spin-wave theory on quasi-one-dimensional heisenberg antiferromagnets

A. Du and G. Z. WEI: Spin-Wave Theory on Quasi-1D Heisenberg Antiferromagnets 495

phys. stat. sol. (b) 191, 495 (1995)

Subject classification: 75.10

Department of Physics, Northeastern University, Shenyang ‘) ( a ) and CCAST (World Laboratory), Beijing’) (b)

Spin-Wave Theory on Quasi-One-Dimensional Heisenberg Antiferromagnets

BY A. Du (a, b) and G. Z. WEI (a)

(Received January 26, 1995)

The low-temperature properties of quasi-one-dimensional Heisenberg antiferromagnets with ferromagnetic coupling between nearest-neighbor spins within a chain and antiferromagnetic coupling between chains are studied by the spin-wave theory. The effects of interchain coupling on the ground state energy, sublattice magnetization, and perpendicular susceptibility at zero temperature are calculated numerically. At low temperatures, asymptotic expressions of sublattice magnetization, internal energy, magnetic specific heat, parallel and perpendicular susceptibilities are given as functions of temperature and interchain coupling strength within two sub-temperature regions. It is found that, if interchain coupling is weak, with increasing temperature the properties of the system change from three- dimensional behavior (Am oc T Z , AE cc T4, C, cc T 3 , xi, cc T5/’, AxI cc T 2 ) to quasi-one-dimensional behavior (Am cc B E cc T3/’, C , cc TI/’, xII a T3/*, Axl a T) .

1. Introduction

Pure one-dimensional Heisenberg systems with finite-range interaction cannot achieve long-range order (LRO) at nonzero temperatures because of the effect of quantum and thermal fluctuations [ 11. When there is some interaction between chains, these fluctuations may be depressed to a certain extent and the system may realize LRO. The transition temperature can be evaluated from (l.JJ‘l)l’z, where J and J’ are interactions between two nearest-neighbor spins within a chain and in two chains, respectively [2, 31.

In the past, the study of low-dimensional Heisenberg systems in theory has mainly concentrated on pure one- and two-dimensional magnetic structures due to their simplicity and similarity to real systems. Actually these studies aimed at the states above the transition temperature. In this case, the LRO of the system is destroyed, only short-range correlation between spins is retained. As most low-dimensional Heisenberg systems discovered in experiments possess weak interchain or interlayer couplings, i.e. have finite transition temperatures, the study of the properties of these systems below their transition temperatures is important [4].

In a previous paper [5] , we studied the low temperature properties of quasi-one- dimensional Heisenberg ferromagnets with ferromagnetic interaction between two nearest- neighbor spins within a chain and between chains, and found that if the interchain interaction is weak, the low temperature magnetization depends linearly on temperature, i.e. Am K T,

I ) Shenyang 110006, People’s Republic of China. ’) POB 8730, Beijing 100080, People’s Republic of China

496 A. Du and G. Z. WEI

in contrast to the T3I2 law in three-dimensional structures and the T In T relation in the quasi-two-dimensional case [6]. In this paper, we study the properties of a quasi-one- dimensional antiferromagnet with ferromagnetic interaction within a chain and antiferromagnetic interaction between chains, which is one kind of quasi-one-dimensional antiferromagnet, and (CH,),NCuCl, is an example [7].

2. Theoretical Model

We adopt the simple cubic chain Heisenberg model with intrachain and interchain lattice parameters a and c, respectively. Supposing the interaction between two nearest-neighbor spins within a chain to be ferromagnetic (J > 0) and the interaction between chains to be antiferromagnetic (J’ > 0), the lattice can be divided into two equivalent sublattices A and B, i E A, j E B. The Hamiltonian for this system is given by

where the first and second summations are over the nearest-neighbor sites within a chain and the third is over the nearest-neighbor sites between chains.

This model is formally the same as that for layered Heisenberg antiferromagnets with antiferromagnetic interaction within a layer and ferromagnetic interaction between layers, only the distributions of coupling strengths are different [S]. As in [8], introducing the Dyson-Maleev transformation, in the linear spin-wave approximation, we obtain the spin-wave energy (h = 1)

Qk = 211 cos (k,a/l/2) cos (kya/l/2). (3 b)

Here In nonlinear spin-wave approximation [9], the P k and Qk in spin-wave energy should

be modified as Pk + Pk(Un) = 2qa’ 4- [1 - COS (k ,C) ] a, Qk + Qk(Un) = 2qa‘ COS (k,a/\/Z) x cos ( k , a / f i ) , where a and a’ are correction factors taking into account interactions between

intrachain spin waves and between interchain spin waves, respectively, which are functions of temperature and interchain coupling strength. At low temperatures, a and a’ approach unity [9], so we do not consider them when calculating the physical quantities except for the perpendicular susceptibility.

represents the relative interchain interaction, q = J’/J.

3. Calculation of Physical Quantities and Discussion

3.1 Sublattice magnetization ( g p , is taken as unit)

The sublattice magnetization of the system is given by

Spin-Wave Theory on Quasi-One-Dimensional Heisenberg Antiferromagnets 497

Fig. 1. The dependences of sublattice magnetization at zero temperature and ground state energy on the relative interchain coupling strength J’/J. The dashed lines are the results for a quasi-one-dimensional ferromagnet; S = 1/2, E , is renormalized by - 4 N J

I . . . . .

o 0 2 a 4 f f i 0 8 in

JXJ- where p = 1/T ( k , = 1) and (S“), is the sublattice magnetization at zero temperature,

The numerical result of ( S ’ ) , as a function of relative interchain coupling strength is shown in Fig. 1 and 2. In the three-dimensional and one-dimensional cases, (S’), is 0.443 and 0.5, respectively. Because of the effect of quantum fluctuations, a small antiferromagnetic

n

4 x

10’

JLJ - Fig. 2. The dependences of perpendicular susceptibility and sublattice magnetization at zero temperature on the relative interchain coupling strength J’ /J ; the full and dashed lines are the results of nonlinear and linear spin-wave approximations, respectively; S = 1/2, ~ ~ ( 0 ) is renormalized by N(gl*d2/(16J’S)


interchain coupling may cause a rapid deviation of the system from its classical Ntel state (see the line in Fig. 2 as r j + 0). The stronger the antiferromagnetic interaction, the more (S'), deviates from its classical value of 0.5.

Although stronger antiferromagnetic interchain coupling induces stronger quantum fluctuations, reduces the sublattice magnetization at zero temperature, it depresses the thermal fluctuations of spins at finite temperatures and makes the system achieve LRO. When the temperature is lower than the transition temperature, which is expressed by T < 7', = 4JS at this moment, according to the depression of interchain coupling to spin thermal fluctuations, we can determine a characteristic temperature To, like in [5, 6, 81, To = 8J 'S = 8JSrj, which may reflect the dimensional crossover of the system. When the interchain coupling J' is weak, J' 4 J , the low temperature region, T < TI, may be divided into two subregions, T 4 To and To 4 T 4 TI. It is clearly seen that the first subregion reflects the three-dimensional characteristics of the system and the second reflects the quasi-one-dimensional characteristics of the system. If J' is not very weak, only the first subregion exists and the system exhibits three-dimensional characteristics at low temperatures.

In the two low temperature regions, the sublattice magnetization (4) can be expressed in asymptotic form as

1 1 c

where C1 = (1/z2) = 1.140; 5 is the Riemann zeta-function.

In the quasi-one-dimensional case, To < T < TI, it is clearly seen that with increasing temperature, the sublattice magnetization changes from three-dimensional temperature behavior ( T 2 ) to quasi-one-dimensional behavior (T ) , and the three-dimensional behavior reflects the antiferromagnetic characteristics of the system and the quasi-one-dimensional behavior reflects the characteristics of chain-like ferromagnets [5]. The temperature dependences of magnetization in quasi-two-dimensional Heisenberg ferromagnets and antiferromagnets are both in the form Tln T, so we deduce that in quasi-one-dimensional Heisenberg systems the temperature dependence of the magnetization is in the T form.

dx dy [l - (cos x cos 0 0

3.2 Internal energy and spec@ heat

The internal energy of the system can be obtained from E = (H), the result is

where E , is the ground state energy, expressed as

E , = -2(2rj + 1) N J S ( S + 1) + 1 w k . k

Fig. 1 shows the dependence of E , on the relative interchain coupling strength; the dashed lines represent the results for ferromagnets with ferromagnetic interchain coupling. When


the interchain coupling is the same for the two cases, the ground state energy of the system with antiferromagnetic interchain coupling is lower than that with ferromagnetic interchain coupling.

At low temperatures, the internal energy can also be expressed in asymptotic form within the two temperature regions as

niil

where C, = (l/n2) j j dx dy [l - (cos x cos y)’I1/’ = 0.842. 0 0

The magnetic specific heat can accordingly be obtained from C, = aE/aT. At low temperatures, its asymptotic expression is given by

In the quasi-one-dimensional case, the temperature dependence of specific heat is of TI’’ form, which is just characteristic of pure one-dimensional Heisenberg ferromagnets [lo], where the contribution of interchain antiferromagnetic interaction to specific heat is very small and appears in the second and other higher terms in (lob). Similarly, the three- dimensional behavior of specific heat exhibits antiferromagnetic characteristics of the system, i.e. T 3 temperature dependence.

3.3 Parallel and perpendicular susceptibilities

In order to discuss the response characteristics of the system to an external magnetic field, we calculate the parallel and perpendicular susceptibilities of the system. When the external field is applied along the direction of the z-axis of spins, we can obtain in linear response approximation the parallel susceptibility of the system with Liu’s method [9] as follows:

XI1 = 2(gPd2 P c exp (P%) b P (PWk) - . (11) k

At zero temperature x,, is zero. At low temperatures the asymptotic expressions for the parallel susceptibility in the two temperature regions are given by

I I X

where C, = (1/n2) j f dx dy [ l - (cos x cos y)2]-3/4 = 2.064. 0 0


In the quasi-one-dimensional case, at To 6 T < TI, the parallel susceptibility depends linearly on temperature. When the interchain coupling is very weak, it approaches infinity which is just characteristic of pure ferromagnets.

When the external field is applied perpendicularly to the z-axis of spins, we find in the linear response and linear spin-wave approximations that the perpendicular susceptibility is independent of temperature [8], which is in agreement with the results of mean-field theory [ 111. In order to find the dependence of perpendicular suceptibility on temperature, the spin-wave interaction should be considered [9]. Then the perpendicular susceptibility is expressed as

where m(un) is the sublattice magnetization obtained by considering the spin-wave interaction. Its numerical result at zero temperature is shown in Fig. 2 by the full line, it is smaller than that obtained in linear spin-wave approximation, when J’ = J , rn, = 0.443, mo(un) = 0.438.

At low temperatures, the asymptotic expressions of perpendicular susceptibility in the two temperature regions are given by

r . 1 8 \ 3 / 2 1

where cio and a; are correction factors at zero temperature taking into account the spin-wave interactions within a chain and between chains, respectively, and ~ ~ ( 0 ) is the perpendicular susceptibility at zero temperature,

The numerical result of ~ ~ ( 0 ) is shown in Fig. 2; when J’ + 0, ~ ~ ( 0 ) + a, which also is characteristic of ferromagnets.

In the above, we used T 4 TI = 4JS to represent the low temperature region, but we know the transition temperature of the system is of the order ([JJ’[)’’z. Thus the low temperature region defined should be modified to make the temperature lower than the transition temperature. For this purpose, we consider the kinematic interaction of spin-waves, i.e. the deviation of a spin from its equilibrium state is not beyond 2s. With this con- straint, we obtain the condition that the sublattice magnetization of the system should satisfy [5]

S m > -

4s + 1

Only with this modification the results of spin-wave theory are valid in the low temperature region (T < TI) and can be used to discuss the dimensional crossover of the system.


4. Conclusions

The low temperature properties of quasi-one-dimensional Heisenberg antiferromagnets with ferromagnetic interaction between two nearest-neighbor spins within a chain and antiferromagnetic interaction between chains are studied with the spin-wave theory. At zero temperature, with increasing interchain coupling strength, the effect of quantum fluctuations on the sublattice magnetization increases. The ground state energy of an antiferromagnet with ferromagnetic interaction within a chain and antiferromagnetic interaction between the chains is lower than that of a ferromagnet with the same intra- and interchain coupling strengths. The perpendicular susceptibility decreases rapidly with the increase of the interchain coupling strength. At low temperatures, for weak interchain coupling strength, sublat tice magnetization, parallel and perpendicular susceptibilites cross over from three- dimensional temperature behavior of the antiferromagnet (Am cc T 2 , xII cc T512, AxI cc T 2 ) to quasi-one-dimensional temperature behavior (Am K 7; x,, cc T312, AxI cc 7'). Internal energy and specific heat cross over from three-dimensional temperature behavior of the antiferromagnet (AE cc T4, C, cc T 3 ) to pure one-dimensional temperature behavior of the ferromagnet (AE K T312, C, K T'I'). If the interchain coupling is not very weak, the properties of the system exhibit only the three-dimensional temperature behavior of antiferrornagnets.

References

N. D. MERMIN and H. WAGNER, Phys. Rev. Letters 17, 1133 (1966). D. WELZ, J. Phys.: Condensed Matter 5, 3643 (1993). S. H. Lru, J. Magnetism magnetic Mater. 82, 294 (1989). L. J. DE JOUGH and A. R. MIEDEMA, Adv. Phys. 23, 1 (1974). A. Du and G. Z. WEI, Phys. Rev. B 49, 15360 (1994). A. Du and G. Z. WEI, Phys. Rev. B 46, 8614 (1992). K . RAVINDRAN, D. N. HAINES, and J. E. DRUMHELLER, J. appl. Phys. 67, 5481 (1990) A. Du and G. Z. WEI, phys. stat. sol. (b) 177, 243 (1993). S. H. LIU, Phys. Rev. 142, 267 (1966). J . C. BONNER and M. E. FISHER, Phys. Rev. 135, A640 (1964). S. V. VONSOVSKII, Magnetism, Vol. 2, John Wiley & Sons, New York 1974.

spin-wave theory on quasi-one-dimensional heisenberg antiferromagnets

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