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On the magnetic structure of a hexagonal Ising antiferromagnet
Solid State Communications, Vol. 46, No. 4, pp. 329-332, 1983. Printed in Great Britain.
0038-1098/83/160329-04503.00/0 Pergamon Press Ltd.
ON THE MAGNETIC STRUCTURE OF A HEXAGONAL ISING ANTIFERROMAGNET F. Matsubara Department of Engineering Science, Tohoku University, Sendai 980, Japan (Received 16 December 1982 by J. Kanamori)
Domain wall free energies are calculated in a hexagonal Ising antiferromagnet and shown to be very small compared with those in an ordinary two-sublattice antiferromagnet. It is suggested that the magnetic structure of its higher temperature ordered phase is one in which many domains are tangled with each other. Some comments are given on experimental results in CsCoC13and CsCoBr3. RECENTLY, AN INTEREST has been given on an unusual magnetic ordering process observed in hexagonal Ising-like antiferromagnets CsCoBr3 [1] and CsCoC13 [2], in that initially only two-thirds of the spins on each basal plane order antiferromagnetically, and the remaining one-third orders at a lower temperature (hereafter, we call these phases the partially disordered antiferromagnetic (pdAF) and ferrimagnetic (FR) phases, respectively). This ordering process has been explained by an Ising model with intraplane antiferromagnetic nearest neighbor and ferromagnetic next nearest neighbor interactions [3-6]. These theoretical results have predicted that in accordance with this ordering process, two or three anomalies of the specific heat occur in these compounds. However, only one anomaly of the specific heat has been experimentally observed [ 1 ]. Recently, a Monte Carlo method has been applied to this model [7]. The results have shown that only one anomaly of the specific heat occurs at a temperature (TN) above which the sublattice magnetizations vanish. The results have also shown that, in a temperature range between Tlv and a lower temperature (TF) below which the FR phase is realized, the sublattice magnetizations fluctuate in Monte Carlo steps and interchange their roles. These Monte Carlo results have explained an experimental result of magnetoelectric effect measurements on CsCoC13 [8]. Both these experimental and theoretical results indicate that these compounds are divided into many magnetic domains. This magnetic structure may be possible only when domain wall free energies are very small. In this paper, we will calculate the wall free energies and discuss a possibility of the occurrence of this magnetic structure. We start with the model described by tin
3£ = - J - q2 y
o xai+ls + J1 tl?l?l
2
~ xu
(1)
where the 1st, 2nd and 3rd terms represent the interplane coupling energy, the intraplane nearest neighbor and next nearest neighbor coupling energies, respectively. The coupling constants Jo, J1 and J2 are assumed to be all positive. This model has been extensively studied under the assumption that the hexagonal lattice is divided into equivalent three magnetic sublattices, which are denoted by a-,/3- and 3'-sublattice, and the results have predicted that this model exhibits the successive phase transitions [3-6]. In Fig. 1, we show the sublattices and realizable phases of this model. We note that in this model there exist six states minimizing the free energy and that barriers separating these states are very small compared with those of an ordinary two-sublattice antiferromagnetic model (see Fig. 3 in [3]). This indicates that the domain wall free energies of this model are very small compared with those of the latter model. We discuss this problem in more detail.
~ c
~
pdAF ~ FR ~
1
®TN>T>T3 l'r>T
Fig. 1. The sublattices in the c-plane and the phases appearing in the hexagonal antiferromagnet. In a small temperature range T 3 > T > TF, a phase called a threesublattice ferrimagnetic (FR3) phase occurs, in which Two types of domains occurring in this model are schematically shown in Fig. 2. We consider both the domain walls perpendicular and parallel to the basal plane (c-plane), the free energies of which are calculated in different ways. Here we use the same method as was used by Scatapino et al. in which the interaction along the c-axis is taken into account exactly and the coupling
329
MAGNETIC STRUCTURE OF A HEXAGONAL ISING ANTIFERROMAGNET
330 DOMAIN
~
WALL
,
We can easily determine
DOMAIN
.... ~
.... ~
TN>T>TF
Fig. 2. Schematic illustrations of the spin configurations in domains and walls in the hexagonal antiferromagnet. Walls of types A and B actually occur but those of type C do not because of having very large wall free energy.
02t
..............
~0
".
.
.
.
.
~
.15-
iW A L L - o
.
~b
. . . .
.l.C
~.
/01
\
l ,I
\ ',",k!
•,
x10 ', \
',
',
", ,,,
o,os
0
,
,
0.2
0.4
",.I/
0.6
,
0.8
Vol. 46, No. 4
1 TIT
Fig. 3. Domain wall free energies as functions of Tin the case Jo/J1 = 10 and J2/J1 = 0.1. Thick curves indicate those of the types A and B, and thin curves those of the type C.
The wall free energy per chain in the c-plane, fw, is determined from fw = (F/N--fo)Na, where fo, N and N a are tile minimum of the free energy per spin, the numbers of the total spins and the spins on the chain, respectively. We calculate lye's of two walls parallel to the a-plane and to the b-plane (see Fig. 1), and show them in Fig. 3. These two curves supply the minimum and maximum of the free energies of the walls perpendicular to the c-plane. We see that fw's of the walls of the type A appearing in the range TN > T > "IF exhibit an unusual temperature dependence, i.e., these fw's initially increase with decreasing the temperature, then decrease and finally vanish as the temperature tends to TF. This temperature dependence is very similar to that of the height of the free energy barrier which is given by the difference in the free energy between the pdAF phase and the FR phase (see Fig. 3 in [3]). We must emphasize that these fw's are much smaller than free energies of walls similar to those in the ordinary antiferromagnet (walls of the type Cin Fig. 2). The latters are calculated under the assumption that (ox)-r = 0 holds over the whole lattice and also shown in Fig. 3. On the other hand, fw's of the walls of the type B appearing in the range T3 > T monotonically increase with decreasing the temperature and reach their boundary energies, which are given as 2J2 for the boundary parallel to the a-plane and as 4.I2/3 for the boundary parallel to the b-plane. (2) Domain walls parallel to the c-plane. In this case the free energy per chain along the c-axis is given by
NeF/N = ~ Z {--kT In Z ~ + ½ ~.,
FINe =
Z
Z { - - k r l n Z ~ + ½
zT,= e~ao/2 cosh/3A~ + A~-
J1
x/e ~J° sinh 2/3A~ + e -~a° ,
nn
r~'4= r~ A+XE~I'
J2
rj
(6)
rl
where <%>~ means the thermal average of the spin on nth c-plane of the rbsublattice, and N c -
Z n = Tr T~ l-I
1
Tnn,n+l TriNe,
(7)
?1=1
(2) (3)
nnn
A'
(5)
(4)
where/3 = 1/kT, Ne is the number of spins per chain along the c-axis and
(8) Here T~n and Tnn,n+l are the well-known transfer matrices described by Ann and by A n, and A,n+ 1, respectively. We assume that near opposite surfaces different spin configurations are realized, i.e., the matrices Tnn,n + 1 for n ~ 1 and n ~ N e are diagonalized by the use of different unitary matrices U and V. Then assuming that the number of spins in the wall, which is denoted by L, is much smaller than Arc, we obtain
Vol. 46, No. 4
MAGNETIC STRUCTURE OF A HEXAGONAL ISING ANTIFERROMAGNET
I\ / \
~', ',
be increased by 2eoS, and the entropy associated with this energy by k In cs, where 2eo is the energy per unit surface and c is a constant of order 1 which is determined from the topology of the lattice. We note that eo and c depend on the direction of the surface, so that these should be replaced by some averages (%) and (c). Thus the free energy gain may be expressed as A f S ( 2 ( e o ) - - k T l n (c)). From A f = 0, we may obtain the N6el temperature a s T N ~-- 2(eo)/k In (c). We apply the same method to obtain an expression of the free energy gain/D resulting from the occurence of a domain. For the domain, the boundary energy 2eo should be replaced by f w and the constant c by cl/(n + 1) 2, where 'n + 1' is the width of the wall. Thus we may obtain the free energy gain/)9 as
I \~ n=5 '.
',
~
WALL-b
x 1
~x
331
NT/T
fD ~-" S ( f w - k T l n c u ( n + l ) : ) 0.4
0.6
0.8
(2(eo)S/((n + 1)2))((fw(n + 1)2/2eo)-- T/TN).
1 T/TN
(12)
Fig. 4. Temperature dependence offwn(n + 1)2/2eo for
This shows that the free energy gain fD depends on
small n in the case Jo/J1 = 10 and J2/Jt = 0.l. The surface energy 2eo is chosen as 2/3J1 + 8/3J2 for the wall parallel to the b-plane and as 2/3Jo for the wall parallel to the c-plane.
Z n "" Tr
x
X V
X~2/
X~.2
U -1
I-I .=M.1
fwn(n + 1)2/2eo rather than fw (~ fw=), where fwn is
Tnn n+l '
~ (X~I)M(x~I) Ne-M-L Zg. (9)
(Tf-I I'[M+L rl Here Z~ -- - t,~ n=M+x T~, n+ 1V)u, and k~ux> knu2, Xvnl> Xvn: and Nc, M >>L are assumed. Thus we obtain the equations for (On)n's as
(on),~ = OZ~I~(3An.)IZ~L,
(10)
and the expression of the wall free energy per chain as
the free energy of the wall with finite width n. We calculate fw,(n + 1)2/2eo and show them in Fig. 4. We estimate fo by using the results for small n, and show it schematically in Fig. 5. We see that fo has a very small value in the range TN > T > TF. Especially, it should be negative for TN >~T and T >~ T F. This indicates that in this range the system is divided into many domains. On the other hand, in the range TF > T, fD monotonically increases with decreasing the temperature resulting that the spin configuration tends to uniform over the whole lattice. The mean size of domains expected from similar arguments is also shown in Fig. 4. W N
Z
fw = ~
--kTlnZ~+½ r/
~
(o,)~A~--L]o
.
n=M+l
(11) We calculate fw's and show them in Fig. 3. These fw's have the same temperature dependence as those of the walls perpendicular to the c-plane. We see that f w of the wall o f the type A is also much smaller than that of the type C. We have shown that the wall free energy o f this model in the range T N > T > T F is very small compared with that in the ordinary antiferromagnet. Now we discuss which magnetic structure is of the most probable. The N6el temperature of this model can be estimated as follows. Suppose that we make a connected region of reverse spins in the pdAF phase. This region will be bounded by a surface of area S (>> 1). The energy will
o
o
I
i
I i I t
,,]
',
\ \x x
,,
T
Fig. 5. Schematic illustrations of the temperature dependences of the domain free energy fo (solid curves) and the mean domain size (broken curve).
332
MAGNETIC STRUCTURE OF A HEXAGONAL ISING ANTIFERROMAGNET
We may conclude that in the range Tlv > T > TF the magnetic structure of this model is one in which many domains with the pdAF spin configuration or the FR3 spin configuration are tangled with each other, whereas in the range TF > T a single domain FR phase is realized. By using this picture of the spin ordering, we may explain some experimental results unsolved so far. In the range TN > T > TF, the spin configuration may markedly fluctuate because the domains are not so large. This may explain an experiment on neutron diffuse scatterings in CsCoC13 [10]. As the temperature is decreased through the range T3 > T > TF, the wall region extends and the domain region reduces, and thus the two regions interchange their roles. This change in the magnetic structure is of continuous, so that no anomaly of the specific heat occurs [1]. The domain size reduces in this range. This may explain anomalous decreases of the magnetic neutron diffraction intensities observed in CsCoBr 3 [1] and CsCoC13 [2, 10].
Vol. 46, No. 4
Acknowledgements - The author would like to thank Professor S. Inawashiro and Dr T. Horiguchi for their useful discussions.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
REFERENCES W.B. Yelon, D.E. Cox & M. Eibschutz, Phys. Rev. B12, 5007 (1975). M. Mekata & K. Adachi, J. Phys. Soc. Japan 44, 806 (1978). M. Mekata, J. Phys. Soc. Japan 42, 76 (1977). H. Shiba, Progr. Theor. Phys. 64, 466 (1980). F. Matsubara, J. Phys. Soc. Japan 51, 2424 (1982). M. Kaburagi, T. Tonegawa & J. Kanamori, to be published in J. Phys. Soc. Japan 52, supplement (1983). F. Matsubara & S. Ikeda (to be published). E. Kita, K. Adachi, M. Mekata & K. Siratori (preprint). D.J. Scalapino, Y. Imry & P. Pincus, Phys. Rev. B l l , 2042 (1975). H. Yoshizawa & K. Hirakawa, J. Phys. Soc. Japan 46,448 (1979).
0038-1098/83/160329-04503.00/0 Pergamon Press Ltd.
ON THE MAGNETIC STRUCTURE OF A HEXAGONAL ISING ANTIFERROMAGNET F. Matsubara Department of Engineering Science, Tohoku University, Sendai 980, Japan (Received 16 December 1982 by J. Kanamori)
Domain wall free energies are calculated in a hexagonal Ising antiferromagnet and shown to be very small compared with those in an ordinary two-sublattice antiferromagnet. It is suggested that the magnetic structure of its higher temperature ordered phase is one in which many domains are tangled with each other. Some comments are given on experimental results in CsCoC13and CsCoBr3. RECENTLY, AN INTEREST has been given on an unusual magnetic ordering process observed in hexagonal Ising-like antiferromagnets CsCoBr3 [1] and CsCoC13 [2], in that initially only two-thirds of the spins on each basal plane order antiferromagnetically, and the remaining one-third orders at a lower temperature (hereafter, we call these phases the partially disordered antiferromagnetic (pdAF) and ferrimagnetic (FR) phases, respectively). This ordering process has been explained by an Ising model with intraplane antiferromagnetic nearest neighbor and ferromagnetic next nearest neighbor interactions [3-6]. These theoretical results have predicted that in accordance with this ordering process, two or three anomalies of the specific heat occur in these compounds. However, only one anomaly of the specific heat has been experimentally observed [ 1 ]. Recently, a Monte Carlo method has been applied to this model [7]. The results have shown that only one anomaly of the specific heat occurs at a temperature (TN) above which the sublattice magnetizations vanish. The results have also shown that, in a temperature range between Tlv and a lower temperature (TF) below which the FR phase is realized, the sublattice magnetizations fluctuate in Monte Carlo steps and interchange their roles. These Monte Carlo results have explained an experimental result of magnetoelectric effect measurements on CsCoC13 [8]. Both these experimental and theoretical results indicate that these compounds are divided into many magnetic domains. This magnetic structure may be possible only when domain wall free energies are very small. In this paper, we will calculate the wall free energies and discuss a possibility of the occurrence of this magnetic structure. We start with the model described by tin
3£ = - J - q2 y
o xai+ls + J1 tl?l?l
2
~ xu
(1)
where the 1st, 2nd and 3rd terms represent the interplane coupling energy, the intraplane nearest neighbor and next nearest neighbor coupling energies, respectively. The coupling constants Jo, J1 and J2 are assumed to be all positive. This model has been extensively studied under the assumption that the hexagonal lattice is divided into equivalent three magnetic sublattices, which are denoted by a-,/3- and 3'-sublattice, and the results have predicted that this model exhibits the successive phase transitions [3-6]. In Fig. 1, we show the sublattices and realizable phases of this model. We note that in this model there exist six states minimizing the free energy and that barriers separating these states are very small compared with those of an ordinary two-sublattice antiferromagnetic model (see Fig. 3 in [3]). This indicates that the domain wall free energies of this model are very small compared with those of the latter model. We discuss this problem in more detail.
~ c
~
pdAF ~ FR ~
1
®TN>T>T3 l'r>T
Fig. 1. The sublattices in the c-plane and the phases appearing in the hexagonal antiferromagnet. In a small temperature range T 3 > T > TF, a phase called a threesublattice ferrimagnetic (FR3) phase occurs, in which Two types of domains occurring in this model are schematically shown in Fig. 2. We consider both the domain walls perpendicular and parallel to the basal plane (c-plane), the free energies of which are calculated in different ways. Here we use the same method as was used by Scatapino et al. in which the interaction along the c-axis is taken into account exactly and the coupling
329
MAGNETIC STRUCTURE OF A HEXAGONAL ISING ANTIFERROMAGNET
330 DOMAIN
~
WALL
,
We can easily determine
DOMAIN
.... ~
.... ~
TN>T>TF
Fig. 2. Schematic illustrations of the spin configurations in domains and walls in the hexagonal antiferromagnet. Walls of types A and B actually occur but those of type C do not because of having very large wall free energy.
02t
..............
~0
".
.
.
.
.
~
.15-
iW A L L - o
.
~b
. . . .
.l.C
~.
/01
\
l ,I
\ ',",k!
•,
x10 ', \
',
',
", ,,,
o,os
0
,
,
0.2
0.4
",.I/
0.6
,
0.8
Vol. 46, No. 4
1 TIT
Fig. 3. Domain wall free energies as functions of Tin the case Jo/J1 = 10 and J2/J1 = 0.1. Thick curves indicate those of the types A and B, and thin curves those of the type C.
The wall free energy per chain in the c-plane, fw, is determined from fw = (F/N--fo)Na, where fo, N and N a are tile minimum of the free energy per spin, the numbers of the total spins and the spins on the chain, respectively. We calculate lye's of two walls parallel to the a-plane and to the b-plane (see Fig. 1), and show them in Fig. 3. These two curves supply the minimum and maximum of the free energies of the walls perpendicular to the c-plane. We see that fw's of the walls of the type A appearing in the range TN > T > "IF exhibit an unusual temperature dependence, i.e., these fw's initially increase with decreasing the temperature, then decrease and finally vanish as the temperature tends to TF. This temperature dependence is very similar to that of the height of the free energy barrier which is given by the difference in the free energy between the pdAF phase and the FR phase (see Fig. 3 in [3]). We must emphasize that these fw's are much smaller than free energies of walls similar to those in the ordinary antiferromagnet (walls of the type Cin Fig. 2). The latters are calculated under the assumption that (ox)-r = 0 holds over the whole lattice and also shown in Fig. 3. On the other hand, fw's of the walls of the type B appearing in the range T3 > T monotonically increase with decreasing the temperature and reach their boundary energies, which are given as 2J2 for the boundary parallel to the a-plane and as 4.I2/3 for the boundary parallel to the b-plane. (2) Domain walls parallel to the c-plane. In this case the free energy per chain along the c-axis is given by
NeF/N = ~ Z {--kT In Z ~ + ½ ~.,
FINe =
Z
Z { - - k r l n Z ~ + ½
zT,= e~ao/2 cosh/3A~ + A~-
J1
x/e ~J° sinh 2/3A~ + e -~a° ,
nn
r~'4= r~ A+XE~I'
J2
rj
(6)
rl
where <%>~ means the thermal average of the spin on nth c-plane of the rbsublattice, and N c -
Z n = Tr T~ l-I
1
Tnn,n+l TriNe,
(7)
?1=1
(2) (3)
nnn
A'
(5)
(4)
where/3 = 1/kT, Ne is the number of spins per chain along the c-axis and
(8) Here T~n and Tnn,n+l are the well-known transfer matrices described by Ann and by A n, and A,n+ 1, respectively. We assume that near opposite surfaces different spin configurations are realized, i.e., the matrices Tnn,n + 1 for n ~ 1 and n ~ N e are diagonalized by the use of different unitary matrices U and V. Then assuming that the number of spins in the wall, which is denoted by L, is much smaller than Arc, we obtain
Vol. 46, No. 4
MAGNETIC STRUCTURE OF A HEXAGONAL ISING ANTIFERROMAGNET
I\ / \
~', ',
be increased by 2eoS, and the entropy associated with this energy by k In cs, where 2eo is the energy per unit surface and c is a constant of order 1 which is determined from the topology of the lattice. We note that eo and c depend on the direction of the surface, so that these should be replaced by some averages (%) and (c). Thus the free energy gain may be expressed as A f S ( 2 ( e o ) - - k T l n (c)). From A f = 0, we may obtain the N6el temperature a s T N ~-- 2(eo)/k In (c). We apply the same method to obtain an expression of the free energy gain/D resulting from the occurence of a domain. For the domain, the boundary energy 2eo should be replaced by f w and the constant c by cl/(n + 1) 2, where 'n + 1' is the width of the wall. Thus we may obtain the free energy gain/)9 as
I \~ n=5 '.
',
~
WALL-b
x 1
~x
331
NT/T
fD ~-" S ( f w - k T l n c u ( n + l ) : ) 0.4
0.6
0.8
(2(eo)S/((n + 1)2))((fw(n + 1)2/2eo)-- T/TN).
1 T/TN
(12)
Fig. 4. Temperature dependence offwn(n + 1)2/2eo for
This shows that the free energy gain fD depends on
small n in the case Jo/J1 = 10 and J2/Jt = 0.l. The surface energy 2eo is chosen as 2/3J1 + 8/3J2 for the wall parallel to the b-plane and as 2/3Jo for the wall parallel to the c-plane.
Z n "" Tr
x
X V
X~2/
X~.2
U -1
I-I .=M.1
fwn(n + 1)2/2eo rather than fw (~ fw=), where fwn is
Tnn n+l '
~ (X~I)M(x~I) Ne-M-L Zg. (9)
(Tf-I I'[M+L rl Here Z~ -- - t,~ n=M+x T~, n+ 1V)u, and k~ux> knu2, Xvnl> Xvn: and Nc, M >>L are assumed. Thus we obtain the equations for (On)n's as
(on),~ = OZ~I~(3An.)IZ~L,
(10)
and the expression of the wall free energy per chain as
the free energy of the wall with finite width n. We calculate fw,(n + 1)2/2eo and show them in Fig. 4. We estimate fo by using the results for small n, and show it schematically in Fig. 5. We see that fo has a very small value in the range TN > T > TF. Especially, it should be negative for TN >~T and T >~ T F. This indicates that in this range the system is divided into many domains. On the other hand, in the range TF > T, fD monotonically increases with decreasing the temperature resulting that the spin configuration tends to uniform over the whole lattice. The mean size of domains expected from similar arguments is also shown in Fig. 4. W N
Z
fw = ~
--kTlnZ~+½ r/
~
(o,)~A~--L]o
.
n=M+l
(11) We calculate fw's and show them in Fig. 3. These fw's have the same temperature dependence as those of the walls perpendicular to the c-plane. We see that f w of the wall o f the type A is also much smaller than that of the type C. We have shown that the wall free energy o f this model in the range T N > T > T F is very small compared with that in the ordinary antiferromagnet. Now we discuss which magnetic structure is of the most probable. The N6el temperature of this model can be estimated as follows. Suppose that we make a connected region of reverse spins in the pdAF phase. This region will be bounded by a surface of area S (>> 1). The energy will
o
o
I
i
I i I t
,,]
',
\ \x x
,,
T
Fig. 5. Schematic illustrations of the temperature dependences of the domain free energy fo (solid curves) and the mean domain size (broken curve).
332
MAGNETIC STRUCTURE OF A HEXAGONAL ISING ANTIFERROMAGNET
We may conclude that in the range Tlv > T > TF the magnetic structure of this model is one in which many domains with the pdAF spin configuration or the FR3 spin configuration are tangled with each other, whereas in the range TF > T a single domain FR phase is realized. By using this picture of the spin ordering, we may explain some experimental results unsolved so far. In the range TN > T > TF, the spin configuration may markedly fluctuate because the domains are not so large. This may explain an experiment on neutron diffuse scatterings in CsCoC13 [10]. As the temperature is decreased through the range T3 > T > TF, the wall region extends and the domain region reduces, and thus the two regions interchange their roles. This change in the magnetic structure is of continuous, so that no anomaly of the specific heat occurs [1]. The domain size reduces in this range. This may explain anomalous decreases of the magnetic neutron diffraction intensities observed in CsCoBr 3 [1] and CsCoC13 [2, 10].
Vol. 46, No. 4
Acknowledgements - The author would like to thank Professor S. Inawashiro and Dr T. Horiguchi for their useful discussions.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
REFERENCES W.B. Yelon, D.E. Cox & M. Eibschutz, Phys. Rev. B12, 5007 (1975). M. Mekata & K. Adachi, J. Phys. Soc. Japan 44, 806 (1978). M. Mekata, J. Phys. Soc. Japan 42, 76 (1977). H. Shiba, Progr. Theor. Phys. 64, 466 (1980). F. Matsubara, J. Phys. Soc. Japan 51, 2424 (1982). M. Kaburagi, T. Tonegawa & J. Kanamori, to be published in J. Phys. Soc. Japan 52, supplement (1983). F. Matsubara & S. Ikeda (to be published). E. Kita, K. Adachi, M. Mekata & K. Siratori (preprint). D.J. Scalapino, Y. Imry & P. Pincus, Phys. Rev. B l l , 2042 (1975). H. Yoshizawa & K. Hirakawa, J. Phys. Soc. Japan 46,448 (1979).