbnˇ2

(

(Ren22 Run2 ) Tu1 2 RunRenTe

)

1 1 2 i , n

!

ˇ_Vn1 12 b3ˇ2

(

(Re1Re22 Ru1Ru2) Tu1 2(Ru1Re21 Ru2Re1) Te

)

, HA24 Hint1 Hvib2 ˇ

!

i , n

g

mRinqin1 1 2Vn

h

1

!

nWn[L(qun, qun, qen, qen) 1 1L(qun, 2ˇRun, qen, 2ˇRen) 1L(2ˇRun, qun, 2ˇRen, qen) ] 2 2W3[L(qu1, qu2, qe1, qe2)1L(qu1, 2ˇRu2, qe1, 2ˇRe2)1L(2ˇRu1, qu2, 2ˇRe1, qe2) ] , (16) where L(r1, r2, r3, r4) is defined by L(r1, r2, r3, r4) 4 (r3r42 r1r2) Tu1 (r2r31 r1r4) Te, (17)

HA1 involves only electronic operators and so can be diagonalized with the Rin as free

parameters.

The approach has the advantage that, by back-transforming the orbital states localized in the wells, new states are obtained which are vibronic in nature. Vibronic behavior need not to be introduced artificially by multiplying the orbital states by simple harmonic-oscillator functions. It also allows the overlap between any two of the

TABLEI.

Number j Ru1 Ru2 Re1 Re2 Eigenstate Energy

1 2 3 2x x/2 x/2 2j j/2 j/2 0 (k3 x)/2 2(k3 x)/2 0 (k3 j)/2 2(k3 j)/2 Nz ; 0 b Ny ; 0 b Nx ; 0 b ETot JT 4 2(ˇV/2 ) x

untransformed states to be calculated in terms of the fundamental constants. The calculation is not restricted to the adiabatic limit.

The values of Rin for each well j which minimize the energy, and the transformed

ground states localized in them, appear summarized in table I, where

.

`

/

`

´

x 4 V ˇ_m 1 14b2 ( 1 14b1)( 1 14b2) 24b23 , j 4 2 V ˇ_m b3 ( 1 14b1)( 1 14b2) 24b23 . (18)

The two-mode JT energy for ETot

JT are given, including linear and quadratic coupling,

as EJTTot4 24

g

K12 ˇ_v1 1 K22 ˇ_v2

h

1 14b2 ( 1 14b1)( 1 14b2) 24b23 . (19) To have ETot

JT lower than those of each mode EJTn (n 41, 2) the following inequality is

needed:

g

K12 ˇ_v1 1 K22 ˇ_v2

h

1 14b2 ( 1 14b1)( 1 14b2) 24b23 D K 2 n ˇ_vn

g

1 132 Wn V2 n Kn2 ˇ_vn

h

21 . (20)

In the limit case when the quadratic magnitudes are smaller than the linear ones

g

32Wn

V2

n

K2

n

ˇ_vn b1

h

, we can define the constant k14 2

VkˇV1

2k2 m corresponding to the effec-tive harmonic oscillator of frequency equal to V1so in this limit the EJTTotbecomes

EJTTot(EQK 0 ) 4 24 k21 ˇ_V1 1 128

g

k21 ˇ_V1

h

2 _W 1 V2 (21)

which has some analogy with that given in the tetragonal case of ideal T 7 e JT system [20].

Untransformed ground states are obtained by multiplying the transformed ones given in table I by U, after substitution of the correct values of Rin for the state

concerned. This specific form of U, which will be called Uj, is given by

Uj4 exp

k

!

i , n

C( j)

in (bin2 bin†)

l

with Ci1( j)4 2

o

ˇ_m 2 V1 xni( j), C ( j) i2 4 2

o

ˇ_m 2 V2 jni( j). (228) The n( j)

i are deduced from table I, where

Rin4 (xdi , u1 jdi , e) ni( j).

(23)

The untransformed states will be written in the form NXo( j)8; 0 b 4UjNXo( j); 0 b ,

(24)

where the zero indicates that there are no phonon excitations present in the trans-formed picture, but the untranstrans-formed one contains phonon excitations as Uj contains

phonon operators.

In the general case, untransformed ground states do not have cubic symmetry, and are not orthogonal to each other. Suitable states are obtained by constructing linear combination of the ground states in the lowest energy set of wells [17] (situation observed in T 7 t and T 7 (e 1t) JT systems). For T7 (e1e) JT problems, the untransformed ground states Nx 8; 0b, Ny 8; 0b and Nz 8; 0b localized in tetragonal wells are appropriate as approximate cubic eigenstates of H because they have mutually orthogonal orbits.

When considering the effect of strains, it is necessary to know the vibronic overlaps between states Nx 8; 0b, Ny 8; 0b and Nz 8; 0b. It can be shown that overlaps as defined by Dunn [17] are the same between all the three states:

S 4exp

y

26

g

K 2 1 (ˇv1)2 1 K 2 2 (ˇv2)

h

1 14b2 ( 1 14b1)( 1 14b2) 24b23

z

. (25)

5. – Energy of the grounds states

In the case of the T 7 (e 1e) JT system, the untransformed states are good eigenstates of the Hamiltonian, and the orbital operators (Tu and Te) are diagonal, so

the matrix elements between states in any well are non-vanishing for the same orbital parts, and the reduction factor [21] referring to those orbital operators is equal to one.

The matrix elements between ground states (see appendix A) are given by aX( j)8 o ; 0 NHNXo( j)8; 0 b 4 (26) 4

!

i 4u e

y

_g

!

n 41 2 ˇVn

g

C( j) 2 in 1 1 2

hh

aIb ( j) 2 2 V

o

ˇ_V1 2 m aTib ( j)_C( j) in

z

1 Hue( j)( 0 , 0 , A , A 8, 0, 0) .

So the energy of the ground state is

E04 ˇV11 ˇV21 EJTTot.

(27)

The triplet has been lowered by an energy equal to the ETot

JT as no combination of

states has been considered. The energy given by (19) is valid when neglecting

H

×

2(containing phonon contribution) in the limit of very strongly coupling. In this

section we evaluate the first-, and second-order perturbation correction to (19) when considering the total Hamiltonian.

6. – Perturbation correction for the two-mode T 7 (e 1e) JT system

The perturbation applied is [20]

V 42

!

_m P0HPmHP0

D Em

, (28)

where P0 is the projection operator for the ground states and Pm is the projection

operator for excited states of relative energy D Em.

Excited states with one and two explicit phonon excitations only will be included in the calculation. In other words, our system is restricted to m 40, 1 and 2.

Details of matrix elements between states in any of the two wells required for calculation of zero energies of perturbation effect are given in the appendix A. Calculation is presented for ground, first- and second-excited states in the well.

6.1. First- and second-excited states. – As seen before, no combination of untransformed cubic ground states is made, so the first and second untransformed states are good excited states.

First excited states are defined as

NXo( j)8; mpb with

u

X( j)8 o 4 Nx 8 b, Ny 8 b and Nz 8 b mp4 u1, e1, u2, e2

v

. (29)

The second ones are

NXo( j)8; mpmlb with (mp; ml4 u1, e1, u2, e2) .

(30)

So, for fixed X( j)8

o there are four first-excited states, and ten second-excited states.

6.2. Contribution to the ground states. – For the ground states, the summation is made over all states with m 41 and m42. After evaluation of matrix elements necessary to do this, we conclude that the contribution to energies due to the four first states vanishes, and that those due to the ten second states lead to a partial lift of degeneracy.

The energy of the ground states has the components

.

`

`

/

`

`

´

EZ 04 E02 ˇV1 2 b2 1 1 24b21 2 ˇV2 2 b2 2 1 24b22 2 2ˇV1

g

t ˇ_V1

h

2 _{2( 1 1V} 2/V1) ( 1 1V2/V1)22 4(b11 b2V2/V1)2 , Exy 0 4 E02 ˇV1 b21 2( 1 2b2 1) 2 ˇV2 b22 2( 1 2b2 2) 2 2ˇV1

g

t ˇ_V1

h

2 _{2( 1 1V} 2/V1) ( 1 1V2/V1)22 (b11 b2V2/V1)2 , (31)

where

t 4 ˇ]

2

2kV1V2

(318)

and where E0is given by (27).

The Nx 8b and Ny 8b ground states remain degenerate.

6._{3. Contribution to the states with m 41. – Calculation of the zero energy}

referring to m 41 shows that for fixed X( j)8

o orbital states there are two degenerate

doublets, with energies equal to ˇV1and ˇV2, respectively.

When evaluating the four first-excited-states energies, we have found that the corrections due to states with m 40 and m42 are equal to zero. The only remained contributions are those due to m 41, but with different zero energies.

The energies become

.

`

`

/

`

`

´

E0 , z un 4 E01 ˇVn1 h t2 1 4 W23r22 4 tW3r ˇ(V12 V2) , E0 , xy un 4 E01 ˇVn1 h t2 1 4 W23r21 2 tW3r ˇ_{(V12 V2)} , E0 , z en 4 E01 ˇVn1 h t2 1 4 W23r21 4 tW3r ˇ(V12 V2) , E0 , xy en 4 E01 ˇVn1 h t2 1 4 a23r22 2 ta3r ˇ_{(V12 V2)} , (32) where h 4./ ´ 1 21 if n 41 if n 42 and r 4 ˇ_kV1V2 2 m . (33)

As for the ground state, the perturbation effect lifts a partial degeneracy for the states with fixed phonon excitation. The Nx 8; inb and Ny 8; inb (i=u , e ; n=1 , 2 ) remain

degenerate.

6._{4. Contribution to the states with m 42. – Calculation of the zero energies}

Fig. 1. – The variation in energy of the T state with 0, 1 and 2 phonon excitations of a T 7 (e 1e) system as a function of K1(with 2 v 42v14 3 v2, V14 4 V2, W14 0.002 V12, and 4 W24 W1) with

respect to the energy of T ground states. The solid lines show the energy levels of z orbital states. The dashed lines show the energy levels of x and y orbital states. At K14 4 ˇv the energy levels

appear in increasing order as: Ez

0, E xy 0 , E0, Euz2, E xy e2, E xy u2, E z e2, E z e1, E xy u1E xy e1E z u1, Eez22, E xy u22, E z u2e2, E xy u2e2, E xy e21, E z u22, E z e1e2, E xy u1u2, E z u2e1, E xy u1e2, E xy u2e1, E z u1e2, E xy e2e1, E z u1u2, E z e21, E xy u21, Euxy1e1, E z u1e1, E xy e21 and E z u2 1.

Ny 8 ; Xpb states with fixed phonon states Xp. The zero energies are

.

`

`

`

/

`

`

`

´

Ei0 , z2 n 4 E01 2 ˇVn( 1 12hbn) , Ei0 , xy2 n 4 E01 2 ˇVn( 1 2hbn) , Ei0 , z1i24 E01 ˇV1( 1 12hb1) 1ˇV2( 1 12hb2) , E0 , xy i1i2 4 E01 ˇV1( 1 2hb1) 1ˇV2( 1 2hb2) , Eu0 , znen4 E xy unen4 E01 2 ˇVn, Eu0 , z1e24 E01 ˇ(V11 V2) 12ˇV1(b12 b2V2/V1) , Eu0 , z2e14 E01 ˇ(V11 V2) 22ˇV1(b12 b2V2/V1) , Eu0 , xy1e24 E01 ˇ(V11 V2) 2ˇV1(b12 b2V2/V1) , E0 , xy u2e14 E01 ˇ(V11 V2) 1ˇV1(b12 b2V2/V1) , (34)

Using the expression of matrix elements given in appendix A and the perturbation Hamiltonian V, we found that there is no effect of the state with m 41, the only non-vanishing contributions were those relative to m 40 and those with the same m, but having different zero energies.

The exact energies of m 42 are presented in appendix B. The main results of our calculations are displayed in fig. 1 . The energies of the ground states, first- and second-excited states are plotted as a function of K1, with 2 v 42v14 3 v2, V14 4 V2,

W14 0.002V12and W14 4 W2. The graph shows the partial lift of degeneracy caused by

the weak magnitude of the quadratic coupling. The plot is for ETot

JT E EJTn so the

two-mode tetragonal wells are energy minima. The T1ground-state energy E0has been

taken as the zero energy.

7. – Strain effect

It is usual to model systems of deep-level impurities in semiconductors by constructing effective or spin Hamiltonian. In many cases, it is found necessary to include random strain terms in the Hamiltonians in order to explain the available experimental data (e.g., [25-27]). However, it is not usually possible to predict which symmetries and magnitudes of strains are active in a given system without reference to the experimental results.

A more fundamental method of modeling deep-level impurities is to use basic Jahn-Teller (JT) theory (e.g., [6, 21]). As strain terms in JT Hamiltonians are functions of the ion-lattice couplings, it is possible to predict which strains will have a large effect on a given system using these models.

The approach we have adopted, valid for most strains in a crystal when expecting that the strain energies are weaker than those due to (JT) effects, to determine the effect of strains upon the ground states of T 7 (e 1e) (JT) systems consists in considering this effect as a perturbation. After doing the transformation (6) to the static contribution of the coordinate of the e-type, the Hamiltonian of strain has the form Hs4 V – E(q–u1Tu1 q–e1Te) 1V – T(Q – 4Tyz1 Q – 5Tzx1 Q – 6Txy) (35)

which will be used to describe strains of e and t symmetries. V–Eand V

–

T are equal to V

and VT, respectively, multiplied by the appropriate JT reduction factor, and q

–

u1 is the

static contribution to qu1 from strain of Tu-type symmetry etc. Tu, Te, Txy, Tyzand Tzx

are orbital operators.

The magnitude of Q–sdetermines the size and symmetry of the strain experienced by

any one ion in crystal. In real crystal, the Qs will vary from ion to ion, giving a

distribution of sites at different strains.

We will study in detail strains along a 001 b, a 111 b, a 110 b axes of the crystal. Strains along any other axis can be resolved into combination of these strains [28], so they need not to be considered separately. The strain Hamiltonian for these three strain directions has the form

Hs4 pV – Eq–E(C1†C11 C2†C22 2 C3†C3) 1sV – TQ – T(C1†C21 C2†C1) 1 (36) 1rV–TQ – T(C2†C31 C3†C21 C3†C11 C1†C3) ,

TABLEII. Strain axis

[001] [111] [110]

(a) (b) (a) (b) (a) (b)

1 2 — Ez 02 2 g Ez 01 g 1 z — 1 1 1 Ez 01 z 1 l J1 J2 1 1 1 Ez 01 g Ez 01 z 2 g/2 1 3 l/2 Ez 01 z 2 g/2 2 3 l/2 where p 4 1 2 , p 40 , p 421 4 , s 4r40 s 4r4231 /2/2 s 4231 /2_{3 /4 ,} r 40 for a 001 b strains , for a 111 b strains , for a 110 b strains and q–E4 q–u14 q–e1; Q – T4 Q – 44 Q – 54 Q – 6. (37)

The usual assumption that in tetragonal problem t modes are quenched, will not be made. For example, trigonal strains can be important in systems where tetragonal modes are quenched, because V–Eis small [29,30].

In the general case, as the strain Hamiltonian Hs does not have cubic symmetry, it

will be coupled to levels outside the triplet ground states (situation confronted in trigonal and orthorhombic problems). In our case, the states are exact eigenstates of cubic symmetry, so no excited states need to be considered. The effects of these three types of strains treated as a perturbation on the tetragonal well are summarized in table II, where the eigenvalues J1_{and J}2_{are given by}

J6 4 E0z1 1 2 (z 2l)6 1 2

k

8 l 2 1 (z 2 l)2, (38)

z is the difference between the ground-state energies Exy

0 and E0zand g 4 1 2 q – EV – E, l 4 k3 2 q – TV – TS . (39)

The columns (a) show the number of states corresponding to the energies in columns (b). In the case of moderate coupling, it is instructive to plot the energy levels shown in table II as a function of the e- and t-type strains. Figure 2 shows the energy levels of

T 7 (e 1e) JT system with 2v42v14 3 v2, K14 1.7 ˇv , V14 4 V2, W14 0.1V12 and

W24 W1/2 as a function of V

–

Fig. 2. – The energy level of a T 7 (e 1e) JT system with K14 1.7 ˇv and W14 0.1 V12, where 2 v 4

2 v14 3 v2, V14 4 V2and W24 W1/2 . The solid lines show the energy levels as a function of V

– Q–4 V–TQ

–

Tfor a 111 b strains. The dashed lines show the energy levels as a function of V

– Q–4 V–Eq – E4 V–TQ –

Tfor a 110 b strain. The energies are relative to E0z.

the crystal. The energy levels have been plotted relative to the T1ground-state energy.

Similar graphs have been plotted for other moderately coupled T 7 (e 1e) JT systems. In all cases, the behavior of the results is very similar to that above. The inversion splitting and the separation of the energy levels both decrease as K1or W1/V12increases.

In very strongly coupling limit and when we neglect the inversion splitting z of the

Ez

0 and E0x , yin zero energy (the quadratic coupling or the effect of m 42 are vanished),

the results have some analogy to those of [31] for T 7 e . In this limit the results are identical to those obtained by considering the effect of Hsupon the ground state in each

of the wells separately. This arises because the states localized in the wells are an alternative set of eigenstates.

8. – Conclusion

This paper has extended the transformation method presented by Bates et al. [20] and Dunn [17] for T 7 e , T 7 t and T 7 (e 1t) JT systems to the two-mode T7 (e1e) JT systems. Calculation has been done taking into account the effect of both linear and quadratic couplings, the latter has been taken exactly in the minimization procedure. We have simplified the Hamiltonian by using a transformation in the phonon coordinate space.

The results show that there is a lift of degeneracy on the ground states when we have included the quadratic coupling and the effect of the states with two-phonon excitation: For fixed phonon operator Xpthere is a partial degeneracy of Nx 8; Xpb and

Ny 8 ; Xpb orbital states.

Concerning the effect of quadratic JT coupling, we found that a weak magnitude of the quadratic JT coupling plays a considerable effect on energy levels as also shown by Sakamoto [19].

We also note that the only non-vanishing effect on the eigenstates is for the states with number of phonons having the same parity as those of the zero eigenstates.

We have also investigated the effect of strain along tetragonal, trigonal, and orthorhombic axis, and shown that the latter have a larger effect on the two-mode

T 7 (e 1e) JT system in the region of moderate coupling.

AP P E N D I X A

A.1. – The Hamiltonian

In order to calculate the matrix elements, it is necessary to write the Hamiltonian in terms of creation and annihilation operators referring to the coordinate and momenta conjugate qinand pin, respectively.

The Hamiltonian has the form

H 4

!

i 4u e

g

!

n 41 2 ˇ_Vn

g

_bin†_bin1 1 2

h

2 t(b † i12 bi1)(bi2†2 bi2)

h

I 2F(bi1†1 bi1) Ti1 (A.1) 1

!

_nbnˇVn

[(

(ben†1 ben)22 (bun†1 bun)2

)

Tu1 2(ben†1 ben)(bun† 1 bun) Te

]

2 2b3ˇkV1V2

[(

(be1†1 be1)(be2†1 be2) 2 (bu1†1 bu1)(bu2†1 bu2)

)

Tu1 1

(

(b† e11 be1)(bu2†1 bu2) 1 (be2†1 be2)(bu1†1 bu1)

)

Te

]

with t 4 ˇ] 2 2kV1V2 and _{F 4V}

o

ˇV1 2 m . A.2. – Matrix elements of H

When calculating the first-order perturbation correction to the ground states we have to evaluate the matrix elements. The only non-vanishing elements are those between the same orbital states. In the below expression, we denote

aIb( j)

4aXo( j)NXo( j)b , aTub( j))4aXo( j)NTuNXo( j)b and aTeb( j)4aXo( j)NTeNXo( j)b .

The contribution due to quadratic term is given by H( j) ue(u , v , X , X8, w, z) 4

!

nbn ˇ_{Vn[XuvunenwzaTub}( j)_{1 2 X8uvunenwzaTeb}( j)_{] 2} (A.3)

2b3ˇkV1V2[ (X 8uve1 e2 wz2 X 8uvu1 u2 wz)aTub( j)1 (X 8uvu1 e2 wz1 X 8uvu2 e1 wz)aTeb( j)] .

A.2.1. Matrix elements between ground states aX( j)8 o ; 0 NHNXo( j)8; 0 b 4 4

!

i 4u e

k

!

n 41 2 ˇ_Vn

g

_Cin( j)2₁ 1 2

h

aIb ( j) 1 2 FaTib( j)C ( j) i1

l

1 Hue( j)( 0 , 0 , A , A 8, 0, 0) , where Aunen4 4 C( j) 2 en 2 4 C( j) 2 un and A 8inrm4 4 Cin( j)Crm( j). (A.4)

A.2.2. Matrix elements between excited states. Evaluation of the perturbation requires knowledge of the energies of first- and second-excited states in the wells.

A.2.2.1. F i r s t - e x c i t e d s t a t e s . Deduced for the matrix elements which have the form aX( j)

o ; nlNHNXo( j); npb. Two cases have to be considered

i) l 4p aX( j)8 o ; nlNHNXo( j)8; nlb 4 aXo( j)8; 0 NHNXo( j)8; 0 b 1

!

i 4u e

!

n 41 2 ˇ_{Vndin , laIb}( j)_. (A.5) ii) l c p aX( j)8 o ; npNHNXo( j)8; nlb 4t

!

i 4u e (di1 , pdi2 , l1 di1 , ldi2 , p)aIb( j)1 Hue( j)(p , 0 , 0 , E 8, l, 0) , where E 8pinrml4 (din , pdrm , l1 din , ldrm , p) . (A.6) A.2.2.2. S e c o n d - e x c i t e d s t a t e s .

– Matrix elements necessary to calculate the zero energies of second-excited states: i) Elements of type aXo( j)8; nl2NHNXo( j)8; nl2b aX( j)8 o ; nl2NHNXo( j)8; nl2b 4 aXo( j)8; 0 NHNXo( j)8; 0 b 1 (A.7) 12

!

n ˇ_Vn

k

!

i din , laIb( j)1 bn(den , l2 dun , l)aTub( j)

l

.

ii) Elements of type aXo( j)8; nlnpNHNXo( j)8; nlnpb aX( j)8 o ; nlnpNHNXo( j)8; nlnpb 4 aXo( j)8; 0 NHNXo( j)8; 0 b 1 (A.8) 1

!

_nˇ_Vn

k

!

i (din , p1 din , l) 1 aIb( j)1 2 bn(den , p2 dun , p1 den , l2 dun , l)aTub( j)

l

.

– Mixed matrix elements of the states with m 42. The matrix elements of type aXo( j)8; npnlNHNXo( j)8; nqnsb. The only non-vanishing terms are

i) l 4q4s and lcp aX( j)8 o ; npnlNHNXo( j)8; nl2b 4 4k2 t

!

i 4u e (di1 , pdi2 , l1 di1 , ldi2 , p)aIb( j)1k2 Hue( j)(p , l , 0 , G8, l, l) , where G8linrmp4 (din , ldrm , p1 din , pdrm , l) . (A.9) ii) l 4q and pcscl (A.10) aXo( j)8; nlnpNHNXo( j)8; nlnsb 4t

!

i 4u e (di1 , pdi2 , s1 di1 , sdi2 , p)aIb( j)1 1Hue( j)(l , p , 0 , D8, l, s) , where D8lpinrmls4 (din , pdrm , s1 din , sdrm , p).

A.2.3. Matrix elements between states with different m

A._{2.3.1. E l e m e n t s b e t w e e n m 40 an d m41} aXo( j)8; 0 NHNXo( j)8; nlb 4 4 2

!

i 4u e

k

!

n 41 2 ˇVnC ( j) in din , laIb ( j) 1 2 FaTib ( j) Ci1( j)

l

1 H ( j) ue ( 0 , 0 , j , j 8, 0, l) , where (A.11) junenl4 8 C( j) 2 en den , l2 8 C ( j)2 un , ldun , l and j 8inrml4 4 C ( j)2 in drm , l1 4 C( j) 2 rm din , l.

A._{2.3.2. E l e m e n t s b e t w e e n m 4 0 an d m 4 2 . The terms of the form}

i) l 4p

aXo( j)8; 0 NHNXo( j)8; nl2b 4Hue( j)( 0 , 0 , z , 0 , l , l)

with

zunenl4 den , l2 dun , l.

ii) l c p aXo( j)8; 0 NHNXo( j)8; nlnpb 42t

!

i 4u e (di1 , pdi2 , l1 di1 , ldi2 , p)aIb( j). (A.13)

A._{2.3.3. E l e m e n t s b e t w e e n m 4 1 an d m 4 2 . Those are the matrix elements of}

type aX( j)8

o ; nlNHNXo( j)8; npnqb. The non-vanishing elements are

i) l 4p4q aX( j)8 o ; nlNHNXo( j)8; nl2b 4 4 2k2

!

i 4u e

k

!

n 41 2 ˇVnCin( j)din , laIb( j)2 FaTib( j)

l

1k2 Hue( j)(l , 0 , F , F8, l, l) , with

Flunen4 4(Cen( j)den , l2 Cun( j)dun , l) and F8linrm4 2(Cin( j)drm , l1 Crm( j)din , l) .

(A.14) ii) l 4qcp aX( j)8 o ; nlNHNXo( j)8; npnlb 4 4 2

!

i 4u e

k

g

!

n 41 2 ˇ_VnCp( j)_{din , p}

h

_aIb( j)_{1 FaTib}( j)_{di1 , p}

l

_{1 Hue}( j)_{(l , 0 , 0 , k 8, l, p) ,} where k 8inrmp4 (Cin( j)drm , p1 Crm( j)din , p) . (A.15) AP P E N D I X B

The energies of the second-excited states

In this section we will give the energies expressions of second excited states when applying the perturbation due to states with m 40, m41 and the states with m42 but with different zero energies, by using the matrix elements appearing before.

We denote EXo

Xp the energy of state with orbital part Xoand phonon part Xp.

In all the following expressions, we note that

h 4./ ´ 1 21 if i 4u if i 4e and x 4 . / ´ 1 21 if Xp4 u1e2, if Xp4 e1u2, E_iz2 n4 E01 2 ˇVn1 4 ˇVnbn1 ˇ_Vnb2_n 1 12hb2n 1 2(t 2hˇkV1V2b3) 2

!

q 41 2 (21)q 2nˇ_{Vq( 1 12hbq)} , (B.1)

E_ixy2 n 4 E01 2 ˇVn2 5 ˇVnbn1 ˇVnb2n 4( 1 2hb2 n) 2 2

(

t 1h(ˇ/2)kV1V2b3

)

2

!

q 41 2 (21)q 2n_ˇV q( 1 2hbq) 2 (B.2) 23 2 ˇ2_V1V2 b 2 3

!

q 41 2 (21)q 2nˇ_Vq

₍

_{1 1 (21)}q_hbq

₎

, Eiz1i24 E01

!

n 41 2 ˇ_{Vn( 1 12hbn) 1} t 2

!

q 41 2 ˇ_{Vq( 1 12hbq)} , (B.3) E_ixy 1i24 E01

!

p 41 2

y

ˇ_Vp

g

_{1 2} 5 2hbp

h

1 3 4 b2 3ˇ2V1V2

!

q 41 2 (21)q 2pˇ_Vq

(

_{( 1 2 (21)}q_hbq

)

z

1 (B.4) 1 t 2

!

q 41 2 ˇ_{Vq( 1 2hbq)} , Ez unen4 E01 2 ˇVn1

!

p 41 2

₍

_{t 1 (21)}p_ˇ kV1V2b3

)

2

!

q 41 2 (21)q 2n_ˇV q

(

1 22(21)pbq

)

, (B.5) Exy unen4 E01 2 ˇVn2

!

p 41 2

y

2

(

t 2 (21)p_{(ˇ/2 )kV} 1V2b3

)

2

!

q 41 2 (21)q 2n_ˇV q

(

1 2 (21)pbq

)

1 (B.6) 13 4 ˇ2V1V2b23

!

q 41 2 (21)q 2nˇ_Vq

(

_{1 1 (21)}p 1q_bq

)

z

, Ez Xp4 E01

!

p 41 2

y

ˇ_Vp

₍

_{1 2 (21)}p_xbp

₎

₁

(

t 2 (21) p_ˇkV 1V2b3

)

2

!

q 41 2 (21)q 2pˇ_Vq

₍

_{1 22(21)}p_bq

₎

z

, (B.7) Exy Xp4 E01

!

p 41 2

y

ˇ_Vp

g

_{1 1} 5 2(21) p_xb p

h

1

(

t 2 (21)p_{(ˇ/2 )} kV1V2b3

)

2

!

q 41 2 (21)q 2pˇ_Vq

₍

_{1 2 (21)}p_bq

₎

z

. (B.8)

R E F E R E N C E S

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