R. P. SINGH and M. SINGH: Effect of Interlayer Coupling on Nee1 Temperature 571

phys. stat. sol. (b) 169, 571 (1992)

Subject classification: 75.10; 75.30; S.10.15

Department of Physics, University of Western Ontario, London ’)

Effect of Interlayer Coupling on NCel Temperature in Antiferromagnets

R. P. SINGH and M. SINGH

The Nee1 temperature is calculated for a 3D Heisenberg model taking into consideration the anisotropy in exchange coupling constant. By virtue of the two-sublattice approach to antiferromagnets the equation of motion is solved for the Green’s function and an expression for Neel temperature is obtained. The higher-order Green’s functions are decoupled by random phase approximation. These expressions are applied to evaluate the Neel temperature of YBa,Cu,O,+, and La,CuO,. The effect of interlayer coupling is examined. The theoretical results are compared with experimental values of Nee1 temperature.

Unter Berucksichtigung der Anisotropie in der Austauschkopplungskonstante wird die NCel-Tem- peratur fur ein 3D-Heisenbergmodell berechnet. Durch Anwendung des Zwei-Untergitter-Verfahrens auf Antiferromagnete wird die Bewegungsgleichung fur die Greensche Funktion gelost und ein Aus- druck fur die Neel-Temperatur erhalten. Durch Random-Phase-Naherung werden die Greenschen Funktionen hoherer Ordnung entkoppelt. Diese Ausdrucke werden benutzt, urn die Neel-Temperatur von YBa,Cu30, +, und La,CuO, zu berechnen. Der EinfluR von Zwischenschichtkopplung wird untersucht. Die theoretischen Ergebnisse werden mit experimentellen Werten der NCel-Temperatur verglichen.

1. Introduction

The anisotropy in antiferromagnetic correlation coupling in certain materials plays an important role in the magnetic properties. For example, high temperature superconductors are antiferromagnetic in normal state and it has already been experimentally established that they have large difference in the antiferromagnetic coupling constant within CuO, planes and between the CuO, planes in the c-direction. They have anomalous magnetic and electronic properties due to this anisotropy [l to 51. It is well established now that antiferromagnetism (AFM) in high temperature superconductors (HTSc) is driven by weak interlayer coupling and disappears by doping. For instance, changing the oxygen stoichiome- try in YBa,Cu,O, + (1-2-3) the system undergoes an antiferromagnetic to superconducting transition. The ground state of these HTSc is known to be an AFM insulator with CU” spins coupled antiferromagnetically along and between CuO, planes when x 5 0.4. It is now well known that the strongest magnetic interactions in these HTSc lie within the CuO, planes. The study of the insulating phase in these high temperature superconductors can give enough evidence about the interplay between magnetism and superconductivity. The ground state of these undoped systems shows antiferromagnetic long-range order [6] . It is found that for x 5 0.4 in YB~,CU,O,,~, Cu2+ spins are coupled antiferromagnetically on the Q O , planes and along the tetragonal axis. In these layered oxides the intraplanar

’) London, Ontario N6A 3K7, Canada.

572 R. P. SINGH and M. SINGH

coupling J , , is much stronger than the interplanar coupling J , [7]. When 2D correla- tions become sufficiently long-ranged in these high temperature superconductors 3D ordering follows due to the presence of this weak but finite interlayer coupling. In the temperature range T < TN (TN is the Neel temperature), where 3D ordering sets in, the interlayer coupling plays a very crucial role in determining the AF correlation in these materials [8].

Recently, there has been some effort [7 to 12, 16 to 191 to study the role of this interlayer coupling in HTSc. 2D and quasi-2D Heisenberg models have been extensively studied to get an insight into the magnetic dynamics of these HTSc [8, 121. According to Tranquada et al. [7], in layered systems the Neel temperature is determined by kBTN = J?&, where <2D is the correlation length in the two-dimensional plane and Jyf is the effective coupling between planes. They have shown that for a TN z 400K in 1-2-3, the effective inter- layer coupling should be of the order of 0.2meV. Chakravarty et al. [8] used the renormalization-group approach to study the quantum nonlinear o-model in two space dimensions and one Euclidean time dimension. They have calculated the correlation length at low temperatures with the p-function calculated in one and two loop order. They have estimated the interlayer coupling in La,CuO, (2-1-4) J , to be of the order of 10-sJ1l . This small value of interlayer coupling has insignificant effect on two-dimensional correla- tion above the 3D Nkel ordering temperature. Further, the zero temperature properties calculated for an isolated CuO, plane are only weakly affected by such a small value of J,.

Singh et al. [12, 131 have pursued the role of the interlayer coupling in magnetic dynamics of copper-based antiferromagnets even further. They have argued that the excitation of spin waves in anisotropic antiferromagnets such as 1-2-3 results in a characteristic dependence of sublattice magnetization on temperature. Further, due to these excitations there is a crossover from 3D to quasi-2D behaviour with temperature dependence of sublattice magnetization varying from - T 2 (3D case) to - T In T (quasi-2D case). According to them, the Niel temperature in copper-based antiferromagnets falls logarithmically with decreasing interlayer coupling. Li et al. [9] have shown that the coupling between layers is essential in case of 1-2-3 to achieve a superconducting transition temperature of 90 K. They have also studied the effect of layer thickness on the interlayer coupling in copper oxide-based superlattices. Most of these results are obtained in a 2D or quasi-2D Heisenberg model. A true generalization of these theoretical results is only possible when a 3D model is intensely examined and the above results are reproduced. The three-dimensional anisotropic Heisenberg model is difficult to study and there has been very limited effort to analyze the magnetic dynamics in such a model of HTSc.

In this paper, we have studied the three-dimensional anisotropic Heisenberg model to calculate the magnetization and Neel temperature of an anisotropic antiferromagnet. We have obtained expressions for the Neel temperature for spin-4 using two-sublattice approaches for antiferromagnets. We have obtained an expression for the magnetization using the equation of motion method for the Green’s function. In this theory the higher-order Green’s functions are decoupled by the random phase approximation. Making some mathematical simplifications we have obtained an expression for the NCel temperature. We have self-consistently calculated the variation of sublattice magnetization with temperature. The effect of interlayer coupling on the Nee1 temperature is also examined. In the last section we have applied these expressions to calculate the Nkel temperature and magnetiza- tion of 1-2-3 and 2-1-4 compounds.

Effect of Interlayer Coupling on Nee1 Temperature in Antiferrornagnets 573

2. Theory

We start with the anisotropic Hamiltonian

H = c 2511 Si. Sj + c 2J,Si . S j , ab C

where J , , and J , are the AFM coupling constant in ab-plane and c-direction, respectively. A similar situation arises in high temperature superconductors. For example, in 1-2-3 and 2-1-4 compounds JII will be the coupling constant between spins within the copper-oxygen planes and J , will be the coupling constant between the copper-oxygen planes, respectively. The sums are taken between nearest neighbours within the ab-plane and between planes in the c-direction. The expression for the doping dependent coupling constant for AFM correlation has already been given by Singh [20]. He has considered four possible mechanisms which contribute towards AFM correlation, viz. kinetic exchange, two superexchange mechanisms, and an indirect correlation exchange in HTSc. We will now make use of the thermodynamical Green’s function method to investigate the Hamiltonian given in (1). This method was initially proposed by Bogolyubov to study ferromagnetism [14]. Later it was used by Hewson and Ter Haar [15] to study the magnetic properties of the antiferrornagnet CuCl, . H,O. We divide the simple cubic lattice into two sublattices denoted by ‘f‘ and ‘g’ where the nearest neighbour of ‘f‘ is on ‘g’ sublattice and vice-versa. We would now develop the equation of motion for the Green’s function given by ((SF, S J ) where S’ = S” isy are the spin raising and lowering operators, respectively. The corresponding equation of motion is written as

Evaluating commutators appearing in the above equation and writing the corresponding equation of motion we find a Green’s function of higher order. In order to avoid complications with the equation of motion for higher-order Green’s functions, we ap- proximate the solution by applying the random phase approximation. We write

<<s;sj’, s,>> = (s;) <is;> s,>> >

<<s;sg, s, >> = c q > (<Sf > s; >> . Making the above approximation we obtain

where G,, is the Green’s function when j’ and g sites are on the same sublattice and G,, when they are on different sublattices. Here G,, is defined as follows:

Here, N is the total number of spins in the lattice and k is the reciprocal lattice vector which runs over the first Brillouin zone. In (3), 5 and y are defined as

j j 5 = 2 5 , , C e i k t i - S ) + 2 5 1 e i k ( j - f )

ab C

574 R. P. SINGH and M. SINGH

and

y 8Jli + 4 J l .

Similar expressions can be obtained when the sites are assumed to be on different sublattices. Finally, GI, has the following form:

where

and

El = < S ; > v m .

The correlation functions are related to the Green’s function by the following equa- tion:

+ m

Using this equation we have evaluated the correlation function (SiSf) as

(S,Sf) = 2(s;) ~ F [ P coth (A) - 11 N 2knT (7)

To calculate the average value of spin in one of the sublattices in c-direction we use the following equation for spin-4:

(SF) = 4 - (sts:). (8)

Putting the value of ( S g - S f ) from (7) in the above equation we get

- 1 = ~ ~ [ P c ~ t h ( ~ ) - I] + 2 . S N 2knT

(9)

Here we have represented ( S i ) by s. The above equation would give us the variation of sublattice magnetization with temperature. It can, however, be reduced to give an explicit expression for the Nee1 temperature. We know that as T + TN, S + 0 and the above expression reduces to

Effect of Interlayer Coupling on NCel Temperature in Antiferromagnets 575

A precise numerical estimate of the Neel temperature can be obtained by integrating the above expression. This is a general expression but it can be used to give us the variation of Nee1 temperature with the intralayer and, more importantly, the interlayer coupling in case of high temperature superconductors.

3. Results and Discussion

We have calculated the Neel temperature in case of YBa,Cu,O,+, and La,CuO, for x = 0 using the expression derived earlier. Using (9) we have calculated self-consistently s against temperature. The numerical results are presented in Fig. 1 for 1-2-3 and 2-1-4 collectively. The sublattice magnetization is related to f by M = gpB$ where g is the g-factor and ,uB the Bohr magneton. It is clear from the figure that the magnetization decreases as the temperature increases and becomes zero at T 2 TN. The values of the Nee1 temperature of 460 K for 1-2-3 and 320 K for 2-1-4 are consistent with the experimental values obtained so far. We have also calculated the values of Nee1 temperature from (10) directly. For the same values of the parameters Jll and J , we found TN = 470 K and 321 K for 1-2-3 and 2-1-4, respectively. It is worth noting that the difference in the values of Nee1 temperature obtained from (9) and (10) is very small. The values of the parameter Jll = 5.5 meV for 1-2-3 and 3.7 meV for 2-1-4 are different from those reported in the literature but the ratio of interlayer to intralayer coupling is taken to be of the order of lo-’ which incidently is the ratio found in 2-1-4 compounds. The different values of Jll may be due to the crudeness of the approximation involved in getting the anisotropic expression for the Neel temperature. However, the nature of the variation of sublattice magnetization with temperature from our theory is consistent with the results obtained by Singh et al. [13]. We have used the random phase approximation to simplify our theoretical calculation of the Green’s function. This difference can be reduced by doing theoretical calculations beyond the random phase approximation.

In conclusion, we have calculated the magnetization and Nee1 temperature by using the anisotropic 3D Heisenberg model for antiferromagnets. We have applied the random phase approximation in the calculation of the Green’s function to get an analytical expression for the Neel temperature. Using these theoretical expressions for the Neel temperature and the magnetization we have calculated the Neel temperature of 1-2-3 and 2-1-4 compounds.

Fig. 1. Variation of Swith tempera- ture for undoped YBa,Cu,O,+, and La,CuO,. The value of J , , = 5.5 meV for 1-2-3 (curve 1) and 3.7meV for 2-1-4 (curve2) com- pounds. The ratio of interplanar to intraplanar coupling is taken to be 1 0 - 5

0 100 200 300 4 00 500 Tfifl-

576 R. P. SINCH and M. SINGH: Effect of Interlayer Coupling on Nee1 Temperature

Acknowledgements

We are thankful to J. Sonier for helpful discussions and going through the theoretical formulations. One of us (MS) is also thankful to Natural Science and Engineering Research Council of Canada for financial support in the form of a Research grant.

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(Received October 1, 1991)