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The influence of defect site statistics on ferroelectric properties
Solid State Communications, Vol. 39, pp. 155-157. Pergamon Press Ltd. 1981. Printed in Great Britain.
0038-109 8/81/010155-03 $02.00/0
THE INFLUENCE OF DEFECT SITE STATISTICS ON FERROELECTRIC PROPERTIES K.H. Weyrich and R. Siems Theoretische Physik, Universitiit des Saarlandes, Federal Republic of Germany
(Received 19 February 1981 by P.H.. Dederichs) Averaged Green's functions are determined for lattices containing point defects with given site-correlations by extending existing averaging procedures. The anharmonicity of ferroelectric crystals is treated in SCPA. The results show a strong influence of the correlations on ferroelectric properties. For comparison MD calculations were performed which show a good agreement with the analytic approximations. 1. INTRODUCTION THE PHYSICAL PROPERTIES of solids depend on their content of lattice defects. In ferromagnets and ferroelectrics point defects influence static qualities like the critical temperature, the Curie constant, the polarization [e.g. l, 2] as well as the dynamic behavior, e.g. the dynamic structure factor [e.g. 3, 4]. So far the interest of physicists was centered mainly on defects distributed at random. Due to interactions, which are present in any case, the defect pofitions will, however, always be correlated. In this paper results of an investigation are presented in which, for a simple model, the influence of defect position correlations on the properties of an anharmonic (ferroelectric) crystal is determined. The microscopic state is described by one coordinate x n (momentum p") per unit cell n (local normal coordinate, [5]). The potential contains a single particle contribution with quadratic and quartic terms and a harmonic interaction with a long range contribution. The defects produce a diagonal perturbation. The Hamiltonian is, thus, givenby pnpn
H = ~ [2m~ -~er~)4"½(A+ AAr;)(x"): + i(B + ABT?)(xn)41 + ~..wnmxnx m. !
(I.I)
cv = ((~))
(unu n) = k r R e
Gnn(¢~ 2 = 0).
(2.1)
G "'n is the Green's function corresponding to a harmonic Hamiltonian with the coupling constants
{<<:,~>>(1 -- 8 . m ) -- c , c , } .
((xn) 2 + (u"2))] + W "m
(2.2)
The system of equations is completed by the relations
(A + AAr~ + 3(B + ABz~)(un')}(x">
(1.2)
CvC~
By means of the SCPA the problem is reduced to the solution of a system of equations for the average displacements (x n) and the harmonic fluctuations (unu ") about them. The autocorrelation function is connected to the Green's function by
x
and k~m =
2. SCPA AND MD
n, I~I
e, AA and AB describe the changes of mass and local potential due to the defects.1"he occupation numbers r~ are 8 vi if there is a defect at n and 8~ otherwise. The concentrations and correlations of defect- (u = 1) and host-atoms (u = 2) are
1
(( >>implies an averaging over the defect configurations and (when applied to fluctuating quantities) in addition a thermal averaging. We treated the system, on the one hand, by analytical methods of approximation: The statistical mechanics of the anharmonic crystal was reduced, by the Self Consistent Phonon Approximation (SCPA), to that of a harmonic crystal. The lattice dynamics of the latter was then handled by extending the existing averaging procedures (VCA, ATA, CPA; cf., e.g. [6]) to the case of correlated defect positions. On the other hand we performed molecular dynamics (MD) calculations which avoid the approximations necessarily employed in the analytical methods.
(I.3) 155
+ (B + aBr?)C~">' + Y w"" ~ >
= o
m
connecting the average displacements with the fluctuations. These equations,which describe a fixed
(2.3)
156
DEFECT SITE STATISTICS ON FERROELECTRIC PROPERTIES
Vol. 39, No. 1
0.6
0.4
0.2 •
1.'2
0.8
1.'6
\b
2.'0 )
Fig. I. Polarization (0~)) o f a ld crystal vs. T: (a) CPA for defects distributed at random; (b) modified ATA for nm defects with an attractive correlation (K 0 1 = 0.06;K 0 2 = 0.04; K 03 = 0.02; kit = (c t ) - 2 K m ); (c) modified ATA for defects with a repulsive correlation (K °t = K °2 = K °3 = - 0.01). [For (b) and (c) the starting point was the CPA crystal with a random defect_distributLon.] e, o: MD results corresponding to (a) and (b) respectively. Potential and mass parameters: ,4 = 10, AA = - - 4 , A = 8, A B = e = 0; c~ = 0.01. configuration of defect positions, are averaged, similar as in [2], over all such configurations, leading to the following set of equations for host and defect cells: o ((u2v)) = k T R e ((G(°)(°)(co 2 = 0)))v° ~ I4/o m {wvp((xl))
rtt
+ w ~ m (0c2))} + {A + A A ~ t v + 3(B + A B l y ) x ((u2v))}((xv)) + {B + ABStv}((xv)) 3 = 0
with
(2.4)
rl rlrl rlrlrl wvu = eU(kvu +1)
and
v = 1,2.
(())vn implies an averaging over confgurations with a v-type cell at n. For a further development of the theory one has to make allowance for the correlation of defect positions. In ATA this can be done by taking k~urn into consideration when decoupling products of one particle T-matrices. This type of k-weighted averaging is performed in the perturbation series solving the Dyson equation for (G)vn upon decoupling products of T-matrices. The general structure of the result is ((G))vn = ((G)) + P
tl
t
~ auav aaa BKp'P. #, a. p'. K
(2.5)
((G)) is the average Green's function defined without specifying the occupation (host or defect atom) of any site. It was already calculated before [7]. The sum extends over subscripts 1 and 2, P is the Green's function of the perfect crystal and Bpd are
matrices having the symmetry of the lattice, v ~ # is a square matrix of dimension 2(z + 1) where z is the number of cells connected to that at n by non-zero correlations. Equations (2.4) and (2.5) allow, e.g. the calculation of the polarization curve of the crystal in SCPA. In the actual calculations we started, in contrast to [7], from the CPA crystal for randomly distributed defects and the corresponding Green's function P. In addition to these analytical calculations, MD-simulations for a linear chain were performed for comparison. The interactions I4trim w e r e split into short range contributions j n m and the long range part X f ( n -- m). The latter was developed into a Fourier series with coefficientsJ~(k). To reduce the computer time only small wave vectors k are considered which is a reasonable approximation for the long range interaction. 3. RESULTS AND DISCUSSION For explicit calculations a nearest neighbor interaction J and the k = 0 term A = ~/(k = O) of the long range interaction were considered. Dimensionless quantities ,4 = - - A / J ; A = -- A/J, 7"= T k B / J 2 and .~ = x / - - B[J x where introduced. In Fig. 1 the polarization of an one-dimensional crystal containing 10% defects is shown for different correlations of the defect positions. The curves represent ,the results of the analytical approximations, the dots and circles are MD results. There is a good agreement between the latter and the former. Both methods yield
Vol. 39, No. 1
DEFECT SITE STATISTICS ON FERROELECTRIC PROPERTIES
0.2
157
¸
0.7
\b \
I )
I
0.06
0.O8
OAO
Y
Fig. 2. Polarization ((:~)) of a cubic crystal vs ~: (a) CPA for defect distributed at random; (b) modified ATA for defects with an attractive correlation (3 K ° 1 = O.04)L(c) modified ATA for defects with a repulsive correlation (3 K °1 = -- 0.01).4 = 2.75; A/] = - 1 ; A = 0.75; AB = e = 0; cl = 0.1. a marked increase of the polarization and of the critical temperature with increasing correlation: i.e., for equal defect concentrations, P and Tc increase when going from repulsive over random to attractive defect distributions. Further we obtained a considerable enhancement of the defect band dispersion with increasing correlations of defect positions. Similar effects were deduced for 3-dimensional models. In Fig. 2 SCPA-polarization curves are shown for a crystal with nearest neighbor interactions and correlations (with cubic symmetry) of various magnitudes. In the case considered the variation of Tc due to a change of the defect site correlations is about 5% (for equal defect concentrations). Details of the calculations and further results will be published elsewhere.
work was performed within the frame of the "Sonderforschungsbereich Ferroelektrika" which is supported by the Deutsche Forschungsgemeinschaft.
Acknowledgement - ~
REFERENCES 1. 2. 3. 4. 5. 6. 7.
R. Liebmann, B. Schaub & H.G. Schuster, Z. Phys. 1337, 69 (1980). V.L Aksenov, K.H. Breter & N. Hakida, Soy. Phys. Solid State 20, 846 (1978). H.G. Schuster, Phys. Lett. 62A, 47 (1977). B.I. Halperin, C.M. Varma, Phys. Rev. BI4, 4030 (1976). H. Thomas, Structural Phase Transitions and S o f t Modes (Edited by E. Samuelsen, E. Andersen & J. Feder). Oslo (1971). R.J. Elliot, J.A. Krumhansl & P.L. Leath, Rev. Mod. Phys. 46,465 (1974). B.L. Gyorffy, Phys. Rev. BI, 3290 (1970).
0038-109 8/81/010155-03 $02.00/0
THE INFLUENCE OF DEFECT SITE STATISTICS ON FERROELECTRIC PROPERTIES K.H. Weyrich and R. Siems Theoretische Physik, Universitiit des Saarlandes, Federal Republic of Germany
(Received 19 February 1981 by P.H.. Dederichs) Averaged Green's functions are determined for lattices containing point defects with given site-correlations by extending existing averaging procedures. The anharmonicity of ferroelectric crystals is treated in SCPA. The results show a strong influence of the correlations on ferroelectric properties. For comparison MD calculations were performed which show a good agreement with the analytic approximations. 1. INTRODUCTION THE PHYSICAL PROPERTIES of solids depend on their content of lattice defects. In ferromagnets and ferroelectrics point defects influence static qualities like the critical temperature, the Curie constant, the polarization [e.g. l, 2] as well as the dynamic behavior, e.g. the dynamic structure factor [e.g. 3, 4]. So far the interest of physicists was centered mainly on defects distributed at random. Due to interactions, which are present in any case, the defect pofitions will, however, always be correlated. In this paper results of an investigation are presented in which, for a simple model, the influence of defect position correlations on the properties of an anharmonic (ferroelectric) crystal is determined. The microscopic state is described by one coordinate x n (momentum p") per unit cell n (local normal coordinate, [5]). The potential contains a single particle contribution with quadratic and quartic terms and a harmonic interaction with a long range contribution. The defects produce a diagonal perturbation. The Hamiltonian is, thus, givenby pnpn
H = ~ [2m~ -~er~)4"½(A+ AAr;)(x"): + i(B + ABT?)(xn)41 + ~..wnmxnx m. !
(I.I)
cv = ((~))
(unu n) = k r R e
Gnn(¢~ 2 = 0).
(2.1)
G "'n is the Green's function corresponding to a harmonic Hamiltonian with the coupling constants
{<<:,~>>(1 -- 8 . m ) -- c , c , } .
((xn) 2 + (u"2))] + W "m
(2.2)
The system of equations is completed by the relations
(A + AAr~ + 3(B + ABz~)(un')}(x">
(1.2)
CvC~
By means of the SCPA the problem is reduced to the solution of a system of equations for the average displacements (x n) and the harmonic fluctuations (unu ") about them. The autocorrelation function is connected to the Green's function by
x
and k~m =
2. SCPA AND MD
n, I~I
e, AA and AB describe the changes of mass and local potential due to the defects.1"he occupation numbers r~ are 8 vi if there is a defect at n and 8~ otherwise. The concentrations and correlations of defect- (u = 1) and host-atoms (u = 2) are
1
(( >>implies an averaging over the defect configurations and (when applied to fluctuating quantities) in addition a thermal averaging. We treated the system, on the one hand, by analytical methods of approximation: The statistical mechanics of the anharmonic crystal was reduced, by the Self Consistent Phonon Approximation (SCPA), to that of a harmonic crystal. The lattice dynamics of the latter was then handled by extending the existing averaging procedures (VCA, ATA, CPA; cf., e.g. [6]) to the case of correlated defect positions. On the other hand we performed molecular dynamics (MD) calculations which avoid the approximations necessarily employed in the analytical methods.
(I.3) 155
+ (B + aBr?)C~">' + Y w"" ~ >
= o
m
connecting the average displacements with the fluctuations. These equations,which describe a fixed
(2.3)
156
DEFECT SITE STATISTICS ON FERROELECTRIC PROPERTIES
Vol. 39, No. 1
0.6
0.4
0.2 •
1.'2
0.8
1.'6
\b
2.'0 )
Fig. I. Polarization (0~)) o f a ld crystal vs. T: (a) CPA for defects distributed at random; (b) modified ATA for nm defects with an attractive correlation (K 0 1 = 0.06;K 0 2 = 0.04; K 03 = 0.02; kit = (c t ) - 2 K m ); (c) modified ATA for defects with a repulsive correlation (K °t = K °2 = K °3 = - 0.01). [For (b) and (c) the starting point was the CPA crystal with a random defect_distributLon.] e, o: MD results corresponding to (a) and (b) respectively. Potential and mass parameters: ,4 = 10, AA = - - 4 , A = 8, A B = e = 0; c~ = 0.01. configuration of defect positions, are averaged, similar as in [2], over all such configurations, leading to the following set of equations for host and defect cells: o ((u2v)) = k T R e ((G(°)(°)(co 2 = 0)))v° ~ I4/o m {wvp((xl))
rtt
+ w ~ m (0c2))} + {A + A A ~ t v + 3(B + A B l y ) x ((u2v))}((xv)) + {B + ABStv}((xv)) 3 = 0
with
(2.4)
rl rlrl rlrlrl wvu = eU(kvu +1)
and
v = 1,2.
(())vn implies an averaging over confgurations with a v-type cell at n. For a further development of the theory one has to make allowance for the correlation of defect positions. In ATA this can be done by taking k~urn into consideration when decoupling products of one particle T-matrices. This type of k-weighted averaging is performed in the perturbation series solving the Dyson equation for (G)vn upon decoupling products of T-matrices. The general structure of the result is ((G))vn = ((G)) + P
tl
t
~ auav aaa BKp'P. #, a. p'. K
(2.5)
((G)) is the average Green's function defined without specifying the occupation (host or defect atom) of any site. It was already calculated before [7]. The sum extends over subscripts 1 and 2, P is the Green's function of the perfect crystal and Bpd are
matrices having the symmetry of the lattice, v ~ # is a square matrix of dimension 2(z + 1) where z is the number of cells connected to that at n by non-zero correlations. Equations (2.4) and (2.5) allow, e.g. the calculation of the polarization curve of the crystal in SCPA. In the actual calculations we started, in contrast to [7], from the CPA crystal for randomly distributed defects and the corresponding Green's function P. In addition to these analytical calculations, MD-simulations for a linear chain were performed for comparison. The interactions I4trim w e r e split into short range contributions j n m and the long range part X f ( n -- m). The latter was developed into a Fourier series with coefficientsJ~(k). To reduce the computer time only small wave vectors k are considered which is a reasonable approximation for the long range interaction. 3. RESULTS AND DISCUSSION For explicit calculations a nearest neighbor interaction J and the k = 0 term A = ~/(k = O) of the long range interaction were considered. Dimensionless quantities ,4 = - - A / J ; A = -- A/J, 7"= T k B / J 2 and .~ = x / - - B[J x where introduced. In Fig. 1 the polarization of an one-dimensional crystal containing 10% defects is shown for different correlations of the defect positions. The curves represent ,the results of the analytical approximations, the dots and circles are MD results. There is a good agreement between the latter and the former. Both methods yield
Vol. 39, No. 1
DEFECT SITE STATISTICS ON FERROELECTRIC PROPERTIES
0.2
157
¸
0.7
\b \
I )
I
0.06
0.O8
OAO
Y
Fig. 2. Polarization ((:~)) of a cubic crystal vs ~: (a) CPA for defect distributed at random; (b) modified ATA for defects with an attractive correlation (3 K ° 1 = O.04)L(c) modified ATA for defects with a repulsive correlation (3 K °1 = -- 0.01).4 = 2.75; A/] = - 1 ; A = 0.75; AB = e = 0; cl = 0.1. a marked increase of the polarization and of the critical temperature with increasing correlation: i.e., for equal defect concentrations, P and Tc increase when going from repulsive over random to attractive defect distributions. Further we obtained a considerable enhancement of the defect band dispersion with increasing correlations of defect positions. Similar effects were deduced for 3-dimensional models. In Fig. 2 SCPA-polarization curves are shown for a crystal with nearest neighbor interactions and correlations (with cubic symmetry) of various magnitudes. In the case considered the variation of Tc due to a change of the defect site correlations is about 5% (for equal defect concentrations). Details of the calculations and further results will be published elsewhere.
work was performed within the frame of the "Sonderforschungsbereich Ferroelektrika" which is supported by the Deutsche Forschungsgemeinschaft.
Acknowledgement - ~
REFERENCES 1. 2. 3. 4. 5. 6. 7.
R. Liebmann, B. Schaub & H.G. Schuster, Z. Phys. 1337, 69 (1980). V.L Aksenov, K.H. Breter & N. Hakida, Soy. Phys. Solid State 20, 846 (1978). H.G. Schuster, Phys. Lett. 62A, 47 (1977). B.I. Halperin, C.M. Varma, Phys. Rev. BI4, 4030 (1976). H. Thomas, Structural Phase Transitions and S o f t Modes (Edited by E. Samuelsen, E. Andersen & J. Feder). Oslo (1971). R.J. Elliot, J.A. Krumhansl & P.L. Leath, Rev. Mod. Phys. 46,465 (1974). B.L. Gyorffy, Phys. Rev. BI, 3290 (1970).