The spin-wave theory of the susceptibility of Heisenberg magnet

Physica l19B (1983) 265-268 North-Holland Publishing Company

THE SPIN-WAVE THEORY OF THE SUSCEPTIBILITY OF HEISENBERG MAGNET II. Perpendicular susceptibility of a ferromagnet with small anisotropy Kazuo T S U R U and Norikiyo U R Y U Department of Applied Science, Faculty of Engineering, Kyushu University, Fukuoka 812, Japan Received 9 November 1982 A general character of the perpendicular susceptibility of ferromagnet and antiferromagnet has been studied by the renormalized spin-wave theory. It is shown that the fluctuation part can be neglected for the perpendicular susceptibility of a ferromagnet as well as of a ferromagnetic layer system with a weak antiferromagnetic interlayer coupling. For these systems, it is sufficient to consider the static part only to explain the experimental data. The discussions are made taking into consideration the results obtained in the previous paper in which the susceptibility of a quadratic layer antiferromagnet K2MnF4 has been analyzed.

1. Introduction In the previous paper [1] (which hereafter is referred to as I) we have analyzed the susceptibilities of a quadratic layer antiferromagnet K2MnF4 and compared the results with those of (CzHsNH3)2CuCI4. In the present paper, we consider the perpendicular susceptibility X± of the Heisenberg ferromagnet with a small uniaxial anisotropy. In the compound (C2HsNH3)2CuCI4, a strong ferromagnetic coupling Jf exists between the nearest-neighbor spin pairs in the c plane and a very weak antiferromagnetic coupling Ja~ between the ferromagnetic sheets [2]. Then we may say that this compound is a typical example of the nearly two-dimensional ferromagnet. In I we have remarked that the fluctuation part )~) is much smaller than the static part ,~) for the perpendicular susceptibility of (C2HsNH3)zCuC14. The ratio ) ~ ) / ~ ) is proportional to Jar~Jr ( ~ 1 0 - 3 ) . However, we have not given any detailed discussions about its reason. Therefore, we show here explicitly that the fluctuation part can be neglected in Xl of the (C2HsNH3)2CuC14 type antiferromagnets as well as ferromagnets with a small anisotropy. Except the dimensionality of the lattice, a pure ferromagnet corresponds to the limit Jar--> 0. Therefore the fluctuation part will show a similar behavior in both the systems stated above. For simplicity, we study the perpendicular suscep-

tibility of a ferromagnet by the same method of renormalized spin-wave (RSW) theory as employed in I. Previously, Velu et al. [3] measured and analyzed the perpendicular susceptibility of the three-dimensional ferromagnet CuRb:Br4.2H20. They considered the static part only and obtained good agreements between the theory and the experiment up to 0.7To (To the critical temperature). The contribution from the fluctuation of spins, however, was not estimated. We elucidate the reason why the fluctuation part could be neglected for this system. Further supplementary comments will be given for I.

2. Theory Here we derive the expression for the perpendicular susceptibility X± of a ferromagnet with a small uniaxial anisotropy. The Hamiltonian is given by = Yt~° + ~ ' ,

(2.1)

with g(o = -2.1 Z St" Sm- K Z S~S~ , {tm)

~ ( ' = - g t z a H x ~ S'[, I

0378-4363/83/0000-0000/$03.00 © 1983 North-Holland

(2.2a)

{lm)

(2.2b)

266

K. Tsuru, N . Ury~ / T h e susceptibility o f H e i s e n b e r g magnet. H

where J is the nearest-neighbor exchange coupling of the isotropic part and K that of anisotropy. Further g is the g-factor, /zs the Bohr magneton and Hx an external magnetic field. All the parameters are assumed to be positive. The unperturbed part of the Hamiltonian equation (2.2a) is a simplification of that in ref. [3]. The Zeeman term 9g' is taken as a perturbation. Using the Holstein-Primakoff method [4] and the Fourier transforms of the Boson operators ak and a ; , the Hamiltonian can be written as follows:

obtained as

k

where n~ = a]ak and Elk = # [ ( 1 -- A l -~-/12)(1 -- ~/k) -~- ~'(1 - / 1 1 -

(2.3a)

~ ' = Yg, + Y(3,

(2.3b)

here we have neglected the terms higher than quartic and

")/kA2)] ,

(2.6a) g(A~,/12) = ~ ¢ N S [ ( A 1 - / 1 2 ) 2+ ~'(a 2+/12)1, /11 = ( N S ) * ~ . (nk),

(2.6b)

/12 = ( N S ) -1 ~ , yk(nk) .

k

y~0 = Yg0+ ~2 + ~4,

(2.5)

= ~o + ~ eknk + g(A,,/12),

k

(2.6c) Here ( - . - ) denotes the thermal average. Then the free energy is given by

F = ~ o + ~ fk(A,,/12)+ g ( A l , / 1 2 ) ,

(2.7a)

k

27go= -z2NSo¢(1 + ~),

(2.4a)

fk(/11,/12) = k B T log[1 - exp(-/3ek)],

~1

(2.4b)

with the Boltzmann constant kB, temperature T and fl = 1/kBT. T h e explicit formula of/11 and A 2 can be obtained from the minimum conditions of the free energy,

=

-X/-N-S/2glzsHx(ao + a ~d),

~=

J(1-

Yk + ~ ) a ; a k ,

(2.4c)

k

gtzuHx ~_~ {a]la~ak36(k ~ - k 2 - k3) 4"X/2--N-S k,k2k3 + a],a*~2ak36(k~ + k2 -- k3)},

(2.4d)

aF = 0 0/1i

(i = 1, 2),

(2.7b)

(2.8)

or the relation ~4 =

4 N S k~k, x a *k,a *k2ak3ak4a(kl

qt_

k 2 - k 3 - k4).

(2.4e)

We have introduced the following quantities J = 2zJS,

~ = K/J,

and

1

Yk = z ~-" eik "p p

where the summation ~ p is taken over the nearest-neighbor spins, z is the number of them, and N is the total number of spins. The diagonal part of gg4 is renormalized into ~2 and subsequently the RSW-Hamiltonian is

~ k f k ( A l ' A2) = (Flk) '

(2.9)

and eq. (2.6c) [1]. If we use eq. (2.9) for the calculation of eq. (2.8), self-evident equations are obtained. For instance, we obtain immediately the following relation: OF

Ofk aek + a

a A l -- k'~ 0Ek 0/11 = - ~

J(1

~1

g ( A l ' A2)

- ~,~ + ~ ' ) < n 0

k + NS~[(1 + ~")A 1 - A2] = O.

These facts guarantee the justification of the

K. Tsuru, N. Uryfi / The susceptibility of Heisenberg magnet. H

decouplings applied to ~4. Conversely speaking, we have chosen the decouplings uniquely to satisfy the above relations. If we introduce A3 as defined by /13 = ( N S ) -1 Z yk(aka-k + a~a*-,) k

for the renormalization of ~(4, the m i n i m u m condition OF/OA3= 0 yields A3 = 0. Then it was not necessary to introduce A 3 in the present case. For a system which has the terms aka_~ and a~a*-k in Y(2, one must take A3 into account. With the use of such a method one can easily obtain the terms of the higher order corrections even for a complicated system. Next, we give the formula of the perpendicular susceptibility at Hx = 0. Along the same procedure as developed in I, the result is obtained as

X± =

N(g/zB)2 0 ~ ) + . ~ ) )

(2.10)

2zJ

where , ~ ) and , ~ ) denote the static and the fluctuation part, respectively. They are given by

~(1 - A 1

1

~?~ = - -~ ( N s ) -2 Z

-

-

A2)'

(2.11a)

#

klk2k3 ~'kl - - e k 2 - - ~ k 3

267

be applied to the off-diagonal part of the even degree terms. For example, the off-diagonal part of eq. (2.4e) can be calculated by employing the procedure, and the Oguchi's correction 0.2/S [5] is obtained in the expression for the spontaneous magnetization of three-dimensional ferromagnet. Secondly, it is noted that the o p e r a t o r G1 (see eq. (3.16a) of I) is proportional to the term of the first degree in ( Z t S ~ - E,,S~). In I we introduced a canonical transformation defined by Ygt = e - i ° ( ~

+ Y(') e i°,

( G = G1 + G3).

The quantity G has been chosen so as to eliminate the odd degree terms. In the present case, G1 is given by

Ol

=

i~/-N--~

glXBHx (ao- a"d). Co

Then, the o p e r a t o r G1 represents a simultaneous rotation of spins around the y axis (cf. ref. [6]). If we omit the terms Yg3 and G3, eq. (3.8) of I yields a cardinal n u m b e r and the terms higher than HZx do not a p p e a r in the transformed Hamiltonian. In this case the canonical transformation introduced in I gives a uniform shift of the origin of the Boson operators at k = 0 . Such a transformation is equivalent to that used by K u b o [7] and Oguchi [5]. T h e r e f o r e the present m e t h o d is a generalization of the previous ones.

× (n,1)a(kl- k2- k~) (MS) -~

P

3. D i s c u s s i o n

klk2k3 ~ k l - - C,k2 - - E k 3

× (nkln,~ + nk~n,~- n,~nk~)8(k~- k2- k3).

(2.11b) For simplicity we have omitted the term of 1/8S which corresponds to that in eq. (3.12) of I. The expression of eq. (2.11a) agrees with that of Velu et al. [3], whose n u m e r a t o r has been approximated as ( 1 - A1). H e r e supplementary c o m m e n t s should be m a d e in connection with the procedure of calculation developed in I. Firstly, the procedure to calculate the second order perturbation energy, which consists of the odd degree terms, can also

In the previous section we have derived the expression of perpendicular susceptibility for the ferromagnet with a uniaxial anisotropy. T h e order of magnitude of the fluctuation part can be estimated as follows: The main contribution for the first t e r m of eq. (2.11b) comes from the vicinity of kl = 0 because of the Bose factor (nkl). Then one m a y approximate e~l = 0 and eke -~ ek3 at very low temperatures. Neglecting the second term, we obtain (3.1)

K. Tsuru, N. Uryft / The susceptibility of Heisenberg magnet. H

268

Noting that

1~ 1 -N

ek

1

1

kB T ~ F

~ '

(3.2)

with T~ F a critical temperature predicted by the Green's function method, the fluctuation part 2~) can be estimated as of an order of unity. We may conclude that the ratio )~)/)?~) is proportional to ~'. Therefore, the fluctuation part can be neglected in the system with ~ 1 or in the (C2HsNH3)zCuC14 type antiferromagnet. In the case of a large sr value the fluctuation part should be considered. These results are consistent with those of Velu et al. [3]. For X± of the ferromagnet CuRb2Br4-2H20, the fluctuation part can be neglected because of the small value of r ( ~ 1 0 3).

Considering the results stated above and those obtained in I, we arrive at the following conclusion. Generally speaking, the static part )?y) can be interpreted as determined by the interactions which suppress the induced magnetic moment along the external field from the viewpoint of the static balance of torques. The antiferromagnetic coupling Jar in (C2H5NH3)2CuC14, the anisotropy K, etc. belong to this kind of interactions. On the other hand, the fluctuation part )?f) is determined by a major interaction which governs the property of the spin-wave dispersion at finite wave numbers. That is the ferromagnetic exchange coupling in the compounds (C2HsNH3)2CuC14 and CuRb2Br4"2H20. For these systems the interactions can be classified into two kinds stated above. In the case of the K2MnF4 type systems, however, both the static and fluctuation parts of X± are governed by a sole interaction, i.e. the antiferromagnetic exchange coupling.

The above situations may be understood more fully by a consideration of the following facts. The susceptibility of the ordered state must transfer continuously to that of the paramagnetic region at the critical temperature. In the K2MnF4 (or MnF2) type antiferromagnets, the static part )?~) alone does not satisfy this condition, because )~) is approximately proportional to the sublatrice magnetization and decreases rapidly with increasing temperature. Then we had to take the fluctuation part into account to explain the experimental data. We may remark that the fluctuation part )?f) has a paramagnetic character in such a system. As for the (C2HsNH3)~CuCI4 type antiferromagnets or the pure ferromagnets, the effects of the minor interactions such as Ja or K decrease rapidly with increasing temperature because of the spin-wave renormafization (see for instance eqs. (2.6a), (2.20b) of I or (2.19) in ref. [8]). They may be neglected in the paramagnetic region. Therefore, the static part )?~) alone will have a large value comparable to the paramagnetic susceptibility at temperatures near To.

References [1] K. Tsuru and N. Uryfi, Physica lI5B (1983) 205. [2] L.J. de Jongh, W.D. Amstel and A.R. Miedema, Physica 58 (1972) 277. [3] E. Velu, J.-P. Renard and B. Lecuyer, Phys. Rev. B14 (1976) 5088. [4] T. Holstein and H. Primakoff, Phys. Rev. 58 (1940) 1098. [5] T. Oguchi, Phys. Rev. 117 (1960) 117. [6] J. Kanamori and K. Yosida, Prog. Theor. Phys. 14 (1955) 423. [7] R. Kuho, Phys. Rev. 87 (1952) 568. 18] K. Tsuru and N. Uryfi, J. Phys. Soc. Jap. 41 (1976) 804.