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Theory of itinerant electron antiferromagnetism
144
Journal of Magnetism and Magnetic Materials 45 (1984) 144 150 North-Holland, Amsterdam
T H E O R Y OF I T I N E R A N T E L E C T R O N A N T I F E R R O M A G N E T I S M M. S H I M I Z U Department of Applied Pl~vsics, Nagoya UniversiO,, Nag¢~va464, Japan
Itinerant electron antiferromagnetism is studied by taking into account the realistic electronic structure and the effect of spin fluctuations within static Gaussian statistics. Expression for the magnetic susceptibility above and below the Ndel temperature are obtained. It is shown that spin fluctuations can stabilize antiferromagnetism in a certain case. The character of magnetic transitions and the comparison with experiment are discussed.
1. Introduction
Itinerant electron antiferromagnetism was first studied by Lidiard [1] as an extension of the Stoner model of ferromagnetism. Since then many theoretical works on itinerant electron antiferromagnetism have been carried out [2-5]. Recently, the effect of spin fluctuations on itinerant electron antiferromagnetism has been taken into account [6]. By making use of the free electron model with umklapp processes, it has been shown that the Ndel temperature T N is lowered from its H a r t r e e - F o c k value and the temperature variation of the uniform and staggered spin susceptibilities has been calculated only above T N. In experiments on metallic antiferromagnets, various kinds of magnetic transitions and their combinations have so far been found, in addition to the ordinary second-order transition at T N, e.g. the first-order transitions between the antiferromagnetic and paramagnetic or ferromagnetic states and between two kinds of antiferromagnetic states and the transitions from the antiferromagnetic state to the spin-density-wave, canted or ferrimagnetic state. In the YMn 2 compound the paramagnetic susceptibility X above the first-order transition temperature still increases with increasing temperature, as in Cr although Cr exhibits a spin-density-wave state, and shows a large magnetovolume effect below this transition temperature [7]. Moreover, some compounds, e.g. TbCu [8,9], HoCu 2 and ErCu 2 [10], Pt 0.678Fe0.322 [11] and NbS [12], PrCd [13], etc. show double peaks in the temperature variation of X, although rare earth compounds may not be good examples of itinerant electron antiferromagnets. In order to explain these experimental results mentioned above qualitatively and quantitatively, a realistic electronic structure of each metallic antiferromagnet and the effect of spin fluctuations should be taken into account. This is the purpose of this paper. The effect of spin fluctuations on the magnetic properties of metallic ferromagnets and paramagnets has been calculated before within static Gaussian statistics by a realistic itinerant electron model [14-16]. Recently, this treatment has been applied to intermetallic compounds of Y and transition metals [17,18]. These calculations have also been extended to the case, where the exchange stiffness constant A is negative so that the wave vector dependent spin susceptibility Xq shows a peak at the cutoff wave vector qm, and this has been applied to Rh and fcc Fe [19]. In the case of fcc Fe it has been found that the averaged square of the fluctuating magnetic moment ~q at ~/I = qm becomes finite at 0 K, although a long-range antiferromagnetic order is not included. In this paper the previous treatment of spin fluctuations in the realistic itinerant electron model for metallic ferromagnets is extended to the case where Xq shows a single maximum at ]ql = Q. Expressions for the uniform and staggered spin susceptibilities above and below T N are derived. Their temperature variation and the character of the magnetic phase transitions are discussed. 0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M. Shirnizu / Itinerant electron antiferrornagnetism
145
2. Averaged magnetic free energy It is assumed that the exchange stiffness energy is given by ½AI ~7MI 2 + ½BI x72M[ 2 with A < 0 and B > 0. The magnetic free energy of an itinerant electron magnet with volume V can be written as a power series of the square of the magnetization density M~(r) (i = x, y, z) as
f = f d 3 r [ ½ ( a l - a ) M ( r ) 2 + ~a3M(r)4 + ~asM(r)6+ ... +½AlUM(r)]2+ ½B[~y2M(r)[2 + . . . ] , (1)
where M(r) 2 = Z,M,(r)2; the coefficients a~, a 3, a 5.... can be given as functions of T in terms of the density of states curve [20] and a is the molecular field coefficient. Mi(r ) is assumed to be given by the sum of the uniform magnetization of M0,, the staggered magnetization MQfir)= f2MQ, cos(Q-r) with wave vector Q and the fluctuating magnetization m~(r) = Zqmq, e iq'r,
M,(r) = Mo, + MQ,(r) + m, (,').
(2)
By the same method as before [14] f in eq. (1) is averaged over the gaussian distribution of mqi. Then the averaged f is given by a function of 34o2,, MQ, and ~z q, = ( m q i m q i ) , which is the average of mq~mq~, as
f = -½k,TEi,q(ln(2rrV21~ 2~qzi ) + 1} + V [½Y~i,q(Aq 2 + Bq")~zq, + ½(al- a ) ~ 2 + ¼a3(~4 q._ 2Y~i~?) +
6+
+
2 2 2 ) + ½(A 0 2 + BQ4)M~
+ ½ { a l - a + a3~ 2 + a5(~4+ 2Y~i,~¢ ) ) ( M 2 + M~)+o3Z,~:2(Mg, + M~,) + 2asZ, (~2,{52 + 2~4)(M2, + M ~ , ) + ~(a3 + 2as~2){M g + ~M~ + 2Mo2M~ + 4(Mo'MQ) 2}
+ 2asY~i~2i { Mo2M~i+ ~22v*OtV*O 3..2~.2 i + M~M~+ MdMd,+4(Mo.MQ)Mo,MQ~ 2 2 } + 1
(
6+ 5
6 + 3MoMd 4 2+
91"21"4~12(M2 3 }1 T + 2yMQ)(M°'MQ)2
(3)
where
M g = ~iMgt,
M ~ = ~.tM~i,
~2= ~ q~qi2
and
The formulae to calculate Mo~, MQ, and ~q2 are obtained from the extremum condition of f w i t h respect to M0,, M0g and ~q2i. The matrix components of the uniform and staggered spin susceptibilities can also be obtained by X~ ~= (OOf/OMo,)/OMoj)/V and XQ)j = (O(af/aMQ,)/OMo./)/V, respectively. As the expressions of these formulae are lengthy, the results are shown below only for usual cases.
3. Spin susceptibility above and below TN
3.1. Pararnagnetic state above TN In this case Moi = MQi = 0 (i = x, y, Z), ~2 = ~2. --~~2..= ½~2,X~I = 6,jX -1 and XQ]j =6,sXQ 1. The uniform
M. Shirnizu / Itinerant electron antiferromagnetism
146
spin susceptibility X is given as before [14] by X
-
I = a l _ o~ +
5 2 ~a3g; + ~5a5~4 +
(4)
...
and the staggered susceptibility Xo is given by X Q 1 = X - 1 -{~ A Q
2+
BQ 4 = X -1 -
X , n 1,
(5)
where Xm 1 = 4 B / A 2 and the value of Q can be determined as
Q = (-A/2B)
1/2
(6)
from the condition that Xq shows a m a x i m u m at Iql = Q. The ~2 is given by
~2i = k , T V - ' ( X
-1 + Aq 2 + Bq 4 ) ~.
(7)
Then, the sum of ~2 over q can be integrated as
~2 = 3 k , T V - , ~ q ( X - 1
+ Aq2 + Bq4)
= (3k,T/4,~){ZB(B/x)I/2
'
+ }AIB )
,/2
(8)
Further, eq. (8) is rewritten as
2=(3k.r/a
,)XmQV2{(x.,/xt lj2 + 1} 'j2
(9)
By solving eqs. (4) and (9) we can get the temperature variation of X and ~2. It should be noted here that the same expression as eq. (8) is obtained also in the ferromagnetic and paramagnetic cases A > 0 and B > 0 and there is no need to introduce the cutoff of the wave vector as in the previous treatment of spin fluctuations for the case A > 0 and B = 0 [14]. In the ferromagnetic case the value of ~2 at the Curie temperature Tc is simply given by ( 3 k B T J 4 , ~ ) ( A B ) 1/2. T N is determined by the condition XQ 1 = 0 at T = T N and we have from eq. (5) X = Xm at Ty and from eq. (9) the value of ~2 at T N is given by
~ = ( 3/8"~ )kBTNXmQ 3/2.
(10)
Therefore, T N can also be defined as the temperature where the value of X-~ in eq. (4) becomes equal to that of X~ ~, as shown schematically in fig. 1. It is easily seen from eq. (4) that in the normal case where a3 > 0 and a 5 > 0 the value of T N is reduced by the effect of spin fluctuations. However, if a 3 < 0 and a s > 0 the spin fluctuations may raise the value of T s. In the case, where the temperature variation of X-~ in eq. (4) shows a m i n i m u m due to that of al or due to the negative sign of a 3 together with the fact that ~2 is proportional to T at lower temperatures, and if the value of X~,1 lies just between the value of 1 / X at 0 K and its m i n i m u m value as shown in fig. lb, antiferromagnetism can appear only in a finite range of temperature, say between T M and T N. The negative value of a 3 and the minimum in the temperature variation of a~ can stabilize the antiferromagnetic order at finite temperature through the effect of spin fluctuations, even if antiferromagnetism is not stable at 0 K, as well as in the case of ferromagnetism [14,16]. It is seen that both values of X 1 at Ty and TM are the same as given by X~,~. We may call this p h e n o m e n a a thermal spontaneous antiferromagnetism corresponding to the thermal spontaneous magnetization in the case of ferromagnetism as observed for Y2Ni 7 [21,22]. This case will be further discussed below.
M. Shimizu / Itinerant electronantiferromagnetism
147
3.2. A ntiferromagnetic state below TN In this case, M q . 4= 0, MQ,. = Moy = 0, M0, = 0 (i = x, y, z). ,~. = ~l and ~+ = ,~, = ~t. T h e parallel spin susceptibility XII when the external field H is applied along the z direction is given by
XII- 1
al-~+a3(3~(+2~:t)+as(15~4+12~2~et +8~4)+3{a3+2as(5~+2~2)}MQ--+ 15a ?~'4"4
(11) and the p e r p e n d i c u l a r spin susceptibility X± when H is applied along the x or y direction is given by 3 4 +(a 3 + 6a5,~( + 8as~2)M~: + ~a5 M~:. (12)
X~_ t = a] - a + a 3 (,~( + 4,~ 2) + a s (3,~ 4 + 8,~,~t2 + 24~ 4)
H e r e M~+, ,~2 = y~+~¢,q 2 and ~2 = )2q~tZq are d e t e r m i n e d , respectively, f r o m -X/,, ~ + a~ - a + a3(3~ ~ + 2~¢2) + a5 (15,~ 4 + 12,~(,~ 2 + 8,~t4) + ~(a33 +
lOas~+aas~2)M~:+ 7asM~.=0,5 4
(13)
~=k.TV-~Y~q(X,I]+Aq2+Bq4)-'=(kBT/4v/2'rr)XmQ3/2{(Xm/XI,)]/2+I}
-~/~
(14)
'/2 + ]} -,/2
(15)
and
+:t = kBTV+'F-.q(X~ t + A q 2 + B q 4)
! = (kBT/4x/2'rr)X,+.Q3/2(CXm/X±)
T h e values of Xll, X±, MQ:, ~2 and ,~ at each t e m p e r a t u r e are o b t a i n e d b y solving eqs. (11)-(15) simultaneously. At T N, M~: b e c o m e s zero and ~2 = ,~2 = 3,~ t : ,sothatxN = X±=X=Xm. Fromeqs.(ll) and (13) Xii ~ is rewritten as
5asM~:.
X~ 1 = Xm I + 3(a3 + 1 0 a s , ~ + 4as,~t2) M~: +
x.'
x-V
-
~
X4
-T
?_ ,o,_i x
"'--s'/
,
~:' (hi
~2 (ol
TH
-T
0
TI,,I
/
"T
,I"/%
~2 (el 2
/
o
TN
gN o
x'/
x_,'?
t<:,,-,1 x+tl ,,
---~
x; Ic)
X-1 (b)
x -~ (a)
(16)
TN
,T
-T
Fig. 1. Schematic curves (solid curves) for the t e m p e r a t u r e variation of X 1 Xtt 1, X l t , XQ 1 and ~-2: (a) in the n o r m a l case with a 3 > 0 and a 5 > 0, (b) in the case where a 3 < 0, a 5 > 0 and the value of Xm I lies between the values of a 1 - a at 0 K and at the m i n i m u m ( ¢ 2 is the value of ¢2 given by eq. (9) at TM) and (c) in the case where a 3 < 0, a 5 > 0 and the value of al a at 0 K is smaller than Xm I.
M. Shimizu / Itinerant electron antiferromagnetism
148
It is noted here that XII is finite at 0 K in c o n t r a s t to the case of the H e i s e n b e r g model. F r o m eqs. (12) and (13) X± 1 is rewritten as X~_' =XT,,' + 2 ( , ~ - ~ 2 ) { a 3
+a5(8~
+6{(+
M~).)}-(½a 3 + 7as,~)M~:-asMS_.
(17)
In eq. (15) ,~2qtshould be positive at all q so that X~ ~ > XnP and then from eqs. (14)-(17) it can be seen that ,~ > ,~2 and X± > Xll in the n o r m a l case where a 3 > 0 and a 5 > 0. F r o m eq. (17) it is seen that '~t should not be zero at 0 K (cf. ref. [19]). These results for the n o r m a l case a 3 > 0 and a 5 > 0 are shown schematically in fig. I a, where ~2 = ,~ is assumed. There are very few m e a s u r e m e n t s which show the difference between the parallel and p e r p e n d i c u l a r magnetic susceptibilities of metallic a n t i f e r r o m a g n e t s [23]-[25].
4. Character of magnetic transitions In order to discuss the character of the magnetic transition, for simplicity we assume that ~ = (~ = ~ 2 even below T y a n d MO: is very small. By defining new variables X = X m / X j I and y = ~ 2 / ~ with ~2r = 3TXmQ3/2/(4v~,~), eqs. (14) and (11) are rewritten, respectively, as
r = ( x '/2 + 1) -~/2
{18)
and X = 3 - 2 ( a 1 - a ) x m - T10a 3 X m ~ - Y~-
7_~asX,n~.y 2
(19)
at T = T N , X = I and Y = l / f 2 . By examining the solutions of eqs. (18) and (19) we can discuss the c h a r a c t e r of the m a g n e t i c transitions. F o r simplicity, a 3, a s, A, B and a are assumed to be i n d e p e n d e n t of T. The t e m p e r a t u r e variation of a~ is taken into account. The qualitative results are shown in the following way. (A) Simplest case where a 3 > 0, a 5 = ... = 0. If a~ - a increases m o n o t o n i c a l l y with increasing temperature, the c o n d i t i o n for itinerant electron a n t i f e r r o m a g n e t i s m is given by [ a I - a ] 0 x m < 1,
(20)
where [ a ~ - a] 0 means the value of a ~ - a at 0 K, irrespective of the sign of a ~ - a (fig. la). W h e n [a~-a]r=T~Xm<-3, the transition b e c o m e s first-order. W h e n a ] - a shows a m i n i m u m at T = T~, a n t i f e r r o m a g n e t i s m a p p e a r s only in a finite range of t e m p e r a t u r e as shown in fig. l b , if the c o n d i t i o n ( a , - a ) X m + 5a3kBTX~,Q3/2/(B~) < 1
(21)
at T = T I is satisfied, even if [al - a]0Xr,, > 1. Schematic curves of 42 are shown in fig. 1. F r o m eq. (16) MS: is p r o p o r t i o n a l to Xii 1 _ X,11 for small MQ.., so that the difference XT 1 - X~, ~ in fig. 1 shows roughly the t e m p e r a t u r e variation of MS:. (B) Case where a 3 < 0, a 5 > 0 a n d a v = ... = 0. In this case X o ( T ) = a l ~ always shows a m a x i m u m [20]. Then, the c o n d i t i o n for a n t i f e r r o m a g n e t i s m is given by eq. (20) and Xll shows a m i n i m u m below TN (fig. lc), as frequently observed. In this case the t e m p e r a t u r e variation of MQ:will show a m a x i m u m . Even if [a I - a]0Xm > 1, (a) if the value of Xm I lies between the values of [a~ - a]o a n d the m i n i m u m value of (a~ - a) at T = T 1, (a 1 - a) .... or (b) if the value of X~, I is lower than ( a l - a)m and the c o n d i t i o n ( a , - a l x , n + S a 3 k_B T x ~ , Q -
- / ( 8 ~ z l + 3 5 a s ( k B T ) 2 X 3m Q 3/ ( 8 q z ) 2 < 1
(22)
M. Shimizu / Itinerant electron antiferromagnetism
149
at T = T~ is satisfied, antiferromagnetism appears in a finite range of temperature as shown in fig. lb. It should be noted that the values of Xll at T N and T M are the same, but if A and B change with temperature these two values of Xll may be different. A good example of this case is not yet found in experiment as far as the author knows.
5. Discussion and conclusions As can be seen from eq. (3) the coefficients of the terms M o and M~ and those of M~, M~, Mo23//~ and ( M o. M ~ ) 2 are not independent of each other. The coefficients of M 4, ~ M ~ , 2 Md2M ~2 and 4( Mo " MQ ) 2 are equal, if ~ = ~ 2 is assumed. In principle, they may be independent, but the difference will be small. Therefore, by the same argument as that given by Moriya and Usami [26] in the phenomenological theory for the coexistence of ferro- and antiferromagnetism in the two peak model of X (q), it is concluded that the coexistence of ferro- and antiferromagnetism, i.e. canted magnetism ( M 0 _1_Mo), is possible in the single peak model of x ( q ) and in the usual case where a 3 > 0 and a 5 > 0 if 1 I X becomes largely negative or 1/X < - 2 / X m at low temperatures, but otherwise the coexistence never happens. In the present model the coefficient of the M~ term in eq. (3) is always smaller than that of the M02 term so that the antiferromagnetic state is always stable. In the present theory the value of the paramagnetic susceptibility has its largest value Xnl at T N (and T M), as shown schematically in fig. 1. Therefore, we cannot expect an increase of X above T N and a maximum in the temperature variation of X in the present model. In order to explain the temperature variation of X in YMn2, perhaps X m 1 (A o r B -1) must decrease with increasing temperature, or a large volume expansion may be necessary to stabilize the antiferromagnetic state and the first-order transition from the paramagnetic state to the antiferromagnetic state will occur at a certain temperature below the temperature where X shows a maximum. When the unenhanced susceptibility X o ( T ) = a~ 1 shows a maximum, as frequently we have, the sign of a 3 will be mostly negative. If a~ is negative we must include a higher-order positive term, say the term of a5 in eq. (3). The transition at TN is usually second-order, but if the value of [a 1 - ~]0 is large and negative, the transition becomes the first-order, as shown above, but the discontinuity at the transition point will be small. However, the transition at T M is always second-order. In the present treatment the effect of anisotropy energy has not been taken into account. To carry out a quantitative comparison between the calculated and observed results it will be necessary to include the anisotropy energy. This is a future problem. At present there are very few calculations on the electronic structure of metallic antiferromagnets. If these calculations become available we can carry out a quantitative comparison between our theory and experiment. It will be very interesting to check the relation given by eq. (10) by measuring the value of (N at TN in neutron diffuse scattering experiments on metallic an ti ferromagnet s. As shown above it is concluded that the temperature variation of the unenhanced susceptibility X o ( T ) = a~ ~ and the higher-order terms, say the term of as, than the term of a 3 for spin fluctuations are very important in determining the character of the transition between the antiferro- and paramagnetic states and also the temperature variation of X above T N. The approximation of static Gaussian statistics for spin fluctuations has been made use of in the present theory, but in this approximation the quantum effect which is important at lower temperatures and the critical fluctuations which are important around the transition temperature are not included and the present theory should be improved at these points. The former point can be improved by replacing eqs. (8), (14) and (15) by the corresponding formula of the fluctuation dissipation theorem [27] and the latter point seems to be a very difficult problem at the moment.
M. Shimizu / Itinerant electron antiferromagnetism
150
Acknowledgements I am very glad to contribute this paper for the occasion of the sixtieth birthday of Prof. Peter Wohlfarth. A part of this paper was written in the Institut f~r Festk0rperforschung der Kernforschungsanlage Jhlich and the author is very grateful to Prof. M. Campagna and Dr. S.F. Alvarado for their kind hospitality.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [lll [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
A.B. Lidiard, Proc. Roy. Soc. A224 (1954) 161. D.R. Penn, Phys. Rev. 142 (1966) 350. T.M. Rice, Phys. Rev. B2 (1970) 3619. S. Asano and J. Yamashita, J. Phys. Soc. Japan 31 (1971) 1000. A.M. O16s, Phys. Rev. B28 (1983) 327. H. Hasegawa and T. Moriya, J. Phys. Soc. Japan 36 (1974) 1542. Y. Nakamura, M. Shiga and S. Kawano, Physica 120B (1983) 212. R.E. Walline and W.E. Wallace, J. Chem. Phys. 42 (1965) 604. J. Pierre, Compt. Rendus B265 (1967) 1169. T.D. Hien, N.H. Luong, N.H. Duc, N.P. Thuy, P.H. Frings and J.J.M. Franse, J. Magn. Magn. Mat. 31-34 (1983) 245. L. Vinokurova, V. Ivanov, L. Sagoyan and M. Pardavi-Horvath, J. Magn. Magn. Mat. 15-18 (1980) 453. G.A. Smolensky, V.E. Adamjan and G.M. Loginov, J. Appl. Phys. 39 (1968) 786. R. Aleonard and P. Morin, J. Magm Magn. Mat. 42 (1984) 151. M. Shimizu, Phys. Lett. 81A (1981) 87; J. Magn. Magn. Mat. 31 34 (1983) 299. M. Shimizu and H. Okada, Phys. lett. 85A (1981) 474. M. Shimizu, A. Kunihara and A. Tamaoki, Phys. Lett. 99A (1983) 107. H. Yamada, J. Inoue, K. Terao, S. Kanda and M. Shimizu, J. Phys. F14 (1984) 1943. M. Shimizu, J. lnoue and S. Nagasawa, J. Phys. F (1984) in press. M. Shimizu and A. Kunihara, Phys. Lett. 100A (1984) 218. M. Shimizu, Rep. Progr. Phys. 44 (1981) 329, D. Gignoux, R. Lemaire, P. Molho and F. Tasset, J. Appl. Phys. 52 (1981) 2087. M. Shimizu, Proc. Phys. Soc. 86 (1965) 147; J. de Phys. 43 (1982) 681. A.R. Pepper and R. Street, Proc. Phys. Soc. 87 (1966) 971. M.O. Steinitz, E. Fawcett, C.E. Burleson, J.A. Schaefer, L.O. Frishman and J.A. Marcus, Phys. Rev. B5 (1972) 3675. R. Mendia-Monterroso, Thesis, University of Grenoble (1984). T. Moriya and K. Usami, Solid State Commun. 23 (1977) 935. R. Kubo, J. Phys. Soc. Japan 12 (1957) 893.
Journal of Magnetism and Magnetic Materials 45 (1984) 144 150 North-Holland, Amsterdam
T H E O R Y OF I T I N E R A N T E L E C T R O N A N T I F E R R O M A G N E T I S M M. S H I M I Z U Department of Applied Pl~vsics, Nagoya UniversiO,, Nag¢~va464, Japan
Itinerant electron antiferromagnetism is studied by taking into account the realistic electronic structure and the effect of spin fluctuations within static Gaussian statistics. Expression for the magnetic susceptibility above and below the Ndel temperature are obtained. It is shown that spin fluctuations can stabilize antiferromagnetism in a certain case. The character of magnetic transitions and the comparison with experiment are discussed.
1. Introduction
Itinerant electron antiferromagnetism was first studied by Lidiard [1] as an extension of the Stoner model of ferromagnetism. Since then many theoretical works on itinerant electron antiferromagnetism have been carried out [2-5]. Recently, the effect of spin fluctuations on itinerant electron antiferromagnetism has been taken into account [6]. By making use of the free electron model with umklapp processes, it has been shown that the Ndel temperature T N is lowered from its H a r t r e e - F o c k value and the temperature variation of the uniform and staggered spin susceptibilities has been calculated only above T N. In experiments on metallic antiferromagnets, various kinds of magnetic transitions and their combinations have so far been found, in addition to the ordinary second-order transition at T N, e.g. the first-order transitions between the antiferromagnetic and paramagnetic or ferromagnetic states and between two kinds of antiferromagnetic states and the transitions from the antiferromagnetic state to the spin-density-wave, canted or ferrimagnetic state. In the YMn 2 compound the paramagnetic susceptibility X above the first-order transition temperature still increases with increasing temperature, as in Cr although Cr exhibits a spin-density-wave state, and shows a large magnetovolume effect below this transition temperature [7]. Moreover, some compounds, e.g. TbCu [8,9], HoCu 2 and ErCu 2 [10], Pt 0.678Fe0.322 [11] and NbS [12], PrCd [13], etc. show double peaks in the temperature variation of X, although rare earth compounds may not be good examples of itinerant electron antiferromagnets. In order to explain these experimental results mentioned above qualitatively and quantitatively, a realistic electronic structure of each metallic antiferromagnet and the effect of spin fluctuations should be taken into account. This is the purpose of this paper. The effect of spin fluctuations on the magnetic properties of metallic ferromagnets and paramagnets has been calculated before within static Gaussian statistics by a realistic itinerant electron model [14-16]. Recently, this treatment has been applied to intermetallic compounds of Y and transition metals [17,18]. These calculations have also been extended to the case, where the exchange stiffness constant A is negative so that the wave vector dependent spin susceptibility Xq shows a peak at the cutoff wave vector qm, and this has been applied to Rh and fcc Fe [19]. In the case of fcc Fe it has been found that the averaged square of the fluctuating magnetic moment ~q at ~/I = qm becomes finite at 0 K, although a long-range antiferromagnetic order is not included. In this paper the previous treatment of spin fluctuations in the realistic itinerant electron model for metallic ferromagnets is extended to the case where Xq shows a single maximum at ]ql = Q. Expressions for the uniform and staggered spin susceptibilities above and below T N are derived. Their temperature variation and the character of the magnetic phase transitions are discussed. 0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M. Shirnizu / Itinerant electron antiferrornagnetism
145
2. Averaged magnetic free energy It is assumed that the exchange stiffness energy is given by ½AI ~7MI 2 + ½BI x72M[ 2 with A < 0 and B > 0. The magnetic free energy of an itinerant electron magnet with volume V can be written as a power series of the square of the magnetization density M~(r) (i = x, y, z) as
f = f d 3 r [ ½ ( a l - a ) M ( r ) 2 + ~a3M(r)4 + ~asM(r)6+ ... +½AlUM(r)]2+ ½B[~y2M(r)[2 + . . . ] , (1)
where M(r) 2 = Z,M,(r)2; the coefficients a~, a 3, a 5.... can be given as functions of T in terms of the density of states curve [20] and a is the molecular field coefficient. Mi(r ) is assumed to be given by the sum of the uniform magnetization of M0,, the staggered magnetization MQfir)= f2MQ, cos(Q-r) with wave vector Q and the fluctuating magnetization m~(r) = Zqmq, e iq'r,
M,(r) = Mo, + MQ,(r) + m, (,').
(2)
By the same method as before [14] f in eq. (1) is averaged over the gaussian distribution of mqi. Then the averaged f is given by a function of 34o2,, MQ, and ~z q, = ( m q i m q i ) , which is the average of mq~mq~, as
f = -½k,TEi,q(ln(2rrV21~ 2~qzi ) + 1} + V [½Y~i,q(Aq 2 + Bq")~zq, + ½(al- a ) ~ 2 + ¼a3(~4 q._ 2Y~i~?) +
6+
+
2 2 2 ) + ½(A 0 2 + BQ4)M~
+ ½ { a l - a + a3~ 2 + a5(~4+ 2Y~i,~¢ ) ) ( M 2 + M~)+o3Z,~:2(Mg, + M~,) + 2asZ, (~2,{52 + 2~4)(M2, + M ~ , ) + ~(a3 + 2as~2){M g + ~M~ + 2Mo2M~ + 4(Mo'MQ) 2}
+ 2asY~i~2i { Mo2M~i+ ~22v*OtV*O 3..2~.2 i + M~M~+ MdMd,+4(Mo.MQ)Mo,MQ~ 2 2 } + 1
(
6+ 5
6 + 3MoMd 4 2+
91"21"4~12(M2 3 }1 T + 2yMQ)(M°'MQ)2
(3)
where
M g = ~iMgt,
M ~ = ~.tM~i,
~2= ~ q~qi2
and
The formulae to calculate Mo~, MQ, and ~q2 are obtained from the extremum condition of f w i t h respect to M0,, M0g and ~q2i. The matrix components of the uniform and staggered spin susceptibilities can also be obtained by X~ ~= (OOf/OMo,)/OMoj)/V and XQ)j = (O(af/aMQ,)/OMo./)/V, respectively. As the expressions of these formulae are lengthy, the results are shown below only for usual cases.
3. Spin susceptibility above and below TN
3.1. Pararnagnetic state above TN In this case Moi = MQi = 0 (i = x, y, Z), ~2 = ~2. --~~2..= ½~2,X~I = 6,jX -1 and XQ]j =6,sXQ 1. The uniform
M. Shirnizu / Itinerant electron antiferromagnetism
146
spin susceptibility X is given as before [14] by X
-
I = a l _ o~ +
5 2 ~a3g; + ~5a5~4 +
(4)
...
and the staggered susceptibility Xo is given by X Q 1 = X - 1 -{~ A Q
2+
BQ 4 = X -1 -
X , n 1,
(5)
where Xm 1 = 4 B / A 2 and the value of Q can be determined as
Q = (-A/2B)
1/2
(6)
from the condition that Xq shows a m a x i m u m at Iql = Q. The ~2 is given by
~2i = k , T V - ' ( X
-1 + Aq 2 + Bq 4 ) ~.
(7)
Then, the sum of ~2 over q can be integrated as
~2 = 3 k , T V - , ~ q ( X - 1
+ Aq2 + Bq4)
= (3k,T/4,~){ZB(B/x)I/2
'
+ }AIB )
,/2
(8)
Further, eq. (8) is rewritten as
2=(3k.r/a
,)XmQV2{(x.,/xt lj2 + 1} 'j2
(9)
By solving eqs. (4) and (9) we can get the temperature variation of X and ~2. It should be noted here that the same expression as eq. (8) is obtained also in the ferromagnetic and paramagnetic cases A > 0 and B > 0 and there is no need to introduce the cutoff of the wave vector as in the previous treatment of spin fluctuations for the case A > 0 and B = 0 [14]. In the ferromagnetic case the value of ~2 at the Curie temperature Tc is simply given by ( 3 k B T J 4 , ~ ) ( A B ) 1/2. T N is determined by the condition XQ 1 = 0 at T = T N and we have from eq. (5) X = Xm at Ty and from eq. (9) the value of ~2 at T N is given by
~ = ( 3/8"~ )kBTNXmQ 3/2.
(10)
Therefore, T N can also be defined as the temperature where the value of X-~ in eq. (4) becomes equal to that of X~ ~, as shown schematically in fig. 1. It is easily seen from eq. (4) that in the normal case where a3 > 0 and a 5 > 0 the value of T N is reduced by the effect of spin fluctuations. However, if a 3 < 0 and a s > 0 the spin fluctuations may raise the value of T s. In the case, where the temperature variation of X-~ in eq. (4) shows a m i n i m u m due to that of al or due to the negative sign of a 3 together with the fact that ~2 is proportional to T at lower temperatures, and if the value of X~,1 lies just between the value of 1 / X at 0 K and its m i n i m u m value as shown in fig. lb, antiferromagnetism can appear only in a finite range of temperature, say between T M and T N. The negative value of a 3 and the minimum in the temperature variation of a~ can stabilize the antiferromagnetic order at finite temperature through the effect of spin fluctuations, even if antiferromagnetism is not stable at 0 K, as well as in the case of ferromagnetism [14,16]. It is seen that both values of X 1 at Ty and TM are the same as given by X~,~. We may call this p h e n o m e n a a thermal spontaneous antiferromagnetism corresponding to the thermal spontaneous magnetization in the case of ferromagnetism as observed for Y2Ni 7 [21,22]. This case will be further discussed below.
M. Shimizu / Itinerant electronantiferromagnetism
147
3.2. A ntiferromagnetic state below TN In this case, M q . 4= 0, MQ,. = Moy = 0, M0, = 0 (i = x, y, z). ,~. = ~l and ~+ = ,~, = ~t. T h e parallel spin susceptibility XII when the external field H is applied along the z direction is given by
XII- 1
al-~+a3(3~(+2~:t)+as(15~4+12~2~et +8~4)+3{a3+2as(5~+2~2)}MQ--+ 15a ?~'4"4
(11) and the p e r p e n d i c u l a r spin susceptibility X± when H is applied along the x or y direction is given by 3 4 +(a 3 + 6a5,~( + 8as~2)M~: + ~a5 M~:. (12)
X~_ t = a] - a + a 3 (,~( + 4,~ 2) + a s (3,~ 4 + 8,~,~t2 + 24~ 4)
H e r e M~+, ,~2 = y~+~¢,q 2 and ~2 = )2q~tZq are d e t e r m i n e d , respectively, f r o m -X/,, ~ + a~ - a + a3(3~ ~ + 2~¢2) + a5 (15,~ 4 + 12,~(,~ 2 + 8,~t4) + ~(a33 +
lOas~+aas~2)M~:+ 7asM~.=0,5 4
(13)
~=k.TV-~Y~q(X,I]+Aq2+Bq4)-'=(kBT/4v/2'rr)XmQ3/2{(Xm/XI,)]/2+I}
-~/~
(14)
'/2 + ]} -,/2
(15)
and
+:t = kBTV+'F-.q(X~ t + A q 2 + B q 4)
! = (kBT/4x/2'rr)X,+.Q3/2(CXm/X±)
T h e values of Xll, X±, MQ:, ~2 and ,~ at each t e m p e r a t u r e are o b t a i n e d b y solving eqs. (11)-(15) simultaneously. At T N, M~: b e c o m e s zero and ~2 = ,~2 = 3,~ t : ,sothatxN = X±=X=Xm. Fromeqs.(ll) and (13) Xii ~ is rewritten as
5asM~:.
X~ 1 = Xm I + 3(a3 + 1 0 a s , ~ + 4as,~t2) M~: +
x.'
x-V
-
~
X4
-T
?_ ,o,_i x
"'--s'/
,
~:' (hi
~2 (ol
TH
-T
0
TI,,I
/
"T
,I"/%
~2 (el 2
/
o
TN
gN o
x'/
x_,'?
t<:,,-,1 x+tl ,,
---~
x; Ic)
X-1 (b)
x -~ (a)
(16)
TN
,T
-T
Fig. 1. Schematic curves (solid curves) for the t e m p e r a t u r e variation of X 1 Xtt 1, X l t , XQ 1 and ~-2: (a) in the n o r m a l case with a 3 > 0 and a 5 > 0, (b) in the case where a 3 < 0, a 5 > 0 and the value of Xm I lies between the values of a 1 - a at 0 K and at the m i n i m u m ( ¢ 2 is the value of ¢2 given by eq. (9) at TM) and (c) in the case where a 3 < 0, a 5 > 0 and the value of al a at 0 K is smaller than Xm I.
M. Shimizu / Itinerant electron antiferromagnetism
148
It is noted here that XII is finite at 0 K in c o n t r a s t to the case of the H e i s e n b e r g model. F r o m eqs. (12) and (13) X± 1 is rewritten as X~_' =XT,,' + 2 ( , ~ - ~ 2 ) { a 3
+a5(8~
+6{(+
M~).)}-(½a 3 + 7as,~)M~:-asMS_.
(17)
In eq. (15) ,~2qtshould be positive at all q so that X~ ~ > XnP and then from eqs. (14)-(17) it can be seen that ,~ > ,~2 and X± > Xll in the n o r m a l case where a 3 > 0 and a 5 > 0. F r o m eq. (17) it is seen that '~t should not be zero at 0 K (cf. ref. [19]). These results for the n o r m a l case a 3 > 0 and a 5 > 0 are shown schematically in fig. I a, where ~2 = ,~ is assumed. There are very few m e a s u r e m e n t s which show the difference between the parallel and p e r p e n d i c u l a r magnetic susceptibilities of metallic a n t i f e r r o m a g n e t s [23]-[25].
4. Character of magnetic transitions In order to discuss the character of the magnetic transition, for simplicity we assume that ~ = (~ = ~ 2 even below T y a n d MO: is very small. By defining new variables X = X m / X j I and y = ~ 2 / ~ with ~2r = 3TXmQ3/2/(4v~,~), eqs. (14) and (11) are rewritten, respectively, as
r = ( x '/2 + 1) -~/2
{18)
and X = 3 - 2 ( a 1 - a ) x m - T10a 3 X m ~ - Y~-
7_~asX,n~.y 2
(19)
at T = T N , X = I and Y = l / f 2 . By examining the solutions of eqs. (18) and (19) we can discuss the c h a r a c t e r of the m a g n e t i c transitions. F o r simplicity, a 3, a s, A, B and a are assumed to be i n d e p e n d e n t of T. The t e m p e r a t u r e variation of a~ is taken into account. The qualitative results are shown in the following way. (A) Simplest case where a 3 > 0, a 5 = ... = 0. If a~ - a increases m o n o t o n i c a l l y with increasing temperature, the c o n d i t i o n for itinerant electron a n t i f e r r o m a g n e t i s m is given by [ a I - a ] 0 x m < 1,
(20)
where [ a ~ - a] 0 means the value of a ~ - a at 0 K, irrespective of the sign of a ~ - a (fig. la). W h e n [a~-a]r=T~Xm<-3, the transition b e c o m e s first-order. W h e n a ] - a shows a m i n i m u m at T = T~, a n t i f e r r o m a g n e t i s m a p p e a r s only in a finite range of t e m p e r a t u r e as shown in fig. l b , if the c o n d i t i o n ( a , - a ) X m + 5a3kBTX~,Q3/2/(B~) < 1
(21)
at T = T I is satisfied, even if [al - a]0Xr,, > 1. Schematic curves of 42 are shown in fig. 1. F r o m eq. (16) MS: is p r o p o r t i o n a l to Xii 1 _ X,11 for small MQ.., so that the difference XT 1 - X~, ~ in fig. 1 shows roughly the t e m p e r a t u r e variation of MS:. (B) Case where a 3 < 0, a 5 > 0 a n d a v = ... = 0. In this case X o ( T ) = a l ~ always shows a m a x i m u m [20]. Then, the c o n d i t i o n for a n t i f e r r o m a g n e t i s m is given by eq. (20) and Xll shows a m i n i m u m below TN (fig. lc), as frequently observed. In this case the t e m p e r a t u r e variation of MQ:will show a m a x i m u m . Even if [a I - a]0Xm > 1, (a) if the value of Xm I lies between the values of [a~ - a]o a n d the m i n i m u m value of (a~ - a) at T = T 1, (a 1 - a) .... or (b) if the value of X~, I is lower than ( a l - a)m and the c o n d i t i o n ( a , - a l x , n + S a 3 k_B T x ~ , Q -
- / ( 8 ~ z l + 3 5 a s ( k B T ) 2 X 3m Q 3/ ( 8 q z ) 2 < 1
(22)
M. Shimizu / Itinerant electron antiferromagnetism
149
at T = T~ is satisfied, antiferromagnetism appears in a finite range of temperature as shown in fig. lb. It should be noted that the values of Xll at T N and T M are the same, but if A and B change with temperature these two values of Xll may be different. A good example of this case is not yet found in experiment as far as the author knows.
5. Discussion and conclusions As can be seen from eq. (3) the coefficients of the terms M o and M~ and those of M~, M~, Mo23//~ and ( M o. M ~ ) 2 are not independent of each other. The coefficients of M 4, ~ M ~ , 2 Md2M ~2 and 4( Mo " MQ ) 2 are equal, if ~ = ~ 2 is assumed. In principle, they may be independent, but the difference will be small. Therefore, by the same argument as that given by Moriya and Usami [26] in the phenomenological theory for the coexistence of ferro- and antiferromagnetism in the two peak model of X (q), it is concluded that the coexistence of ferro- and antiferromagnetism, i.e. canted magnetism ( M 0 _1_Mo), is possible in the single peak model of x ( q ) and in the usual case where a 3 > 0 and a 5 > 0 if 1 I X becomes largely negative or 1/X < - 2 / X m at low temperatures, but otherwise the coexistence never happens. In the present model the coefficient of the M~ term in eq. (3) is always smaller than that of the M02 term so that the antiferromagnetic state is always stable. In the present theory the value of the paramagnetic susceptibility has its largest value Xnl at T N (and T M), as shown schematically in fig. 1. Therefore, we cannot expect an increase of X above T N and a maximum in the temperature variation of X in the present model. In order to explain the temperature variation of X in YMn2, perhaps X m 1 (A o r B -1) must decrease with increasing temperature, or a large volume expansion may be necessary to stabilize the antiferromagnetic state and the first-order transition from the paramagnetic state to the antiferromagnetic state will occur at a certain temperature below the temperature where X shows a maximum. When the unenhanced susceptibility X o ( T ) = a~ 1 shows a maximum, as frequently we have, the sign of a 3 will be mostly negative. If a~ is negative we must include a higher-order positive term, say the term of a5 in eq. (3). The transition at TN is usually second-order, but if the value of [a 1 - ~]0 is large and negative, the transition becomes the first-order, as shown above, but the discontinuity at the transition point will be small. However, the transition at T M is always second-order. In the present treatment the effect of anisotropy energy has not been taken into account. To carry out a quantitative comparison between the calculated and observed results it will be necessary to include the anisotropy energy. This is a future problem. At present there are very few calculations on the electronic structure of metallic antiferromagnets. If these calculations become available we can carry out a quantitative comparison between our theory and experiment. It will be very interesting to check the relation given by eq. (10) by measuring the value of (N at TN in neutron diffuse scattering experiments on metallic an ti ferromagnet s. As shown above it is concluded that the temperature variation of the unenhanced susceptibility X o ( T ) = a~ ~ and the higher-order terms, say the term of as, than the term of a 3 for spin fluctuations are very important in determining the character of the transition between the antiferro- and paramagnetic states and also the temperature variation of X above T N. The approximation of static Gaussian statistics for spin fluctuations has been made use of in the present theory, but in this approximation the quantum effect which is important at lower temperatures and the critical fluctuations which are important around the transition temperature are not included and the present theory should be improved at these points. The former point can be improved by replacing eqs. (8), (14) and (15) by the corresponding formula of the fluctuation dissipation theorem [27] and the latter point seems to be a very difficult problem at the moment.
M. Shimizu / Itinerant electron antiferromagnetism
150
Acknowledgements I am very glad to contribute this paper for the occasion of the sixtieth birthday of Prof. Peter Wohlfarth. A part of this paper was written in the Institut f~r Festk0rperforschung der Kernforschungsanlage Jhlich and the author is very grateful to Prof. M. Campagna and Dr. S.F. Alvarado for their kind hospitality.
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