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Magnetic transitions in layered triangular antiferromagnets
4 November
CQs .
1996
mm
PHYSICS LETTERS A
em
~
ELSEVIER
Physics Letters A 222 (1996) 269-274
Magnetic transitions in layered triangular antiferromagnets I.N. Bondarenko, R.S. Gekht, V.I. Ponomarev Kirensky Institute of Physics, Russian Academy of Sciences, Siberian Branch, Krasnoyarsk Received
660036, Russian Federation
4 July 1996; accepted for publication 5 August 19% Communicated by V.M. Agmnovich
Abstract Magnetic states and phase transitions of the layered dollar ~tife~omagn~~ in an applied field are studied. It is shown that in compounds like VBr, and VCl, quantum effects change the ground-state structure and cause successive phase transitions as the magnetic field increases. Coplanar structures of different spin configuration are realized far from the saturation field and a noncoplanar structure of umbrella-type configuration is realized near this field. The ground-state phase diagram is constructed, and a finite region of fields where the collinear phase is also possible is indicated. PACS: 75.30X2; 75.50.Ee Ke~or~: Triangular ~tife~orna~ne~;
~~~
and thermal fluctuations;
1. Recently much attention has been paid to the study of frustrated systems on a triangular lattice. Results of this study, both theoretical and experimental, are presented in recent reviews [l-4]. It is known that in nonfrustrated 3D systems quantum fluctuations modify only the spin structure but do not change the nature of the ground state. However, in frustrated systems quantum fluctuations are very important since they, as first pointed out by Nagaev [5], can cause a new phase transition mechanism. The effects of a magnetic field were also explored lately and many interesting results have been obtained [6-81. The goal of the present paper is to determine the spin structure of VX,-type compounds (X = Br, Cl) in an external magnetic field. We shall see below that quantum fluctuations induce successive phase transitions between different coplanar configurations in such substances. VX,-type compounds have the crystal structure of CdI, (space group P ?ml). The V2+ ions are ar0375-9~i/Q6/$12.~ Copyright PIi SO37.5-9601(96)00635-4
Successive
phase tmnsitions
ranged in the c-plane on the triangular layers which form a hexagonal lattice in the c-direction [9]. According to neutron-scattering data, below 40 K spins order with 120” structure [IO- 131. The interactions within and between layers are antiferromagnetic, however, since the neighbour layers are separated by two Br- layers the intra-plane exchange coupling J is two orders of magnitude larger than the inter-plane coupling J’. The interactions in the presence of a magnetic field are well described by the following Hamiltonian (the Heisenberg character of the spin system is reflected in the isotropic g-factors of the V2’ ion [14]), z%T==2JCSi,
l
sj,
ijn
+2J’cSi, in
l
Sin+, - pH
l
xSin,
(1)
in
where the summation i, j is taken over all nearestneighbour pairs in the ath layer; we assume here
0 1996 Elsevier Science B.V. All rights
reserved.
270
LN. ~~nda~e~~o et al. / Physics Leners A 222 (19961269-274
that the magnetic field is applied along the c-axis (axis z). In the pure 2D Heisenberg systems (J’ = O), the minimum-energy spin structures satisfy the condition s, + s, + s, = $$ (Si is the spin of the ith sublattice). It is evident that the system with coplanar or noncoplanar spin configuration has a nontrivial degeneracy: for a field H < 6JS/p all values of spin angles are accessible by rotating the vector S, and its position uniquely determines the direction of the other two vectors S,, S,. For H > 6 JS/p, the arbitrary orientations of Si are confined in a definite angular region. The forbidden angular region grows with H until all spins are ahgned along the field at H = 18 JS/p. The interplane interaction (J’ + 0) removes the continuous degeneracy and a state whose spin components perpendicular to the field form a 120” structure is selected from all possible configurations. However, for small J’ the energy difference between various states is also small and can be compensated by quantum effects. 2. The ground state of the layered triangular antiferromagnets VBr,, VCl, consists of six sublattices. Let us consider in the large-S limit six possible structures: one with noncoplanar spin configuration and five structures with coplanar configuration (Fig. 1). The canting angles and the energy of the noncoplanar umbrella-type structure (Fig. la> obtained by m~i~zation (1) are given by e,=
$&=2(a-l)7r/3,
a=1,2,3,
=(2a-5)~/3,
a=4,5,6,
E* -= N
(2)
b
C
d
e
f
Fig. 1. Spin configurations of the layered triangular autiferromagnet in the external magnetic field H: (a) noncoplanar umbrella-type configuration; (b), (c), cd>, (e), (0 coplanar configurations. The arrows with solid and dashed lines correspond to the spin directions in the two diffe~nt-TV layers.
The canting
angles for the b-configuration:
pH+2(3+j)JS cos 8=
’
4(3+j)JS
for the c-configuration: 1 cos t?=$hk+4hk, (k=
(9 I- 4j)/(9
1 cos x=;hk-=
(6)
+ 3j)) and for the f-configuration:
/.LH- 2(3 -j)JS cos @=
8,
a
2(3+j)JS
The minimum d-configuration
(7)
*
energy and the angles 8 and x of the (Fig. Id) in the case of small j are
E,=E, -(3Jf2J’)S2+SH~h2.
(3)
+ jJ( 1 - h2 - [( 1 - h2)( 1 - 9h2)] I/Z)S2N.
Here cos 19=/z, where h=H/H, (H,=(18J+ SJ’)S/p is the saturation field). For J’ Z 0 the energy of coplanar configurations is larger than that of the umbrella-type configuration. So, the minimum energy of co~igurations b, c and f (Figs. lb, lc, If) corrected to first order in j = J’/J (j +c 1) is
3hfl cos 0 = 2
E, = E,
Although the difference between this energy and the umbrella-type one is positive for H > 0, E, in (8) is
+jJ( 1 - h2)S2N.
(4
(8) - -ijh,
3hCQS x= 2
1 - fjh. (91
I.N. Bondarenko
et al./ Physics Letters A 222 (19961269-274
smaller than the energy of configurations b and f (h < f> and at H = 0 coincide with the energy E, of the noncoplanar structure a. Finally, the minimum energy of the coplanar configuration e is E =E 0
+ytl
-@)(5k-
*
1)
2h
S2N.
(10)
The equilibrium solutions for the angles at j = 0
coincide with the co~sponding ones in (6) and (8). We note that this configuration is degenerate with respect to the change of the angle between two planes including the spins 1, 2, 3 and 4, 5, 6. 3. Let us consider now the effect of quantum fluctuations. We take the local coordinate system in which the quantized axis is chosen along the spin direction. To describe the spin deviations from the equilibrium classical states we introduce six Bose operators ukLy. Restricting to cubic terms in the Holstein-Primakoff transformation we obtain .Z’ = E, + S?‘(l) + A@) + Zc3)_
271
Below we will compare the zero-point energy of different configurations with the energy of the structures due to small but finite j. The diagonal elements e(k) and flk) of the 6 X 6 matrices by putting j = 0 equal unity and zero, respectively; the nondiagonal elements are presented as e&(k)
=e&(k)
=B$‘(k),
&3(k)
=&L(k)
=$p)(k)~
(15)
where B$)(k)
= fzQ[(l f cos 0, cos 0s) X cos( rp, - ‘pp) + sin 0, sin 0, - i(cos 8, -t- cos Sp) sin( ‘p, - ipp)]
and l/k = f [exp(ik.)
+exp(i
-kx:fiky)
(16)
(11)
The linear term in 8’ is given by
(12)
Nere the indices cy, /!I are either 1, 2, 3 or 4, 5, 6. The remaining elements of e_ i s and f,,, rt 3 type are zero. We have calculated the excitation spectrum in the zero-point energy E@‘= -3JSN+
&(k).
(17)
kn
Here for k If 0 A,@(k) =2J,s(k){sin
rJp sin((p,-
For noncoplanar configuration a e,(k) (8, = EJ3.E)
cpp)
+ i[sin 0, cos @@ - cos @, sin @! Xcos(%-
cpp)]}7
e,(k) (13)
where J,@(k) are the Fourier components of the exchange interactions between the spins of distinct sublattices. As it should, Z?(‘) vanishes for such equilibrium configurations. The quadratic term in ZC2) can be written as ZC2) = - 3JSN + 3J.S~ c+V, CQ,
(14)
is given by
= ‘%+3(k) ={[I-h,(k)][1+(2-3hZ)h,(k)]]“Z -6hpJ
k),
(18)
where h,(k)
=j(cos[k,+$r(cu-
I)]
+2 cos[+k, - $?I-( cy- l)] cos$‘%k,},
k
where CY:= (a;, aud Mk=
e(k) f*(-k)
a,:, . . . , ale, a-k,. u-~~,.-. , u-~,+)
~a(k)=$(sin[k,+?j~(~-l)] -2 sin[+k, - $r( o - l)] cos~fik,).
f(k) e*(-k)
I .
It is known that in zero field the low-lying excitations of the triangular structures have a two spin-wave
I.N. Bon&we&o et al.,/ Physics Letters A 222 (1996) 269-274
272
1
0.75 0.50
-0.25-
0.25 0 M
0.5
0
I-
K
Fig. 2. The energy spectmm e,(k) at N = 0.3H, for eonfigurations a and b. The solid line corresponds to the noncoplanar configuration a and the dashed line corresponds to the coplanar configuration b.
velocity differing by a numerical multiplier fi [ 151. However, when h is nonzero the umbrella-ty~ structure has only one spin-wave velocity c= $G-XJS. The lowest branch of the spin-wave spectrum softens additionally near symmetric points of the Brillouin zone and has in the vicinity of them quadratic dependence. coplanar structures have similar properties if h # 0 holds. The spectra of configurations a and b at h = 0.3 are shown in Fig. 2 for a comparison. It is irn~~t that except for symmetric points of the Brillouin zone the lowest branch of configuration b is always lower than the corresponding one of configuration a. Moreover, in the largest region of k the second branch of configuration b is ako lower than that for the a-configuration. A similar situation is seen by comp~ng the spectra of umbrella-~~ structure with the other coplanar structures. We calculated the l/S quantum corrections to the ground-state energy for these six structures. Results displayed in Fig. 3 show that the energy of noncoplanar structure a is higher than that of coplanar struc-
1
h
Fig. 3. Quantnm corrections to the ground-state energy (Ae== AE/3JSN) for six spin configurations (solid line) and the energy difference of the classical structures measured from noncop~~~ configuration a at jS = 0.1 (dashed line).
tures, and the quantum corrections are the most significant for configurations b and c. The latter, however, do not realize at any H if jS > 0.096 holds. Instead, due to the competition between quantum quotations and inte~l~n coupling, the coplanar d-structure (If C H,> and umbrella-type structure (H > H,> are stabilized. The plot of total energy at jS = 0.1 for different structures is presented in
e 0
-0.10
a
f
-0.15 ~
-0.20 c
h
“X? j’
-025 i ’
0
0.5
1
h
Fig. 4. The total reduced ground-state energy e = E/SJSN for the six spin con~gumtions at jS= 0.1. e as a function of h is measured from e, = E, /3JSN for noncoplanar structure a.
I.N. Bondarenko
et al./Physics
Letters A 222 (1996) 269-274
273
classical equilibrium states. The cubic term in (11) is given by
XUfk,a ak,a ak,o +4CA,p(k,)afkjpak2Buk~*
+H.c-
018
I f20>
Fig. 5. The ground-state phase diagram in the jS- h plane (the l/S correction).
Fig. 4. Additional states b and c are stabilized if JS < 0.096 holds. As jS decreases, the region of these phases becomes larger and in the limit jS = 0 it covers the entire range 0 < H < EZ,(Fig. 5). In ?&-type compounds, V2+ ions are associated with an isotropic S = 3 moment, and the exchange parameters, for instance, for VBr, .Z= 16 K, J’ = 0.2 K were obtained [lo]. Hence, jS = 0.019. Thus, one can expect that the structure changes at h, = 0.18, h, = 3 and h, = 0.7 or at the fields H, = 58 T, ZZ,= 107 T and ZZ3= 227 T (g-factor for VBr, equals 2). Such an order of H can be reached in pulsed-field installations [16]. In contrast to the case of h = h,, the transitions at h = h, and h = h, are discontinuous. The magnitudes of the magnetization jumps at H, and H, are given by AM, = fjh,M,,
AM, = $jh,M,,
(1%
where Ma = /IS is the saturation magnetization. Substituting j, h, and h, for VBr, in (19) yields A~~/~~ = 0.75 x 10-3, A~~/~~ = 0.97 x 10-3. In nonzero field quantum fluctuations modify the different classical structures. Configurational changes are to be especially noticeable near the critical field (h = f > since the collinear configurations are strongly favored by quantum fluctuations. Let us calculate the qu~~rn corrections to the canting angles of the
Replacing two of the three operators in (20) with the expectation values and setting GE@(‘) + GYc3)= 0 we obtain the equations for 0, and cp,. In the case of coplanar configurations they have the following form, C.Z,,(O) P
sin( 6, - eP) - gsin
=-
Imp sin( Oz”)- ep’o)),
8$
0,
(21)
where Zaus=
RS-[L/3(“)(u:,ak,)
+Ja;(k)((a,+,dkp)
-
b,:,ak,&)]
(22)
and f#‘) is the canting angle of the classical structure. &aluating the expectation values in (22) for b and, c configurations by the usual means, we found that the collinear structure (0, = 0, = rr, o2 = 8, = 0, = I!?,= 0) emerges in the region between h 2.1 = f - 0.018/S
and h,,, = f f 0.043/S.
(23) For VBr, they correspond to the critical fields Hz,, = 104 T and Z& = 117 T. Table 1 Temperature dependence of the reduced field regions Ah,, Ah,, Ah,, Ah, of the co~pond~g con~gurations a, b, c, d at jS= 0.1 (I = T/ZE,)
t
4
Ah,
Ah,
Aha
0
0.31 0.3 1 0.30 0.29
-
_ 0.017 0.036 0.061
0.69 0.64 0.56 0.47
0.05 0.10 0.15
0.011 0.024 0.037
I.N. Barren
274
et a~./Fhysics
We considered also the effect of thermal fluctuations. A contribution to the free energy F=E*-33JSNf
L&rem A 222 (1996)
269-274
and c are still more stabilized by quantum fluctuations.
CE*(k) References [ll R.S. Gekht, Usp. Fiz. Nauk 159 (1989) 261 [Sov. Phys. Usp.
from the entropy was found numerically. The results show that, as the temperature is raised, the range of noncoplanar structure a of the phase diagram jS - h decreases both due to reno~alization of the saturation field and by reason of the stabilization of the structures b and c. In Table 1 we presented the stability region hh of structures d, b, c and a at various values of reduced temperature t = T/2e,, where E, = 3JS is the magni~de of e,(k) at the intersection of dispersion curves. It is clear that at j,S = 0.1 configurations b and c emerge if t > tc = 0.01 holds, and the stability region of configuration d, as in the case of configuration a, decreases with temperature increasing. In conclusion, we note that CsCuCl, appears to be the only compound known to date where quantum fluctuations really remove degeneracy in a frustrated spin system [6,7]. However, unlike the situation considered here, the successive phase transitions between distinct coplanar states are absent in CsCuCl,. We note also that the series compounds AGO, (A = H, Li, Na) consist of triangular layers with 120” spin structure, too, [17,1,8,12] but they are stacked rhombohedrally (space group R ?m). In view of this, the system is frustrated not only in the plane but also in the third direction which gives rise to an additional degeneracy of the classical states 119,201. We believe that in AGO,-type compounds the d configuration which in part is stabilized by interplane exchange coupling will be hardly realized at any values of j while the coplanar configurations b
32 (1989) 8711; R.S. Gekht and V.I. Ponomarev, Phase Trans. A 20 (1990) 27. 121L.P. Regnault and J. Rossat-Mignod, in: Magnetic properties of layered transition metal compounds, ed. L.J. De Jongh (Kluwer, Dordrecht, 1990) p. 271. r31 A. Harrison, Ann. Rep. Prog. Chem. (R. Sot. Cbem.) 87 A (1992) 211; 88 A (1992) 447. 141 M.L. Plumer, A. Caille, A. Mailhot and H.T. Diep, in: Magnetic systems with competing interactions, ed. H.T. Diep (World Scientific, Singapore, 1994). [51 E.L. Nagaev, Pis’ma Zh. Eksp. Tear. Fiz. 39 (1984) 402. [61 H. Nojiri, Y. Tokunaga and M. Motokawa, J. Phys. (Paris) 49 (1988) 1459. [71 H. Shiba and T. Nikuni in: Recent advances in magnetism of transition metal compounds, ed. A. Kotani and N. Suzuki (World Scientific, Singapore, 1993) p. 372. E.L. Nagaev and E.V. Zhasinas, Phys. Bl A.V. Mi~eenkov, Lett. A 205 (1995) 101. 191 KS. Aleksandrov, N.V. Fedoseeva and I.P. Spevakova, in: Magnetic phase transitions in crystals (Nauka, Novosibirsk, 1983). [lOI H. Kadowaki, K. Ubukoshi and K. Hirakawa, J. Phys. Sot. Japan 54 (1985) 363. 1111 H. Kadowaki et al., J. Pbys. Sot. Japan 56 (1987) 4027. 1121 N. Kojima et al., J. Phys. Sot. Japan 62 (1993) 4137. H.V. Loneysen and R.K. El31 J. Wosnitza, R. Deutschmann, Kremer, J. Phys. Condens. Matter 6 (1994) 8045. t141 I. Yamada, K. Ubukoshi and K. Hirakawa, J. Phys. Sot. Japan 53 (1984) 381. Il51 T. Dombre and N. Read, Phys. Rev. B 39 (1989) 6797. D61 J.J.M. Franse, J. Magn. Magn. Mater. 90/91 (1990) 20. iI71 J.L. Soubeyroux et al., Phys. Status Soiidi A 67 (1981) 633. El81 J. Ajiro et al., J. Phys. Sot. Japan 57 (1988) 2268. [191 E. Rastelli and A. Tassi, J. Phys. C 19 (1986) L423. [201 S.S. Aplesnin and R.S. Gekbt, Zh. Eksp. Tear. Fiz. 96 (1989) 2163 [Sov. Phys. JETP 68 (1989) 12501.
CQs .
1996
mm
PHYSICS LETTERS A
em
~
ELSEVIER
Physics Letters A 222 (1996) 269-274
Magnetic transitions in layered triangular antiferromagnets I.N. Bondarenko, R.S. Gekht, V.I. Ponomarev Kirensky Institute of Physics, Russian Academy of Sciences, Siberian Branch, Krasnoyarsk Received
660036, Russian Federation
4 July 1996; accepted for publication 5 August 19% Communicated by V.M. Agmnovich
Abstract Magnetic states and phase transitions of the layered dollar ~tife~omagn~~ in an applied field are studied. It is shown that in compounds like VBr, and VCl, quantum effects change the ground-state structure and cause successive phase transitions as the magnetic field increases. Coplanar structures of different spin configuration are realized far from the saturation field and a noncoplanar structure of umbrella-type configuration is realized near this field. The ground-state phase diagram is constructed, and a finite region of fields where the collinear phase is also possible is indicated. PACS: 75.30X2; 75.50.Ee Ke~or~: Triangular ~tife~orna~ne~;
~~~
and thermal fluctuations;
1. Recently much attention has been paid to the study of frustrated systems on a triangular lattice. Results of this study, both theoretical and experimental, are presented in recent reviews [l-4]. It is known that in nonfrustrated 3D systems quantum fluctuations modify only the spin structure but do not change the nature of the ground state. However, in frustrated systems quantum fluctuations are very important since they, as first pointed out by Nagaev [5], can cause a new phase transition mechanism. The effects of a magnetic field were also explored lately and many interesting results have been obtained [6-81. The goal of the present paper is to determine the spin structure of VX,-type compounds (X = Br, Cl) in an external magnetic field. We shall see below that quantum fluctuations induce successive phase transitions between different coplanar configurations in such substances. VX,-type compounds have the crystal structure of CdI, (space group P ?ml). The V2+ ions are ar0375-9~i/Q6/$12.~ Copyright PIi SO37.5-9601(96)00635-4
Successive
phase tmnsitions
ranged in the c-plane on the triangular layers which form a hexagonal lattice in the c-direction [9]. According to neutron-scattering data, below 40 K spins order with 120” structure [IO- 131. The interactions within and between layers are antiferromagnetic, however, since the neighbour layers are separated by two Br- layers the intra-plane exchange coupling J is two orders of magnitude larger than the inter-plane coupling J’. The interactions in the presence of a magnetic field are well described by the following Hamiltonian (the Heisenberg character of the spin system is reflected in the isotropic g-factors of the V2’ ion [14]), z%T==2JCSi,
l
sj,
ijn
+2J’cSi, in
l
Sin+, - pH
l
xSin,
(1)
in
where the summation i, j is taken over all nearestneighbour pairs in the ath layer; we assume here
0 1996 Elsevier Science B.V. All rights
reserved.
270
LN. ~~nda~e~~o et al. / Physics Leners A 222 (19961269-274
that the magnetic field is applied along the c-axis (axis z). In the pure 2D Heisenberg systems (J’ = O), the minimum-energy spin structures satisfy the condition s, + s, + s, = $$ (Si is the spin of the ith sublattice). It is evident that the system with coplanar or noncoplanar spin configuration has a nontrivial degeneracy: for a field H < 6JS/p all values of spin angles are accessible by rotating the vector S, and its position uniquely determines the direction of the other two vectors S,, S,. For H > 6 JS/p, the arbitrary orientations of Si are confined in a definite angular region. The forbidden angular region grows with H until all spins are ahgned along the field at H = 18 JS/p. The interplane interaction (J’ + 0) removes the continuous degeneracy and a state whose spin components perpendicular to the field form a 120” structure is selected from all possible configurations. However, for small J’ the energy difference between various states is also small and can be compensated by quantum effects. 2. The ground state of the layered triangular antiferromagnets VBr,, VCl, consists of six sublattices. Let us consider in the large-S limit six possible structures: one with noncoplanar spin configuration and five structures with coplanar configuration (Fig. 1). The canting angles and the energy of the noncoplanar umbrella-type structure (Fig. la> obtained by m~i~zation (1) are given by e,=
$&=2(a-l)7r/3,
a=1,2,3,
=(2a-5)~/3,
a=4,5,6,
E* -= N
(2)
b
C
d
e
f
Fig. 1. Spin configurations of the layered triangular autiferromagnet in the external magnetic field H: (a) noncoplanar umbrella-type configuration; (b), (c), cd>, (e), (0 coplanar configurations. The arrows with solid and dashed lines correspond to the spin directions in the two diffe~nt-TV layers.
The canting
angles for the b-configuration:
pH+2(3+j)JS cos 8=
’
4(3+j)JS
for the c-configuration: 1 cos t?=$hk+4hk, (k=
(9 I- 4j)/(9
1 cos x=;hk-=
(6)
+ 3j)) and for the f-configuration:
/.LH- 2(3 -j)JS cos @=
8,
a
2(3+j)JS
The minimum d-configuration
(7)
*
energy and the angles 8 and x of the (Fig. Id) in the case of small j are
E,=E, -(3Jf2J’)S2+SH~h2.
(3)
+ jJ( 1 - h2 - [( 1 - h2)( 1 - 9h2)] I/Z)S2N.
Here cos 19=/z, where h=H/H, (H,=(18J+ SJ’)S/p is the saturation field). For J’ Z 0 the energy of coplanar configurations is larger than that of the umbrella-type configuration. So, the minimum energy of co~igurations b, c and f (Figs. lb, lc, If) corrected to first order in j = J’/J (j +c 1) is
3hfl cos 0 = 2
E, = E,
Although the difference between this energy and the umbrella-type one is positive for H > 0, E, in (8) is
+jJ( 1 - h2)S2N.
(4
(8) - -ijh,
3hCQS x= 2
1 - fjh. (91
I.N. Bondarenko
et al./ Physics Letters A 222 (19961269-274
smaller than the energy of configurations b and f (h < f> and at H = 0 coincide with the energy E, of the noncoplanar structure a. Finally, the minimum energy of the coplanar configuration e is E =E 0
+ytl
-@)(5k-
*
1)
2h
S2N.
(10)
The equilibrium solutions for the angles at j = 0
coincide with the co~sponding ones in (6) and (8). We note that this configuration is degenerate with respect to the change of the angle between two planes including the spins 1, 2, 3 and 4, 5, 6. 3. Let us consider now the effect of quantum fluctuations. We take the local coordinate system in which the quantized axis is chosen along the spin direction. To describe the spin deviations from the equilibrium classical states we introduce six Bose operators ukLy. Restricting to cubic terms in the Holstein-Primakoff transformation we obtain .Z’ = E, + S?‘(l) + A@) + Zc3)_
271
Below we will compare the zero-point energy of different configurations with the energy of the structures due to small but finite j. The diagonal elements e(k) and flk) of the 6 X 6 matrices by putting j = 0 equal unity and zero, respectively; the nondiagonal elements are presented as e&(k)
=e&(k)
=B$‘(k),
&3(k)
=&L(k)
=$p)(k)~
(15)
where B$)(k)
= fzQ[(l f cos 0, cos 0s) X cos( rp, - ‘pp) + sin 0, sin 0, - i(cos 8, -t- cos Sp) sin( ‘p, - ipp)]
and l/k = f [exp(ik.)
+exp(i
-kx:fiky)
(16)
(11)
The linear term in 8’ is given by
(12)
Nere the indices cy, /!I are either 1, 2, 3 or 4, 5, 6. The remaining elements of e_ i s and f,,, rt 3 type are zero. We have calculated the excitation spectrum in the zero-point energy E@‘= -3JSN+
&(k).
(17)
kn
Here for k If 0 A,@(k) =2J,s(k){sin
rJp sin((p,-
For noncoplanar configuration a e,(k) (8, = EJ3.E)
cpp)
+ i[sin 0, cos @@ - cos @, sin @! Xcos(%-
cpp)]}7
e,(k) (13)
where J,@(k) are the Fourier components of the exchange interactions between the spins of distinct sublattices. As it should, Z?(‘) vanishes for such equilibrium configurations. The quadratic term in ZC2) can be written as ZC2) = - 3JSN + 3J.S~ c+V, CQ,
(14)
is given by
= ‘%+3(k) ={[I-h,(k)][1+(2-3hZ)h,(k)]]“Z -6hpJ
k),
(18)
where h,(k)
=j(cos[k,+$r(cu-
I)]
+2 cos[+k, - $?I-( cy- l)] cos$‘%k,},
k
where CY:= (a;, aud Mk=
e(k) f*(-k)
a,:, . . . , ale, a-k,. u-~~,.-. , u-~,+)
~a(k)=$(sin[k,+?j~(~-l)] -2 sin[+k, - $r( o - l)] cos~fik,).
f(k) e*(-k)
I .
It is known that in zero field the low-lying excitations of the triangular structures have a two spin-wave
I.N. Bon&we&o et al.,/ Physics Letters A 222 (1996) 269-274
272
1
0.75 0.50
-0.25-
0.25 0 M
0.5
0
I-
K
Fig. 2. The energy spectmm e,(k) at N = 0.3H, for eonfigurations a and b. The solid line corresponds to the noncoplanar configuration a and the dashed line corresponds to the coplanar configuration b.
velocity differing by a numerical multiplier fi [ 151. However, when h is nonzero the umbrella-ty~ structure has only one spin-wave velocity c= $G-XJS. The lowest branch of the spin-wave spectrum softens additionally near symmetric points of the Brillouin zone and has in the vicinity of them quadratic dependence. coplanar structures have similar properties if h # 0 holds. The spectra of configurations a and b at h = 0.3 are shown in Fig. 2 for a comparison. It is irn~~t that except for symmetric points of the Brillouin zone the lowest branch of configuration b is always lower than the corresponding one of configuration a. Moreover, in the largest region of k the second branch of configuration b is ako lower than that for the a-configuration. A similar situation is seen by comp~ng the spectra of umbrella-~~ structure with the other coplanar structures. We calculated the l/S quantum corrections to the ground-state energy for these six structures. Results displayed in Fig. 3 show that the energy of noncoplanar structure a is higher than that of coplanar struc-
1
h
Fig. 3. Quantnm corrections to the ground-state energy (Ae== AE/3JSN) for six spin configurations (solid line) and the energy difference of the classical structures measured from noncop~~~ configuration a at jS = 0.1 (dashed line).
tures, and the quantum corrections are the most significant for configurations b and c. The latter, however, do not realize at any H if jS > 0.096 holds. Instead, due to the competition between quantum quotations and inte~l~n coupling, the coplanar d-structure (If C H,> and umbrella-type structure (H > H,> are stabilized. The plot of total energy at jS = 0.1 for different structures is presented in
e 0
-0.10
a
f
-0.15 ~
-0.20 c
h
“X? j’
-025 i ’
0
0.5
1
h
Fig. 4. The total reduced ground-state energy e = E/SJSN for the six spin con~gumtions at jS= 0.1. e as a function of h is measured from e, = E, /3JSN for noncoplanar structure a.
I.N. Bondarenko
et al./Physics
Letters A 222 (1996) 269-274
273
classical equilibrium states. The cubic term in (11) is given by
XUfk,a ak,a ak,o +4CA,p(k,)afkjpak2Buk~*
+H.c-
018
I f20>
Fig. 5. The ground-state phase diagram in the jS- h plane (the l/S correction).
Fig. 4. Additional states b and c are stabilized if JS < 0.096 holds. As jS decreases, the region of these phases becomes larger and in the limit jS = 0 it covers the entire range 0 < H < EZ,(Fig. 5). In ?&-type compounds, V2+ ions are associated with an isotropic S = 3 moment, and the exchange parameters, for instance, for VBr, .Z= 16 K, J’ = 0.2 K were obtained [lo]. Hence, jS = 0.019. Thus, one can expect that the structure changes at h, = 0.18, h, = 3 and h, = 0.7 or at the fields H, = 58 T, ZZ,= 107 T and ZZ3= 227 T (g-factor for VBr, equals 2). Such an order of H can be reached in pulsed-field installations [16]. In contrast to the case of h = h,, the transitions at h = h, and h = h, are discontinuous. The magnitudes of the magnetization jumps at H, and H, are given by AM, = fjh,M,,
AM, = $jh,M,,
(1%
where Ma = /IS is the saturation magnetization. Substituting j, h, and h, for VBr, in (19) yields A~~/~~ = 0.75 x 10-3, A~~/~~ = 0.97 x 10-3. In nonzero field quantum fluctuations modify the different classical structures. Configurational changes are to be especially noticeable near the critical field (h = f > since the collinear configurations are strongly favored by quantum fluctuations. Let us calculate the qu~~rn corrections to the canting angles of the
Replacing two of the three operators in (20) with the expectation values and setting GE@(‘) + GYc3)= 0 we obtain the equations for 0, and cp,. In the case of coplanar configurations they have the following form, C.Z,,(O) P
sin( 6, - eP) - gsin
=-
Imp sin( Oz”)- ep’o)),
8$
0,
(21)
where Zaus=
RS-[L/3(“)(u:,ak,)
+Ja;(k)((a,+,dkp)
-
b,:,ak,&)]
(22)
and f#‘) is the canting angle of the classical structure. &aluating the expectation values in (22) for b and, c configurations by the usual means, we found that the collinear structure (0, = 0, = rr, o2 = 8, = 0, = I!?,= 0) emerges in the region between h 2.1 = f - 0.018/S
and h,,, = f f 0.043/S.
(23) For VBr, they correspond to the critical fields Hz,, = 104 T and Z& = 117 T. Table 1 Temperature dependence of the reduced field regions Ah,, Ah,, Ah,, Ah, of the co~pond~g con~gurations a, b, c, d at jS= 0.1 (I = T/ZE,)
t
4
Ah,
Ah,
Aha
0
0.31 0.3 1 0.30 0.29
-
_ 0.017 0.036 0.061
0.69 0.64 0.56 0.47
0.05 0.10 0.15
0.011 0.024 0.037
I.N. Barren
274
et a~./Fhysics
We considered also the effect of thermal fluctuations. A contribution to the free energy F=E*-33JSNf
L&rem A 222 (1996)
269-274
and c are still more stabilized by quantum fluctuations.
CE*(k) References [ll R.S. Gekht, Usp. Fiz. Nauk 159 (1989) 261 [Sov. Phys. Usp.
from the entropy was found numerically. The results show that, as the temperature is raised, the range of noncoplanar structure a of the phase diagram jS - h decreases both due to reno~alization of the saturation field and by reason of the stabilization of the structures b and c. In Table 1 we presented the stability region hh of structures d, b, c and a at various values of reduced temperature t = T/2e,, where E, = 3JS is the magni~de of e,(k) at the intersection of dispersion curves. It is clear that at j,S = 0.1 configurations b and c emerge if t > tc = 0.01 holds, and the stability region of configuration d, as in the case of configuration a, decreases with temperature increasing. In conclusion, we note that CsCuCl, appears to be the only compound known to date where quantum fluctuations really remove degeneracy in a frustrated spin system [6,7]. However, unlike the situation considered here, the successive phase transitions between distinct coplanar states are absent in CsCuCl,. We note also that the series compounds AGO, (A = H, Li, Na) consist of triangular layers with 120” spin structure, too, [17,1,8,12] but they are stacked rhombohedrally (space group R ?m). In view of this, the system is frustrated not only in the plane but also in the third direction which gives rise to an additional degeneracy of the classical states 119,201. We believe that in AGO,-type compounds the d configuration which in part is stabilized by interplane exchange coupling will be hardly realized at any values of j while the coplanar configurations b
32 (1989) 8711; R.S. Gekht and V.I. Ponomarev, Phase Trans. A 20 (1990) 27. 121L.P. Regnault and J. Rossat-Mignod, in: Magnetic properties of layered transition metal compounds, ed. L.J. De Jongh (Kluwer, Dordrecht, 1990) p. 271. r31 A. Harrison, Ann. Rep. Prog. Chem. (R. Sot. Cbem.) 87 A (1992) 211; 88 A (1992) 447. 141 M.L. Plumer, A. Caille, A. Mailhot and H.T. Diep, in: Magnetic systems with competing interactions, ed. H.T. Diep (World Scientific, Singapore, 1994). [51 E.L. Nagaev, Pis’ma Zh. Eksp. Tear. Fiz. 39 (1984) 402. [61 H. Nojiri, Y. Tokunaga and M. Motokawa, J. Phys. (Paris) 49 (1988) 1459. [71 H. Shiba and T. Nikuni in: Recent advances in magnetism of transition metal compounds, ed. A. Kotani and N. Suzuki (World Scientific, Singapore, 1993) p. 372. E.L. Nagaev and E.V. Zhasinas, Phys. Bl A.V. Mi~eenkov, Lett. A 205 (1995) 101. 191 KS. Aleksandrov, N.V. Fedoseeva and I.P. Spevakova, in: Magnetic phase transitions in crystals (Nauka, Novosibirsk, 1983). [lOI H. Kadowaki, K. Ubukoshi and K. Hirakawa, J. Phys. Sot. Japan 54 (1985) 363. 1111 H. Kadowaki et al., J. Pbys. Sot. Japan 56 (1987) 4027. 1121 N. Kojima et al., J. Phys. Sot. Japan 62 (1993) 4137. H.V. Loneysen and R.K. El31 J. Wosnitza, R. Deutschmann, Kremer, J. Phys. Condens. Matter 6 (1994) 8045. t141 I. Yamada, K. Ubukoshi and K. Hirakawa, J. Phys. Sot. Japan 53 (1984) 381. Il51 T. Dombre and N. Read, Phys. Rev. B 39 (1989) 6797. D61 J.J.M. Franse, J. Magn. Magn. Mater. 90/91 (1990) 20. iI71 J.L. Soubeyroux et al., Phys. Status Soiidi A 67 (1981) 633. El81 J. Ajiro et al., J. Phys. Sot. Japan 57 (1988) 2268. [191 E. Rastelli and A. Tassi, J. Phys. C 19 (1986) L423. [201 S.S. Aplesnin and R.S. Gekbt, Zh. Eksp. Tear. Fiz. 96 (1989) 2163 [Sov. Phys. JETP 68 (1989) 12501.