Magnetic model for Ba 2Cu3O4Cl2

Journal of Magnetism and Magnetic Materials 177 181 (1998) 737 738

~i ELSEVIER

Journalof magnetism and magnetic materials

Magnetic model for Ba2Cu304C12 J. Richter a'*, A. Voigt a, J. Schulenburg a, N.B. Ivanov b, R.

Hayn c

Institut Jiir Theoretische Phvsik, Universitiit Magdeburg, P. O. Box 4120. D-39016 Magdeburg, German), h Institute Jbr Solid State Physics, Sofia, Bulgaria ~MPAG 'Elektronensysteme ', TU Dresden, Germany

Abstract Ba2Cu304C12 consists of two types of copper atoms, Cu(A) and Cu(B). We study the corresponding Heisenberg model with three antiferromagnetic couplings, JAA, JBB and JAB- We find interesting frustration effects due to the coupling JAB" ((') 1998 Elsevier Science B.V. All rights reserved. Keywords: Antiferromagnetism; Quantum fluctuations; Frustration

The exciting collective magnetic properties of layered cuprates have attracted much attention over the last decade. Recent experiments on BazCusO4C12 [1,2] show magnetic properties of this quasi-2d quantum antiferromagnet which differ from the well-studied antiferromagnetism of the parent cuprates like LaeCuO4. Most interesting is the observation of two magnetic critical temperatures (T~ ~ 330 K, T~ ~ 40 K) and of a weak ferromagnetic moment [1,2], where a simple explanation of the ferromagnetic moment by a Dzyaloshinsky-Moriya exchange can be ruled out [3]. The important difference between the parent cuprates and BazCusO4C12 is the existence of additional Cu(B) atoms located at the centre of every second Cu(A)-O2 square. The coupling between the A spins JAA is strongly antiferromagnetic. The observation of a smaller second critical temperature T~ [1-3] indicates a weaker antiferromagnetic coupling "/BB between the B spins. Additionally, there is a competing coupling JAB between A and B spins giving rise for interesting frustration effects. We consider the classical and the quantum (spin 1) version of the Heisenberg model H = JAA

Z

(i~A, jeA)

+JAB

SiSj nt- JBB

~

SiSj,

~ SiSj (i~B, j~B)

(1)

(/cA, j~B)

* Corresponding author. Tel.:49 391 671 8841;fax: 49 391 671 1131; e-mail: j [email protected]

where the sums run over nearest neighbour bonds of type A-A, B-B, A-B. Tight binding calculations [4] and the large T~ indicate a strong antiferromagnetic JAA. Because the couplings JBB and JAB are of higher order of the hopping integrals [4] they can be assumed as (much) smaller than JAA- In this paper we focus our interest on antiferrornagnetic JBB, JAB < JAA and discuss the magnetic properties as a function of JAB and JBB for fixed JAA = 1. Since the copper spin is ½the quantum nature of the spins is of great importance. To take into account full quantum fluctuations we use an exact diagonalization algorithm to calculate data for two finite lattices of N = 12 sites (8 A spins and 4 B spins) N = 24 sites (16 A spins and 8 B spins). For comparison we present also some data for the corresponding classical systems. The ground state results for small JBB and JAB are as follows: • Classical model: Starting from the point JAB = 0 we have N6el ordering in the subsystems A and B. These two N~el states are decoupled and can rotate freely with respect to each other, i.e. the ground state is highly degenerated. Increasing the frustrating JAB there is a first-order transition to a non-planar canted state at J~B = 2 ~ (see Fig. 1). In this non-planar state the A spins form a slightly tilted N6el state, where the sublatrice magnetization M~,A is lowered to about 90% for JAB~>J~B and decreases with increasing JAB. In the B subsystem the angle between neighbouring spins is ~z/2 and the spins built a planar state perpendicular to the sublattice magnetization axis of the A subsystem. However, due to the canting of A spins there is

0304-8853/98/$19.00 ,? 1998 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 2 8 3 - 7

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J. Richter et al. / Journal of Magnetism and Magnetic Materials 177-181 (1998) 737-738 1.5

,

i

o

0.4

i

•.~-

JAB

t

x.--x

i

.~. x

I

i

-..x ,. x.

,.,

i

X

0.3

I--El--D--E].-

E]--~3- (~]._.[3._

"..X.

0.2

0.5

M~ j''

o

qllall~ILr~l

0.1

M~,a -D- M;,s ×

XX-. ~+z N-+-

00

011

01.2

0,3 JBB

Fig. 1. Transition line between the N~el phase and the canted phase (see text) for small JAB and JBB (JAA = 1, N = 24). Notice that for larger JAB several other phases appear which are not considered here.

a correlation between A and B spins. There is no ferromagnetic m o m e n t in the non-planar canted state. However, an almost degenerated classical planar state exists with a weak ferromagnetic moment which increases with JAB. Q u a n t u m fluctuations may change this situation and could select a planar state instead of a non-planar ground state. F o r larger JAB, further transitions to other ground states take place, which are not considered here. • Quantum model: We focus our consideration on the lattice with N = 24 sites. The results for N = 12 are comparable supposing a relation of J ~ = 2 J ~ . (Notice that for N = 12 the B-spins have only two nearest B neighbours instead of four.) First, we consider the antiferromagnetic ground state ordering for small JAB, where the two subsystems order in a quantum N6el state. In contrast to the classical case the quantum fluctuations cause a typical 'order from disorder' effect and lift the classical degeneracy by selecting a state with a collinear structure of two sublattice magnetizations. This is accompanied by the development of a magnetic coupling between A and B subsystems. A similar 'order from disorder' phenomenon is well known from the J1 -- J2 antiferromagnet on square lattice for J z / J 1 ~ 0.65 [5, 6]. The stability of the N6el state is supported by quantum fluctuations (Fig. 1). The transition line to a canted state lies well above the classical instability line and for small JBB < 0.07 JAA the line follows approximately the relation J~,B ~ 2 . 7 ~ . A detailed analysis of the canted state gives indications for planar structure which is, however, slightly different from that classical planar state which is almost degenerated with the non-planar classical ground state• The ground state order parameters for N = 24 are shown in Fig. 2 for JaB = 0.1JAA which corresponds to the relation of critical temperatures. Obviously, both systems are antiferromagnetically ordered. However, the antiferromagnetic order in the B system is weakened by JAB and drops down dramatically at the critical line. This is accompanied by an increase of the square of the total

0.2

0.4

0.6

0.8jAB 1

1.2

Fig. 2. Square of magnetic order parameters v e r s u s JAB for = 0.1 (JAA = 1, N = 24). M s, 2 A(Ms. 2 B) staggered magnetic moment of subsystem A (B) MA2(M~) total magnetic moment of subsystem A (B). JBB

3.5 3 2.5 2 1.5 1

JahJA"= 0.5 0.0

--

0.5

00

I

I

I

I

0.25

0.5

0.75

1

I

1.25 kT

1.5

Fig. 3. Specific heat versus temperature for JaB = 0.25 (JAA = 1, N = 12) and two different JAB.

magnetic moments MB, z MA2 indicating the possibility of a weak ferromagnetic moment in the q u a n t u m system. Let us finally present the temperature dependence of the specific heat c(T) for the q u a n t u m case (Fig. 3). The exact calculation of e(T) needs the complete diagonalization of the Hamiltonian and is restricted to very small systems, i.e. to N = 12 in our case. We find two peaks in c(T) indicating the two transition temperatures. The peak positions TA and TB correspond to the coupling strengths JAA and JBB. The frustrating coupling JAB causes a decrease of TA and TB. F o r the parameters used in Fig. 3 we have TA(JAB = 0.5)/TA(JAB = 0) = 0.99 and T B ( J A B = 0 . 5 ) / T B ( J A B = 0) = 0.87. This work was supported by the D F G (Ri 615/1-2). References

[1] I-2] [3] [4] I-5] I-6]

S. Noro et al., Mater. Sci. Eng. B 25 (1994) 167. K. Yamada et al., Physica B 213-214 (1995) 191. F.C. Chou et al., Phys. Rev. Lett. 78 (1997) 535. H. Rosner, R. Hayn, Physica B (1997), to be published. K. Kubo, T. Kishi, J. Phys. Soc. Japan 60 (1991) 567. J. Richter, Phys. Rev. B 47 (1993) 5794.