Temperature (K)
2. Theoretical approaches to itinerant electron metamagnetism (IEM) 1. Itinerant electron system
2.1.1. Magnetically uniform state
2.1.1.1. The Stoner model. The itinerant electron model of magnetism in metals is based on the simple assumption that the magnetic electrons collectively obey the Fermi-Dirac statistics. In addition, the essential electron-electron interactions are included by the use of the molecular field hypothesis. With these assumptions, it is possible to write down the total energy per atom of an itinerant ferromagnet as the sum of two terms. The first term is the total molecular field energy (Fm) which may be expressed as
F m = -½)~M 2, (1)
where M is the magnetization and )~ is the coefficient of the molecular field related to the effective exchange energy between the electrons, I, given by I = 2/~,~.
Introducing the standard notation N(e), n± and e ± for the density of the states at a given energy e, the number of + and - spins per atom and the chemical potential in the subbands, respectively, the magnetization can be written as
M = ~tB(n + -- n_), (2)
where
/[±
n± = N ( e ) de. (3)
Letting @ be the Fermi energy for an itinerant paramagnet, then the total one-electron (kinetic) energy per atom at 0 K is given by
f e = eN(e) de - eN(e) de. (4)
182 N.H. DUC and T. GOTO
The total energy is given by the sum o f Fm and Fe which, when minimized, gives I M
e + - e- - - Ae, (5)
where Ae is the so-called exchange splitting. The second derivative 02F/OM 2 < 0 leads to the following condition for the appearance o f ferromagnetism (Stoner criterion):
7 = IN(e~) > 1. (6)
At 0 K, the corresponding equilibrium value o f the relative magnetization/z (=M/nl~B) is given by
(i) # = 0 if [N(eF) < 1,
(ii) 0 < # < 1 if IN(eF) >1 but N(eF) is not very small, (iii) /~ = 1 if IN(eF) > 1 and N(ev) is very small.
Correspondingly, there exist (i) paramagnetic, (ii) weakly itinerant ferromagnetic, and (iii) strongly itinerant ferromagnetic materials.
2.1.1.2. Wohlfarth-Rhodes-Shimizu (WRS) model for IEM. Using a Landau-type expression o f the flee energy o f the d-electrons, Wohlfarth and Rhodes (1962) have pointed out that IEM occurs in a paramagnet if there exists a m a x i m u m in the temperature dependence o f its magnetic susceptibility. Shimizu (1982), however, has obtained a more detailed condition for the appearance o f IEM. We refer to their works together as the Wohlfarth-Rhodes-Shimizu (WRS) theory and briefly review it as follows.
The behaviour o f an itinerant paramagnet in an applied magnetic field depends on the form o f N(e) near the Fermi level. For a paramagnetic system, where the Stoner condition is close to being satisfied, magnetic ordering can appear if it is possible to have a situation that an applied magnetic field increases the density o f states at the Fermi level. Figure 1 illustrates schematically sections o f N(E) near e = eF for some characteristic cases where N(eF) decreases (case a) or increases (cases b - d ) when spin subbands are split in the magnetic field (B). As one can see from this figure, the condition
[N(ev) = N+(eF) + N (eF)]B- 0 < [N(ev) = N+(ev) + N-(6F)]B > 0 (7) holds when the function N(e) has a positive curvature near ev, i.e., N ' ( e F ) > 0. As we will discuss below, this is also the condition o f the MT for an itinerant electron system.
In the case when the resulting splitting o f the subbands is much smaller than their bandwidth, the band-splitting Ae, and therefore the magnetization, are small parameters.
The free energy F(M) can be expanded in powers o f M as
F(M) = ~(all _ )~)M 2 + la3M4 + lasM6 - B M , (8) where the last term is the magnetostatic energy in the presence o f the external magnetic field. The Landau expansion coefficients al, a 3 and a5 are determined by the behaviour
IEM OF Co SUBLATTICE IN R-Co INTERMETALLICS 183
(a)
(b)
(c)
(d)
N(~) 4 N ( s ) ,
N+(~) I _ _ _ ~ ~ N+(u)
i <i,, --
N_(e) t . . . ~ N_(c)
N ( s ) ~ N ( c )
___~. N+(0
N+(~) _ . _
, - -
-
~ N_(a)
N(~),
N+(c) - - ~ N_(c) -'-'i-,~ ' ~
/
N ( e ) ,
1 N+(0
. - -' I
N_(c)
N(0,
N+(c)
N-(tO N ( ~ ) , N+(~) !
N_(~)
. . . !
!
!
B=O B>O
Fig. 1. Schematic diagrams of the dependence of the density of states on the energy N(~) near ~ = e F for cases of (a) negative curvature and (b-d) positive curvature of N(e) for (left) B = 0 and (right) B > 0. (After Levitin
and Markosyan 1988.)
o f the N ( e ) c u r v e n e a r e = eF and, indeed, c a n b e r e p r e s e n t e d as series in even powers o f the temperature (Wohlfarth a n d R h o d e s 1962, Edwards a n d W o h l f a r t h 1968, de Chfitel a n d de B o e r 1970 a n d B l o c h et al. 1975):
1
al (T) - 2tJgN(ev ) + aT2
+/~T4)'
(9.a)a/l ( T ) = al ( T ) - ~ - ~ ~lvL ) ~/~9 ~ ~'~ F" (1 - 7 + a T 2
1 +/iT4),
(9.b)N.H. DUC and T. GOTO 1
a3(T) - 2#2N(eF) (y + 6T2), (10)
1
as(T) - 2#2N(eF) t/, (11)
where a,/3, y, 6 and ~/depend only on the second and fourth derivatives of N(e) and are given by
~2k~
a = - T v 2 , (12)
3-g4 ]C 4
/3 = ~ ( 1 0 v 2 - 7v4), (13)
1
Y - 24g2N(eF) a g2, (14)
srek2 (4v 2 -- V4), (15)
6 - 144#2N(ev) 2 1
tl - 1920#4 N(eF)4 (lOv2 -- V4), (16)
where
N(n)(eF)
V, - N(eF) (17)
As X(T) = 1/a~(T), the temperature dependence of the paramagnetic susceptibility is given as (see eq. 9.b):
z(T) = 2~t~N(eF)
1 - 7 + a T 2 + f i t 4" (18)
At 0 K,
2#~N(ev) (19a)
Z(0)= 1 - I N ( e v )
is the Pauli susceptibility0(o =2g~N(eF)),which is enhanced by the Stoner enhancement factor S = [ 1 - I N ( e F ) ] 1.
Z(0) =SZo. (19b)
At low temperatures, z(T) can be written as
z ( T ) = Sx0[1 - aST2]. (19c)
1EM OF Co SUBLATTICE IN R-Co INTERMETALLICS 185
F (b)
T Mo
Fig. 2. Magnetic part of the free energy as a function of magnetization: the curves corresponding to (a) alas/a ~ < ~6 and (b) 3 <a~aja23 < ¼. (After Yamada 1993.)
The equation of state can be determined from the condition for the minimum of the free energy (8):
B = a'I(T)M + a3(T)M 3 + as(T)M 5. (20)
According to eq. (20), the field dependences of the magnetization and susceptibility of an itinerant paramagnet can be represented, respectively, as
1 B _ a3(T) B3 + - - .
M(T,B) = a~(r) @ (21)
= X ( T , O)B - a 3 ( T ) x ( T , 0)4B 3 + . . . ,
x(T, B) = X(T, 0)[1 - a3(T)x(T, 0)3B 2] + . . . . (22)
The character of these field dependences is determined by the sign of the a3(T) coefficient.
One can see from here that the susceptibility of the itinerant paramagnet can increase as the field intensifies. In this case a3(T) must be negative. This corresponds to the condition of v2 > 0 (see eqs. 10 and 14), i.e., corresponds to the case of the positive curvature of the DOS near eF as already mentioned at the beginning of this section. In this case, as can be seen from eqs. (12), (13) and (18), z(T) will also show a maximum. Thus, the experimental observation of a susceptibility maximum was usually regarded as an indication of the possibility of the MT in the itinerant paramagnet (Wohlfarth and Rhodes 1962). However, as discussed below, the appearance of the MT still depends strictly on the values of the coefficients a~, a3 and a5 (Shimizu 1982).
' >0, a3 <0, a5 > 0 and a~as/a~ < ¼, it can be seen from eq. (8) that In the case of a 1
the free energy F(M) has two minima at M = 0 and at a finite value of M (= Mo), and a maximum between the two (see fig. 2). When a~as/a 2 < 3 , F(Mo) is negative as shown by curve (a). Then, the ferromagnetic state at M =M0 is most stable without the external field. When ~4 >a~as/a~ > -~, F(Mo) is positive as shown by curve (b) and the state at M=Mo is metastable. However, this metastable state can be stabilized by the external magnetic field and the MT from the paramagnetic to the ferromagnetic state occurs at a critical field Bo. To see this fact more clearly we discuss the magnetization curve M(B) as follows.
186 N.H. DUC and T. GOTO M
1 B
Fig. 3. Schematic curves of M(B) for the (1) ferromagnetic, (2) metamagnetic and (3) paramagnetic states.
M
, f
! -%
b°%
J
It
Bc B
Fig. 4. Schematic magnetization curve for a weakly ferromag- netic (WF) system undergoing a MT into the strongly ferromag- netic (SF) state in an external magnetic field.
The magnetization curves obtained from eq. (21) are presented in fig. 3. In the case o f atl > 0, a 3 < 0, a5 > 0 and dlae/a 2 > 9 , M(B) increases monotonically with increasing B (curve 3). On the other hand, when 9 > a~as/a 2 > 3 , M(H) is S-shaped (curve 2). That is a metamagnetic first-order transition from the paramagnetic into the ferromagnetic state with hysteresis in the magnetization curve. Then the complete condition for the appearance o f the MT is given as
, 9 a~a5 3 (23)
a l > 0 , a3 < 0 , a5 > 0 , ~ > a - T > 1-~.
When a]as/a~ < 3 , the system becomes ferromagnetic even at B = 0 , as shown by curve (1) in fig. 3.
The MT discussed above has been described for an itinerant electron paramagnet with a~ > 0. It should be noted that in the model under consideration, the FOT induced by an applied magnetic field can also arise when the starting state is magnetically ordered, i.e., in the case o f a~l < 0. This is the field-induced transition from a weakly ferromagnetic (WF) into strongly ferromagnetic (SF) state. Shimizu (1982) has described such a transition by including the term o f 1 ga7M 8 in the expression o f the free energy (8). Then, if a3 > 0, a5 < 0 and a7 > 0 for a definite ratio o f the magnitude o f these coefficients, this transition will be a FOT. The behaviour o f the magnetization o f such a weakly ferromagnetic system is shown systematically in fig. 4. This is the case for the Y(Co,A1)2 and Lu(Co,A1)2 compounds which will be discussed in sect. 3.1.1.2.
IEM OF Co SUBLATTICE 1N R~Co INTERMETALLICS 187 2.1.1.3. A new approach to itinerant electron metamagnetism (Duc et al. 1992b). In the well-known approach for IEM, volume effects are not considered. However, large volume increases ( A V / V ~ 5%) are systematically observed to occur at the transition towards ferromagnetism (Givord and Shah 1972, Wada et al. 1988), indicating that the contribution from the elastic-energy term may be important. This has been confimled by band structure calculations on YCo2 (Schwarz and Mohn 1984), showing that YCo2 is a Pauli paramagnet at the equilibrium volume but at a slightly larger volume, ferromagnetism is stable. Subsequently, Hathaway and Cullen (1991) have incorporated the effect of volume variations to discuss the magnetic properties of the RCo2 compounds. A new approach to IEM in which the mechanism for the FOT is connected with the volume discontinuity at this transition has been proposed by Duc et al. (1992b). In this approach, the MT results simply from the interplay between the magnetic energy and the elastic energy.
2.1.1.3.1. Description o f the model. A simple band structure cannot describe the intermetallic compounds, but it has the advantage of giving IEM without considering the structure of the DOS. Here, an elliptic density of states whose curvature is always negative is chosen:
10 v / ~ 5 _ e2 ' N ( e ) = ~ 5
where 2 W is the bandwidth.
In the Stoner model at 0 K the total energy in the presence of a magnetic field is written as a summation of the electronic (Fd) and elastic (Flat) parts. The electronic energy can be given as:
Eel = eN(e) de + eN(e) de - - - - M B . (24)
w ~ 4
The total number of electrons, n, and the magnetization, M, are obtained by solving two self-consistent equations.
The lattice energy can be written as (Ohkawa 1989, Duc et al. 1992b)
Fla t = C~'~ ÷ K ~ 2, (25)
where g2 = ( V - Vo)/Vo, and C and K are the expansion coefficients. K is the contribution to the elastic constant due to all terms other than the band contribution. Here, the reference volume V0 was chosen in order to get a minimum of the total energy at this volume in the paramagnetic state. C is determined by minimizing eq. (25) at £2 = 0, which gives
OFel 092 ~=o + C = 0. (26)
The elastic constant K of the system in the paramagnetic state is then given by
- - 1 0 2 F e l D = 0
K = K + ~ - ~ , (27)
which is smaller than K.
I f the d-electrons are treated in the tight-binding approximation, the volume dependence of the bandwidth is given by W = W0 exp(-qf2), where q is of the order of 1 to 5_ 3 for 3d electrons (Slater and Koster 1954). In this approximation, the energy of the paramagnetic state also varies exponentially with volume. From eq. (26), the value of C is deduced as
C : qFe~(f2 : O , M : 0).
In the following sections, different aspects of IEM are discussed.
>
- 2 . 0 0 3 0 . 5
- 2 . 0 0 4
- 2 . 0 0 5
- 2 . 0 0 6
-2.007 -0.05
I
n = 9.55 K = 2.6
I I I
M ~ 0.4
0.2
~m -
0.1 l
i~,/nc
I I
0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2
f2
0.3
Fig. 5. Variation of the total energy and magnetic moment as a function of volume for the electrons in the elliptic band with n = 9.55, K = 2.6 eV, I = 2.5 eV and W 0 = 5 eV (see text). (After Duc et al. 1992b.)
2.1.1.3.2. M a g n e t i c state at B = 0. For B = 0 , as mentioned above, a magnetic state is obtained if the inverse of the Stoner parameter, S -1 = 1 - I N ( e ) , is negative at f 2 = 0 . With regard to the IEM, we restrict our consideration to the case Sq(g2 = 0 ) > 0. By increasing f2, the bandwidth decreases and s-l(g2) may become negative at g2 > g2o. For small values of the Stoner parameter, f2o is given by
1
g2c = - - l l n ( 1 - q S(f21= 0 ) ) ~ S(I2 = 0) q"
I f K is not too large, the competition between the elastic energy which increases with volume and the magnetic energy which decreases when the volume increases will give the total energy F t o t ( g 2 , M) a second minimum at f2 = f2M (>g2c), with M ~ 0 as shown in fig. 5. However, in the present case the magnetic state is not a stable one, since
f t o t ( $ ' 2 0 ) < f t o t ( ~ 2 M ) : the stable state is paramagnetic but a MT can be induced by an applied magnetic field or a negative pressure.
2.1.1.3.3. The induced-field m e t a m a g n e t i c transition. The total energy in different magnetic fields is presented in fig. 6a for the electron system described above (see also
IEM OF Co SUBLATTICE IN R~2o INTERMETALLICS 189
Fig. 6.
- 2 . 0 0 2 - 2 . 0 0 4
~" - 2 . 0 0 6 - 2 . 0 0 8 - 2 . 0 1 0 - 2 . 0 1 2 - 2 . 0 1 4 - 2 . 0 1 6
0 . 5
0 . 4
0 . 3
0 . 2
0 . 1
0
I l l I
(a) / B=O T
55 T
~
I 0 0 T0 0 T
00 T
0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5
( b ) r
I n = 9.55
I I I I
loo 200 300 400
B (Y)
i , I l
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0
0 . 2 0
0 . 1 6
0 . 1 2
f2
0.08 0.04 0
B ( T )
(a) Total energy for the same parameters as in fig. 5 in the presence of a magnetic field.
(b) The corresponding magnetization and magnetostriction curves. (After Duc et al. 1992b.)
fig. 5). As the metastable state for £2 = £2M is magnetic, the decrease in energy due to magnetic fields is larger for this state than the paramagnetic state at g2 = 0. For this case, i.e., when Ftot(£20, B ) = Ftot(£2M, B), a metamagnetic transition occurs at a field o f Bc = 55 T, and a large volume discontinuity (£2 = 0.15) is found (see fig. 6b). Such an order o f magnitude o f the critical field o f metamagnetism was found for the YCo2 and LuCo2 compounds.
2.1.1.3.4. Metamagnetic transition induced by pressure. The effect o f pressure on IEM can be discussed by introducing a contribution p£2 to the total energy. Starting from the non-magnetic state, an 1EM is obtained for negative values o f p . For IPl <Pc, the volume increases as £2 = - p / K . But for