- Email: [email protected]
Electron-phonon coupling in dielectrics
J. Phys.
Chem. Solids.
Pergamon
Press 1966. Vol. 27, pp. 1837-1847.
ELECTRON-PHONON
COUPLING
21 April
IN DIELECTRICS*
and K. P. SINHA
G. C. SHUKLA
National Chemical Laboratory, (Received
Printed in Great Britain.
Poona 8, India
1966; in revisedform
17 June 1966)
Abstract-The instability of certain vibrational modes in ionic dielectrics such as BaTiOs is theoretically studied by taking account of electron-vibration coupling. It is found that the vibronic interactions in the elementary units e.g. Ti-0-Ti or TiOs octahedra render the polar modes unstable. This provides a microscopic description of the origin of ferroelectric phase transition. The temperature dependence of the frequency of the soft (the unstable transverse optical) mode is then investigated by considering the collective oscillations of a lattice of such elementary units after including anharmonic interactions. The stabilization of the soft mode, beyond a certain temperature T,, occurs as a result of fourth order anharmonic interaction involving the acoustic and the transverse optical mode. The temperature dependence of the effective frequency, namely, weea (transverse optical) CC(T- T,) is similar to the relation found by Silverman. The agreement with available experimental results is satisfactory.
1. INTRODUCTION
FOLLOWING the suggestions of ANDERSON(~)and COCHRAN,(~) several workers(3p4) have studied the ferroelectric phase transition in relation to the instability of certain lattice vibration modes of the system. These studies are confined to ionic dielectrics such as barium titanate. These treatments start with the assumption that the transverse optical frequency is imaginary in the harmonic approximation. c4) They could render the frequency real (positive) by including certain anharmanic terms in the potential function on the basis of a renormalization procedure. The temperature dependence of the frequency wr is then given by wr2 oc (T- Z’,), where T, is a characteristic temperature. This could also explain the CurieWeiss behaviour of the dielectric constant. This approach seems to be in the right direction; however, the assumption that the transverse frequency is imaginary in the harmonic approximation remains ad hoc. Also, the nuclear potential function was taken on the assumption that the electronic ground state is far removed from the excited states.(*) This is questionable and the role of electron-vibration coupling cannot be ignored. * Communication No. 881 from National Laboratory, Poona 8, India.
Chemical
This coupling may furnish a description of polarizability of ions as considered by Slater on phenomenological basis.t5) In fact, it was shown in a recent paper that electron-vibration coupling effects are indeed important.(s~‘) When one takes account of some low lying electronic states which mix with the ground electronic states via some vibrational mode a distortion of the basic ionic unit takes place resulting in a permanent dipole. In the paper referred to, the electron vibration problem involving the potential function up to some anharmonic terms for the unit Ti-0-Ti was considered. Kinetic energy terms were not included in the analysis.@) In the present paper, we intend to include the kinetic energy terms explicitly and study the lattice dynamics of such systems by recognizing the electronic as well as anharmonic interactions involved in the problem. We shall see that the Jahn-Teller like terms do indeed lead to expressions suggesting instability of certain modes (optical) which are eventually stabilized as a result of anharmonic interactions. The instability appears for individual units such as Ti-0-Ti or TiO, (octahedra) owing to Jahn-Teller like effects. Furthermore, the inclusion of electronic effects incorporates the covalency terms which may
1837
1838
G. C. SHUKLA
be necessary units.@)
in the
chemical
bonding
of the
2. VIBRONIC COUPLING IN ELEMENTARY UNITS In this section, we consider the coupling of the low-lying excited electronic states with the ground state of the elementary systems owing to certain vibrational perturbations. The calculation is confined to the perovskite or WO, type ferro-electrics. The crystal structure of perovskite type system is discussed at length in many references.(gJO) We shall make note of the nearest neighbour coordination of 02- and M (i.e. Ti*+ or W6+) tons. In the cubic phase we have the collinear units e.g. Ti-0-Ti with Ti-0 distance the same on both sides of 0 and in the regular octahedron TiOs. Our mission is to show that vibronic effects (electron-vibration coupling) will lead to the distortion of the units either Ti-0-Ti or TiO, resulting in a permanent dielectric dipole. The problem of the triatomic units e.g. M-O-M in the static approximation was solved in a previous paper.@) We shall briefly consider the vibrational problem for this and then take up the octahedral unit MO,. 2.1 The case of triatomic unit We consider the situation when we have two electronic levels for the system namely, the symmetrical state I/, and the antisymmetrical state #a. As before the energy separation is defined as 25 = &/a,,) - J%U
where H, is the bare electronic part of the Hamiltonian, H, the vibrational part which in the harmonic approximation is given by
ViQ!,
is the electron-vibration coupling. V” is the first derivative of the electron-lattice potential function with respect to the nuclear displacement coordinate Qi evaluated at equilibrium position. The problem is to diagonalize the Hamiltonian (2.2). It is expedient to rearrange the Hamiltonian (2.2) so that its relationship with the manifold of symmetrical state $, (ground) and antisymmetrical (excited state) #, is clear. We make use of Dirac’s bra (1 and ket )> notations.
For the triatomic unit under consideration, the vibrational modes of interest are the symmetrical vibration (Q,) and antisymmetrical (Qs) modes. The electron-vibration coupling terms which survive and couple the symmetrical (I/J and antisymmetrical state (&I,) through V”Qo where V” = (~V/CYQO)~ and has odd parity,
We shall now make use of a canonical transformation in order to remove the electron-vibration coupling term occurring in first order of Qe. The transformed Hamiltonian is given by H, = edisHeiS = H + i[H, S] -$d[H,
Sl, Sl+
. . .
(24
where we have taken S = ig(I~~~>~slQ~-I~s>(~alQ~)
(2.7)
The quantity g is so chosen that the linear term in Qs is eliminated from the transformed Hamiltonian. This is easily done by making use of the relation
(2.3)
where QI and Pi are normal coordinates and conjugate momentum of the ith vibrational mode; wi is the corresponding frequency;
He, = 2
K. P. SINHA
(2.1)
The problem then is to consider the coupling between the states #s and $, owing to antisymmetrical vibration. The Hamiltonian for the system is H = H,+H,+H,, (2.2)
HtJ= 8 2 (PF+ Wi2QF),
and
(2.4)
i[H,, S] + H,,
= 0.
(24
We get A ___g = (E,-IsE,)
(2.9)
ELECTRON-PHONON
COUPLING
HamilThus with A,, = A,,, the transformed tonian up to harmonic and diagonal terms is
IN
DIELECTRICS
1839
direction.@) In the earlier paper, it was estimated that for the unit Ti-0-Ti the condition A2 wa2 < J
HT = I#, )Es (&I + I&z )Kz <&I
QS2)+W'02+~02Q02) is obtained and one expects the instability. _ I~s>(&s>2<4slQo2 We shall now analyse the electron-vibration (%--Es) coupling for the octahedral unit namely ML,, where M is the metal ion and L stands for the +lW(&.)2~+a/Qo2 (2.10) ligand ion. (&z--Es) +&(P,2+~,82
As we are henceforth interested in the vibrational problem, we shall take the expectation value of the Hamiltonian with respect to I#,) or I$,). Thus, taking
= Es + B(Ps’ + us2
Qs2>
$$-)Qs2] s a where we find that the frequency tion is modified as
(2.11) of the vibra-
.
(2.12)
If the average is taken with respect to the state I#=) we would see that the effective frequency relation will have a plus sign in place of (12). The analysis given above shows that if the system is in the ground state having symmetrical electronic state (E,-Es) > 0, the effective frequency of the odd vibration decreases [cf. equation (2.12)]. On the other hand, when it is in the excited state E,, the frequency will be pushed up. This is a consequence of the electron-vibration coupling incorporated in the treatment. The electronic ground state is indeed totally symmetrical I&.>. Let us now examine the effective frequency wo2(eff)
=
(wo2 - g)
The condition as2 < A,,2/jJj makes the effective frequency imaginary which implies that instability develops in the system. It has been pointed out before that this leads to the distortion of the system in which the central atom moves in one direction and the two extreme atoms move in the opposite
2.2 The case of octahedral unit It is expedient to consider the problem with the help of a specific example. The most appropriate unit is the octahedral cluster TiOs with the Ti ion at the centre and each oxygen ion at the corners. The various electronic configurations of the complex (TiOJshave been discussed by NELSON@) following the molecular orbital approach. However, he has not considered the electron-vibration coupling which seems to us to be of central importance in the present problem. To understand this, a discussion of the one electron molecular orbitals and many electron states for the unit along with the vibrational modes will be essential. The definitions and notations used here are the same as those given by VAN VLECK;(~~)OPIK and PRYCE(~~)and PRYCEet aZ.(13) in a different context. The two ligand ions (e.g. 02-) on the t_ x axis are respectively numbered 1 and 4, the two on the + y axis 2 and 5 and the two on the fz axis 3 and 6. If we started from the ionic picture i.e. Ti4+ and 02- then the total number of electrons to be accounted for are as follows: 18(Ti*+)+6x 10(02-) = 78. Of these, 18 (in ls22s22p63s23p6) can be regarded with sufficient accuracy as completely located on Ti4+ centre and 6 x 4 = 24 are located on oxygens (in Isa, u_~).* The remaining electrons (36) would occupy orbitals which can approximately be regarded as built up from the combination of 2s, 2p orbitals on oxygen and 3s, 4s and 4p atomic orbitals on Ti4 + . We shall refer to the appropriate hybrids (sp) of 2s and 2p orbitals on 02- which have end on * The hybridization of 2s and 2p orbitals of the ligand ion will give two orbit&, which are designated as + and - where, the former points towards the metal ion and the latter in the diagonally opposite direction.
1840
G.
C.
SHUKLA
overlap with the metal ions as ui orbitals and the remaining 2~ orbitals as r orbitals. Here D and n respectively refer to those orbitals which have even and odd symmetry under rotation by 180” about the axis joining the metal and ligand ions. It should be noted that the bonding molecular orbitals involving the u orbitals of the ligand are expected to be of lower energy than those involving the 7~ orbitals because the u orbital function overlaps more on to the Ti and makes use of its attractive field than the n orbitals. Then in the increasing order of energy the molecular orbitals are given as b qgb,
tlub,
e!?
h7n*
tzga,
ega
’
tb 29
’
alga,
2 lub,
t2un>
tlua,
t’lua
Here the superscripts, b, n and a refer to bonding, non-bonding and antibonding orbitals respectively; al, e and t are the group theoretical notations of the orbitals (see GRIFFITHS~*). In the absence of electron-vibration interactions and for the regular octahedron, the ground electronic state has the configuration
and
K.
P.
SINHA
with a similar expression Likewise we will have
for transition
I * - * (tzu”>6; (lAl,)
from
> + l(tzu”)5(t2g”); lAlu
or lT1,
or
IE,, or (2.15a)
ITa, > I . . . (tzl(‘%
(t’l,b)“.
(1-4,) > --f I . . . (tz,“)“(ega>; ‘Tlu
or ‘Tzu > (2.15b)
We now consider the various vibrational modes for the octahedral unit MO,. There are fifteen normal coordinates of vibration which belong to the following types of representations : A 157, E,,
T,,,
Tlu’,
T2,
and
T2,.
From symmetry considerations alone we can see that the various excited electronic states can couple with the ground state [#a; IAl,) via the vibrational perturbation having appropriate symmetry. The Hamiltonian involving electron-vibration interaction for the present problem is similar to
(2.13) I~g(14g)> -+ ~(~19b>2(esb>4(t~~b>6(tZOb)606(tZun~6~~~gn~61
This accounts for all the electrons and the states until bonding and non-bonding orbitals are completely filled. Let us now consider the excited states in which a single electron is transferred from one of the bonding or non-bonding (predominantly ligand orbitals) t, orbitals to one of the antibonding orbitals. Group theoretically various symmetry states are possible;@) I . . - (t,,b)6; PA,,)
Hi = 2
(2.5). The general form can be written IN’
or
or lTlu
IE,. or (2.14a)
IT,,) + l(tlUE)5(eg”); lT1, lTall>
or (2.14b)
“) )&I
(+~(r,“)l
CPia2
+ wi2 Qia?
as
ia + S 2 i.n +
2 (Lw(jJ) (k.?‘)
x
bvf)
> -+ l(tl,b)5(t2,a); lA2,
1 . . . (t1,b)6; (lA,,)>
equation
i
Id4ri”)
1a2
)Ar,%,‘(r,‘)
(2.16)
where ri refers to the irreducible representation of the octahedral group. Thus j#(I’ioL)) denotes the crth base function belonging to the Fi irreducible representation. Qrla likewise represents a similar vibrational mode
ELECTRON-PHONON
COUPLING
IN
with
with
A g.ua=Aa u .g
(2.18) It is clear from equation (2.17) that only those matrix elements will survive for which the corresponding products representation Ft X rk x r, will contain the identity representation I’, on decomposition. In the present problem we are not interested in a general situation of equation (2.16). Our interest is to see as to how this ground electronic state [$,,(lAl,)) couples with the excited states via equation (2.17). Furthermore only low lying excited electronic states will have any significant vibronic interaction. The low lying electronic state of symmetry T,, is of particular interest in the present problem. Such states, will couple with the ground denoted by j$,(Tl,)), configuration j+!~,(~/l~,)) through the vibrational modes of symmetry T,,. Of these, the mode in which the positive ion moves against two of the ligand ions in opposite directions gives rise to the dipole moment. It is referred to as wlc mode with one of the Q’s called as Q1s (see KOIDE and tiYCE(15)). This describes the rattling of the positive ion in the octahedron of negative ions. The importance of coupling via this mode has also been stressed by ORGEL. We shall therefore confine our attention to the vibronic coupling of states \I#,(~A~,)) and jt,!QT~J) via the appropriate vibrational modes T,,. The Hamiltonian (2.16) can be written as: H red
1841
DIELECTRICS
=
I&
+t
)-%
c
a
<&I
+
&LQ?
+9 C
cu.f(# Td
2
l&i=
)-%t
(that
1
m,?Qm,oL?)
P'i.a2 + wi2Qi,a2)
(2.21)
Thus, the transformed vibrational Hamiltonian after the expectation value over the ground electronic state is expressed as
Hv,,(E,) = E,++
2 (%a2+ QQi.a2) a#(Tld
(2.22) Thus, we see that for the elementary octahedron also the vibronic interaction can render the effective frequency Weft
2=
WTlU2 -
24.ua2
--
(&
imaginary
-
(2.23)
4,)
i.e. wrl,a < This instabilitywilllead to a distortionofthe system in which the metal ions moves in either of the x, y, z directions with the movement of the two appropriate ligands in the opposite direction. This will give a permanent dipole moment. This seems to be possible in any of the three directions for the elementary octahedron. However, as has been discussed in Appendix A, the environment of oxygen ions in BaTiOs is such that only one of the directions (say z) will be preferred. So far we have discussed the vibronic problem for the elementary units, namely, Ti-0-Ti and TiO,. In the next section we consider a lattice of such units and discuss the vibrational problem of the lattice including anharmonic terms as well as vibronic interactions in the elementary units.
3. COLLECTIVE OSCILLATIONS In the previous sections, we have analysed the a vibrational problem of elementary units (tri-ionic (2.19) + C.C.) or octahedral) by taking account of the vibronic coupling caused by some specific modes. This gave The transformation analogous to equation (2.6) rise to instability of the modes in question. Let us can be used now, where now consider an assembly of such units forming a S(oct) = ic G”(l#,“) <~,IQ~u~--l~o)lattice. In BaTiO,, the TiO, octahedra are joined by their corners. This forms a three dimensional (2.20) <#ualQ~ua) network and the large holes are occupied by the + ( c
I#,
>4,ua(T1.“)
(#,“I
Qcq,~
1842
G.
C.
SHUKLA
and
Ba2+ ions. Of course, in BaTiO, the Ba2+ ions do play some role in view of the large size and overlap with oxygen ions. Perhaps in WO, type ferroelectrics (or from structural considerations in ReOs) one has to consider the lattice of octahedra alone. We confine our attention to the vibrational problem of the lattice made up of these units. For N such units and including anharmonic terms the lattice Hamiltonian is
K. P. SINHA c __
2
H2=&;
02
a.0
WO OaJoa
x(b,O+ +boo)(boo-!-boo+) ~(b,=+b_.=+)(b_,~+b,~+) (3.7) H32i
2 (I[
co.0 (woO)a(boO+boO+)
x (boo+ boo +)(boo + boo +)
almQIAQm,,
x(boo+boo+)+~
bnnQ,~QmuQnv Ci,mnQisQ~~QmrrQnv (3.1) where the frequency, wrL2 refers to the Ith unit and its hth mode. P,, = f&and QIn are the normal coordinates of the Zth unit and its Xth mode. In the above, we have included the translational mode of the individual units also in order that the acoustical mode for the whole crystal is taken properly into account. For present purposes we shall be interested in a detailed study of the modes which involve the vibronic instability [cf. equation (2.23)] and their interactions with the acoustic mode contained in the third and fourth order anharmonic terms. As shown in Appendix B, in second quantization representation the Hamiltonian (3.1) is transformed to HT = H,+Ho+Hoo+H,+H,+H,,
(3.2)
where
c wooco~o
x (boo + boo+)(boo + boo +) x (boo +b_,O+)(b_,O
+b,‘+)
1 (34
The external field Eayt _.__has not been introduced so far in our Hamiltonian. We are for the moment interested in the lattice dynamical problem without the influence of such a field. For the present our primary interest is to see the effect of interaction between acoustic modes on the long wavelength optical modes. The third order anharmonic terms [cf. equation (3.6)] can be eliminated, to a large extent, by an appropriate canonical transformation (see Appendix C). We shall thus consider the effect of the acoustic and optical modes with (a0 # 0) on the long wavelength polar mode woo by taking thermal averages as explained below. We write the diagonal part of the Hamiltonian explicitly
Ha =
2 fi w,“( boa+ boa + 3)
(3.3)
Ho =
c fi w,“( b,’ + boo + 4) a#0
(3.4)
+ 2 ?iwo0(b,0+b,o+3) O#O
(3.5)
++‘o”‘+(wooz+;( 5 C,.,o
Ho0 = fiwoO(boO+boo+i) ti 3’s Boo,a _ HI = -1/N 0 2 OJoowo~~‘a
HTa = 2 fh,“(b,“+b,a+~)
1
-t- 1 c,,,,CQoo2)))Qoo’], O#O
x (boo + + boo)(boo + + boo) x(boa+boa+)
(3 ~5)
where we have introduced Ofi0 etc. for convenience.
(3.9)
the variable PO0 and We shall. however,
ELECTRON-PHONON
restrict the coupling with the acoustical whose thermal average is given by
mode
IN
1843
DIELECTRICS
which can be written as H = B 2 (PO,2 +
0.
= kBT/hoaa.
Thus the effective frequency QJeff
COUPLING
is
+3(G2
+
~o,,~Qc,,~)
%ff2
bom,2)
(4.2)
where now
a=
+;c
3)
echo
(3.10)
where Jgu = &(E,- E,) and wU is the frequency of the optical mode of one unit [i.e. equation (2.23)]. 2 is the number of the nearest units coupled to unit at 1. The term may have its origin in the electrostatic interaction between dipoles. For a long range interaction it may have the form -$ne2L where L is the local Lorentz field parameter for u = 0. For cubic system, L = 47r/3 for transverse mode and L = -&r/3 for longitudinal mode. This will give a negative contribution for the transverse mode. Our expression (3.10) differs from a corresponding relation of Silverman in many ways.(4) First mu2 is positive. The instability arises because of the terms - Agu2/1J,,] owing to vibronic interaction. This type of terms are discussed for the first time here. The dipole-dipole interaction may also give negative contribution. In the event u,,~ is outweighed by these negative terms, the stabilization comes from (1/N)C,C,~~okBT/(woa)2. The expression (3.10) also gives the same temperature dependence as obtained by SILVERMAN(~)although the factors are different W,tr2 0~ (T- T,)a 4. DISPERSION OF THE SOFT MODE In this section, we study the wave vector dependence of the soft optical mode. This is carried out following a simplified picture. We have seen that only fourth order anharmonic terms are significant. Thus we consider the two modes, the soft optical and the acoustic modes and their interactions. The Hamiltonian is
If we confine our attention to the interaction between the nearest neighbours only, then after taking the thermal average (QGa2), we get
Weff2(cTO) = [ f.!J,2--+y PU
+2D 2 (cos csOZR 00 + cos cr,,YR+ cos a,“R)
where R is the distance between two octahedra and D denotes the coefficient for the dipole-dipole interaction. For transverse oscillation it will be negative. The summation over us, aa in the last term of equation (4.4) is rather complicated for a general three dimensional case. We shall therefore write this result for a one dimensional chain of elementary octahedra. Now w, 2 = 4a sin2(uaa) with a as the force constant, R” = 2a, a being the distance between a metal ion and a ligand ion. Further, we get,
=-
4c N 82 60 x
sin2( aaa) u.
kBT a sin2( oaaa) I
cos2(uoa)
= 4c 2 uo
F
cosy aoa)
(4.5)
1844
G.
Finally, putting model gives
Weff
2=
SHUKLA
20 = -1~1 the one dimensional
zF
-___
A, 2 -
wu2
-I-
C.
y
ii
cos(2 oaa)
1 go
1
2 /3T COS~(G~~)
(4.6)
J
go
where p = 4Ck,/a. The temperature dependence of the frequency now be written for os = 0 as
can
\ where T, =
A 2 --=+
(4.7)
y-wu2
l-&1 The variation of weff2 on ue at any particular temperature is given by equation (4.6). This is given in Fig. 1, for arbitrary values of the parameters involved. We get the same sort of depenACos(26oa)f
1
BCzs.(doa)=f(da)
A--B,
8
E-12
f
I .I
I .2
I .3
I .4
boa -__, Tl-. FIG. 1, Wave-vector and temperature dependence of the soft optical mode where A = Iy] and B = @T [cf. equation (4.6)].
dence as obtained by Silverman although our functions are different from his.c4) As in his case, the agreements with the results of cOwLEY’18) is reasonably good.
and
K. P. SINHA
5. CONCLUDING REMARKS In the preceding sections, we have taken account of the low lying electronic states in the vibrational problem of some ionic dielectrics. To incorporate the vibronic interaction, we have chosen appropriate elementary units e.g. Ti-0-Ti or TiO, octahedra. The ground electronic states of such systems interact with the low lying odd states via the odd vibrational modes. On eliminating such terms by a renormalization procedure, we get an effective frequency for such units which contains vibronic interaction effects and can render the frequency imaginary or zero. This effect is attributed to be the cause of the instability of the mode giving rise to the electric dipole moment. Starting with such unstable units we then studied the collective oscillations of a system of elementary units. Anharmonic terms were also included. The approach is similar to those adopted by ANDERSON(~) and SILVERMAN;(~) however, there are important
differences
in the starting
model.
The we
fourth
have
coupling optical
of
the
to the differences
anharmonic main
acoustic
mode,
frequency.
order
taken
owing
mode
terms,
interaction and
the
give rise to temperature
Interactions
between
the
where from
in the
transverse dependent elementary
units are of the dipolar
type. An explicit form of the dependence of frequency on the wave vector could be given only for a one-dimensional chain. Both the predicted temperature and wave-vector dependences show the right trend. The main purpose of the present paper has been to invoke the role of vibronic interaction towards the instability of certain lattice modes. This has done away with the ad hoc assumptions made by earlier workers. REFERENCES 1. ANDERSON P. W., Paper presented at All-Union
2. 3. 4.
5. 6.
Conference Dielectrics, Moscow, 1958 [Russian Version Published in &zika Dielektrik& (editor G. I. Skanavi). Akad. Nauk SSSR MOSCOW, (1960)]. ” COCHRAN W., Phys. Rev. Lett. 3, 412 (1959). COWLEY R. A., Pkys. Rew. 134, A981 (1964). SILVERMAN B. D., Phys. Rev. 135A, 1596 (1964); This contains references to earlier papers by him and collaborators. SLATER J. C., Pkys. Rev. 78, 748 (1950). SINHA K. P. and SINHA A. P. B., Indias J. Pure Appl. Phy., 2, 91 (1964).
ELECTRON-PHONON
COUPLING
7. SHUKLA G. C. and SINHA K. P., Indian J. Pure Appl. Phys. 3, 430 (1965). 8. NELSON C. W., Technical Report 179, Laboratory of Insulation Research, MIT, 1963. 9. JONA F. and SHIRANE G., Ferroelectric Crystals. Pergamon Press, Oxford (1962). 10. KANZIC W., Solid State Physics, Volume 4, (editors F. Seitz and D. Tumbull). Academic Press, New York (1957). 11. VAN VLFCK J. H., J. Chem. Phys. 7, 72 (1939). 12. ~PIK U. and PFtYCE M. H. L., Proc. R. Sot. 238A, 425 (1957). 13. PRYCE M. H. L., SINHA K. P. and TANABE Y., Molec. Phys. 9, 33 (1965). 14. GRIFFITHS J. S., The Theory of Transition Metal Ions. Cambridge University Press, London (1961). 15. KOIDE S. and PRYCE M. H. L., Phil. Mug. 3, 607 (1958). 16. ORCEL L. E., Discuss. Faraday Sot. 2, 138 (1958). 17. SZICETI B., Proc. R. Sot. 252A, 217 (1959). 18. COWLEY R. A., Phys. Rev. Lett. 9, 159 (1962).
APPENDIX
A
We consider the explicit forms of the normal coordinates. Since, there are two representations of type T,, their interactions have to be considered. This evaluation involves certain assumptions about the forces between atoms. Such calculations based on valence force and central force models have been carried out by KOIDE and PRYCE.“~) We shall use the final results. The ratio of the masses of oxygen and Ti i.e. MO/MT, = 3. Thus the normal coordinates of present interest are
“(&s +Z, -WA Qu"(TaJ= d(;)(G+ Y5-2Yd
IN
1845
DIELECTRICS
I
(A-2)
-$gz,Y,4-l- . . .
where R. is the distance between the metal and ligand ion in the undistorted octahedron.
(A-3)
The summation is over the k electrons of the central ion. The other components namely, P,, and V,, can be obtained by a cyclic permutation of equation (2.20). Although, the quantities V,,, etc. have been deduced on an electrostatic model, the general form given in equation (2.20) will remain the same on other models also, apart from the change in the definitions of the coefficients a, 6 and c etc. The polynomials involving electronic coordinates etc. clearly indicate that the vibronic perturbations will mix excited electronic states of T,, symmetry with the ground state.
&,"(TI,) =
&‘(Td
( 1(X,+x,
= 2/ ;
APPENDIX
B
Let us introduce the transformation normal coordinates, namely,
to lattice wave
-2x,,)
(A-1)
The corresponding derivative of the potential function [cf. equation (2.18)] can be easily deduced for an electrostatic interaction between the Iigand point ions and an electron of the metal ion at distance rr,, apart, namely q/rot where p = Ze. The method of Van Vleck is adopted here. We get
vu2
(av_ ) = +Ro5
=
8QU2
b --(
Zk3 - g+J~2)
where Qo, and Q, are lattice modes corresponding to the optical and acoustic modes of vibrations respectively; oa, cro are the propagation vectors and R, is the vector spanning the lth unit. The Hamiltonian (3.1) is
I846
G. C. SHUKLA
transformed
and
H = 2 iiPoaPo,* + Wo.2Qo,Qo,*>
x A(c,
+
cs,’ + cst,” -K)
+ r+‘+
where K is a reciprocal
D, = 2
c+”
+ os,,,“’
P.
SINHA
In proceeding further, we shall be guided by some physical conditions (selection rules) and approximations. First we consider those cases where the reciprocal lattice vector involved in the momentum conservation relations (A functions) are zero (K = 0). Thus it should be noted that only those operator products will have non-zero finite averages which represent processes with zero total momentum (see ANDJGXSON~).In view of the selection rules derived by SZICETI(~~) the third order terms with coefficients such as Boo,,Boo,,Bgo,will not survive. Also, for a system having centre of symmetry, as in the present case, (boo +bOo +) i.e. QOo cannot enter in odd powers (1). This implies that the coefficients B,,,,, should be zero.
to
x A ( cs
K.
APPENDIX
(B.2)
-K)
lattice vector and w3s = w~~-I-D,,
C
HTr = exp( - is) H, exp( is) i2 = H, + i[H,, S] +$Hr,
Q.hexP(-i%*Rh)
[H,, s]]
h
B u*.u,’’ *b.”” = 2 bl,,+ht,,+hff
exp[i(a,,‘*Rh’+a,,,“.Rh”)]
C b,,O.’’ ,o,” ” .fJ#“’I” = 2 Cl,l+ltt,l+h~~,l+h~t,
exp[i(a,.‘.
R,‘+
csr,“. Rh” + cstsj”‘* Rh”‘)]
R, = R, -R,
03.3)
The sue s is used as a general symbol for optical (o) or acoustical (a) modes. It is convenient at this stage to go over to the second quantization representation by using the following transformations
where S=2-
d,wo=G(bOa+
-boa)
x (boo+f boo)(boo+ + boo) The relevant commutation
[Ho,
(B-4)
relations
(C.l)l
arc
Sl = 0, [[Ha, S], S] =;-
2/N
x ~~~~~~~~~~~ + + boo)
i
x (boo+ +boO)(boO+ $.b,O) We get
-
H=
x (boo + + b,O)
c
x (bonS”+b_.,,S”)A(a,+o,,‘+a,,,“-K)
x (b,.. s” + b_o,,~“+)(b,,,,s”’
+ S_,,,,,s”+)A(
cr, -+ os,’ + os,,” + c+”
-K)
(B.5)
COUPLING
ELECTRON-PHONON
If429Sl
=
ifi@ dN
0
a+ boa+)(boO++ boo)
x (boo+ -i- boo)w,~“G
IN
transformed HT
=
Hamiltonian 2
1847
DIELECTRICS reduces to
hoJO”(b,“+b,=
+*)
+
2 hw,O a#0
:(b,O+b,OSt)+hw,O(boO+~oO+l)
---
1
0
x
2N 2
2 Boo2
----+o”+ wOoo~Oa
+b,O)
x (boo++ boO)(boOf + 6,O) x BooG(boo+ +boo)(boo++boo)
x (boo+ + boo)
x (boo++ boo)(boo++ boo)
+zi
*
(C-2) The commutator [Hoc’, S] is non-diagonal. The commutators of S with [H,, S], [Hoe, S] and fourth order anharmonic terms give terms of the order l/N2/N. These can be safely neglected. We derive the value of G from the consideration HI -I- i[H,, which gives
G = B,,d
S]
fi 0 i
= 0
1 2-ooa51z
(
C-3)
oc fi
x (b_oa+b,a+)
+f(9”z --(hJOf c~~~~ + boo) f.ooO w,O
x (boo++boO)(b,O+ +b_.o) x (b_,“+b,o+) 1 ?i 2coo ---( b,O+ + b,O)
0
(C-4)
where B oo = Boa+ It may be noted that the canonical transformation used here is different from that of SILVERMAN.(*) In his expression, the coefficient similar to G contains a factor l/(w,a)a-(woO)a. Making use of (C.2) and (C.4) the
Go.,
+ boo) --Q---$@o”
x (boo++bo”)(b,a+b,_a+)
+22
WOO
2
(WOO)2
x (boo+ + boo)(boo+-t-boo) x (boo+ -I- b,O)
WI
The terms involving Boo2 and COO do not involve any summation and are of order l/N. Their signs are opposite and since COO, is expected to be positive they will largely cancel each other.
Chem. Solids.
Pergamon
Press 1966. Vol. 27, pp. 1837-1847.
ELECTRON-PHONON
COUPLING
21 April
IN DIELECTRICS*
and K. P. SINHA
G. C. SHUKLA
National Chemical Laboratory, (Received
Printed in Great Britain.
Poona 8, India
1966; in revisedform
17 June 1966)
Abstract-The instability of certain vibrational modes in ionic dielectrics such as BaTiOs is theoretically studied by taking account of electron-vibration coupling. It is found that the vibronic interactions in the elementary units e.g. Ti-0-Ti or TiOs octahedra render the polar modes unstable. This provides a microscopic description of the origin of ferroelectric phase transition. The temperature dependence of the frequency of the soft (the unstable transverse optical) mode is then investigated by considering the collective oscillations of a lattice of such elementary units after including anharmonic interactions. The stabilization of the soft mode, beyond a certain temperature T,, occurs as a result of fourth order anharmonic interaction involving the acoustic and the transverse optical mode. The temperature dependence of the effective frequency, namely, weea (transverse optical) CC(T- T,) is similar to the relation found by Silverman. The agreement with available experimental results is satisfactory.
1. INTRODUCTION
FOLLOWING the suggestions of ANDERSON(~)and COCHRAN,(~) several workers(3p4) have studied the ferroelectric phase transition in relation to the instability of certain lattice vibration modes of the system. These studies are confined to ionic dielectrics such as barium titanate. These treatments start with the assumption that the transverse optical frequency is imaginary in the harmonic approximation. c4) They could render the frequency real (positive) by including certain anharmanic terms in the potential function on the basis of a renormalization procedure. The temperature dependence of the frequency wr is then given by wr2 oc (T- Z’,), where T, is a characteristic temperature. This could also explain the CurieWeiss behaviour of the dielectric constant. This approach seems to be in the right direction; however, the assumption that the transverse frequency is imaginary in the harmonic approximation remains ad hoc. Also, the nuclear potential function was taken on the assumption that the electronic ground state is far removed from the excited states.(*) This is questionable and the role of electron-vibration coupling cannot be ignored. * Communication No. 881 from National Laboratory, Poona 8, India.
Chemical
This coupling may furnish a description of polarizability of ions as considered by Slater on phenomenological basis.t5) In fact, it was shown in a recent paper that electron-vibration coupling effects are indeed important.(s~‘) When one takes account of some low lying electronic states which mix with the ground electronic states via some vibrational mode a distortion of the basic ionic unit takes place resulting in a permanent dipole. In the paper referred to, the electron vibration problem involving the potential function up to some anharmonic terms for the unit Ti-0-Ti was considered. Kinetic energy terms were not included in the analysis.@) In the present paper, we intend to include the kinetic energy terms explicitly and study the lattice dynamics of such systems by recognizing the electronic as well as anharmonic interactions involved in the problem. We shall see that the Jahn-Teller like terms do indeed lead to expressions suggesting instability of certain modes (optical) which are eventually stabilized as a result of anharmonic interactions. The instability appears for individual units such as Ti-0-Ti or TiO, (octahedra) owing to Jahn-Teller like effects. Furthermore, the inclusion of electronic effects incorporates the covalency terms which may
1837
1838
G. C. SHUKLA
be necessary units.@)
in the
chemical
bonding
of the
2. VIBRONIC COUPLING IN ELEMENTARY UNITS In this section, we consider the coupling of the low-lying excited electronic states with the ground state of the elementary systems owing to certain vibrational perturbations. The calculation is confined to the perovskite or WO, type ferro-electrics. The crystal structure of perovskite type system is discussed at length in many references.(gJO) We shall make note of the nearest neighbour coordination of 02- and M (i.e. Ti*+ or W6+) tons. In the cubic phase we have the collinear units e.g. Ti-0-Ti with Ti-0 distance the same on both sides of 0 and in the regular octahedron TiOs. Our mission is to show that vibronic effects (electron-vibration coupling) will lead to the distortion of the units either Ti-0-Ti or TiO, resulting in a permanent dielectric dipole. The problem of the triatomic units e.g. M-O-M in the static approximation was solved in a previous paper.@) We shall briefly consider the vibrational problem for this and then take up the octahedral unit MO,. 2.1 The case of triatomic unit We consider the situation when we have two electronic levels for the system namely, the symmetrical state I/, and the antisymmetrical state #a. As before the energy separation is defined as 25 = &/a,,) - J%U
where H, is the bare electronic part of the Hamiltonian, H, the vibrational part which in the harmonic approximation is given by
ViQ!,
is the electron-vibration coupling. V” is the first derivative of the electron-lattice potential function with respect to the nuclear displacement coordinate Qi evaluated at equilibrium position. The problem is to diagonalize the Hamiltonian (2.2). It is expedient to rearrange the Hamiltonian (2.2) so that its relationship with the manifold of symmetrical state $, (ground) and antisymmetrical (excited state) #, is clear. We make use of Dirac’s bra (1 and ket )> notations.
For the triatomic unit under consideration, the vibrational modes of interest are the symmetrical vibration (Q,) and antisymmetrical (Qs) modes. The electron-vibration coupling terms which survive and couple the symmetrical (I/J and antisymmetrical state (&I,) through V”Qo where V” = (~V/CYQO)~ and has odd parity,
We shall now make use of a canonical transformation in order to remove the electron-vibration coupling term occurring in first order of Qe. The transformed Hamiltonian is given by H, = edisHeiS = H + i[H, S] -$d[H,
Sl, Sl+
. . .
(24
where we have taken S = ig(I~~~>~slQ~-I~s>(~alQ~)
(2.7)
The quantity g is so chosen that the linear term in Qs is eliminated from the transformed Hamiltonian. This is easily done by making use of the relation
(2.3)
where QI and Pi are normal coordinates and conjugate momentum of the ith vibrational mode; wi is the corresponding frequency;
He, = 2
K. P. SINHA
(2.1)
The problem then is to consider the coupling between the states #s and $, owing to antisymmetrical vibration. The Hamiltonian for the system is H = H,+H,+H,, (2.2)
HtJ= 8 2 (PF+ Wi2QF),
and
(2.4)
i[H,, S] + H,,
= 0.
(24
We get A ___g = (E,-IsE,)
(2.9)
ELECTRON-PHONON
COUPLING
HamilThus with A,, = A,,, the transformed tonian up to harmonic and diagonal terms is
IN
DIELECTRICS
1839
direction.@) In the earlier paper, it was estimated that for the unit Ti-0-Ti the condition A2 wa2 < J
HT = I#, )Es (&I + I&z )Kz <&I
QS2)+W'02+~02Q02) is obtained and one expects the instability. _ I~s>(&s>2<4slQo2 We shall now analyse the electron-vibration (%--Es) coupling for the octahedral unit namely ML,, where M is the metal ion and L stands for the +lW(&.)2~+a/Qo2 (2.10) ligand ion. (&z--Es) +&(P,2+~,82
As we are henceforth interested in the vibrational problem, we shall take the expectation value of the Hamiltonian with respect to I#,) or I$,). Thus, taking
= Es + B(Ps’ + us2
Qs2>
$$-)Qs2] s a where we find that the frequency tion is modified as
(2.11) of the vibra-
.
(2.12)
If the average is taken with respect to the state I#=) we would see that the effective frequency relation will have a plus sign in place of (12). The analysis given above shows that if the system is in the ground state having symmetrical electronic state (E,-Es) > 0, the effective frequency of the odd vibration decreases [cf. equation (2.12)]. On the other hand, when it is in the excited state E,, the frequency will be pushed up. This is a consequence of the electron-vibration coupling incorporated in the treatment. The electronic ground state is indeed totally symmetrical I&.>. Let us now examine the effective frequency wo2(eff)
=
(wo2 - g)
The condition as2 < A,,2/jJj makes the effective frequency imaginary which implies that instability develops in the system. It has been pointed out before that this leads to the distortion of the system in which the central atom moves in one direction and the two extreme atoms move in the opposite
2.2 The case of octahedral unit It is expedient to consider the problem with the help of a specific example. The most appropriate unit is the octahedral cluster TiOs with the Ti ion at the centre and each oxygen ion at the corners. The various electronic configurations of the complex (TiOJshave been discussed by NELSON@) following the molecular orbital approach. However, he has not considered the electron-vibration coupling which seems to us to be of central importance in the present problem. To understand this, a discussion of the one electron molecular orbitals and many electron states for the unit along with the vibrational modes will be essential. The definitions and notations used here are the same as those given by VAN VLECK;(~~)OPIK and PRYCE(~~)and PRYCEet aZ.(13) in a different context. The two ligand ions (e.g. 02-) on the t_ x axis are respectively numbered 1 and 4, the two on the + y axis 2 and 5 and the two on the fz axis 3 and 6. If we started from the ionic picture i.e. Ti4+ and 02- then the total number of electrons to be accounted for are as follows: 18(Ti*+)+6x 10(02-) = 78. Of these, 18 (in ls22s22p63s23p6) can be regarded with sufficient accuracy as completely located on Ti4+ centre and 6 x 4 = 24 are located on oxygens (in Isa, u_~).* The remaining electrons (36) would occupy orbitals which can approximately be regarded as built up from the combination of 2s, 2p orbitals on oxygen and 3s, 4s and 4p atomic orbitals on Ti4 + . We shall refer to the appropriate hybrids (sp) of 2s and 2p orbitals on 02- which have end on * The hybridization of 2s and 2p orbitals of the ligand ion will give two orbit&, which are designated as + and - where, the former points towards the metal ion and the latter in the diagonally opposite direction.
1840
G.
C.
SHUKLA
overlap with the metal ions as ui orbitals and the remaining 2~ orbitals as r orbitals. Here D and n respectively refer to those orbitals which have even and odd symmetry under rotation by 180” about the axis joining the metal and ligand ions. It should be noted that the bonding molecular orbitals involving the u orbitals of the ligand are expected to be of lower energy than those involving the 7~ orbitals because the u orbital function overlaps more on to the Ti and makes use of its attractive field than the n orbitals. Then in the increasing order of energy the molecular orbitals are given as b qgb,
tlub,
e!?
h7n*
tzga,
ega
’
tb 29
’
alga,
2 lub,
t2un>
tlua,
t’lua
Here the superscripts, b, n and a refer to bonding, non-bonding and antibonding orbitals respectively; al, e and t are the group theoretical notations of the orbitals (see GRIFFITHS~*). In the absence of electron-vibration interactions and for the regular octahedron, the ground electronic state has the configuration
and
K.
P.
SINHA
with a similar expression Likewise we will have
for transition
I * - * (tzu”>6; (lAl,)
from
> + l(tzu”)5(t2g”); lAlu
or lT1,
or
IE,, or (2.15a)
ITa, > I . . . (tzl(‘%
(t’l,b)“.
(1-4,) > --f I . . . (tz,“)“(ega>; ‘Tlu
or ‘Tzu > (2.15b)
We now consider the various vibrational modes for the octahedral unit MO,. There are fifteen normal coordinates of vibration which belong to the following types of representations : A 157, E,,
T,,,
Tlu’,
T2,
and
T2,.
From symmetry considerations alone we can see that the various excited electronic states can couple with the ground state [#a; IAl,) via the vibrational perturbation having appropriate symmetry. The Hamiltonian involving electron-vibration interaction for the present problem is similar to
(2.13) I~g(14g)> -+ ~(~19b>2(esb>4(t~~b>6(tZOb)606(tZun~6~~~gn~61
This accounts for all the electrons and the states until bonding and non-bonding orbitals are completely filled. Let us now consider the excited states in which a single electron is transferred from one of the bonding or non-bonding (predominantly ligand orbitals) t, orbitals to one of the antibonding orbitals. Group theoretically various symmetry states are possible;@) I . . - (t,,b)6; PA,,)
Hi = 2
(2.5). The general form can be written IN’
or
or lTlu
IE,. or (2.14a)
IT,,) + l(tlUE)5(eg”); lT1, lTall>
or (2.14b)
“) )&I
(+~(r,“)l
CPia2
+ wi2 Qia?
as
ia + S 2 i.n +
2 (Lw(jJ) (k.?‘)
x
bvf)
> -+ l(tl,b)5(t2,a); lA2,
1 . . . (t1,b)6; (lA,,)>
equation
i
Id4ri”)
1a2
)Ar,%,‘(r,‘)
(2.16)
where ri refers to the irreducible representation of the octahedral group. Thus j#(I’ioL)) denotes the crth base function belonging to the Fi irreducible representation. Qrla likewise represents a similar vibrational mode
ELECTRON-PHONON
COUPLING
IN
with
with
A g.ua=Aa u .g
(2.18) It is clear from equation (2.17) that only those matrix elements will survive for which the corresponding products representation Ft X rk x r, will contain the identity representation I’, on decomposition. In the present problem we are not interested in a general situation of equation (2.16). Our interest is to see as to how this ground electronic state [$,,(lAl,)) couples with the excited states via equation (2.17). Furthermore only low lying excited electronic states will have any significant vibronic interaction. The low lying electronic state of symmetry T,, is of particular interest in the present problem. Such states, will couple with the ground denoted by j$,(Tl,)), configuration j+!~,(~/l~,)) through the vibrational modes of symmetry T,,. Of these, the mode in which the positive ion moves against two of the ligand ions in opposite directions gives rise to the dipole moment. It is referred to as wlc mode with one of the Q’s called as Q1s (see KOIDE and tiYCE(15)). This describes the rattling of the positive ion in the octahedron of negative ions. The importance of coupling via this mode has also been stressed by ORGEL. We shall therefore confine our attention to the vibronic coupling of states \I#,(~A~,)) and jt,!QT~J) via the appropriate vibrational modes T,,. The Hamiltonian (2.16) can be written as: H red
1841
DIELECTRICS
=
I&
+t
)-%
c
a
<&I
+
&LQ?
+9 C
cu.f(# Td
2
l&i=
)-%t
(that
1
m,?Qm,oL?)
P'i.a2 + wi2Qi,a2)
(2.21)
Thus, the transformed vibrational Hamiltonian after the expectation value over the ground electronic state is expressed as
Hv,,(E,) = E,++
2 (%a2+ QQi.a2) a#(Tld
(2.22) Thus, we see that for the elementary octahedron also the vibronic interaction can render the effective frequency Weft
2=
WTlU2 -
24.ua2
--
(&
imaginary
-
(2.23)
4,)
i.e. wrl,a < This instabilitywilllead to a distortionofthe system in which the metal ions moves in either of the x, y, z directions with the movement of the two appropriate ligands in the opposite direction. This will give a permanent dipole moment. This seems to be possible in any of the three directions for the elementary octahedron. However, as has been discussed in Appendix A, the environment of oxygen ions in BaTiOs is such that only one of the directions (say z) will be preferred. So far we have discussed the vibronic problem for the elementary units, namely, Ti-0-Ti and TiO,. In the next section we consider a lattice of such units and discuss the vibrational problem of the lattice including anharmonic terms as well as vibronic interactions in the elementary units.
3. COLLECTIVE OSCILLATIONS In the previous sections, we have analysed the a vibrational problem of elementary units (tri-ionic (2.19) + C.C.) or octahedral) by taking account of the vibronic coupling caused by some specific modes. This gave The transformation analogous to equation (2.6) rise to instability of the modes in question. Let us can be used now, where now consider an assembly of such units forming a S(oct) = ic G”(l#,“) <~,IQ~u~--l~o)lattice. In BaTiO,, the TiO, octahedra are joined by their corners. This forms a three dimensional (2.20) <#ualQ~ua) network and the large holes are occupied by the + ( c
I#,
>4,ua(T1.“)
(#,“I
Qcq,~
1842
G.
C.
SHUKLA
and
Ba2+ ions. Of course, in BaTiO, the Ba2+ ions do play some role in view of the large size and overlap with oxygen ions. Perhaps in WO, type ferroelectrics (or from structural considerations in ReOs) one has to consider the lattice of octahedra alone. We confine our attention to the vibrational problem of the lattice made up of these units. For N such units and including anharmonic terms the lattice Hamiltonian is
K. P. SINHA c __
2
H2=&;
02
a.0
WO OaJoa
x(b,O+ +boo)(boo-!-boo+) ~(b,=+b_.=+)(b_,~+b,~+) (3.7) H32i
2 (I[
co.0 (woO)a(boO+boO+)
x (boo+ boo +)(boo + boo +)
almQIAQm,,
x(boo+boo+)+~
bnnQ,~QmuQnv Ci,mnQisQ~~QmrrQnv (3.1) where the frequency, wrL2 refers to the Ith unit and its hth mode. P,, = f&and QIn are the normal coordinates of the Zth unit and its Xth mode. In the above, we have included the translational mode of the individual units also in order that the acoustical mode for the whole crystal is taken properly into account. For present purposes we shall be interested in a detailed study of the modes which involve the vibronic instability [cf. equation (2.23)] and their interactions with the acoustic mode contained in the third and fourth order anharmonic terms. As shown in Appendix B, in second quantization representation the Hamiltonian (3.1) is transformed to HT = H,+Ho+Hoo+H,+H,+H,,
(3.2)
where
c wooco~o
x (boo + boo+)(boo + boo +) x (boo +b_,O+)(b_,O
+b,‘+)
1 (34
The external field Eayt _.__has not been introduced so far in our Hamiltonian. We are for the moment interested in the lattice dynamical problem without the influence of such a field. For the present our primary interest is to see the effect of interaction between acoustic modes on the long wavelength optical modes. The third order anharmonic terms [cf. equation (3.6)] can be eliminated, to a large extent, by an appropriate canonical transformation (see Appendix C). We shall thus consider the effect of the acoustic and optical modes with (a0 # 0) on the long wavelength polar mode woo by taking thermal averages as explained below. We write the diagonal part of the Hamiltonian explicitly
Ha =
2 fi w,“( boa+ boa + 3)
(3.3)
Ho =
c fi w,“( b,’ + boo + 4) a#0
(3.4)
+ 2 ?iwo0(b,0+b,o+3) O#O
(3.5)
++‘o”‘+(wooz+;( 5 C,.,o
Ho0 = fiwoO(boO+boo+i) ti 3’s Boo,a _ HI = -1/N 0 2 OJoowo~~‘a
HTa = 2 fh,“(b,“+b,a+~)
1
-t- 1 c,,,,CQoo2)))Qoo’], O#O
x (boo + + boo)(boo + + boo) x(boa+boa+)
(3 ~5)
where we have introduced Ofi0 etc. for convenience.
(3.9)
the variable PO0 and We shall. however,
ELECTRON-PHONON
restrict the coupling with the acoustical whose thermal average is given by
mode
IN
1843
DIELECTRICS
which can be written as H = B 2 (PO,2 +
0.
= kBT/hoaa.
Thus the effective frequency QJeff
COUPLING
is
+3(G2
+
~o,,~Qc,,~)
%ff2
bom,2)
(4.2)
where now
a=
+;c
3)
echo
(3.10)
where Jgu = &(E,- E,) and wU is the frequency of the optical mode of one unit [i.e. equation (2.23)]. 2 is the number of the nearest units coupled to unit at 1. The term may have its origin in the electrostatic interaction between dipoles. For a long range interaction it may have the form -$ne2L where L is the local Lorentz field parameter for u = 0. For cubic system, L = 47r/3 for transverse mode and L = -&r/3 for longitudinal mode. This will give a negative contribution for the transverse mode. Our expression (3.10) differs from a corresponding relation of Silverman in many ways.(4) First mu2 is positive. The instability arises because of the terms - Agu2/1J,,] owing to vibronic interaction. This type of terms are discussed for the first time here. The dipole-dipole interaction may also give negative contribution. In the event u,,~ is outweighed by these negative terms, the stabilization comes from (1/N)C,C,~~okBT/(woa)2. The expression (3.10) also gives the same temperature dependence as obtained by SILVERMAN(~)although the factors are different W,tr2 0~ (T- T,)a 4. DISPERSION OF THE SOFT MODE In this section, we study the wave vector dependence of the soft optical mode. This is carried out following a simplified picture. We have seen that only fourth order anharmonic terms are significant. Thus we consider the two modes, the soft optical and the acoustic modes and their interactions. The Hamiltonian is
If we confine our attention to the interaction between the nearest neighbours only, then after taking the thermal average (QGa2), we get
Weff2(cTO) = [ f.!J,2--+y PU
+2D 2 (cos csOZR 00 + cos cr,,YR+ cos a,“R)
where R is the distance between two octahedra and D denotes the coefficient for the dipole-dipole interaction. For transverse oscillation it will be negative. The summation over us, aa in the last term of equation (4.4) is rather complicated for a general three dimensional case. We shall therefore write this result for a one dimensional chain of elementary octahedra. Now w, 2 = 4a sin2(uaa) with a as the force constant, R” = 2a, a being the distance between a metal ion and a ligand ion. Further, we get,
=-
4c N 82 60 x
sin2( aaa) u.
kBT a sin2( oaaa) I
cos2(uoa)
= 4c 2 uo
F
cosy aoa)
(4.5)
1844
G.
Finally, putting model gives
Weff
2=
SHUKLA
20 = -1~1 the one dimensional
zF
-___
A, 2 -
wu2
-I-
C.
y
ii
cos(2 oaa)
1 go
1
2 /3T COS~(G~~)
(4.6)
J
go
where p = 4Ck,/a. The temperature dependence of the frequency now be written for os = 0 as
can
\ where T, =
A 2 --=+
(4.7)
y-wu2
l-&1 The variation of weff2 on ue at any particular temperature is given by equation (4.6). This is given in Fig. 1, for arbitrary values of the parameters involved. We get the same sort of depenACos(26oa)f
1
BCzs.(doa)=f(da)
A--B,
8
E-12
f
I .I
I .2
I .3
I .4
boa -__, Tl-. FIG. 1, Wave-vector and temperature dependence of the soft optical mode where A = Iy] and B = @T [cf. equation (4.6)].
dence as obtained by Silverman although our functions are different from his.c4) As in his case, the agreements with the results of cOwLEY’18) is reasonably good.
and
K. P. SINHA
5. CONCLUDING REMARKS In the preceding sections, we have taken account of the low lying electronic states in the vibrational problem of some ionic dielectrics. To incorporate the vibronic interaction, we have chosen appropriate elementary units e.g. Ti-0-Ti or TiO, octahedra. The ground electronic states of such systems interact with the low lying odd states via the odd vibrational modes. On eliminating such terms by a renormalization procedure, we get an effective frequency for such units which contains vibronic interaction effects and can render the frequency imaginary or zero. This effect is attributed to be the cause of the instability of the mode giving rise to the electric dipole moment. Starting with such unstable units we then studied the collective oscillations of a system of elementary units. Anharmonic terms were also included. The approach is similar to those adopted by ANDERSON(~) and SILVERMAN;(~) however, there are important
differences
in the starting
model.
The we
fourth
have
coupling optical
of
the
to the differences
anharmonic main
acoustic
mode,
frequency.
order
taken
owing
mode
terms,
interaction and
the
give rise to temperature
Interactions
between
the
where from
in the
transverse dependent elementary
units are of the dipolar
type. An explicit form of the dependence of frequency on the wave vector could be given only for a one-dimensional chain. Both the predicted temperature and wave-vector dependences show the right trend. The main purpose of the present paper has been to invoke the role of vibronic interaction towards the instability of certain lattice modes. This has done away with the ad hoc assumptions made by earlier workers. REFERENCES 1. ANDERSON P. W., Paper presented at All-Union
2. 3. 4.
5. 6.
Conference Dielectrics, Moscow, 1958 [Russian Version Published in &zika Dielektrik& (editor G. I. Skanavi). Akad. Nauk SSSR MOSCOW, (1960)]. ” COCHRAN W., Phys. Rev. Lett. 3, 412 (1959). COWLEY R. A., Pkys. Rew. 134, A981 (1964). SILVERMAN B. D., Phys. Rev. 135A, 1596 (1964); This contains references to earlier papers by him and collaborators. SLATER J. C., Pkys. Rev. 78, 748 (1950). SINHA K. P. and SINHA A. P. B., Indias J. Pure Appl. Phy., 2, 91 (1964).
ELECTRON-PHONON
COUPLING
7. SHUKLA G. C. and SINHA K. P., Indian J. Pure Appl. Phys. 3, 430 (1965). 8. NELSON C. W., Technical Report 179, Laboratory of Insulation Research, MIT, 1963. 9. JONA F. and SHIRANE G., Ferroelectric Crystals. Pergamon Press, Oxford (1962). 10. KANZIC W., Solid State Physics, Volume 4, (editors F. Seitz and D. Tumbull). Academic Press, New York (1957). 11. VAN VLFCK J. H., J. Chem. Phys. 7, 72 (1939). 12. ~PIK U. and PFtYCE M. H. L., Proc. R. Sot. 238A, 425 (1957). 13. PRYCE M. H. L., SINHA K. P. and TANABE Y., Molec. Phys. 9, 33 (1965). 14. GRIFFITHS J. S., The Theory of Transition Metal Ions. Cambridge University Press, London (1961). 15. KOIDE S. and PRYCE M. H. L., Phil. Mug. 3, 607 (1958). 16. ORCEL L. E., Discuss. Faraday Sot. 2, 138 (1958). 17. SZICETI B., Proc. R. Sot. 252A, 217 (1959). 18. COWLEY R. A., Phys. Rev. Lett. 9, 159 (1962).
APPENDIX
A
We consider the explicit forms of the normal coordinates. Since, there are two representations of type T,, their interactions have to be considered. This evaluation involves certain assumptions about the forces between atoms. Such calculations based on valence force and central force models have been carried out by KOIDE and PRYCE.“~) We shall use the final results. The ratio of the masses of oxygen and Ti i.e. MO/MT, = 3. Thus the normal coordinates of present interest are
“(&s +Z, -WA Qu"(TaJ= d(;)(G+ Y5-2Yd
IN
1845
DIELECTRICS
I
(A-2)
-$gz,Y,4-l- . . .
where R. is the distance between the metal and ligand ion in the undistorted octahedron.
(A-3)
The summation is over the k electrons of the central ion. The other components namely, P,, and V,, can be obtained by a cyclic permutation of equation (2.20). Although, the quantities V,,, etc. have been deduced on an electrostatic model, the general form given in equation (2.20) will remain the same on other models also, apart from the change in the definitions of the coefficients a, 6 and c etc. The polynomials involving electronic coordinates etc. clearly indicate that the vibronic perturbations will mix excited electronic states of T,, symmetry with the ground state.
&,"(TI,) =
&‘(Td
( 1(X,+x,
= 2/ ;
APPENDIX
B
Let us introduce the transformation normal coordinates, namely,
to lattice wave
-2x,,)
(A-1)
The corresponding derivative of the potential function [cf. equation (2.18)] can be easily deduced for an electrostatic interaction between the Iigand point ions and an electron of the metal ion at distance rr,, apart, namely q/rot where p = Ze. The method of Van Vleck is adopted here. We get
vu2
(av_ ) = +Ro5
=
8QU2
b --(
Zk3 - g+J~2)
where Qo, and Q, are lattice modes corresponding to the optical and acoustic modes of vibrations respectively; oa, cro are the propagation vectors and R, is the vector spanning the lth unit. The Hamiltonian (3.1) is
I846
G. C. SHUKLA
transformed
and
H = 2 iiPoaPo,* + Wo.2Qo,Qo,*>
x A(c,
+
cs,’ + cst,” -K)
+ r+‘+
where K is a reciprocal
D, = 2
c+”
+ os,,,“’
P.
SINHA
In proceeding further, we shall be guided by some physical conditions (selection rules) and approximations. First we consider those cases where the reciprocal lattice vector involved in the momentum conservation relations (A functions) are zero (K = 0). Thus it should be noted that only those operator products will have non-zero finite averages which represent processes with zero total momentum (see ANDJGXSON~).In view of the selection rules derived by SZICETI(~~) the third order terms with coefficients such as Boo,,Boo,,Bgo,will not survive. Also, for a system having centre of symmetry, as in the present case, (boo +bOo +) i.e. QOo cannot enter in odd powers (1). This implies that the coefficients B,,,,, should be zero.
to
x A ( cs
K.
APPENDIX
(B.2)
-K)
lattice vector and w3s = w~~-I-D,,
C
HTr = exp( - is) H, exp( is) i2 = H, + i[H,, S] +$Hr,
Q.hexP(-i%*Rh)
[H,, s]]
h
B u*.u,’’ *b.”” = 2 bl,,+ht,,+hff
exp[i(a,,‘*Rh’+a,,,“.Rh”)]
C b,,O.’’ ,o,” ” .fJ#“’I” = 2 Cl,l+ltt,l+h~~,l+h~t,
exp[i(a,.‘.
R,‘+
csr,“. Rh” + cstsj”‘* Rh”‘)]
R, = R, -R,
03.3)
The sue s is used as a general symbol for optical (o) or acoustical (a) modes. It is convenient at this stage to go over to the second quantization representation by using the following transformations
where S=2-
d,wo=G(bOa+
-boa)
x (boo+f boo)(boo+ + boo) The relevant commutation
[Ho,
(B-4)
relations
(C.l)l
arc
Sl = 0, [[Ha, S], S] =;-
2/N
x ~~~~~~~~~~~ + + boo)
i
x (boo+ +boO)(boO+ $.b,O) We get
-
H=
x (boo + + b,O)
c
x (bonS”+b_.,,S”)A(a,+o,,‘+a,,,“-K)
x (b,.. s” + b_o,,~“+)(b,,,,s”’
+ S_,,,,,s”+)A(
cr, -+ os,’ + os,,” + c+”
-K)
(B.5)
COUPLING
ELECTRON-PHONON
If429Sl
=
ifi@ dN
0
a+ boa+)(boO++ boo)
x (boo+ -i- boo)w,~“G
IN
transformed HT
=
Hamiltonian 2
1847
DIELECTRICS reduces to
hoJO”(b,“+b,=
+*)
+
2 hw,O a#0
:(b,O+b,OSt)+hw,O(boO+~oO+l)
---
1
0
x
2N 2
2 Boo2
----+o”+ wOoo~Oa
+b,O)
x (boo++ boO)(boOf + 6,O) x BooG(boo+ +boo)(boo++boo)
x (boo+ + boo)
x (boo++ boo)(boo++ boo)
+zi
*
(C-2) The commutator [Hoc’, S] is non-diagonal. The commutators of S with [H,, S], [Hoe, S] and fourth order anharmonic terms give terms of the order l/N2/N. These can be safely neglected. We derive the value of G from the consideration HI -I- i[H,, which gives
G = B,,d
S]
fi 0 i
= 0
1 2-ooa51z
(
C-3)
oc fi
x (b_oa+b,a+)
+f(9”z --(hJOf c~~~~ + boo) f.ooO w,O
x (boo++boO)(b,O+ +b_.o) x (b_,“+b,o+) 1 ?i 2coo ---( b,O+ + b,O)
0
(C-4)
where B oo = Boa+ It may be noted that the canonical transformation used here is different from that of SILVERMAN.(*) In his expression, the coefficient similar to G contains a factor l/(w,a)a-(woO)a. Making use of (C.2) and (C.4) the
Go.,
+ boo) --Q---$@o”
x (boo++bo”)(b,a+b,_a+)
+22
WOO
2
(WOO)2
x (boo+ + boo)(boo+-t-boo) x (boo+ -I- b,O)
WI
The terms involving Boo2 and COO do not involve any summation and are of order l/N. Their signs are opposite and since COO, is expected to be positive they will largely cancel each other.