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Steady configurations of a square dipole lattice in an external field
Volume 150,number 3,4
PHYSICSLETTERSA
5 November i 990
Steady configurations of a square dipole lattice in an external field V.E. Klymenko, V.V. Kukhtin, V.M. Ogenko and V.M. Rosenbaum Institutefor Surface Chemistry, Academy of Sciences of the UkrainianSSR, ProspektNauki 31, Kiev 252028, USSR Received9 May 1990;acceptedfor publication 31 August 1990 Communicatedby D. Bloch
The conditions of existence of the ground and metastable states are obtained for a dipole system on a square lattice in an external co-planarfield. The type of transition to the ferroelectricphase is elucidated.
Analysis of planar dipole systems is usually made without taking into account an external electric field (see, e.g., refs. [ 1,2] and the references therein). For the simple cubic dipole lattice it was found in ref. [3] that even infinitesimal fields break the degeneracy of the ground state, force the dipoles to orient along some lattice axis and lead to the appearance of metastable states. Below we show that similar conclusions also hold for the square dipole lattice. In the k-representation the Hamiltonian of the dipole system is as follows [4,5 ], H=Nh,
h=½ ~ f f ' ~ ( k ) ~ B _ , - I z E . ~ o ,
(1)
k
where N is the number of dipoles, a, fl =x, y; I~ is the common value of the dipole moment, E=E(cos~oE, sin ~0E) is the strength of a static external electric field, ~0Eis the angle between E and the lattice axis x, which will be further confined to the interval 0 ° ~<~0E~<45 °, owing to the underlying lattice symmetry, PUP(k)= Z V~P(r) e x p ( - i k - r ) ,
~,= 1 --Z ~, e x p ( - i k . r ) ,
r
where V~P(r) is the dipole interaction tensor and 4, is the unit dipole vector placed at the point r. Expanding ( 1 ) with the eigenvectors of VaP(k), defined for the ground state with zero field E, we obtain with the relevant parameterization (e.g., ~ o - ~/o(COS~0o,sin q~o)), h= ½VFr/02+ ½VA(t/2 + ~/2) + ½( VA' -- VA) 0/2 sin2~0~ + t/22COS2¢2) --/zEt/o cos (~0o- ¢ e ) ,
(2)
where VA, VA' and VF are the eigenvalues of the tensor V ~ ( k ) , which were found in ref. [4] to be -5.099, 6.033 and -4.517, respectively, in units of Iz2/a 3 (a is the lattice constant). The parameters ~/i, ~0~are constrained by 2 2 2 /~0"~/~1 " J ~ 2 =
1,
~/I ~/2 c o s ( ~ l
- ~ 7 2 ) ~-.'~0 ,
~0/~1 c o s ( ~ 9 0 - ~ l )
=0
,
/~0~2 c 0 s ( ~ 9 0 - ~ 9 2 ) - ~ 0
•
(3)
Minimization of the Hamiltonian (2) over r/i, ~0igives, first, a ferroelectric state (~h =1/2=0, ~/o= 1 ) which is characterized by a linear dependence of its energy on the field, h~ = ½Vv-/xE, and the ground state with quadratic behaviour, ~ 1 f12E2 cos (tpE --~70) h2 = ½VA+ ½(VA,- PA) sin2~o- ~ VF-- VA--(VA'- VA) sin2~o '
(4)
ElsevierSciencePublishers B.V. (North-Holland)
2 13
Volume 150, number 3,4
0
PHYSICS LETTERS A
13,22 °
,
05*
~oE
5 November 1990
Fig. 1. The phase diagram of a square dipole lattice in an external field: region I corresponds to the ground ferroelectric state; II to the ground antiferroelectric one; III to the metastable ferroelectric one.
where {Upis defined by
#Ecos(~E-~o) V F " VA -- ( VA, -- PA) sin2~ao
----- 1+
y-sin2~o sin {ao cos ~ao
tg(cpE--f)O),1-1/2 y =
PF- p.
~A,_~A =0.0523 .
(5)
For weak fields the ground state keeps the features o f an antiferroelectric structure with the dipoles oriented along the axis y (when 0 ° ~<{ae~<45 ° ). Increasing the field results in a transition to the ferroelectric phase at and ~ae=45 ° - 0 ° (the continuous curve of fig. 1 ). We find that this transition is smooth for 0 ° <{ae < a r c s i n ( 1 ~ 1 ) = 1.5 o and jump-like, otherwise. Let us now consider the metastable states o f a dipole system. For this purpose it is convenient to use the Luttinger-Tisza approach [6] with four sublattices (fig. 2). By introducing angles 0i, formed by the dipoles and the axis x, we get the interaction energy as
Ea3/#=0.341-0.582
h = 1 ( ! ~ - (]VF + ½17A)(COS 01 COS 02 +COS 03 COS 04 + s i n 01 sin 04 + s i n 02 sin 03) 4\' F~"1- ( ~ VF --
½VA-- ½VA' ) (COS 01 COS 03 "{-COS02 COS04 "{-sin 01 sin 03 + sin 02 sin 04)
+(3VF+½VA')(c°sOIC°SO4+c°sO2c°sO3+sinOIsinO2+sinO3sinO4)--#E i=n ~ COS(01--{ae)) "
(6)
Applying the standard procedure for determining the extrema o f expression (6), we obtain, in the limit E--,0, that the second antiferroelectric dipole configuration is realized with dipole orientations along the axis x when /'VF +4VA + 2 V A . ~ 5 45°>~ae>arctg~ VA-Vr ] - .14 ° .
(7)
¥ /
/
/
_/'0t
\
Q@ ®® /
X'X 0
214
cl
Fig. 2. Part of a square lattice (a is the lattice constant) with a four'sublattice dipole structure.
Volume 150, number 3,4
PHYSICS LETI'ERS A
5 November 1990
Note that an investigation of the stability of this metastable state for finite external fields E requires computer calculations. Eq. (6) permits us also to treat a metastable state of the ferroelectric type ( 0i = (oe, i = 1, 2, 3, 4 ); the condition of its stability is given by the following inequality, sin20>~ 7 - ]zg/ ( VA' --VA)
(8)
(the equality sign in (8) corresponds to the dashed line of a phase boundary in fig. 1 ). It should be mentioned that the metastable ferroelectric phase exists in the presence of a vanishing field E under the condition 0>t arcsin x/~ ~ 13.22 °. Thus, the external static electric field, directed parallel to the square dipole lattice, removes the known orientational degeneracy of the ground state, sets out dipoles along some lattice axis in the weak-field case and then changes the dipole configurations into the ferroelectric phase continuously or by a j u m p as one goes to large fields. The existence of metastable states gives one the possibility to carry out the energy-releasing transitions by changing the value a n d / o r direction of an external field. Note that the latter is a rather weak one as compared to the characteristic value of the lattice field: E,,, ( VF-- VA)/~t << VA//~-
References [ 1] S. Romano, Nuovo Cimento 9D (1987) 409. [2] J.G. Brankov and D.M. Danchev, PhysicaA 144 (1987) 128. [3] V.V. Kukhtin and O.V. Shrarnko, Phys. Lett. A 128 (1988) 271. [4] V.M. Rosenbaum and V.M. Ogenko, Fiz. Tverd. Tela 26 (1984) 1448. [5] V.M. Rosenbaum, E.V. Artamonova and V.M. Ogenko, Ukr. Fiz. Zh. 33 (1988) 625. [6] J.M. Luttinger and L. Tisza, Phys. Rev. 70 (1946) 954.
215
PHYSICSLETTERSA
5 November i 990
Steady configurations of a square dipole lattice in an external field V.E. Klymenko, V.V. Kukhtin, V.M. Ogenko and V.M. Rosenbaum Institutefor Surface Chemistry, Academy of Sciences of the UkrainianSSR, ProspektNauki 31, Kiev 252028, USSR Received9 May 1990;acceptedfor publication 31 August 1990 Communicatedby D. Bloch
The conditions of existence of the ground and metastable states are obtained for a dipole system on a square lattice in an external co-planarfield. The type of transition to the ferroelectricphase is elucidated.
Analysis of planar dipole systems is usually made without taking into account an external electric field (see, e.g., refs. [ 1,2] and the references therein). For the simple cubic dipole lattice it was found in ref. [3] that even infinitesimal fields break the degeneracy of the ground state, force the dipoles to orient along some lattice axis and lead to the appearance of metastable states. Below we show that similar conclusions also hold for the square dipole lattice. In the k-representation the Hamiltonian of the dipole system is as follows [4,5 ], H=Nh,
h=½ ~ f f ' ~ ( k ) ~ B _ , - I z E . ~ o ,
(1)
k
where N is the number of dipoles, a, fl =x, y; I~ is the common value of the dipole moment, E=E(cos~oE, sin ~0E) is the strength of a static external electric field, ~0Eis the angle between E and the lattice axis x, which will be further confined to the interval 0 ° ~<~0E~<45 °, owing to the underlying lattice symmetry, PUP(k)= Z V~P(r) e x p ( - i k - r ) ,
~,= 1 --Z ~, e x p ( - i k . r ) ,
r
where V~P(r) is the dipole interaction tensor and 4, is the unit dipole vector placed at the point r. Expanding ( 1 ) with the eigenvectors of VaP(k), defined for the ground state with zero field E, we obtain with the relevant parameterization (e.g., ~ o - ~/o(COS~0o,sin q~o)), h= ½VFr/02+ ½VA(t/2 + ~/2) + ½( VA' -- VA) 0/2 sin2~0~ + t/22COS2¢2) --/zEt/o cos (~0o- ¢ e ) ,
(2)
where VA, VA' and VF are the eigenvalues of the tensor V ~ ( k ) , which were found in ref. [4] to be -5.099, 6.033 and -4.517, respectively, in units of Iz2/a 3 (a is the lattice constant). The parameters ~/i, ~0~are constrained by 2 2 2 /~0"~/~1 " J ~ 2 =
1,
~/I ~/2 c o s ( ~ l
- ~ 7 2 ) ~-.'~0 ,
~0/~1 c o s ( ~ 9 0 - ~ l )
=0
,
/~0~2 c 0 s ( ~ 9 0 - ~ 9 2 ) - ~ 0
•
(3)
Minimization of the Hamiltonian (2) over r/i, ~0igives, first, a ferroelectric state (~h =1/2=0, ~/o= 1 ) which is characterized by a linear dependence of its energy on the field, h~ = ½Vv-/xE, and the ground state with quadratic behaviour, ~ 1 f12E2 cos (tpE --~70) h2 = ½VA+ ½(VA,- PA) sin2~o- ~ VF-- VA--(VA'- VA) sin2~o '
(4)
ElsevierSciencePublishers B.V. (North-Holland)
2 13
Volume 150, number 3,4
0
PHYSICS LETTERS A
13,22 °
,
05*
~oE
5 November 1990
Fig. 1. The phase diagram of a square dipole lattice in an external field: region I corresponds to the ground ferroelectric state; II to the ground antiferroelectric one; III to the metastable ferroelectric one.
where {Upis defined by
#Ecos(~E-~o) V F " VA -- ( VA, -- PA) sin2~ao
----- 1+
y-sin2~o sin {ao cos ~ao
tg(cpE--f)O),1-1/2 y =
PF- p.
~A,_~A =0.0523 .
(5)
For weak fields the ground state keeps the features o f an antiferroelectric structure with the dipoles oriented along the axis y (when 0 ° ~<{ae~<45 ° ). Increasing the field results in a transition to the ferroelectric phase at and ~ae=45 ° - 0 ° (the continuous curve of fig. 1 ). We find that this transition is smooth for 0 ° <{ae < a r c s i n ( 1 ~ 1 ) = 1.5 o and jump-like, otherwise. Let us now consider the metastable states o f a dipole system. For this purpose it is convenient to use the Luttinger-Tisza approach [6] with four sublattices (fig. 2). By introducing angles 0i, formed by the dipoles and the axis x, we get the interaction energy as
Ea3/#=0.341-0.582
h = 1 ( ! ~ - (]VF + ½17A)(COS 01 COS 02 +COS 03 COS 04 + s i n 01 sin 04 + s i n 02 sin 03) 4\' F~"1- ( ~ VF --
½VA-- ½VA' ) (COS 01 COS 03 "{-COS02 COS04 "{-sin 01 sin 03 + sin 02 sin 04)
+(3VF+½VA')(c°sOIC°SO4+c°sO2c°sO3+sinOIsinO2+sinO3sinO4)--#E i=n ~ COS(01--{ae)) "
(6)
Applying the standard procedure for determining the extrema o f expression (6), we obtain, in the limit E--,0, that the second antiferroelectric dipole configuration is realized with dipole orientations along the axis x when /'VF +4VA + 2 V A . ~ 5 45°>~ae>arctg~ VA-Vr ] - .14 ° .
(7)
¥ /
/
/
_/'0t
\
Q@ ®® /
X'X 0
214
cl
Fig. 2. Part of a square lattice (a is the lattice constant) with a four'sublattice dipole structure.
Volume 150, number 3,4
PHYSICS LETI'ERS A
5 November 1990
Note that an investigation of the stability of this metastable state for finite external fields E requires computer calculations. Eq. (6) permits us also to treat a metastable state of the ferroelectric type ( 0i = (oe, i = 1, 2, 3, 4 ); the condition of its stability is given by the following inequality, sin20>~ 7 - ]zg/ ( VA' --VA)
(8)
(the equality sign in (8) corresponds to the dashed line of a phase boundary in fig. 1 ). It should be mentioned that the metastable ferroelectric phase exists in the presence of a vanishing field E under the condition 0>t arcsin x/~ ~ 13.22 °. Thus, the external static electric field, directed parallel to the square dipole lattice, removes the known orientational degeneracy of the ground state, sets out dipoles along some lattice axis in the weak-field case and then changes the dipole configurations into the ferroelectric phase continuously or by a j u m p as one goes to large fields. The existence of metastable states gives one the possibility to carry out the energy-releasing transitions by changing the value a n d / o r direction of an external field. Note that the latter is a rather weak one as compared to the characteristic value of the lattice field: E,,, ( VF-- VA)/~t << VA//~-
References [ 1] S. Romano, Nuovo Cimento 9D (1987) 409. [2] J.G. Brankov and D.M. Danchev, PhysicaA 144 (1987) 128. [3] V.V. Kukhtin and O.V. Shrarnko, Phys. Lett. A 128 (1988) 271. [4] V.M. Rosenbaum and V.M. Ogenko, Fiz. Tverd. Tela 26 (1984) 1448. [5] V.M. Rosenbaum, E.V. Artamonova and V.M. Ogenko, Ukr. Fiz. Zh. 33 (1988) 625. [6] J.M. Luttinger and L. Tisza, Phys. Rev. 70 (1946) 954.
215