PHYSICAL REVIEW B VOLUME 49, NUMBER 21 1 JUNE 1994-I

Low-temperature properties of quasi-one-dimensional Heisenberg ferromagnets

A. Du and G. Z. WeiDepartment ofPhysics, Northeastern Uniuersity, Shenyang iI0006, People's Republic of China

(Received 10 November 1993)

Spin-wave theory is used to study the effects of interchain coupling on the properties of quasi-one-dimensional Heisenberg ferromagnets. The asymptotic expressions of magnetization, internal energy,and specific heat with temperature and interchain coupling strength are given at low temperatures. It isfound that, for a weak interchain interaction, with the increase of temperature from zero, the magnetiza-tion and specific heat change from exhibiting three-dimensional behavior (Am ~ —T', C ~ T' ') toquasi-one-dimensional behavior (b,m ~ —T, C ~ T' ').

One-dimensional (1D) and two-dimensional (2D)Heisenberg systems do not show long-range magnetic or-der (LRO) at nonzero temperatures, ' while quasi-1D fer-romagnets with weak interchain interaction in three-dimensional space and quasi-2D systems with weak inter-layer coupling can achieve 3D LRO at finite tempera-tures. ' At low temperatures in 3D ferromagnets, thereduction of magnetization due to the spin thermal exci-tations follows the Bloch T law, and the magneticspecific heat also follows the T law. In 3D antifer-romagnets, the reduction of sublattice magnetization fol-lows a T law, and the specific heat follows a T law. Inquasi-2D Heisenberg systems with weak interlayer cou-pling, the reduction of magnetization in both ferro- andantiferromagnets follows a TlnT law, and specificheats for them follow Tand T laws, respectively. Thetemperature behavior of the specific heat is the same asthat in pure 2D systems. As to quasi-1D systems withweak interchain coupling, the reduction of magnetizationand specific heat due to the spin thermal excitations forferro- and antiferromagnets deserve further study. So, inthis paper, we take a quasi-1D Heisenberg ferromagnet asa prototype to study its low-temperature properties, in-cluding magnetization, internal energy, and specific heat,and we give asymptotic expressions for these physicalquantities at low temperatures.

For simplicity, we start with a simple-cubic-latticeHeisenberg model with interchain and intrachain lattice

parameters, respectively, a, and a and c. The modelHamiltonian is given by

H = —2(1+2')NJS + g toi, al, ar,k

where N is the total number of lattice sites, and

col,=4JS [ 1 —cos( k, c ) + rI [2—cos( k„a ) —cos( k» a ) ] ] .

(2)

(3)

Thus we can calculate such physical quantities of the sys-tem as the magnetization, internal energy, specific heat,and so on.

The magnetization per site m (the unit is taken to be

gp~) is given by

H= —J$S;Sl—J'$S;S),( ij ) ( ij )

where the first sum is over two nearest neighbors withinthe chain and the second sum is over the four nearest-neighbor interchain interactions. We define ri= J'/J todenote the relative strengths of the interchain interac-tions.

Introducing spin deviation operators al and al andperforming the Holstein-Primakoff transformation on S;and S., within the linear approximation, we rewrite Eq.(1) in terms of spin-wave operators al, and al„

m =S——g [exp(Pco„)—1]1

k

, f f f tIk„cIk»tIk, g exp[ —4npJS[1+2ri —ri(cosk„+cosk ) —cosk, ]I,n=1

(4)

where p= 1/k~T. Using (1/m) J oexp(x cos8)dO=Io(x), which is the zeroth-order Bessel function of imaginary argu-Inent, we have

m =S—g exp[ —4(1+2')nPJS][Io(4gnPJS)] Io(4nPJS) .n=1

Similarly, using the definition E =(H ) /N and (1 /) focos0exp(x cos8)19=I,(x), we may obtain the internal energyper site,

0163-1829/94/49(21)/15360(3)/$06. 00 49 15 360 1994 The American Physical Society

BRIEF REPORTS 15 361

E/4JS =— + g exp[ —4(1+2ri)n PJS)Io(4rinPJS) I Io(4rtnPJS) [Io(4n PJS) I—, (4n PJS)](1+2'�)S

n=1

+2rtIo(4nPJS)[Io(4rInPJS) I—, (4rln13JS)]) . (6)

Z~oo Z

Io(x) =2rrx

1+ + . +1

we obtain the asymptotic expression of the magnetizationat low temperatures as follows,

We now discuss the low-temperature behavior of themagnetization and internal energy using Eqs. (5) and (6).At low temperatures, 4PJS »1 (or T «T, =4JS/ktt),and the temperature is also much smaller than the inter-chain coupling strength J'( =rIJ), that is, 4rlPJS »1 (orT«To=4rtJS/ktt}. Using the asymptotic expressionfor the Bessel function,

I

matter how small the interchain coupling strength is, andthe system exhibits 3D temperature characteristics.

When the temperature increases and becomes muchhigher than the interchain coupling strength, that is,To «T«T, (it is clear that such a case exists onlywhen the interchain coupling strength J' is much smallerthan J), we can discuss the quasi-1D properties of the sys-tem. Suppose v=4qPJS(2 —cosk„—cosk~), then fromEq. (4) we have

m =S— f f dk„dk& —f dk, &~s~& k ~+„e ' —1

(10}1 T Tm=S—

(2~) ~ T) To

3 2+g( 5

2 8 2+.. +

TQ

U is a very small quantity, so the long-wave part of k,contributes mostly to the integral. Thus we take1 —cosk, =k, /2, and after simplifying the integral, wehave

( T« To & Ti ), (7)

where g is the Riemann zeta function. The leading termdue to thermal spin-wave excitations is just the well-

known T ~2 law of Bloch describing the spin deviationfrom the maximum value at T =0. Similarly, from Eq.(6},we obtain the internal energy per site as follows,

E/4JS ( 1 +2Y) )S2

' 1/2

Using the Bose-Einstein integral function F(a, v) forsmall u (Refs. 9 and 10),

+ 3 T'" T2(27r) T] To

4

5 5(2+q) 72 24 2

TTQ

tF(a, u)= f, dtI a o e'+' —1

p( I )Q —]+ g ( u} g(a n

)n=Q 7l .

After integrating over k„and k„, we obtain the magneti-zation as follows,

+ + (T «T T, ) . (8)

1/2

C.rk, =4(2~)'"

The magnetic speci6c heat per site is given byC =BE/BT, and it is

'1/2 ' '1/2m=S—

v2 T,

X C+ 1 1

7r 2

1/2

+ . . - +

X 5 +7(2+~) 7 T2 24 2 To

( T« To & T, }, (9}

of which the leading term in T also reflects the 3Dmagnetic character of the Heisenberg system. So, in thetemperature regime T &( TQ T„ the temperature ismuch smaller than the interchain coupling strength, no

(T «oT «Ti), (13)

where C =(1/n)f f dx dy1. /v'2 —cosx —cosy =0.909(Ref. 3). It is found that, for a quasi-1D Heisenberg fer-romagnet, the reduction of the magnetization due tothermal spin-wave excitations follows the T law at lowtemperatures. In the quasi-2D case, it follows the T ln1law, and in the 3D case, it follows the T law.

In a similar manner, from Eq. (6), we obtain the inter-nal energy per site in the quasi-1D case,

15 362 BRIEF REPORTS

/4JS(1+271}S2

1 T2(2~)' T)

X g—+2(3 1

2 2

TQ +. . +

X3

22 1+—

g3 2

Tp + +

(TQ &(T« TI ) . (15)

The leading term of the specific heat in the T' law isthe same as that in a pure 1D system, "but the contribu-tion of the interchain interaction is smaller and expressedin the second- and other higher-order terms.

For 1D Heisenberg ferromagnets, there is no LRO atfinite temperatures, but a finite interchain coupling in 3Dspace can guarantee that the system achieves LRO. Atlow temperatures, T « T, (or ks T « J), when the tem-perature increases initially from zero, T (& To (orks T«J'), intra- and interchain couplings can suppressspin thermal fluctuations sufficiently that the system ex-hibits 3D temperature characteristics [Eqs. (7)—(9)].When interchain coupling is much weaker (J'«J), theincreasing temperature may cross over To (or ks T ))J').In this case, spin thermal fluctuations are so much biggerthat interchain coupling may no longer suppress themsuSciently, and thus the system exhibits quasi-1D tem-

( To &( T « Ti ), (14)

and the specific heat per site is given by1/2

C /k~ =m 8

4(2 )]/P

bn, =g(n, }—(n&} =Q(n&)((n, )+1) . (16)

Because m =m& =S —(n, ), the restriction condition is

thus changed into

Sm& 4S+1 (17)

With such a restriction condition and T, «T «T, , Eq.(13) can be used to discuss the magnetization in quasi-1Dferromagnets, and so can Eqs. (14) and (15).

perature characteristics [Eqs. (13)—15}] rather than 3Dcharacteristics, although the system has 3D LRO. Ofcourse, when the interchain coupling J' is not very small,only the temperature regime T «Tp T, exists, and thesystem exhibits only 3D low-temperature characteristics.

Finally, we discuss the restriction condition of Eq. (13).For a given temperature within the temperature regimeTo «T &(T, , the magnetization m [Eq. (13)] may beequal to or smaller than zero when the interchain cou-pling strength approaches zero (To~0}. This is clearlyincorrect, for spin-wave theory is usually valid at lowtemperatures, T &0.6Tc, where Tc is the Curie tempera-ture. This indicates that the low-temperature regimeT «T, we defined does not reflect completely theeffective range of spin-wave theory as interchain couplingbecomes very small, for in this case ks Tc = 1.556(JJ')'(Ref. 3). However, we can use the restriction conditionof a spin deviation from its equilibrium position at zerotemperature to modify the applicable range of Eq. (13).For a Heisenberg spin system, the spin deviation operatorn, =a&a& is always restricted by n, 2S. Beyond thatrange, unphysical states may be introduced. As in Ref.12, we approximate this restriction condition by

( n&) +b n, & 2S, where ( n, ) is the thermodynamic

average value of n „and An&

is the fluctuation of the spindeviation per site given by

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