- Email: [email protected]
Spin fluctuation theory of ferromagnetic metals
Journal of Magnetism and Magnetic Materials 20 (1980) 171-194 o North-Holland Publishing Company
SPIN FLUCTUATION THEORY OF FERROMAGNETIC METALS
Ran USAMI and T&-u MORIYA Institute for Solid State Physics, Universityof Tokyo, Roppongi, Tokyo Received 15 October 1979
Ferromagnetic metals at finite temperatures are discussed using the functional integral method along the lines of the unitied spin fluctuation theory of Moriya and Takahashi. We develop a simple approach which makes use of only the density of states but still takes account of the nonlocal nature of the spin fluctuations with the aid of a shrgle-site coherent potential approximation. The effect of the charge density fluctuations is also taken into account within the saddle point approximation. The results of numerical calculations for various model densities of states including those for bee and fee d-metals are presented. Particularly, the Curie temperatures and the Curie-Weiss magnetic susceptibilities for Fe, Co and Ni seem to be fairly well reproduced.
1. Introduction
Recently Moriya and Takahashi [ 1,2] (lrereafter referred to as MT) presented a rather general spin fluctuation theory of itinerant electron ferromagnetism which interpolates between the local moment limit and the weakly ferromagnetic limit. They made use of a functional integral method and a model functional which can be regarded as a generalization of the Heisenberg model to permit the variation in amplitude of the local spin fluctuation @SF). They discussed ferromagnetism of metals from a general point of view making use of a relatively small number of physical parameters which can in principle be deduced from a given band structure. They also pointed out an important possibility of strong variation with temperature of the amplitude of LSF and its saturation above a certain temperature, leading to a new concept known as the “temperature-induced local magnetic moments” [3]. This theory has attained considerable success not only in giving a unified picture to itinerant electron ferromagnetism but also in clarifying several previously unexplained phenomena. In order to make detailed theoretical discussions on each specific substance, however, it is essential to calculate the one-body free energy functional from the band structure. There are several possible methods for this purpose. One is to make use of a local density of states. This method has been applied successfully to the problem of nearly ferromagnetic semiconductors by Takahashi and Moriya [4]. The other possible method is the local approach, i.e. to expand the free energy functional in terms of the local field variables such as the terms related with single site only, those related with two sites, etc. This type of approach has recently been developed by Moriya and Hasegawa [5,6]. In order to calculate the nonlocal terms in the free energy functional with this method we need almost full information of the band structure and the calculation for a realistic band structure seems to be generally rather heavy. In this paper we wish to develop a rather simple approach which makes use of the density of states only. The local term in the free energy functional can always be calculated from the density of states and in place of calculating all the nonlocal terms we just calculate the average of the coefficients of the nonlocal terms, which are expressed by a quadratic form in the Fourier q-components of the field variables. Thus, by assuming a simple form for the distribution function for the coefficients we can perform calculations for the free energy and various physical quantities. This is a significant simplification, although at the expense of accuracy, and we will see that the method is useful enough for qualitative and semiquantitative purposes. Because of this simplicity we can also deal 171
K. Usami and T. Moriya / Spin jluctuation theory of fewomagnetic metals
172
easily with the effect of charge density fluctuations. A local approach of a different type has been presented recently by Hasegawa [7] who simply substituted the effect of the nonlocal terms by a molecular field. He made the further simplification of using a local saddle point approximation [8] (LSPA) at the expense of the interpolating nature of the original MT theory. Furthermore, this approximation is rather limited ln its applicability. The present theory, following more closely the work by MT, is expected to be free from these drawbacks. In section 2 we briefly summarize the necessary formulae in the functional integral formalism and derive the formulae for various physical quantities with the use of a model functional which is a generalization of the one by MT; we consider the external magnetic field so that the theory applies both above and below the Curie temperature, Tc . The method of calculating the free energy functional taking account of the charge density fluctuations explicitly with the use of the saddle point approximation is discussed in section 3 and the results of. numerical calculations are presented in section 4. We discuss in section 5 the relation between the present theory and the one with the use of LSPA. In section 6 the conclusions are summarized and discussions are given.
2. Formalism and model functional Our starting point is the single band Hubbard Hamiltonian. Taking account of the effects of band-degeneracy, we use the effective interaction Hamiltonian including both U and J, the effective Coulomb and exchange energies, respectively. The total Hamiltonian is given by
Q.
= C jQ
C
D
tjQa&aQo
,
(1)
where is a hopping matrix element, a& is a creation operator of an electron with spin u at the lattice site j, and nj and Si are the charge and spin density operators, respectively. The effective interaction term is derived by considering the case of five-fold degeneracy with the rotational invariance in the spin space [9]. The last term represents the effect of the external magnetic field applied along the z axis. The free energy of the system is given in a functional integral form through the Strantnovich-Hubbard transformation [lo]. Within the static approximation we have e-PF = ,-NFo+AF) = Trte -N%-&] ,-oU tjQ
,
Fr = (47rJ/@1’2;
c2 = (--nU//3)“2,
K. Usamiand T. Mon>a /Spin fluctuation theory of ferromagnetic metals
173
where p is the chemical potential, /.3= l/T, T being the temperature expressed in energy units, Nis the number operator, and the other notation is usual. With the use of a standard technique [lo], $r [!$,n] is written as
$1ko1
=-$Trln[l -90V1,
with ,ik(Rj--R cj, n ISo
IQ, n’)
= s,,s
Q)
__!_ c
where Tr means the diagonal sum over j, n and in the spin space. Thus we have the problem of a noninteracting electron system moving under the spin and charge fields fluctuating arbitrarily in space. We need to calculate J/r and then to perform the functional integral. Then the suceptibility and the magnetization, etc. can be calculated in a standard way. Here we use the approximation of taking the saddle point value for nj for each sj, so that the functional depends explicitly on cj only. In the case of ferromagnetism or under an external magnetic field, 50 takes a finite value and we have to take account of it to all orders in the expression of $ r, where and in what follows & and Se are the Fourier transforms of Q and Sj, respectively. In addition to to, we introduce variables corresponding to the average squared local amplitude of e-fields: (4) where the prime means to omit the term with 4 = 0 from the summation and No is the number of lattice sites. This variable is to be related with the local amplitude of the spin fluctuations as follows:
We adopt the following model functional for J, r just as in MT [ 1,5]:
with ~X*,]x.
c;ol=O9
(6)
where it is assumed that the coupling between the spin fluctuation modes is mainly local and the higher order terms are expressed in terms of x and to. The reasonableness of this assumption is indicated by the fact that, as MT have shown, this model functional properly interpolates between the weakly ferromagnetic limit and the local moment limit. After evaluating the integral in eq. (2) with the use of this model functional we finally get the following results
ISI: 1 XQ =_ lc’ 25VoP4 1- w(x,, t j&j ’
(74
174
K. Usamiand T. Moriya /Spin fluctuation theory of ferromagnetic metals
x0x
=xoy =-(S,)/lv&,
G’b)
xqa =&Y/(1 - w&a),
where j$ is measured in units of No(er-le)2 and c, = C,/dN,-,. In the-case of the paramagnetic phase the system is isotropic and we have
3 -Its 27rP( &+a
x=&,=(Y a=1
(8)
) ’
-Z&O,
with
(f(a)) = ~P(a)f(a) da,
P(a) =Nf’
G’S[a-
(1 -$)I.
(9)
Here we need to calculate To and P(a), the distribution function for a, from the band structure.
3. Single-site coherent potential approximation Let us consider the actual calculation of $ r, which is similar to the problem of random alloys where the electrons see a potential fluctuating from site to site. In the latter case the coherent potential approximation (CPA) [ 11,121 is known to be a useful method. So we apply the CPA to the present problem. First we rewrite eq. (3) by using the Green’s function, Q = (9;’
- 8)-r,
w9
where C is the site-independent self-energy to be determined later. We get $r[E,s] =-kTr[ln(l
-~oZ)tln{l
- Q(V-
Z:))].
(11)
If we split Q into a site-diagonal part F and site-nondiagonal part g ,
$?=Ftg ,
(12)
we have $r[,$nJ=-a’Ir[ln(l
- Qo~)tln(l
-F(V-Z))tln{l
-@(V--Z)[l
-F(V-E)]-l)].
(13)
K. Usamiand T. Motiya /Spin fluctuation theory of ferromagnetic metals
The first two terms and the last term are the local and the nonlocal parts, respectively. mined by minimizing the first two terms of eq. (13) with respect to Z [12]. We have
5; CjI(V-
Z)[l
-F(V-
IQ]-‘lj?=O.
175
The self-energy is deter-
(14)
This is nothing but the CPA condition. The model functional discussed in the previous section is given by the following steps. First, in the expression for $ i we leave the local part or the terms related with a single site only and among the nonlocal part those related with the relative orientation of the spin fields at different sites. Next vi is determined as a function of ki by minimizing the sum of $0 and the local part of 9 i with respect to nj. We then have the form given in eq. (6). The CPA calculation of the local term is straightforward if we make the approximation of assuming an isotropic distribution in the direction of (15) and take an average value for IS&l. As for the term related with the pa.ir of different sites, in place of calculating the coefficient of each term we calculate only jzi, or the average of yr .,Assuming for example the following form for the distribution function P(u,-J,
(16) where ( u,) is given by (a,>
=l -&J-k
F&=1
-2,
(17)
we can dispense with the calculation of each X, from the full band structure. Thus, the problem can be solved approximately if only the density of states is given. Let us first consider the paramagnetic phase. For later convenience we introduce the following notation: B2 =&3x;
Be =crto;
y =czq.
(18)
In order to calculate x0 and Tsr, for given x, we consider such conditions that a random field with a magnitude B is applied along an arbitrary direction at each site. First we consider x0, the uniform susceptibility for the noninteracting system under the random magnetic field. In this case we assume that an infinitesimal uniform field B. is applied along the z axis. We calculate j&, by the following simple approximation:
(1% L(B, Bo) is given by the sum of the charge field part of J/e and the local part of $Jr. From eqs. (2) and (13) we have I
UB. Bo) = -
&j
Wr(412t
i
$
Im g
ln(1 - So@, e)(E + iBoo,))
-1
1
t;
s
dcImlndet[l
-1
with Qo(k e) = (e - ek + cc)-’ ,
-F(V-
Z)]
(20)
176
K. Usamiand T. Mot-&a/Spin fluctuation theory of ferromagnetic metals
where f(c) is the Fermi distribution function and ): and Pare shifted by iBoo,from those in eqs. (lo)-(14). The direction of the random field B is expressed by the polar angles 0 and 9 with respect to the z axis and c stands for cos 8. IZ, F and V are given by z=[Zt
F=
“,J
(2la)
Ft 0 1’ 0 FJ
Wb)
[
B
ysin
Be-‘@
v=
(2lc)
-$B cos e + ~(~0s e) 1 with
(22) where the coherent potential 2 is a diagonal matrix because of a rotational (14) we have the following CPA equation:
symmetry around the z axis. From
eq.
1 is 0 The
dc(V- Z)[l -F(V- IQ]-' =O.
d#
(23)
-1
charge field is determined
aL ar(c)=
0
by the condition
’
and we get r(c) = g
jdef(e)
Im $)
&
A(c),
with A(c)=det[l
-F(V- Z)],
(24) where the variational property of the free energy in the CPA, aLlaI: = 0,is employed. In the case of vanishing Bo the system has full rotational Z,=
T-Z;
(
F=lx ' No k
1
symmetry.
Thus, the coherent potential
is a scalar quantity
and we have
F,,, (25) 1
E-+-~p-7p+~’
where the subscript p denotes the paramagnetic phase, the coherent potential is again shifted by.y,, stant, and the chemical potential for each B is determined by the following relation: - ijde
f(e) Im
Fp =$ .
yp is a con-
P-9
K. Usami and T. Moriya / Spin fluctuation theory of ferromagnetic metals
177
The expression for Fe given in eq. (19) is rewritten as
with 1
=Lc
H
’
No
(27)
(E-+~p-7p+C()2’
k
Differentiating eqs. (23) and (24) with respect to Bo, we get simultaneous equations for dZ+ /dBo and 1
s
c
dc
MC) -. @O
-1
Solving these equations we get x0 = & sdej(e)
Im
’ f~~)Hp]_dB2~[~~de~(e)
X
Imy]
t $ldef(E)
Im
with R,=l+;=$+%.
(28)
3
P
As for the local susceptibility &, there exists a simple approximation from eq. (7a) FL
1
q--
aL
-.
ax
2nJ
In this paper, however, we calculate it directly by considering such an impurity problem that a local field is applied at one site in the medium characterized by Z,. The potential at the impurity site is expressed by f@L
tB
COS0,) t yi(COSOi)
B
2 sin tIie-i’i
Vi = B _ sb di eNi
i 2
(29) -$(BL
t B
COS ei)
+ yi(COS
ei)
1.
where Bi and @iare the polar angles of the random field with respect to the z axis and BL is an infinitesimal field locally applied along the z axis. The part of the free energy functional associated with the impurity site is given by
Fi = - ,& j
d(cos fIi) [ri(cos ei)]’ + & IdefcE) 1 d(cos Oi) Im In det [ 1 -. (Vi - Zp)Fp] .
-1
From this expression we get the following relation:
j7L’d dBL
(_3_ 1 aBL 9
-1
(30)
K. Usami and T. Moriya /Spin fluctuation theory of ferromagnetic metals
178
with --
aFi
3yi(COS
=o.
(31)
0i)
The local susceptibility
is calculated as
(32) From eqs. (8), (9) and (16)-(18) temperature:
of random fields and the
1+6
B2=12JT
(33)
6 +([email protected]
The average squared amplitude (S,%=$
we get the following relation between the amplitude
(
;-3T.
of the local spin density is given from eq. (5) by (34)
)
On the other hand, considering the magnitude of the magnetic moment along the direction of the random field at each site, we get the following CPA expression for the amplitude of the local moment within the present approximation:
ClS,l> =&-def((e)Im
1 +Ft F P
(35)
. P
Next let us consider the ordered phase with He = 0. Although the system is no longer isotropic, we adopt the model of the isotropic distribution of random fields with a fixed amplitude. Then we can use eq. (23) with nonvanishing BO. It is still difficult to solve this problem. In view of the fact that the static approximation itself is considered to be poor at low temperatures, we here content ourselves with a qualitative feature and consider a very simple case. First we neglect the effect of charge flucutations. Furthermore we expand the CPA condition in terms of the following quantity: B S =? {FJ - Ft - FtFst%
(36)
- &)I,
where S is proportional to B - Bo. Therefore the results is expected to be good in the region where T/Tc 5 1 and also T/TC << 1. We get
with &=I
tFt&
+FIZJ
The chemical potential
-FtF,g
(38)
-Z&).
and the uniform magnetization
as a function
of B are determined
by the following simul-
K. Usami and T. Moriya /Spin fluctuation theory of ferromagnetic metals
179
taneous equations:
s
-N =_- 1 de./@)In@‘?+ FL>,
No
IT
(39) = & Jdef(E)
MO =z
where the equations field with ponent of wool
Im(Ft
-
F+) ,
magnetization MO is measured in units of NogpB. At T = 0 K the random field vanishes and the above reduce to the Hartree-Fock ones. Since we have used a simplified model for B, namely the random an isotropic distribution, we determine the relation between B and T by using only the transverse comthe susceptibilities. We make use of the relation (40)
= 1,
corresponding to the fact that the uniform transverse susceptibility is shown by eq. (7b). We have B2 =6T
I[
$
- X-u(Bt Bo)
diverges in the ordered phase with He = 0, as
1
(41)
,
where we have used eq. (33). The transverse component of the local susceptibility can be calculated in a way similar to the case of the paramagnetic phase. The medium is characterized by the coherent potential given by eqs. (36) (38). The potential in the impurity site is given by B
-sin
2
.;‘,
Bje -% + fB,,
Vi =
(42)
,
i’i + ;BLX 1
where BLx is the infinitesimal local field. The local susceptibility is given by differentiating the magnetic moment of the impurity site along the x-axis with respect to B Lx. The result includes S and Q. We expand it with respect to S, as is done in the calculation of the coherent potential, up to the second order and get ~~,=~Sde~(e)~m~(l+~)+~
Using this expression with eqs. (39) and (41) we can get the magnetization The average squared amplitude of the local spin density is given by (Sf) EM;: + B2’g
(43)
FfFt(l+i$)]. in the ordered phase.
- 3T.
The CPA expression for this quantity
(44) corresponding
to eq. (35) is calculated by 2
(45) Similar to the above calculation the integration integrand with respect to S cos Bj.
over 6Ij is performed by an expansion of the denominator
of the
K. Usamiand T. Moriya / Spin fluctuation theory of ferromagnetic metals
180 4.
Numerical calculations
4.1. Semielliptic density of states In order to see the general features which depend little on the details of the density of states, for strongly magnetic cases at least, we first present the results of numerical calculations for the following semielliptic density of states per atom per spin: p&)
=Z(l
- 2)?
(46)
Here and in what follows the energy is measured in units of half the band width. For simplicity we replace the Fermi distribution function by a step function in the energy integral, since the temperature under consideration is smaller than the band width. Fig. 1 shows ( u), given by eq. (17), as a function of n, the electron number per atom. In the case of vanishing B, fo and FL are independent of U and we simply have the following expressions;
X0 = ;Pool);
XL = -jdepo(e)
s duP* E--W’ _m
(47)
Since the case with vanishing B corresponds to the ground state, we note that the present calculation agrees with the Hartree-Fock (HF) one at T = 0 K. In the case of nonvanishing B the random field tends to suppress the effect of the additional external (uniform or local) field on the moment. Furthermore, since both x0 and XL for given B decreases with an increase of U, ( u.) depends on U. Roughly speaking, in the region with positive ( u) the ferromagnetic state has the lowest energy. On the other hand, in the region with negative ( u) the ferromagnetic state has higher energy than the other states, say the antiferromagnetic state. The present result therefore is consistent with the well-known simple rule that the system tends to antiferromagnetism if the band is nearly half-filled, but to ferromagnetism if the band is mostly filled or empty [13]. It is shown that the critical number corresponding to the vanishing value of W varies with increasing B. Thus, in the case of about a quarter-filled (empty) band there exists the possibility of coexistence of and transi-
0 2
0.5 1.5
in
Fig. 1. The electron number dependence of (0) as a function of B and U for the semielliptic density of states.
K. Usamiand T. Moriya /Spin fluctuation theory of ferromagnetic metals
181
- 0.4
Fig. 2. The random field dependence of
( u 1for
various values of n and Cr.
Fig. 3. The amplitude of local spin fluctuations as a function of B given in eq. (35). The dashed line expresses the result of the local density of states approximation.
tion between ferro- and antiferromagnetism
accompanied
by a change in the amplitude
of the spin fluctuations
1141. In fig. 2 we plot (u) as a function of B for various values of n. In the case where n is less than 1 .O it is shown that (a) with large U is larger than that with small U. Since the Fermi energy for the half-filled case (n = 1) always lies at the center of the symmetric band, (o> is independent of U. And also in this half-filled case ~Q.J) vanishes when B is larger than 1 because of the band splitting given by the CPA. The decrease of ( o> for very small B,as is shown in fig. 2, reflects the rapid decrease of To for small B resulting from the fact that apo/& diverges at the band edge. The values for ,( ISjI> as calculated from eq. (35) are shown in fig. 3 as a function of B,where the values are normalized by the upper bound for the magnetization per atom given by M, = n/2 (n < 1) or MC = (2 - n)/2 (n > 1). For comparison the dashed line plots the locally induced magnetization calculated by assuming B as a uniform field. This corresponds to the local density of states approximation adopted by Takahashi and Moriya [4] if we regard the density of states as an effective local one. Although all the solid lines tend to 1 .O for large B,the values of ( ISj 1)/i& for each fixed B decreases with increasing n. The CPA calculation for the local density of states always gives finite values for the minority spin band as well as the majority spin band for each energy value. Thus ( ISjI) reaches its upper limit only asymptotically with increasing B.On the other hand the dashed line reaches 1 .O at a certain critical value of B for each n. In the calculation that uses the assumed local density of states and the long wave approximation, the saturation occurs much more easily than in the calculation that uses CPA. The Curie temperature for fixed Jis plotted in fig. 4 as a function of n. For comparison the Curie temperature calculated by the HF approximation (HFA) and the magnetization in units of No@n in the ground state are shown in the same figure. The Curie temperature of the present approximation is much lower than the HF value. Although this is qualitatively a desired result, the absolute value of Tc seems to be too much smaller than TF". This results mainly from our use of a single band model and the static approximation where the quantum effect is totally neglected. The effect of degeneracy of the band, which generally tends to increase Tc,will be discussed in the following subsection. An especially important role of the quantum effect for a single band case may be seen from the following consideration. If the local amplitude of the spin fluctuations changes little in the ordered
K. Usamiand. T. Moriya /Spin fluctuation theory of ferromagnetic metals
182
Fig. 4. The Curie temperature plotted against the electron number. The dashed line stands for the HF value and MO is the magnetization per atom in the ground state. T is in units of half the band width.
phase, the critical temperature have
is proportional
to the square of the magnetization
in the ground state [ 11. Thus we
Tc a Wf>T,oK.
(48)
Denoting the number of uncompensated (S$”
= (iAn)*
=
electrons by An, we have the following relation in the classical limit:
&AYz)* .
On the other hand, if we use the operator identity (Sf)qu = i(tZj+ + YljJ- 2njf nj$) = $Atl.
(49) nfO = nj,, , we get
(50)
We then get Qu/@=3/An.
(51)
Thus, the quantum effect, if properly taken account of, makes the critical temperature much higher than the present calculation. With increasing n the critical temperature increases at first and then decreases, although MO shows a monotonic increase. Incidentally MO has its maximum value, n/2, if n is larger than no, shown by an arrow in fig. 4. Since 6 vanishes at the Curie temperature, we have the following relation from eq. (33): Tc a(u).
(52)
Thus, in our calculation Tc is small for small (a> and vanishes at the critical number where (u) becomes zero, indicating instability for ferromagnetism. Near this point our approximation is not necessarily good and we have to take account of the variables corresponding to other magnetic orderings, antiferromagnetism for example. Anyhow, even in the case of large MO we have low Tc if (u.) is small, as is usually the case where the band is nearly quarterfiled or quarter-empty [ 131. Fig. 5 shows the temperature dependences of the inverse susceptibility for various values of n, J and U. The up
183,
K. Usamiand T. Moriya /Spin fluctuation theory of ferromagnetic metals 3 x-l
a
n-
0.2
J = 2.18
/ I' I
"rcw /
1!-1 2'5 -
C
l-l= 0.5
I 1'
J = 2.2 /'
IJZO
Fig. 5. The temperature dependence of the inverse susceptibility: (a) n = 0.2, J = 2.18, (b) n = 0.2, J = 4.0, and (c) n = 0.5, J = 2.2. The up and down spin bands in the ground state are also shown. The dashed line expresses the slope of the local moment case
(CLM).
and down spin bands in the ground state are also shown. Here we confine ourselves to the case of positive GJ). For comparison the dashed line shows the slope corresponding to the local moment case corresponding to the Curie constant CLM =M,2/3.
(53)
The present calculation is applicable to the weak ferromagnets (fig. Sa) as well as to the strong ones (fig. Sb). The susceptibility shows an approximate Curie-Weiss behavior. With an increase of J the slope of x-’ becomes larger and approaches that of the local moment case. This trend is most significant for the case of the quarter-filled band (fig. 5c), as expected [3]. In the case of small U the linearity of x-l is not so good and x-’ has an upward convexity. In the case of large U the linearity of x -’ becomes generally better. This may be due to the fact that the larger U tends to keep the local charge neutrality better, thus also’suppressing the spin fluctuations. It is also shown that the Curie constant, though not necessarily well defined, generally decreases with increasing U. Fig. 5b and c show that the effect of the charge fluctuations, or the dependence on U, becomes less significant as the distance between the Fermi energy and the band edge increases.
K. Usami and T. Moriya /Spin fluctuation theory of ferromagnetic metals
184
\-I
I
m0.2,
n.*
0
’
0
I
‘S-V..,,
J--2.1 8
I
0.005
I
0.25
al
I
I
0.01
01
,
I
0.015
T
.
L M.,M,
I
I
0
1
i
0.02
0.01
I
I
I
0.03
T
I
0.04
Fig. 6. The temperature dependence of the amplitude of local spin fluctuations. The solid and dashed lines are calculated by eqs. (34) and (35), respectively: (a) n = 0.2, J = 2.18, (b) n = 0.5, J = 2.2.
The temperature dependence of the amplitude of the local spin fluctuations is shown in fig. 6. The solid and dashed lines show the results calculated by eqs. (34) and (39, respectively. In fig. 6a all lines show an increase with a T112behavior at low temperatures and then gradually increase with increasing temperature. At high temperatures it is shown that the effect of U is to suppress both ( lS,i) and (S,?)i~2. In fig. 6b the Curie temperature, corresponding to the left ends of lines, is low because of the smallness of(u) in spite of a relatively large amplitude of spin fluctuations at Tc. In our calculation ( ISi I) never exceeds &, but (S~)1’2 can exceed this value. This is related with an approximate nature of our CPA solution for L [go,x] , where the longitudinal fluctuation in the Q field is neglected. We note here that ( IS,l> and L!S~)1/2are expected to show a maximum and then decrease at higher temperatures if we include the temperature effect in the functional due to the Fermi distribution function
1
0
I
0.002
I
I
QoO4
I
I
0.006
1 T
Fig. 7. The temperature dependence of the magnetization for U = 0. For comparison tSj)1/2 calculated by eqs. (44) and (45) are shown by the solid and dashed lines, respectively.
185
K. Usami and T. Moriya /Spin fluctuation theory of ferromagnetic metals
In fig. 7 we plot the temperature dependence of the uniform magnetization. Although we have made the expansion which is valid in the region for T 2 0 or T 5 Tc, the result is a smooth interpolation between the two limits. For small B the magnetization changes proportionally to B2, which in the static approximation increases linearly in Tat low temperature. Thus the magnetization shows a relatively rapid decrease with a T-linear dependence for T<< TC. If the dynamic effect is properly taken into account, the result will be improved. For example, in order to include the contribution due to the spin waves, we have to rewrite eq. (7) by taking account of the frequency dependence:
xff
=‘c
j
41rNe q _.
dw(othF-l)Iml_Ui
(4 o), & 9
IT
where the zero point fluctuations are subtracted. The contribution from the poles due to the spin waves yields x,,~ 0: T312,
(55)
and at low temperatures the magnetization shows the well-known T312 behavior. In the same figure the temperature dependence of - n plot for the
1.0 -1.0
0
1.0
xi
0.5 1.2 0.0
1.0 0.8 0.8 0.4 0.2
cc
Fe
Mn
G 0
2.0
1.6
1.2
aa
n
0
0.01
0.02
Fig.
8. The electron number dependence of ( CT)for the bee metals.
Fig.
9. The temperature dependence of the inverse susceptibility for the bee iron for various values of U.
0.03
T
0 )4
K. Usami and T. Motiya /Spin fluctuation theory of ferromagnetic metals
186 0.30
M, c
CM,
0.30 -
0.25 * I+,
--MC (siy ____________________---------
"Zo3 0.15 0.15O.lO-
-
($j'2
J= 1.317 0.10-
u= 0
---
0 0
Fig. 10. The
0.05I
I 0.01
I
I 0.02
I T
t 0.03
0
0
acm
a.l%x
I 0.003
w.04
T
temperature dependence of the amplitude of local spin fluctuations for the bee iron.
Fig. 11. The temperature dependence of the magnetization for iron. For comparison CSi21l/2 calculked by eqs. (44) and (45) are shown by the solid and,dashed lines, respectively.
case of vanishing B correspond to those in the density of states, as shown by the arrows in the figure. Here we confine ourselves to the case of iron. We adopt the following parameters: n = 1.45 and Me (T = 0 K) = 0.25. These values yield J = 1.317. Fig. 9 shows the temperature dependence of the susceptibility for various values of U. The susceptibilities show an approximate Curie-Weiss behavior. Although the inverse susceptibility becomes straighter with increasing U, the effect of the charge field is relatively small. The Curie constant is very close to that for the local moment case in accordance with experiment. The calculated Curie temperatures are 0.0017,0.0023 and 0.0027 in the reduced units for U = 0, 1.3 17 and 00, respectively. We estimate the d-band width to be eO.5 Ry. Also, we roughly consider the effect of five-fold degeneracy of the d-band [since all the bands contribute additively to the low energy modes of spin fluctuations, we simply make the following substitutions: P(E) + 5$(e) and J + $1 and get a multiplicative factor of 5, since TC is proportional to JM: [ 11. The quantum effect will give another multiplicative factor of 2 assuming S = 1. Thus we get TC = 900 - 1060 (U = J ‘v =J), in good agreement with the observed value Tzbs = 1044 K [ 181. The HF theory gives TsF = 5200 K. In fig. 10 we show the values for (Sf)1/2 and ( ISjI) calculated from eqs. (34) and (35), respectively. They are smaller than MO (T = 0) at T = TC and increase gradually with temperature. I
EF (T/Tc=0.615
1
Fig. 12. The density of states for up and down spin bands in the ferromagnetic phase. The dashed line stands for that in the ground state.
K. Usamiand 7’. Motiya /Spin fluctuation theory of ferromagnetic metals
187
In fig. 11 we plot the magnetization versus the temperature. The magnetization shows a T-linear decrease at low temperatures because of the static approximation. The temperature dependence of (Sf)1/2 calculated by eqs. (44) and (45) are plotted by the solid and dashed lines, respectively. The difference between the two lines is relatively small compared with fig. 7. The density of states for the ordered phase at T/TC = 0.615 is shown in fig. 12 together with that in the ground state as plotted by the dashed line for comparison. At the ground state the up and down spin bands are independent of each other and have the same shape. At finite temperatures in the ordered phase the random fields with arbitrary directions connect the up and down bands, as is shown by eq. (37). Thus, the density of states does not have the same shape and each band has a tail on the other’s side. 4.3. i%e fee metals Next we consider the case of fee metals. The model density of states [ 191 and the electron number dependence of (u.) are shown in fig. 13. For B = 0, the (u) - n curve shows a rapid decrease corresponding to a sharp peak at the band edge at about n x 1.8. Compared with the case of bee metals the critical electron number, where (0) vanishes, varies more strongly with the amplitude of the random field. The region with positive (0) is small compared with the case of bee metals, which is consistent with the more detailed calculation (for B = 0) made by Asano and Yamashita [20] based on a realistic band structure. It is interesting to see from this figure that Fe is expected to be weakly antiferromagnetic while, in agreement with observation, Co and Ni are ferromagnetic. We now consider nickel, putting J = 1.35 and n = 1.88 in the calculation. Since the electron number is small, Fe is much larger than zL. In this case the simple form of the distribution function for u given in eq. (16) may not be a good approximation. The susceptibilities j&r with wave vectors around 4 = 0 have large values and the distribution function is not expected to be symmetric about the center. Nevertheless, for the sake of brevity and for qualitative purposes we adopt eq. (16) for P(u). The temperature dependence of the inverse susceptibility is shown in fig. 14. The Curie-Weiss law holds approximately. With increasing U the Curie constant rapidly decreases. Although the square of the ratio of the magnetization at T = 0 K between iron and nickel is (0.25/0.06)2 = 17.4 in our calculation, the Curie temperature of nickel is nearly the same as that of iron. This results mainly from the much larger difference between j& and
n- 1.88,
Fig. 13. The electron number dependence of (01 far the fee metals. Fii. 14. The temperature dependence of the inverse susceptibility for the fee nickel.
J=1.35
188
K. Usami and T. Moriya /Spin fluctuation theory of ferromagnetic metals
Fig. 15. The temperature dependence of the amplitude of local spin fluctuations for the fee nickel. Fig. 16. The temperature dependence of the inverse susceptibility for the fee cobalt.
&_ for nickel than for iron, and partly from the fact that the amplitude of local spin fluctuations at Tc calculated by eq. (34) in Fe is relatively more reduced from the saturation moment at T = 0 than in Ni. In fig. 15 the values of (S,$1’2 and ( ISj 1)are plotted against temperature. The former exceeds the value of saturation moment MO,indicating that the local charge neutrality is not warranted in this approximation. The relatively large discrepancy between (S~)1/2 and ( ISj I) for Ni indicates that the approximation is poorer in Ni than in Fe. If we simply assume that (S$ = Mz for Ni, the value of Tc is reduced by half. In the case of nickel we expect the important part of the low energy excited states to have a long wave character and it would be more appropriate to make a long wave approximation than to make a local approximation, as in the present approach. In view of this fact and the rough way of considering the effect of band-degeneracy, etc. the above-estimated value for Tc is considered to be reasonable. A recent theory of Korenman and Prange [21] seems to belong essentially to a long wave approximation, though it is necessary for the present purpose to extend it to the case of variable amplitude of the local spin fluctuations. In fig. 16 we plot the temperature dependence of the inverse susceptibility with n = 1.66 and J = 1.6, bearing cobalt in mind. The Curie temperature is nearly the same as that of Fe, a roughly reasonable result, and the susceptibility shows an approximate Curie-Weiss behavior. It is interesting to note that according to the present calculation the ratio of the Curie constant to the one deduced from the saturation moment (assuming the existence of the local moment) is nearly equal to 1 for Fe and successively increases for Co and Ni. This is qualitatively consistent with observation [22]. 4.4. Other examples Quite a few materials show a maximum structure in the susceptibility as the temperature increases. This phenomenon can be realized in our calculation by adopting, for example, such a density of states as given in the top of fig. 17. It is essential that in the ground state the Fermi energy lies in a region of relatively low state density which is, however, very close to the region of high state density. By increasing the amplitude of the random field the peak in the density of states becomes less significant and the height of the low state density close to the peak increases. Thus by increasing the temperature or the amplitude of the random field, j& first increases. This is due to the negative coupling between the spin fluctuation modes. It then decreases as usual since the mode coupling becomes positive when the amplitude of the spin fluctuation becomes large. So there exists a minimum in the inverse susceptibility curve plotted against the temperature, as shown in the figure.
K. Usamiand T. Mqriya /Spin fluctuation theory of ferromagnetic metals
I
0
I
I
0.01
1
0.02
I
I
0.03
189
I
T
Fig. 17. The inverse susceptibility against temperature with a minimum structure.
5. Comparison with the local saddle point approximation It is of some interest to point out the relation between the present theory and the local approximation recently presented by Hasegawa [7]. He made use of the following interaction Hamiltonian:
with J = U and calculated the magnetic moment at an atomic site as an impurity moment under the influence of a uniform molecular field taking account of the random exchange field in the surrounding medium. Use was made of a single site CPA and the saddle point approximation was employed to evaluate the statistical average value for the impurity moment. Then the problem was solved by equating the impurity moment with that in the medium. This result can be derived easily with the use of a variational procedure in evaluating L(B, Bo) in the present theory and by making an additional simplification of the local saddle point approximation. Let us consider a CPA calculation of L(B, Bo) which is somewhat different from the one in section 3. In place of considering the distribution in the direction of the local field &, we consider here only the up and down sites with the spin and charge fields Br, B2 and yr, 72, respectively. The numbers of sites are Nr and N2, respectively, and we define the parameter Nr-N2 5 =-=-~ Nr+Nz
Nr -N2 (57)
No
We have the following relations for the field variables: B. = (B]), = F’
Br + yB2,
B2 = (BfzJ - (,)2 = y2(Bf
-
231B2
+Bi),
(58)
where (Al& denotes the average over sites defined by bli),
=
l-5 y41 t -y&42.
In the calculation of the partition function the division into two kinds of sites, and thus the reduction of the number
190
K. Usami and T. Moriya / Spin jluctuation theory of ferromagnetic metals
of variables for integration, yields the factor Ne!/Nr !Nz!. This leads to the entropy term of the random fields. Thus the expression for L(B, Be) is given by
1+5,1-5 +!_!...51n2
(
P2
2
1-l In 2
)’
where
(60) The CPA equation
reduces to
x:0 = ( Vjo)a - WI0 - 20) F~(vZ~
- x0)*
(61)
Provided we neglect the c-dependence in the term X, in eq. (6), we can determine function of B and Bo by the equations, aL/ayj = 0 and aL/a{ = 0. We get rj = - g
Jde./(e)
C Im 0 1 - (I$
FL7 - &,)F,
-
the values of yr, yz and 5 as a
(62)
Eq. (61) leads to eq. (25) in the case of vanishing B,, and f. The expressions for the uniform and the local susceptibilities are given in the appendix. The result of LSPA by Hasegawa can easily be derived if we put $ 1 = L (B, B,-,) and J = U, neglecting the nonlocal term in eq. (6), and determine the value of B from the following saddle point equation: =O.
(63)
The paramagnetic susceptibility is calculated by inserting this value of B into (A.2) while in the present calculation we take account of the nonlocal term through the caIculation of the local susceptibility TL and determine the value of B from eq. (A.4) taking account of the thermal excitations of nonlocal spin fluctuation modes. The main difference can easily be seen in the weakly ferromagnetic limit or for small B where the present approximation leads to a reasonable result while the LSPA does not. In other words, when the saddle point is at B = 0 the LSPA reduces to the Stoner theory while the original MT theory takes account of the important nonlocal thermal spin fluctuations with the mode-mode coupling effect even in such a case. This means that the LSPA neglects part of the thermal spin fluctuations and the mode-mode coupling effects which seem to be of general importance as will also be seen in the following numerical example. In order to compare the two approximations, we present here the results of numerical calculations for bee iron. In fig. 18 we plot the Curie temperature as a function of J. For comparison the Curie tdmperatures calculated by the HFA and by the LSPA are shown by the dashed and dot-dashed lines, respectively. Although the present approximation agrees with the HF one at T = 0 K, the LSPA does not. If we put J = U = 1.317, the Curie temperature by the present approximation is about 0.09 of that by the HFA and 0.57 of that by the LSPA. Since our calculation includes the effect of the nonlocal term X,, we have a smaller Tc than that by.the LSPA. The latter with J = CJand the band width of 0.5 Ry yields Tc = 800 K [7] without considering the degeneracy of the band. If we take account of the five-fold band-degeneracy and also correct the overestimation of the quantum effect due to the Ising-type model by multiplying by a factor of $, we get Tc = 1500 K for the present calculation and 2700 K for Hasegawa’s LSPA calculation. The numerical values should not be taken so seriously in this Ising-type model. Fig. 19 shows the temperature dependence of the susceptibility. The dot .dashed line shows the result by the LSPA;cForcomparison the slope of the local moment case corresponding to theCurie constant C’LM=Mz is
191
K. Usami and T. iUon)a / Spin fluctuation theory of ferromagnetic metals
Tc
~sp*(l_k1.317) Tc I
n =1.45 I ! I
0.03-
0.02-
i
i
! I I ! 1 i I i
TCHF/10 / / ! ! / ! ,I' u=1.317 ! ! ! / I "Zca i I
I i f iij’4
/
P
n- 1.45
x-1
M~0.2
2.0-
IA
5
J= 1.317
1.6 1.2 0.6 0.4 -
o-
0
0.02
0.04
0.06
0.08
T
0.1
Fig. 18. The Curie temperature for the bee iron as a function of J as calculated’with the use of the interaction Hamiltonian of eq. (56). The dashed and dot-dashed lines show those calculated by the HF theory and the local saddle point approximation (LSPA), respectively. Fig. 19. The temperature dependence of the inverse susceptibility for iron calculated by the present theory (solid line) with the use of the interaction Hamiltonian of eq. (56) and the LSPA (dot-dashed line).
shown by the dashed line. The susceptibility shows an approximate Curie-Weiss behavior. The effect of the charge field is small, as in the previous case of fig. 9. The Curie constant is almost equal to CL~. On the other hand the linearity of the inverse susceptibility calculated by the LSPA is not good and its Curie constant is much larger than CL~. The 1/xLspA - T curve has a tendency towards saturation at high temperatures due to the above-mentioned neglect of part of the mode-mode coupling effect. These comparisons seem to suggest that the local saddle point approximation is not satisfactory even for iron and we have to take account of the nonlocal character of the spin fluctuations with the mode-mode coupling. As was pointed out by the MT theory, the concept of temperature variation of the amplitude of LSF controlled by the mode-mode coupling is an essentially important feature of itinerant electron magnetism. However, the MT theory also points out that the Curie temperature may be evaluated properly without considering this effect seriously, since Tc is independent of the longitudinal stiffness constant for the spin fluctuations. This of course does not apply to the magnetic susceptibility which is sensitive to the longitudinal stiffness constant. Thus, the calculation for the magnetic susceptibility provides a more crucial test for the approximation than that for the Curie temperature. In other words an explanation of the values of T, only would not be enough to justify the approximation. We mention here two recent papers that deal only with the Curie temperature. Hubbard [23] has presented a calculation for the free energy functional for a special configuration where the local spin direction is rotated by an arbitrary angle Q from the direction of the ferromagnetic moment in the ground state. Comparing the curve for the saddle point energy AE vs. r$to the corresponding plot for the Heisenberg model, he estimated Tc = 1200 K. Capellmann [24] has estimated the Curie temperature for Fe and Ni with the use of an effective Heisenberg model with the coupling constant deduced from the band structure. This theory makes use of a long wave approximation with the emphasis on the short range order assuming a fixed amplitude for the local spin fluctuation. Although his use of a molecular field approximation in evaluating TC seems to be inconsistent with his assumption
192
K. Usami and T. MorQa / Spin fluctuation theory of ferromagnetic
metals
of the strong short range order in deriving the effective Heisenberg exchange, reasonable values for TC have been reported for Fe and Ni.
6. Conclusions
and discussions
Along the lines of the unified spin fluctuation theory of itinerant electron magnetism developed by Moriya and Takahashi, we have developed a simple approach which makes use of only the density of states with the use of the single-site coherent potential approximation. It is demonstrated that when the occupied (or empty) fraction of the band is small there is a tendency toward ferromagnetism while the nearly half-filled bands tend to be antiferromag netic in agreement with previous results and also observation. In the weakly ferromagnetic case the Curie constant is much larger than is expected from the saturation moment through the local moment mechanism, while in the strongly ferromagnetic case with the nearly quarter-filled band the Curie constant is close to the local moment case with the ferromagnetic saturation moment. The effect of charge density fluctuations was also treated within the saddle point approximation and the linearity of the inverse susceptibility vs. T was shown to improve generally when the effective Coulomb energy increases. The calculation has been extended to simple models for bee and fee d-metals and the results for Fe, Co and Ni are generally reasonable. We have also demonstrated that a maximum structure in the susceptibility vs. temperature curve can be realized under certain conditions for the density of states. We have discussed the relation between the present theory and that of Hasegawa which makes use of the local saddle point approximation and the molecular field approximation. In the latter theory the thermal excitation of the spin fluctuations of nonlocal character is totally neglected and the interpolating nature of the MT theory is lost; the results do not seem to be satisfactory even for iron. The present theory includes this effect through the calculation of both the uniform and the local susceptibilities, Fe and &, and the results seem to be improved significantly. Our local approach is best applicable to the case where the characteristic wavelength of the system is short or, in other words, the energy difference between the ferro- and antiferromagnetic states is small. When the short range order is strong, as is expected to be the case for example in Ni, the best approach is from the opposite side, i.e. the long wave approximation may be useful. Nevertheless, it is interesting to find that the present theory gives a fairly reasonable result even in this case. We should finally mention that in order to improve the present results, especially at low temperatures, we have to take account of the dynamic effect of spin fluctuations. This should be the subject of future investigation.
The authors wish to thank Dr. H. Hasegawa and Dr. Y. Takahashi for a number of stimulating
discussions.
Appendix Here we show the expressions for the uniform and the local susceptibilities within the approximation discussed in section 5. The uniform susceptibility is calculated by the same expression as eqs. (19) or (27). With the use of the relation
(A.11
K. Usami and T. Moriya /Spin fluctuation theory of ferromagnetic metals
193
we get TO
= x01 +TToz,
64.2)
with - ZpHp/Fp ,
K, = 1 t 2ZpFp IV1 = 1 t gldef(E)
Im
7l
(1 + 2ZpFp)Hp C
W2 =g
JdefcE)
+
KP
F; -HP KpU
+ EpFp)
1’
Im ~~~+~~~~~)].
The local susceptibility is given by eq. (3 1). As in the case of the uniform susceptibility, we take account of the change in distribution of the parallel and antiparallel local &fields due to the external local field. We get: ZL
=rsLl
+xL2,
(A.31
XL2
=
B2 W23 T-B2W3’
with
The amplitude B2 = 4JT
of the random iield is given by 1+6
6 t (a) + (62 + 26(a))“2
*
References [l] T. Moriya and Y. Takahashi, J. Phys. Sot. Japan 45 (1978) 397. [2] T. Moriya and Y. Takahashi, J. de Phys. 39 (1978) C6-1466. [ 31 T. Moriya, Solid State Commun. 26 (1978) 483. [4] Y. Takahashi and T. Moriya, J. Phys. Sot. Japan 46 (1979) 1451. [S] T. Moriya, J. Magn. Magn. Mat. 14 (1979) 1. [6] T. Moriya and H. Hasegawa, J. Phys. Sot. Japan, to be published. [7] H. Hasegawa, J. Phys. Sot. Japan 46 (1979) 1504. [8] M. Cyrot, J. de Phys. 33 (1972) 125;Phil. Mag. 25 (1972) 1031. [9] L. Dworin and A. Narath, Phys. Rev. Lett. 25 (1970) 1287.
(A.41
194
K. Usamiand T. Moriya / Spin fluctuation theory of ferromagnetic metals
[lo] W.E. Evenson, J.R. Schrieffer and S.Q. Wang, J. Appl. Phys. 41 (1970) 1199.
[ 1 l] B. Velicky, S. Kirkpatrick and H. Ehrenreich, Phys. Rev. 175 (1968) 747. [ 121 F. Ducastelle, J. Phys. C8 (1975) 3297. [13] T. Moriya, Progr. Theor. Phys. 33 (1965) 157; M. Inoue and T. Moriya, Progr. Theor. Phys. 38 (1967) 41.
[ 141 T. Moriya and K. Usami, Solid State Commun. 23 (1977) 935. [ 151 K. Levin and K.H. Bennemann, Phys. Rev. B5 (1972) 3770. [ 161 S. Wakoh and J. Yamashita, J. Phys. Sot. Japan 21 (1966) 1712. [ 171 R.A. Tawil and J. Callaway, Phys. Rev. B7 (1973) 4242. [ 181 J. Crangle ana GM. Goodman, Proc. Roy. Sot. (London) A321 (1971) 477. [ 191 J.W.D. Connolly, Phys. Rev. 159 (1967) 415. [20] [21] [22] (231 [24]
S. Asano and J. Yamashita, Progr. Theor. Phys. 49 (1973) 373. V. Korenman and R.E. Prange, Phys. Rev. B19 (1979) 4691,4698. P. Rhodes and E.P. Wohlfarth, Proc. Roy. Sot. A273 (1963) 247. J. Hubbard, Phys. Rev. B 19 (1979) 2626. H. Capellmann, Z. Phys. B34 (1979) 29.
SPIN FLUCTUATION THEORY OF FERROMAGNETIC METALS
Ran USAMI and T&-u MORIYA Institute for Solid State Physics, Universityof Tokyo, Roppongi, Tokyo Received 15 October 1979
Ferromagnetic metals at finite temperatures are discussed using the functional integral method along the lines of the unitied spin fluctuation theory of Moriya and Takahashi. We develop a simple approach which makes use of only the density of states but still takes account of the nonlocal nature of the spin fluctuations with the aid of a shrgle-site coherent potential approximation. The effect of the charge density fluctuations is also taken into account within the saddle point approximation. The results of numerical calculations for various model densities of states including those for bee and fee d-metals are presented. Particularly, the Curie temperatures and the Curie-Weiss magnetic susceptibilities for Fe, Co and Ni seem to be fairly well reproduced.
1. Introduction
Recently Moriya and Takahashi [ 1,2] (lrereafter referred to as MT) presented a rather general spin fluctuation theory of itinerant electron ferromagnetism which interpolates between the local moment limit and the weakly ferromagnetic limit. They made use of a functional integral method and a model functional which can be regarded as a generalization of the Heisenberg model to permit the variation in amplitude of the local spin fluctuation @SF). They discussed ferromagnetism of metals from a general point of view making use of a relatively small number of physical parameters which can in principle be deduced from a given band structure. They also pointed out an important possibility of strong variation with temperature of the amplitude of LSF and its saturation above a certain temperature, leading to a new concept known as the “temperature-induced local magnetic moments” [3]. This theory has attained considerable success not only in giving a unified picture to itinerant electron ferromagnetism but also in clarifying several previously unexplained phenomena. In order to make detailed theoretical discussions on each specific substance, however, it is essential to calculate the one-body free energy functional from the band structure. There are several possible methods for this purpose. One is to make use of a local density of states. This method has been applied successfully to the problem of nearly ferromagnetic semiconductors by Takahashi and Moriya [4]. The other possible method is the local approach, i.e. to expand the free energy functional in terms of the local field variables such as the terms related with single site only, those related with two sites, etc. This type of approach has recently been developed by Moriya and Hasegawa [5,6]. In order to calculate the nonlocal terms in the free energy functional with this method we need almost full information of the band structure and the calculation for a realistic band structure seems to be generally rather heavy. In this paper we wish to develop a rather simple approach which makes use of the density of states only. The local term in the free energy functional can always be calculated from the density of states and in place of calculating all the nonlocal terms we just calculate the average of the coefficients of the nonlocal terms, which are expressed by a quadratic form in the Fourier q-components of the field variables. Thus, by assuming a simple form for the distribution function for the coefficients we can perform calculations for the free energy and various physical quantities. This is a significant simplification, although at the expense of accuracy, and we will see that the method is useful enough for qualitative and semiquantitative purposes. Because of this simplicity we can also deal 171
K. Usami and T. Moriya / Spin jluctuation theory of fewomagnetic metals
172
easily with the effect of charge density fluctuations. A local approach of a different type has been presented recently by Hasegawa [7] who simply substituted the effect of the nonlocal terms by a molecular field. He made the further simplification of using a local saddle point approximation [8] (LSPA) at the expense of the interpolating nature of the original MT theory. Furthermore, this approximation is rather limited ln its applicability. The present theory, following more closely the work by MT, is expected to be free from these drawbacks. In section 2 we briefly summarize the necessary formulae in the functional integral formalism and derive the formulae for various physical quantities with the use of a model functional which is a generalization of the one by MT; we consider the external magnetic field so that the theory applies both above and below the Curie temperature, Tc . The method of calculating the free energy functional taking account of the charge density fluctuations explicitly with the use of the saddle point approximation is discussed in section 3 and the results of. numerical calculations are presented in section 4. We discuss in section 5 the relation between the present theory and the one with the use of LSPA. In section 6 the conclusions are summarized and discussions are given.
2. Formalism and model functional Our starting point is the single band Hubbard Hamiltonian. Taking account of the effects of band-degeneracy, we use the effective interaction Hamiltonian including both U and J, the effective Coulomb and exchange energies, respectively. The total Hamiltonian is given by
Q.
= C jQ
C
D
tjQa&aQo
,
(1)
where is a hopping matrix element, a& is a creation operator of an electron with spin u at the lattice site j, and nj and Si are the charge and spin density operators, respectively. The effective interaction term is derived by considering the case of five-fold degeneracy with the rotational invariance in the spin space [9]. The last term represents the effect of the external magnetic field applied along the z axis. The free energy of the system is given in a functional integral form through the Strantnovich-Hubbard transformation [lo]. Within the static approximation we have e-PF = ,-NFo+AF) = Trte -N%-&] ,-oU tjQ
,
Fr = (47rJ/@1’2;
c2 = (--nU//3)“2,
K. Usamiand T. Mon>a /Spin fluctuation theory of ferromagnetic metals
173
where p is the chemical potential, /.3= l/T, T being the temperature expressed in energy units, Nis the number operator, and the other notation is usual. With the use of a standard technique [lo], $r [!$,n] is written as
$1ko1
=-$Trln[l -90V1,
with ,ik(Rj--R cj, n ISo
IQ, n’)
= s,,s
Q)
__!_ c
where Tr means the diagonal sum over j, n and in the spin space. Thus we have the problem of a noninteracting electron system moving under the spin and charge fields fluctuating arbitrarily in space. We need to calculate J/r and then to perform the functional integral. Then the suceptibility and the magnetization, etc. can be calculated in a standard way. Here we use the approximation of taking the saddle point value for nj for each sj, so that the functional depends explicitly on cj only. In the case of ferromagnetism or under an external magnetic field, 50 takes a finite value and we have to take account of it to all orders in the expression of $ r, where and in what follows & and Se are the Fourier transforms of Q and Sj, respectively. In addition to to, we introduce variables corresponding to the average squared local amplitude of e-fields: (4) where the prime means to omit the term with 4 = 0 from the summation and No is the number of lattice sites. This variable is to be related with the local amplitude of the spin fluctuations as follows:
We adopt the following model functional for J, r just as in MT [ 1,5]:
with ~X*,]x.
c;ol=O9
(6)
where it is assumed that the coupling between the spin fluctuation modes is mainly local and the higher order terms are expressed in terms of x and to. The reasonableness of this assumption is indicated by the fact that, as MT have shown, this model functional properly interpolates between the weakly ferromagnetic limit and the local moment limit. After evaluating the integral in eq. (2) with the use of this model functional we finally get the following results
ISI: 1 XQ =_ lc’ 25VoP4 1- w(x,, t j&j ’
(74
174
K. Usamiand T. Moriya /Spin fluctuation theory of ferromagnetic metals
x0x
=xoy =-(S,)/lv&,
G’b)
xqa =&Y/(1 - w&a),
where j$ is measured in units of No(er-le)2 and c, = C,/dN,-,. In the-case of the paramagnetic phase the system is isotropic and we have
3 -Its 27rP( &+a
x=&,=(Y a=1
(8)
) ’
-Z&O,
with
(f(a)) = ~P(a)f(a) da,
P(a) =Nf’
G’S[a-
(1 -$)I.
(9)
Here we need to calculate To and P(a), the distribution function for a, from the band structure.
3. Single-site coherent potential approximation Let us consider the actual calculation of $ r, which is similar to the problem of random alloys where the electrons see a potential fluctuating from site to site. In the latter case the coherent potential approximation (CPA) [ 11,121 is known to be a useful method. So we apply the CPA to the present problem. First we rewrite eq. (3) by using the Green’s function, Q = (9;’
- 8)-r,
w9
where C is the site-independent self-energy to be determined later. We get $r[E,s] =-kTr[ln(l
-~oZ)tln{l
- Q(V-
Z:))].
(11)
If we split Q into a site-diagonal part F and site-nondiagonal part g ,
$?=Ftg ,
(12)
we have $r[,$nJ=-a’Ir[ln(l
- Qo~)tln(l
-F(V-Z))tln{l
-@(V--Z)[l
-F(V-E)]-l)].
(13)
K. Usamiand T. Motiya /Spin fluctuation theory of ferromagnetic metals
The first two terms and the last term are the local and the nonlocal parts, respectively. mined by minimizing the first two terms of eq. (13) with respect to Z [12]. We have
5; CjI(V-
Z)[l
-F(V-
IQ]-‘lj?=O.
175
The self-energy is deter-
(14)
This is nothing but the CPA condition. The model functional discussed in the previous section is given by the following steps. First, in the expression for $ i we leave the local part or the terms related with a single site only and among the nonlocal part those related with the relative orientation of the spin fields at different sites. Next vi is determined as a function of ki by minimizing the sum of $0 and the local part of 9 i with respect to nj. We then have the form given in eq. (6). The CPA calculation of the local term is straightforward if we make the approximation of assuming an isotropic distribution in the direction of (15) and take an average value for IS&l. As for the term related with the pa.ir of different sites, in place of calculating the coefficient of each term we calculate only jzi, or the average of yr .,Assuming for example the following form for the distribution function P(u,-J,
(16) where ( u,) is given by (a,>
=l -&J-k
F&=1
-2,
(17)
we can dispense with the calculation of each X, from the full band structure. Thus, the problem can be solved approximately if only the density of states is given. Let us first consider the paramagnetic phase. For later convenience we introduce the following notation: B2 =&3x;
Be =crto;
y =czq.
(18)
In order to calculate x0 and Tsr, for given x, we consider such conditions that a random field with a magnitude B is applied along an arbitrary direction at each site. First we consider x0, the uniform susceptibility for the noninteracting system under the random magnetic field. In this case we assume that an infinitesimal uniform field B. is applied along the z axis. We calculate j&, by the following simple approximation:
(1% L(B, Bo) is given by the sum of the charge field part of J/e and the local part of $Jr. From eqs. (2) and (13) we have I
UB. Bo) = -
&j
Wr(412t
i
$
Im g
ln(1 - So@, e)(E + iBoo,))
-1
1
t;
s
dcImlndet[l
-1
with Qo(k e) = (e - ek + cc)-’ ,
-F(V-
Z)]
(20)
176
K. Usamiand T. Mot-&a/Spin fluctuation theory of ferromagnetic metals
where f(c) is the Fermi distribution function and ): and Pare shifted by iBoo,from those in eqs. (lo)-(14). The direction of the random field B is expressed by the polar angles 0 and 9 with respect to the z axis and c stands for cos 8. IZ, F and V are given by z=[Zt
F=
“,J
(2la)
Ft 0 1’ 0 FJ
Wb)
[
B
ysin
Be-‘@
v=
(2lc)
-$B cos e + ~(~0s e) 1 with
(22) where the coherent potential 2 is a diagonal matrix because of a rotational (14) we have the following CPA equation:
symmetry around the z axis. From
eq.
1 is 0 The
dc(V- Z)[l -F(V- IQ]-' =O.
d#
(23)
-1
charge field is determined
aL ar(c)=
0
by the condition
’
and we get r(c) = g
jdef(e)
Im $)
&
A(c),
with A(c)=det[l
-F(V- Z)],
(24) where the variational property of the free energy in the CPA, aLlaI: = 0,is employed. In the case of vanishing Bo the system has full rotational Z,=
T-Z;
(
F=lx ' No k
1
symmetry.
Thus, the coherent potential
is a scalar quantity
and we have
F,,, (25) 1
E-+-~p-7p+~’
where the subscript p denotes the paramagnetic phase, the coherent potential is again shifted by.y,, stant, and the chemical potential for each B is determined by the following relation: - ijde
f(e) Im
Fp =$ .
yp is a con-
P-9
K. Usami and T. Moriya / Spin fluctuation theory of ferromagnetic metals
177
The expression for Fe given in eq. (19) is rewritten as
with 1
=Lc
H
’
No
(27)
(E-+~p-7p+C()2’
k
Differentiating eqs. (23) and (24) with respect to Bo, we get simultaneous equations for dZ+ /dBo and 1
s
c
dc
MC) -. @O
-1
Solving these equations we get x0 = & sdej(e)
Im
’ f~~)Hp]_dB2~[~~de~(e)
X
Imy]
t $ldef(E)
Im
with R,=l+;=$+%.
(28)
3
P
As for the local susceptibility &, there exists a simple approximation from eq. (7a) FL
1
q--
aL
-.
ax
2nJ
In this paper, however, we calculate it directly by considering such an impurity problem that a local field is applied at one site in the medium characterized by Z,. The potential at the impurity site is expressed by f@L
tB
COS0,) t yi(COSOi)
B
2 sin tIie-i’i
Vi = B _ sb di eNi
i 2
(29) -$(BL
t B
COS ei)
+ yi(COS
ei)
1.
where Bi and @iare the polar angles of the random field with respect to the z axis and BL is an infinitesimal field locally applied along the z axis. The part of the free energy functional associated with the impurity site is given by
Fi = - ,& j
d(cos fIi) [ri(cos ei)]’ + & IdefcE) 1 d(cos Oi) Im In det [ 1 -. (Vi - Zp)Fp] .
-1
From this expression we get the following relation:
j7L’d dBL
(_3_ 1 aBL 9
-1
(30)
K. Usami and T. Moriya /Spin fluctuation theory of ferromagnetic metals
178
with --
aFi
3yi(COS
=o.
(31)
0i)
The local susceptibility
is calculated as
(32) From eqs. (8), (9) and (16)-(18) temperature:
of random fields and the
1+6
B2=12JT
(33)
6 +([email protected]
The average squared amplitude (S,%=$
we get the following relation between the amplitude
(
;-3T.
of the local spin density is given from eq. (5) by (34)
)
On the other hand, considering the magnitude of the magnetic moment along the direction of the random field at each site, we get the following CPA expression for the amplitude of the local moment within the present approximation:
ClS,l> =&-def((e)Im
1 +Ft F P
(35)
. P
Next let us consider the ordered phase with He = 0. Although the system is no longer isotropic, we adopt the model of the isotropic distribution of random fields with a fixed amplitude. Then we can use eq. (23) with nonvanishing BO. It is still difficult to solve this problem. In view of the fact that the static approximation itself is considered to be poor at low temperatures, we here content ourselves with a qualitative feature and consider a very simple case. First we neglect the effect of charge flucutations. Furthermore we expand the CPA condition in terms of the following quantity: B S =? {FJ - Ft - FtFst%
(36)
- &)I,
where S is proportional to B - Bo. Therefore the results is expected to be good in the region where T/Tc 5 1 and also T/TC << 1. We get
with &=I
tFt&
+FIZJ
The chemical potential
-FtF,g
(38)
-Z&).
and the uniform magnetization
as a function
of B are determined
by the following simul-
K. Usami and T. Moriya /Spin fluctuation theory of ferromagnetic metals
179
taneous equations:
s
-N =_- 1 de./@)In@‘?+ FL>,
No
IT
(39) = & Jdef(E)
MO =z
where the equations field with ponent of wool
Im(Ft
-
F+) ,
magnetization MO is measured in units of NogpB. At T = 0 K the random field vanishes and the above reduce to the Hartree-Fock ones. Since we have used a simplified model for B, namely the random an isotropic distribution, we determine the relation between B and T by using only the transverse comthe susceptibilities. We make use of the relation (40)
= 1,
corresponding to the fact that the uniform transverse susceptibility is shown by eq. (7b). We have B2 =6T
I[
$
- X-u(Bt Bo)
diverges in the ordered phase with He = 0, as
1
(41)
,
where we have used eq. (33). The transverse component of the local susceptibility can be calculated in a way similar to the case of the paramagnetic phase. The medium is characterized by the coherent potential given by eqs. (36) (38). The potential in the impurity site is given by B
-sin
2
.;‘,
Bje -% + fB,,
Vi =
(42)
,
i’i + ;BLX 1
where BLx is the infinitesimal local field. The local susceptibility is given by differentiating the magnetic moment of the impurity site along the x-axis with respect to B Lx. The result includes S and Q. We expand it with respect to S, as is done in the calculation of the coherent potential, up to the second order and get ~~,=~Sde~(e)~m~(l+~)+~
Using this expression with eqs. (39) and (41) we can get the magnetization The average squared amplitude of the local spin density is given by (Sf) EM;: + B2’g
(43)
FfFt(l+i$)]. in the ordered phase.
- 3T.
The CPA expression for this quantity
(44) corresponding
to eq. (35) is calculated by 2
(45) Similar to the above calculation the integration integrand with respect to S cos Bj.
over 6Ij is performed by an expansion of the denominator
of the
K. Usamiand T. Moriya / Spin fluctuation theory of ferromagnetic metals
180 4.
Numerical calculations
4.1. Semielliptic density of states In order to see the general features which depend little on the details of the density of states, for strongly magnetic cases at least, we first present the results of numerical calculations for the following semielliptic density of states per atom per spin: p&)
=Z(l
- 2)?
(46)
Here and in what follows the energy is measured in units of half the band width. For simplicity we replace the Fermi distribution function by a step function in the energy integral, since the temperature under consideration is smaller than the band width. Fig. 1 shows ( u), given by eq. (17), as a function of n, the electron number per atom. In the case of vanishing B, fo and FL are independent of U and we simply have the following expressions;
X0 = ;Pool);
XL = -jdepo(e)
s duP* E--W’ _m
(47)
Since the case with vanishing B corresponds to the ground state, we note that the present calculation agrees with the Hartree-Fock (HF) one at T = 0 K. In the case of nonvanishing B the random field tends to suppress the effect of the additional external (uniform or local) field on the moment. Furthermore, since both x0 and XL for given B decreases with an increase of U, ( u.) depends on U. Roughly speaking, in the region with positive ( u) the ferromagnetic state has the lowest energy. On the other hand, in the region with negative ( u) the ferromagnetic state has higher energy than the other states, say the antiferromagnetic state. The present result therefore is consistent with the well-known simple rule that the system tends to antiferromagnetism if the band is nearly half-filled, but to ferromagnetism if the band is mostly filled or empty [13]. It is shown that the critical number corresponding to the vanishing value of W varies with increasing B. Thus, in the case of about a quarter-filled (empty) band there exists the possibility of coexistence of and transi-
0 2
0.5 1.5
in
Fig. 1. The electron number dependence of (0) as a function of B and U for the semielliptic density of states.
K. Usamiand T. Moriya /Spin fluctuation theory of ferromagnetic metals
181
- 0.4
Fig. 2. The random field dependence of
( u 1for
various values of n and Cr.
Fig. 3. The amplitude of local spin fluctuations as a function of B given in eq. (35). The dashed line expresses the result of the local density of states approximation.
tion between ferro- and antiferromagnetism
accompanied
by a change in the amplitude
of the spin fluctuations
1141. In fig. 2 we plot (u) as a function of B for various values of n. In the case where n is less than 1 .O it is shown that (a) with large U is larger than that with small U. Since the Fermi energy for the half-filled case (n = 1) always lies at the center of the symmetric band, (o> is independent of U. And also in this half-filled case ~Q.J) vanishes when B is larger than 1 because of the band splitting given by the CPA. The decrease of ( o> for very small B,as is shown in fig. 2, reflects the rapid decrease of To for small B resulting from the fact that apo/& diverges at the band edge. The values for ,( ISjI> as calculated from eq. (35) are shown in fig. 3 as a function of B,where the values are normalized by the upper bound for the magnetization per atom given by M, = n/2 (n < 1) or MC = (2 - n)/2 (n > 1). For comparison the dashed line plots the locally induced magnetization calculated by assuming B as a uniform field. This corresponds to the local density of states approximation adopted by Takahashi and Moriya [4] if we regard the density of states as an effective local one. Although all the solid lines tend to 1 .O for large B,the values of ( ISj 1)/i& for each fixed B decreases with increasing n. The CPA calculation for the local density of states always gives finite values for the minority spin band as well as the majority spin band for each energy value. Thus ( ISjI) reaches its upper limit only asymptotically with increasing B.On the other hand the dashed line reaches 1 .O at a certain critical value of B for each n. In the calculation that uses the assumed local density of states and the long wave approximation, the saturation occurs much more easily than in the calculation that uses CPA. The Curie temperature for fixed Jis plotted in fig. 4 as a function of n. For comparison the Curie temperature calculated by the HF approximation (HFA) and the magnetization in units of No@n in the ground state are shown in the same figure. The Curie temperature of the present approximation is much lower than the HF value. Although this is qualitatively a desired result, the absolute value of Tc seems to be too much smaller than TF". This results mainly from our use of a single band model and the static approximation where the quantum effect is totally neglected. The effect of degeneracy of the band, which generally tends to increase Tc,will be discussed in the following subsection. An especially important role of the quantum effect for a single band case may be seen from the following consideration. If the local amplitude of the spin fluctuations changes little in the ordered
K. Usamiand. T. Moriya /Spin fluctuation theory of ferromagnetic metals
182
Fig. 4. The Curie temperature plotted against the electron number. The dashed line stands for the HF value and MO is the magnetization per atom in the ground state. T is in units of half the band width.
phase, the critical temperature have
is proportional
to the square of the magnetization
in the ground state [ 11. Thus we
Tc a Wf>T,oK.
(48)
Denoting the number of uncompensated (S$”
= (iAn)*
=
electrons by An, we have the following relation in the classical limit:
&AYz)* .
On the other hand, if we use the operator identity (Sf)qu = i(tZj+ + YljJ- 2njf nj$) = $Atl.
(49) nfO = nj,, , we get
(50)
We then get Qu/@=3/An.
(51)
Thus, the quantum effect, if properly taken account of, makes the critical temperature much higher than the present calculation. With increasing n the critical temperature increases at first and then decreases, although MO shows a monotonic increase. Incidentally MO has its maximum value, n/2, if n is larger than no, shown by an arrow in fig. 4. Since 6 vanishes at the Curie temperature, we have the following relation from eq. (33): Tc a(u).
(52)
Thus, in our calculation Tc is small for small (a> and vanishes at the critical number where (u) becomes zero, indicating instability for ferromagnetism. Near this point our approximation is not necessarily good and we have to take account of the variables corresponding to other magnetic orderings, antiferromagnetism for example. Anyhow, even in the case of large MO we have low Tc if (u.) is small, as is usually the case where the band is nearly quarterfiled or quarter-empty [ 131. Fig. 5 shows the temperature dependences of the inverse susceptibility for various values of n, J and U. The up
183,
K. Usamiand T. Moriya /Spin fluctuation theory of ferromagnetic metals 3 x-l
a
n-
0.2
J = 2.18
/ I' I
"rcw /
1!-1 2'5 -
C
l-l= 0.5
I 1'
J = 2.2 /'
IJZO
Fig. 5. The temperature dependence of the inverse susceptibility: (a) n = 0.2, J = 2.18, (b) n = 0.2, J = 4.0, and (c) n = 0.5, J = 2.2. The up and down spin bands in the ground state are also shown. The dashed line expresses the slope of the local moment case
(CLM).
and down spin bands in the ground state are also shown. Here we confine ourselves to the case of positive GJ). For comparison the dashed line shows the slope corresponding to the local moment case corresponding to the Curie constant CLM =M,2/3.
(53)
The present calculation is applicable to the weak ferromagnets (fig. Sa) as well as to the strong ones (fig. Sb). The susceptibility shows an approximate Curie-Weiss behavior. With an increase of J the slope of x-’ becomes larger and approaches that of the local moment case. This trend is most significant for the case of the quarter-filled band (fig. 5c), as expected [3]. In the case of small U the linearity of x-l is not so good and x-’ has an upward convexity. In the case of large U the linearity of x -’ becomes generally better. This may be due to the fact that the larger U tends to keep the local charge neutrality better, thus also’suppressing the spin fluctuations. It is also shown that the Curie constant, though not necessarily well defined, generally decreases with increasing U. Fig. 5b and c show that the effect of the charge fluctuations, or the dependence on U, becomes less significant as the distance between the Fermi energy and the band edge increases.
K. Usami and T. Moriya /Spin fluctuation theory of ferromagnetic metals
184
\-I
I
m0.2,
n.*
0
’
0
I
‘S-V..,,
J--2.1 8
I
0.005
I
0.25
al
I
I
0.01
01
,
I
0.015
T
.
L M.,M,
I
I
0
1
i
0.02
0.01
I
I
I
0.03
T
I
0.04
Fig. 6. The temperature dependence of the amplitude of local spin fluctuations. The solid and dashed lines are calculated by eqs. (34) and (35), respectively: (a) n = 0.2, J = 2.18, (b) n = 0.5, J = 2.2.
The temperature dependence of the amplitude of the local spin fluctuations is shown in fig. 6. The solid and dashed lines show the results calculated by eqs. (34) and (39, respectively. In fig. 6a all lines show an increase with a T112behavior at low temperatures and then gradually increase with increasing temperature. At high temperatures it is shown that the effect of U is to suppress both ( lS,i) and (S,?)i~2. In fig. 6b the Curie temperature, corresponding to the left ends of lines, is low because of the smallness of(u) in spite of a relatively large amplitude of spin fluctuations at Tc. In our calculation ( ISi I) never exceeds &, but (S~)1’2 can exceed this value. This is related with an approximate nature of our CPA solution for L [go,x] , where the longitudinal fluctuation in the Q field is neglected. We note here that ( IS,l> and L!S~)1/2are expected to show a maximum and then decrease at higher temperatures if we include the temperature effect in the functional due to the Fermi distribution function
1
0
I
0.002
I
I
QoO4
I
I
0.006
1 T
Fig. 7. The temperature dependence of the magnetization for U = 0. For comparison tSj)1/2 calculated by eqs. (44) and (45) are shown by the solid and dashed lines, respectively.
185
K. Usami and T. Moriya /Spin fluctuation theory of ferromagnetic metals
In fig. 7 we plot the temperature dependence of the uniform magnetization. Although we have made the expansion which is valid in the region for T 2 0 or T 5 Tc, the result is a smooth interpolation between the two limits. For small B the magnetization changes proportionally to B2, which in the static approximation increases linearly in Tat low temperature. Thus the magnetization shows a relatively rapid decrease with a T-linear dependence for T<< TC. If the dynamic effect is properly taken into account, the result will be improved. For example, in order to include the contribution due to the spin waves, we have to rewrite eq. (7) by taking account of the frequency dependence:
xff
=‘c
j
41rNe q _.
dw(othF-l)Iml_Ui
(4 o), & 9
IT
where the zero point fluctuations are subtracted. The contribution from the poles due to the spin waves yields x,,~ 0: T312,
(55)
and at low temperatures the magnetization shows the well-known T312 behavior. In the same figure the temperature dependence of
0
1.0
xi
0.5 1.2 0.0
1.0 0.8 0.8 0.4 0.2
cc
Fe
Mn
G 0
2.0
1.6
1.2
aa
n
0
0.01
0.02
Fig.
8. The electron number dependence of ( CT)for the bee metals.
Fig.
9. The temperature dependence of the inverse susceptibility for the bee iron for various values of U.
0.03
T
0 )4
K. Usami and T. Motiya /Spin fluctuation theory of ferromagnetic metals
186 0.30
M, c
CM,
0.30 -
0.25 * I+,
--MC (siy ____________________---------
"Zo3 0.15 0.15O.lO-
-
($j'2
J= 1.317 0.10-
u= 0
---
0 0
Fig. 10. The
0.05I
I 0.01
I
I 0.02
I T
t 0.03
0
0
acm
a.l%x
I 0.003
w.04
T
temperature dependence of the amplitude of local spin fluctuations for the bee iron.
Fig. 11. The temperature dependence of the magnetization for iron. For comparison CSi21l/2 calculked by eqs. (44) and (45) are shown by the solid and,dashed lines, respectively.
case of vanishing B correspond to those in the density of states, as shown by the arrows in the figure. Here we confine ourselves to the case of iron. We adopt the following parameters: n = 1.45 and Me (T = 0 K) = 0.25. These values yield J = 1.317. Fig. 9 shows the temperature dependence of the susceptibility for various values of U. The susceptibilities show an approximate Curie-Weiss behavior. Although the inverse susceptibility becomes straighter with increasing U, the effect of the charge field is relatively small. The Curie constant is very close to that for the local moment case in accordance with experiment. The calculated Curie temperatures are 0.0017,0.0023 and 0.0027 in the reduced units for U = 0, 1.3 17 and 00, respectively. We estimate the d-band width to be eO.5 Ry. Also, we roughly consider the effect of five-fold degeneracy of the d-band [since all the bands contribute additively to the low energy modes of spin fluctuations, we simply make the following substitutions: P(E) + 5$(e) and J + $1 and get a multiplicative factor of 5, since TC is proportional to JM: [ 11. The quantum effect will give another multiplicative factor of 2 assuming S = 1. Thus we get TC = 900 - 1060 (U = J ‘v =J), in good agreement with the observed value Tzbs = 1044 K [ 181. The HF theory gives TsF = 5200 K. In fig. 10 we show the values for (Sf)1/2 and ( ISjI) calculated from eqs. (34) and (35), respectively. They are smaller than MO (T = 0) at T = TC and increase gradually with temperature. I
EF (T/Tc=0.615
1
Fig. 12. The density of states for up and down spin bands in the ferromagnetic phase. The dashed line stands for that in the ground state.
K. Usamiand 7’. Motiya /Spin fluctuation theory of ferromagnetic metals
187
In fig. 11 we plot the magnetization versus the temperature. The magnetization shows a T-linear decrease at low temperatures because of the static approximation. The temperature dependence of (Sf)1/2 calculated by eqs. (44) and (45) are plotted by the solid and dashed lines, respectively. The difference between the two lines is relatively small compared with fig. 7. The density of states for the ordered phase at T/TC = 0.615 is shown in fig. 12 together with that in the ground state as plotted by the dashed line for comparison. At the ground state the up and down spin bands are independent of each other and have the same shape. At finite temperatures in the ordered phase the random fields with arbitrary directions connect the up and down bands, as is shown by eq. (37). Thus, the density of states does not have the same shape and each band has a tail on the other’s side. 4.3. i%e fee metals Next we consider the case of fee metals. The model density of states [ 191 and the electron number dependence of (u.) are shown in fig. 13. For B = 0, the (u) - n curve shows a rapid decrease corresponding to a sharp peak at the band edge at about n x 1.8. Compared with the case of bee metals the critical electron number, where (0) vanishes, varies more strongly with the amplitude of the random field. The region with positive (0) is small compared with the case of bee metals, which is consistent with the more detailed calculation (for B = 0) made by Asano and Yamashita [20] based on a realistic band structure. It is interesting to see from this figure that Fe is expected to be weakly antiferromagnetic while, in agreement with observation, Co and Ni are ferromagnetic. We now consider nickel, putting J = 1.35 and n = 1.88 in the calculation. Since the electron number is small, Fe is much larger than zL. In this case the simple form of the distribution function for u given in eq. (16) may not be a good approximation. The susceptibilities j&r with wave vectors around 4 = 0 have large values and the distribution function is not expected to be symmetric about the center. Nevertheless, for the sake of brevity and for qualitative purposes we adopt eq. (16) for P(u). The temperature dependence of the inverse susceptibility is shown in fig. 14. The Curie-Weiss law holds approximately. With increasing U the Curie constant rapidly decreases. Although the square of the ratio of the magnetization at T = 0 K between iron and nickel is (0.25/0.06)2 = 17.4 in our calculation, the Curie temperature of nickel is nearly the same as that of iron. This results mainly from the much larger difference between j& and
n- 1.88,
Fig. 13. The electron number dependence of (01 far the fee metals. Fii. 14. The temperature dependence of the inverse susceptibility for the fee nickel.
J=1.35
188
K. Usami and T. Moriya /Spin fluctuation theory of ferromagnetic metals
Fig. 15. The temperature dependence of the amplitude of local spin fluctuations for the fee nickel. Fig. 16. The temperature dependence of the inverse susceptibility for the fee cobalt.
&_ for nickel than for iron, and partly from the fact that the amplitude of local spin fluctuations at Tc calculated by eq. (34) in Fe is relatively more reduced from the saturation moment at T = 0 than in Ni. In fig. 15 the values of (S,$1’2 and ( ISj 1)are plotted against temperature. The former exceeds the value of saturation moment MO,indicating that the local charge neutrality is not warranted in this approximation. The relatively large discrepancy between (S~)1/2 and ( ISj I) for Ni indicates that the approximation is poorer in Ni than in Fe. If we simply assume that (S$ = Mz for Ni, the value of Tc is reduced by half. In the case of nickel we expect the important part of the low energy excited states to have a long wave character and it would be more appropriate to make a long wave approximation than to make a local approximation, as in the present approach. In view of this fact and the rough way of considering the effect of band-degeneracy, etc. the above-estimated value for Tc is considered to be reasonable. A recent theory of Korenman and Prange [21] seems to belong essentially to a long wave approximation, though it is necessary for the present purpose to extend it to the case of variable amplitude of the local spin fluctuations. In fig. 16 we plot the temperature dependence of the inverse susceptibility with n = 1.66 and J = 1.6, bearing cobalt in mind. The Curie temperature is nearly the same as that of Fe, a roughly reasonable result, and the susceptibility shows an approximate Curie-Weiss behavior. It is interesting to note that according to the present calculation the ratio of the Curie constant to the one deduced from the saturation moment (assuming the existence of the local moment) is nearly equal to 1 for Fe and successively increases for Co and Ni. This is qualitatively consistent with observation [22]. 4.4. Other examples Quite a few materials show a maximum structure in the susceptibility as the temperature increases. This phenomenon can be realized in our calculation by adopting, for example, such a density of states as given in the top of fig. 17. It is essential that in the ground state the Fermi energy lies in a region of relatively low state density which is, however, very close to the region of high state density. By increasing the amplitude of the random field the peak in the density of states becomes less significant and the height of the low state density close to the peak increases. Thus by increasing the temperature or the amplitude of the random field, j& first increases. This is due to the negative coupling between the spin fluctuation modes. It then decreases as usual since the mode coupling becomes positive when the amplitude of the spin fluctuation becomes large. So there exists a minimum in the inverse susceptibility curve plotted against the temperature, as shown in the figure.
K. Usamiand T. Mqriya /Spin fluctuation theory of ferromagnetic metals
I
0
I
I
0.01
1
0.02
I
I
0.03
189
I
T
Fig. 17. The inverse susceptibility against temperature with a minimum structure.
5. Comparison with the local saddle point approximation It is of some interest to point out the relation between the present theory and the local approximation recently presented by Hasegawa [7]. He made use of the following interaction Hamiltonian:
with J = U and calculated the magnetic moment at an atomic site as an impurity moment under the influence of a uniform molecular field taking account of the random exchange field in the surrounding medium. Use was made of a single site CPA and the saddle point approximation was employed to evaluate the statistical average value for the impurity moment. Then the problem was solved by equating the impurity moment with that in the medium. This result can be derived easily with the use of a variational procedure in evaluating L(B, Bo) in the present theory and by making an additional simplification of the local saddle point approximation. Let us consider a CPA calculation of L(B, Bo) which is somewhat different from the one in section 3. In place of considering the distribution in the direction of the local field &, we consider here only the up and down sites with the spin and charge fields Br, B2 and yr, 72, respectively. The numbers of sites are Nr and N2, respectively, and we define the parameter Nr-N2 5 =-=-~ Nr+Nz
Nr -N2 (57)
No
We have the following relations for the field variables: B. = (B]), = F’
Br + yB2,
B2 = (BfzJ - (
-
231B2
+Bi),
(58)
where (Al& denotes the average over sites defined by bli),
=
l-5 y41 t -y&42.
In the calculation of the partition function the division into two kinds of sites, and thus the reduction of the number
190
K. Usami and T. Moriya / Spin jluctuation theory of ferromagnetic metals
of variables for integration, yields the factor Ne!/Nr !Nz!. This leads to the entropy term of the random fields. Thus the expression for L(B, Be) is given by
1+5,1-5 +!_!...51n2
(
P2
2
1-l In 2
)’
where
(60) The CPA equation
reduces to
x:0 = ( Vjo)a - WI0 - 20) F~(vZ~
- x0)*
(61)
Provided we neglect the c-dependence in the term X, in eq. (6), we can determine function of B and Bo by the equations, aL/ayj = 0 and aL/a{ = 0. We get rj = - g
Jde./(e)
C Im 0 1 - (I$
FL7 - &,)F,
-
the values of yr, yz and 5 as a
(62)
Eq. (61) leads to eq. (25) in the case of vanishing B,, and f. The expressions for the uniform and the local susceptibilities are given in the appendix. The result of LSPA by Hasegawa can easily be derived if we put $ 1 = L (B, B,-,) and J = U, neglecting the nonlocal term in eq. (6), and determine the value of B from the following saddle point equation: =O.
(63)
The paramagnetic susceptibility is calculated by inserting this value of B into (A.2) while in the present calculation we take account of the nonlocal term through the caIculation of the local susceptibility TL and determine the value of B from eq. (A.4) taking account of the thermal excitations of nonlocal spin fluctuation modes. The main difference can easily be seen in the weakly ferromagnetic limit or for small B where the present approximation leads to a reasonable result while the LSPA does not. In other words, when the saddle point is at B = 0 the LSPA reduces to the Stoner theory while the original MT theory takes account of the important nonlocal thermal spin fluctuations with the mode-mode coupling effect even in such a case. This means that the LSPA neglects part of the thermal spin fluctuations and the mode-mode coupling effects which seem to be of general importance as will also be seen in the following numerical example. In order to compare the two approximations, we present here the results of numerical calculations for bee iron. In fig. 18 we plot the Curie temperature as a function of J. For comparison the Curie tdmperatures calculated by the HFA and by the LSPA are shown by the dashed and dot-dashed lines, respectively. Although the present approximation agrees with the HF one at T = 0 K, the LSPA does not. If we put J = U = 1.317, the Curie temperature by the present approximation is about 0.09 of that by the HFA and 0.57 of that by the LSPA. Since our calculation includes the effect of the nonlocal term X,, we have a smaller Tc than that by.the LSPA. The latter with J = CJand the band width of 0.5 Ry yields Tc = 800 K [7] without considering the degeneracy of the band. If we take account of the five-fold band-degeneracy and also correct the overestimation of the quantum effect due to the Ising-type model by multiplying by a factor of $, we get Tc = 1500 K for the present calculation and 2700 K for Hasegawa’s LSPA calculation. The numerical values should not be taken so seriously in this Ising-type model. Fig. 19 shows the temperature dependence of the susceptibility. The dot .dashed line shows the result by the LSPA;cForcomparison the slope of the local moment case corresponding to theCurie constant C’LM=Mz is
191
K. Usami and T. iUon)a / Spin fluctuation theory of ferromagnetic metals
Tc
~sp*(l_k1.317) Tc I
n =1.45 I ! I
0.03-
0.02-
i
i
! I I ! 1 i I i
TCHF/10 / / ! ! / ! ,I' u=1.317 ! ! ! / I "Zca i I
I i f iij’4
/
P
n- 1.45
x-1
M~0.2
2.0-
IA
5
J= 1.317
1.6 1.2 0.6 0.4 -
o-
0
0.02
0.04
0.06
0.08
T
0.1
Fig. 18. The Curie temperature for the bee iron as a function of J as calculated’with the use of the interaction Hamiltonian of eq. (56). The dashed and dot-dashed lines show those calculated by the HF theory and the local saddle point approximation (LSPA), respectively. Fig. 19. The temperature dependence of the inverse susceptibility for iron calculated by the present theory (solid line) with the use of the interaction Hamiltonian of eq. (56) and the LSPA (dot-dashed line).
shown by the dashed line. The susceptibility shows an approximate Curie-Weiss behavior. The effect of the charge field is small, as in the previous case of fig. 9. The Curie constant is almost equal to CL~. On the other hand the linearity of the inverse susceptibility calculated by the LSPA is not good and its Curie constant is much larger than CL~. The 1/xLspA - T curve has a tendency towards saturation at high temperatures due to the above-mentioned neglect of part of the mode-mode coupling effect. These comparisons seem to suggest that the local saddle point approximation is not satisfactory even for iron and we have to take account of the nonlocal character of the spin fluctuations with the mode-mode coupling. As was pointed out by the MT theory, the concept of temperature variation of the amplitude of LSF controlled by the mode-mode coupling is an essentially important feature of itinerant electron magnetism. However, the MT theory also points out that the Curie temperature may be evaluated properly without considering this effect seriously, since Tc is independent of the longitudinal stiffness constant for the spin fluctuations. This of course does not apply to the magnetic susceptibility which is sensitive to the longitudinal stiffness constant. Thus, the calculation for the magnetic susceptibility provides a more crucial test for the approximation than that for the Curie temperature. In other words an explanation of the values of T, only would not be enough to justify the approximation. We mention here two recent papers that deal only with the Curie temperature. Hubbard [23] has presented a calculation for the free energy functional for a special configuration where the local spin direction is rotated by an arbitrary angle Q from the direction of the ferromagnetic moment in the ground state. Comparing the curve for the saddle point energy AE vs. r$to the corresponding plot for the Heisenberg model, he estimated Tc = 1200 K. Capellmann [24] has estimated the Curie temperature for Fe and Ni with the use of an effective Heisenberg model with the coupling constant deduced from the band structure. This theory makes use of a long wave approximation with the emphasis on the short range order assuming a fixed amplitude for the local spin fluctuation. Although his use of a molecular field approximation in evaluating TC seems to be inconsistent with his assumption
192
K. Usami and T. MorQa / Spin fluctuation theory of ferromagnetic
metals
of the strong short range order in deriving the effective Heisenberg exchange, reasonable values for TC have been reported for Fe and Ni.
6. Conclusions
and discussions
Along the lines of the unified spin fluctuation theory of itinerant electron magnetism developed by Moriya and Takahashi, we have developed a simple approach which makes use of only the density of states with the use of the single-site coherent potential approximation. It is demonstrated that when the occupied (or empty) fraction of the band is small there is a tendency toward ferromagnetism while the nearly half-filled bands tend to be antiferromag netic in agreement with previous results and also observation. In the weakly ferromagnetic case the Curie constant is much larger than is expected from the saturation moment through the local moment mechanism, while in the strongly ferromagnetic case with the nearly quarter-filled band the Curie constant is close to the local moment case with the ferromagnetic saturation moment. The effect of charge density fluctuations was also treated within the saddle point approximation and the linearity of the inverse susceptibility vs. T was shown to improve generally when the effective Coulomb energy increases. The calculation has been extended to simple models for bee and fee d-metals and the results for Fe, Co and Ni are generally reasonable. We have also demonstrated that a maximum structure in the susceptibility vs. temperature curve can be realized under certain conditions for the density of states. We have discussed the relation between the present theory and that of Hasegawa which makes use of the local saddle point approximation and the molecular field approximation. In the latter theory the thermal excitation of the spin fluctuations of nonlocal character is totally neglected and the interpolating nature of the MT theory is lost; the results do not seem to be satisfactory even for iron. The present theory includes this effect through the calculation of both the uniform and the local susceptibilities, Fe and &, and the results seem to be improved significantly. Our local approach is best applicable to the case where the characteristic wavelength of the system is short or, in other words, the energy difference between the ferro- and antiferromagnetic states is small. When the short range order is strong, as is expected to be the case for example in Ni, the best approach is from the opposite side, i.e. the long wave approximation may be useful. Nevertheless, it is interesting to find that the present theory gives a fairly reasonable result even in this case. We should finally mention that in order to improve the present results, especially at low temperatures, we have to take account of the dynamic effect of spin fluctuations. This should be the subject of future investigation.
The authors wish to thank Dr. H. Hasegawa and Dr. Y. Takahashi for a number of stimulating
discussions.
Appendix Here we show the expressions for the uniform and the local susceptibilities within the approximation discussed in section 5. The uniform susceptibility is calculated by the same expression as eqs. (19) or (27). With the use of the relation
(A.11
K. Usami and T. Moriya /Spin fluctuation theory of ferromagnetic metals
193
we get TO
= x01 +TToz,
64.2)
with - ZpHp/Fp ,
K, = 1 t 2ZpFp IV1 = 1 t gldef(E)
Im
7l
(1 + 2ZpFp)Hp C
W2 =g
JdefcE)
+
KP
F; -HP KpU
+ EpFp)
1’
Im ~~~+~~~~~)].
The local susceptibility is given by eq. (3 1). As in the case of the uniform susceptibility, we take account of the change in distribution of the parallel and antiparallel local &fields due to the external local field. We get: ZL
=rsLl
+xL2,
(A.31
XL2
=
B2 W23 T-B2W3’
with
The amplitude B2 = 4JT
of the random iield is given by 1+6
6 t (a) + (62 + 26(a))“2
*
References [l] T. Moriya and Y. Takahashi, J. Phys. Sot. Japan 45 (1978) 397. [2] T. Moriya and Y. Takahashi, J. de Phys. 39 (1978) C6-1466. [ 31 T. Moriya, Solid State Commun. 26 (1978) 483. [4] Y. Takahashi and T. Moriya, J. Phys. Sot. Japan 46 (1979) 1451. [S] T. Moriya, J. Magn. Magn. Mat. 14 (1979) 1. [6] T. Moriya and H. Hasegawa, J. Phys. Sot. Japan, to be published. [7] H. Hasegawa, J. Phys. Sot. Japan 46 (1979) 1504. [8] M. Cyrot, J. de Phys. 33 (1972) 125;Phil. Mag. 25 (1972) 1031. [9] L. Dworin and A. Narath, Phys. Rev. Lett. 25 (1970) 1287.
(A.41
194
K. Usamiand T. Moriya / Spin fluctuation theory of ferromagnetic metals
[lo] W.E. Evenson, J.R. Schrieffer and S.Q. Wang, J. Appl. Phys. 41 (1970) 1199.
[ 1 l] B. Velicky, S. Kirkpatrick and H. Ehrenreich, Phys. Rev. 175 (1968) 747. [ 121 F. Ducastelle, J. Phys. C8 (1975) 3297. [13] T. Moriya, Progr. Theor. Phys. 33 (1965) 157; M. Inoue and T. Moriya, Progr. Theor. Phys. 38 (1967) 41.
[ 141 T. Moriya and K. Usami, Solid State Commun. 23 (1977) 935. [ 151 K. Levin and K.H. Bennemann, Phys. Rev. B5 (1972) 3770. [ 161 S. Wakoh and J. Yamashita, J. Phys. Sot. Japan 21 (1966) 1712. [ 171 R.A. Tawil and J. Callaway, Phys. Rev. B7 (1973) 4242. [ 181 J. Crangle ana GM. Goodman, Proc. Roy. Sot. (London) A321 (1971) 477. [ 191 J.W.D. Connolly, Phys. Rev. 159 (1967) 415. [20] [21] [22] (231 [24]
S. Asano and J. Yamashita, Progr. Theor. Phys. 49 (1973) 373. V. Korenman and R.E. Prange, Phys. Rev. B19 (1979) 4691,4698. P. Rhodes and E.P. Wohlfarth, Proc. Roy. Sot. A273 (1963) 247. J. Hubbard, Phys. Rev. B 19 (1979) 2626. H. Capellmann, Z. Phys. B34 (1979) 29.