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Ferroelectrical modes at high concentrations of elementary excitations
Volume 93A, number 1
PHYSICS LETTERS
20 December 1982
FERROELECTRJCAL MODES AT HIGH CONCENTRATIONS OF ELEMENTARY EXCITATIONS 1’ D.I.~j. MI1UANICa andBanjaluka, B.S. TO~IC a Technological Faculty, Yugoslavia b Institute of Physics, Novi Sad, Yugoslavia Received 12 July 1982
The presence of soft-mode ferroelectrics of the KDP type was investigated. The type of soft-mode ferroelectrics and the corresponding critical temperature were also analysed.
The possibifity of the existence of soft modes in KDP ferroelectrics [12] is analysed in this paper. According to ref. [2], the soft-mode frequency, having the critical dependence on temperature (i.e. ~ (T Tc)cx, a >0), can be found from the equation ~
—
(I) where e(i) is the dielectric constant of the system. The frequency ~ is a complex variable, in the general case. The relation between its real and imaginary parts characterizes the type of a soft mode. For resonant modes the relation IRe Z~I I Im ~I w 2 is valid. In the opposite case, i.e. when ~ ~ we have the so-called relaxative modes. The soft modes are not necessarily of a phonon character, there is the possibility of the appearance of soft modes in order—disorder ferroelectrics (see refs. [2,3]). In order to examine the last type of soft modes we shall find an expression for the dielectric permeability of ferroelectrics at high concentrations of excitations, i.e. in the domain of the transition temperature. The analogous problem for the exciton case as well as for the spin-waves has been analysed [4—61. Therefore, the mathematical details will be omitted, and only the final results, adopted for KDP ferroelectrics, wifi be given here. The paulian Green function 1 ~ [‘(K,
w)
=
(2ir)
f dt ((P(n, t)IP~(m,0))) exp(—ik.I
+
iwt),
1= n
—
m
(2)
can be expressed in terms of the boson Green functions G(K,
w)(2ir)~
~I)f
dt((B(n, t)IB~(m,0))) exp(—ik .1 +
I=n
iwt),
—
m
(3)
,
in the following way: F(x, w)
=
(1
+
2~
2h)1 [f 1(n)G(K, w) +f2(~)~Z
ql,q2
f —00
dw~dw
0~(q 2 G~
0)(q 1, w1)G(O~q2,w2)G(
3,
(o)3)]~
(4)
where 44
0 031-9l63/82/0000—0000/$02.75 © 1982 North-Holland
Volume 93A, number 1
PHYSICS LETTERS
20 December 1982
f1(n)=(l +4ñ +8ñ2 +8ñ3 +4ñ4)/(1 +2h)4, f2(n)= 21(1 +2ñ)6, {exp[J(0)/20] l}_1 20/J(0), q 3 = K—q1 +q2, w3 = w 00
—
—
(5)
W1 +
and ~)Zrepresents the number of molecules in the crystal. Using the harmonical approximation for the boson Green function G(x, w) in (4), i.e. 0)(K, w) = (i/2ir) [w ~2h(K)]1 (i/2ir)(w f2~)~ ~ =J(0)/2h (6) G(ic, w) G( where ~2h (K) is the frequency of noninteracting excitations, and neglecting spatial dispersion in the final result, we obtain the following expression for the paulian Green function I’: —
—
,
,
i
[fi(~) f 2Qi) 2ir I + —--(w—~2 )C~_2 27Tw—fl~L1+2~1+2h 1
x E q1,q2
f
dw1 dw2G(q1,w2)G(q2,w2)G(q1 _q2,w3)].
(7)
The dielectric permeability, taken in the isotropic approximation, can be expressed over F(w) in the following way
(see refs. [4,5]):
1/e(w)’~1 +(r0E~/8ith)[’(w)+F(—~.~)]
(8)
,
where r is the elementary cell volume and E0 + iw2 in eq. (8), we obtain finally r0E1~f2,~A(O)
—
5 (w~
—
e(w) where =1+ A(O)
=
I
+
4irh(1 +
4ñ
+
8h2
—
is
the electric field per elementary cell. After the substitution t~
fl2)2
+
4w~w~ 4irh(1
+ 2n)5
2h) +
r
r0E~
—
-
8h3
+
4ff4
+
1/2(1
+
2ff)2
,
B(O)
3ir/64(1
+
L (w2 i3(O)+
2~L~A(O)w1w2 —
—
fl2)2
+
~
2n)2fl~(0)
(9)
(10)
and fl~(0)= x(0)/Ii is the transfer frequency of excitations in the ferroelectric. As is seen from eq. (9), the relation l/e(w) = 0 can be satisfied for w 1 and w2 which are functions of temperature. Equating the right-hand side of eq. (9) with zero, we obtain a very complicated system of algebraic equations which cannot be solved analytically. In order to estimate the behaviour of the system in the critical temperature region we shall assume
I~4—(wi —f2~)I’~2w1Iw2I. The relation 1/6(W)
0 reduces in this case to 5fl~4(O)12 ~A2(0) fl~A(O) I 2 4irII(1 +2~) r 2(O) 4B2(O) = W 0E~B 2 = 2B(O) w1 42Hz1, A(8) 1, fl~ 5 X 1013 Hz, r 3 we obtain TakingB(O) 10 0 10—24 cm~,E~ 1012 erg cm fl~A2(O) 4irh(1 +21i)5fl A(O) 1027 Hz2 l051Hz4 1026 Hz2 4B2(O) r 2(O) 0E~B
~r —
=
ji
—
—
,
—
,
,
(11)
(12)
45
Volume 93A, number 1
PHYSICS LETTERS
20 December 1982
so that the approximate solutions of (11) are A(O) 2ith(l +2ñ)5A(O) r 0E~B~(O) w2 It is easily concluded that w1 1w2 I. The condition defining the critical temperature w1 approximately to w1 = 0. So we obtain
13
—
—
—
—
OckBTc=0.192J(0),
( )
—~.
~-
J(0)=2F1fl~
,
+ iw2
=
0, reduces
T~=l39K.
(14)
Let us consider now the case Iw~—(wi + fl~)!~’ 2w11w2 I. Using this condition one can reduce the equation l/e(w)=Oto 4 8A2(O)B2(O)fl~ Ifl~ 1w 2 r 0EO 4 1 2 r0E0 fl~A(O)l 5 J 1 322ir4h4(l +2h)20 0. (15) L 4irh(1 +2h) The estimations, analogous to eq. (12), give —
—
—
_____—-
1027Hz2
T 26Hz2 r4 2(O)B2(O)fl~ 0E~fl~A(6)10 04A 4irh(1 +2ñ)5 322ir4h4(l +2ñ)20 and this leads to the following approximate solutions of eq. (16): ,
1052Hz4
(16)
,
r 0E~A(U)
5
r~E~’A(O)B(O) w2— 327r2112(l + 2ñ)10
O~=kcTB
0.121J(0),
(17)
.
We have w 8irh(1 + 2ñ) 1 ~ 1w2 i again, and the critical temperature can be found from the equation w1
=
0. So we obtain
J(0)= 2hfl~~ Tc = 87K. ,
Concluding these analyses we can say the following: (a) The soft mode exists in ferroelectrics of KDP type. (b) This soft mode is of resonant type I Re wI I Im w I. (c) The critical temperature of the soft mode is of the order of 100 K. References Ill
F. Jona and G. Shirane, Ferroelectric crystals (Pergamon Press, Oxford, 1962). [2] R. Blinc and B. Zeks, Soft modes in ferroelectrics and antiferroelectrics (North-Holland, Amsterdam, 1974) pp. 20—26. [31H.E. Stanley, Introduction to the phase transitions and critical phenomena (Clarendon Press, Oxford, 1971). [4] D. Mirjani~and D. Had~iahmetovi~, Phys. Lett. 90A (1982) 264. [5]U.F. Kozmidis-Luburié and B.S. To~ié,Physica 112B (1982) 331. [6] B.S. To~ié,M.M. Marinkovié and S. Berar, Phys. Stat. SoL 81(1977) 245.
46
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PHYSICS LETTERS
20 December 1982
FERROELECTRJCAL MODES AT HIGH CONCENTRATIONS OF ELEMENTARY EXCITATIONS 1’ D.I.~j. MI1UANICa andBanjaluka, B.S. TO~IC a Technological Faculty, Yugoslavia b Institute of Physics, Novi Sad, Yugoslavia Received 12 July 1982
The presence of soft-mode ferroelectrics of the KDP type was investigated. The type of soft-mode ferroelectrics and the corresponding critical temperature were also analysed.
The possibifity of the existence of soft modes in KDP ferroelectrics [12] is analysed in this paper. According to ref. [2], the soft-mode frequency, having the critical dependence on temperature (i.e. ~ (T Tc)cx, a >0), can be found from the equation ~
—
(I) where e(i) is the dielectric constant of the system. The frequency ~ is a complex variable, in the general case. The relation between its real and imaginary parts characterizes the type of a soft mode. For resonant modes the relation IRe Z~I I Im ~I w 2 is valid. In the opposite case, i.e. when ~ ~ we have the so-called relaxative modes. The soft modes are not necessarily of a phonon character, there is the possibility of the appearance of soft modes in order—disorder ferroelectrics (see refs. [2,3]). In order to examine the last type of soft modes we shall find an expression for the dielectric permeability of ferroelectrics at high concentrations of excitations, i.e. in the domain of the transition temperature. The analogous problem for the exciton case as well as for the spin-waves has been analysed [4—61. Therefore, the mathematical details will be omitted, and only the final results, adopted for KDP ferroelectrics, wifi be given here. The paulian Green function 1 ~ [‘(K,
w)
=
(2ir)
f dt ((P(n, t)IP~(m,0))) exp(—ik.I
+
iwt),
1= n
—
m
(2)
can be expressed in terms of the boson Green functions G(K,
w)(2ir)~
~I)f
dt((B(n, t)IB~(m,0))) exp(—ik .1 +
I=n
iwt),
—
m
(3)
,
in the following way: F(x, w)
=
(1
+
2~
2h)1 [f 1(n)G(K, w) +f2(~)~Z
ql,q2
f —00
dw~dw
0~(q 2 G~
0)(q 1, w1)G(O~q2,w2)G(
3,
(o)3)]~
(4)
where 44
0 031-9l63/82/0000—0000/$02.75 © 1982 North-Holland
Volume 93A, number 1
PHYSICS LETTERS
20 December 1982
f1(n)=(l +4ñ +8ñ2 +8ñ3 +4ñ4)/(1 +2h)4, f2(n)= 21(1 +2ñ)6, {exp[J(0)/20] l}_1 20/J(0), q 3 = K—q1 +q2, w3 = w 00
—
—
(5)
W1 +
and ~)Zrepresents the number of molecules in the crystal. Using the harmonical approximation for the boson Green function G(x, w) in (4), i.e. 0)(K, w) = (i/2ir) [w ~2h(K)]1 (i/2ir)(w f2~)~ ~ =J(0)/2h (6) G(ic, w) G( where ~2h (K) is the frequency of noninteracting excitations, and neglecting spatial dispersion in the final result, we obtain the following expression for the paulian Green function I’: —
—
,
,
i
[fi(~) f 2Qi) 2ir I + —--(w—~2 )C~_2 27Tw—fl~L1+2~1+2h 1
x E q1,q2
f
dw1 dw2G(q1,w2)G(q2,w2)G(q1 _q2,w3)].
(7)
The dielectric permeability, taken in the isotropic approximation, can be expressed over F(w) in the following way
(see refs. [4,5]):
1/e(w)’~1 +(r0E~/8ith)[’(w)+F(—~.~)]
(8)
,
where r is the elementary cell volume and E0 + iw2 in eq. (8), we obtain finally r0E1~f2,~A(O)
—
5 (w~
—
e(w) where =1+ A(O)
=
I
+
4irh(1 +
4ñ
+
8h2
—
is
the electric field per elementary cell. After the substitution t~
fl2)2
+
4w~w~ 4irh(1
+ 2n)5
2h) +
r
r0E~
—
-
8h3
+
4ff4
+
1/2(1
+
2ff)2
,
B(O)
3ir/64(1
+
L (w2 i3(O)+
2~L~A(O)w1w2 —
—
fl2)2
+
~
2n)2fl~(0)
(9)
(10)
and fl~(0)= x(0)/Ii is the transfer frequency of excitations in the ferroelectric. As is seen from eq. (9), the relation l/e(w) = 0 can be satisfied for w 1 and w2 which are functions of temperature. Equating the right-hand side of eq. (9) with zero, we obtain a very complicated system of algebraic equations which cannot be solved analytically. In order to estimate the behaviour of the system in the critical temperature region we shall assume
I~4—(wi —f2~)I’~2w1Iw2I. The relation 1/6(W)
0 reduces in this case to 5fl~4(O)12 ~A2(0) fl~A(O) I 2 4irII(1 +2~) r 2(O) 4B2(O) = W 0E~B 2 = 2B(O) w1 42Hz1, A(8) 1, fl~ 5 X 1013 Hz, r 3 we obtain TakingB(O) 10 0 10—24 cm~,E~ 1012 erg cm fl~A2(O) 4irh(1 +21i)5fl A(O) 1027 Hz2 l051Hz4 1026 Hz2 4B2(O) r 2(O) 0E~B
~r —
=
ji
—
—
,
—
,
,
(11)
(12)
45
Volume 93A, number 1
PHYSICS LETTERS
20 December 1982
so that the approximate solutions of (11) are A(O) 2ith(l +2ñ)5A(O) r 0E~B~(O) w2 It is easily concluded that w1 1w2 I. The condition defining the critical temperature w1 approximately to w1 = 0. So we obtain
13
—
—
—
—
OckBTc=0.192J(0),
( )
—~.
~-
J(0)=2F1fl~
,
+ iw2
=
0, reduces
T~=l39K.
(14)
Let us consider now the case Iw~—(wi + fl~)!~’ 2w11w2 I. Using this condition one can reduce the equation l/e(w)=Oto 4 8A2(O)B2(O)fl~ Ifl~ 1w 2 r 0EO 4 1 2 r0E0 fl~A(O)l 5 J 1 322ir4h4(l +2h)20 0. (15) L 4irh(1 +2h) The estimations, analogous to eq. (12), give —
—
—
_____—-
1027Hz2
T 26Hz2 r4 2(O)B2(O)fl~ 0E~fl~A(6)10 04A 4irh(1 +2ñ)5 322ir4h4(l +2ñ)20 and this leads to the following approximate solutions of eq. (16): ,
1052Hz4
(16)
,
r 0E~A(U)
5
r~E~’A(O)B(O) w2— 327r2112(l + 2ñ)10
O~=kcTB
0.121J(0),
(17)
.
We have w 8irh(1 + 2ñ) 1 ~ 1w2 i again, and the critical temperature can be found from the equation w1
=
0. So we obtain
J(0)= 2hfl~~ Tc = 87K. ,
Concluding these analyses we can say the following: (a) The soft mode exists in ferroelectrics of KDP type. (b) This soft mode is of resonant type I Re wI I Im w I. (c) The critical temperature of the soft mode is of the order of 100 K. References Ill
F. Jona and G. Shirane, Ferroelectric crystals (Pergamon Press, Oxford, 1962). [2] R. Blinc and B. Zeks, Soft modes in ferroelectrics and antiferroelectrics (North-Holland, Amsterdam, 1974) pp. 20—26. [31H.E. Stanley, Introduction to the phase transitions and critical phenomena (Clarendon Press, Oxford, 1971). [4] D. Mirjani~and D. Had~iahmetovi~, Phys. Lett. 90A (1982) 264. [5]U.F. Kozmidis-Luburié and B.S. To~ié,Physica 112B (1982) 331. [6] B.S. To~ié,M.M. Marinkovié and S. Berar, Phys. Stat. SoL 81(1977) 245.
46
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