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Metallic magnetism and magnetic volume collapse
Journal of Magnetism and Magnetic Materials 87 (1990) 163-176 North-Holland
163
METALLIC MAGNETISM AND MAGNETIC VOLUME COLLAPSE M. SCHROTER, P. ENTEL and S.G. MISHRA t Theoretische Physik, Universitiit Duisburg and SFB 166, Postfach 10 15 03, D-4100 Duisburg 1, FRG Received 4 0 c t o b e i !989; in revised form 7 November 1989
We present a new description of possible magnetovolume instabilities in itinerant electron systems based on a generalized Cfinzburg-Landau expression. The Landau coefficients are input data and are obtained by reproducing binding surfaces (total energies as function of volume and the magnetic moment), which are the result of spin-polarized band structure calculations performed with the fixed-spin-moment(FSM) method. Finite temperature results are obtained by constructing a classical field theory. It is shown that essentially two Landau parameters (the fourth-order coefficient and the critical volume) may serve to classify itinerant magnetic systems as stable or potentially instable systems. In the first category the magnetic phase transition is of second order while, in the latter, the transition can be of first order, associated with a large magnetic volume collapse. We show that all INVAR systems are potentially instable.
1. Introduction
In metallic systems there is general interrelation between loss of magnetism and a decrease in the atomic volume. This tendency is very pronounced in transition-metal compounds and is responsible for magnetoelastic anomalies like the INVAR effect. For a recent review of the physical properties of INVAR systems and further references see ref. [1]. Of particular interest are hypothetical elemental systems such as bcc and fee manganese and fee iron for which the computed magnetic moment shows a discontinuous zero-moment to high-moment transition at a critical lattice spacing [2,3]. The close relation between magnetic and lattice properties in a metal was subject of intensive research during the last decade. For a recent theoretical review with special emphasis on the importance of the electron-phonon interaction in itinerant magnetic systems see ref. [4]. In this paper a new theory is presented which can account for many particularities of stable and unstable metallic systems. When trying to describe magnetism and volume changes simultaneously, one meets two obstacles. I Perraanent address: Institute of Physics, Bhnbaneshwar 752005, India.
The first is connected with the fact that neither the Heisenberg model of localized spins nor the Stoner theory of band magnetism is able to describe adequately the magnetic properties of transition metals. This is only possible in the frame of a unified theory of magnetism [5,6] (spin fluctuation theory) which in a quantitative respect is still in leading strings. On the other hand the coupling of spin to lattice degrees of freedom is a principle difficulty, since both degrees of freedom can be of equal relevance. In view ,~f these difficulties we use in this work a phenomenological model Harrd~tonian in form of a generalized Landau expansion which correctly reproduces total energy curves of transitionmetal compounds at zero temperature and which, in the frame of a classical field theory, leads to a general phase diagram for metallic systems. The drawbacks of the pheno.~enological theory at finite temperatures will be critically discussed. As stated above, the close connection of volume and magnetization changes is at the origin of low-spin to high-spin transitions, which can be viewed as a magnetovolume instability. We show that this instability can be discussed by starting from a Landau theory for metallic magnetism. The microscopic origin of such instabilities and
0304-8853/90/$03.50 © 1990 - El~vier Science Publishers B.V. (North-Holland)
M. SchrOter et al. / Magnetism and magnetic volume collapse
164
their relation to the INVAR effect is briefly discussed. Preliminary results were reported in ref. [71.
2. Ginzburg-Landau theory of metallic magnetism
Here, we briefly indicate how the generalized Landau expansion for transition metals can be derived from a microscopic quantum model. For simplicity, we first neglect lattice degrees of freedom. Then, it can be shown that the multiband hubbard model is suited for an evaluation of quasi-particle bands. An appropriate parametrization of the Hamiltonian in the Hartree-Fock approximation leads to renormalized on-site potentials for s, p and d electrons whose explicit form is given in ref. [8]. The solution of the corresponding self-consistency equations leads to realistic ~i:.~polarized band structures for transition metals, where transfer integrals are fixed at high-symmetry points and Uaa is chosen to give the correct amount of polarization. Shimizu showed that, on the basis of such a one-electron theory, the magnetic free energy can be written as [91
~[m I = ~
din' A(m', T)
-
½am2,
(1)
where za is the exchange splitting of the polarized bands and ct the coefficient of the molecular field. The exchange splitting can be expanded in a power g~ries of m by inverting a low-temperature expansion obtained from the relation for the number of spin-up and -down electrons and their corresponding chemical potentials (restricted microcanonical ensemble), i.e. +oo
.,,=f_
oo
d, o ( , ) / ( , - , , ,),
(2)
to Zi/gp. B = aim + a3 m3 + asm 5 + ... +O([kBT/EF]2), where the coefficients a~
leading
are given in terms of band density of states and its derivatives at the Fermi energy. It is generally bOieved that one can write down this expansion o~!.y for weak itinerant ferromagnets with small exchange splitting. However, in metallic ferromagnets we find that the expansion
is valid for any m, since the expansion parameter is, in principle, the ratio of magnetization density to the electron number density, m = m/(nPB), a quantity always less than unity since each electron can contribute, at most, one /xB to the itinerant magnetic moment. To give an example, we consider the free elect~on density of states and find, for the maglletic free energy, that =
-
2 +
+
nEF + ,#83,~ 8 + - . ' ,
(3)
a fast converging series. The only serious limitation is that this series expansion for metallic systems is, in principle, a low-temperature expansion and any use of it in the whole temperature range must be justified in the frame of a spin fluctuation theory. In the present work, this will be done by constructing a classical field theory which allows the evaluation of renormalized Landau coefficients. With respect to the spin-wave spectrum, only an expansion around k = 0 will be taken into account. It should be understood that m is, in principle, a vector and leads at finite temperatures to longitudinal and transverse fluctuations. Also a decomposition of m according to different electron orbitals has been left out in eqs. (1)-(3) for brevity. The coupling to the lattice is straightforward. In the original multiband Hubbard model the corresponding transfer integrals are taken as volume-dependent quantities. This is a well-defined procedure [10] and leads to volume-dependent Landau coefficients. We will neglect the dynamics of resulting electron-phonon interactions and only take into account the most essential static volume-dependencies. This procedure leads to the following phenomenological model Hamiltonian, where gradient terms have been added,
°~[ m, rO] = V o ~ fvo(r) d3r { g~( Vc° )2 + ½xr°2 + y~3 + 8~o4 + . . .
+ g , , E ( V , m # ) 2 + a ( % -- ~o)m 2 a#
+a'a~Zm2 + bm4 + cm6 + ... ~. ]
(4)
M. Schrrter et a L / Magnetism and magnetic volume colh~pse
The thermodynamic variables are magnetization and volume which are split into static and fluctuating parts
60(r, T ) = V ° ( T ) - ~2° V'(r, T) ~2o + j2o = 60o(T) + 60'(r, T),
( d ( r , r)> = o, re(r, T ) = t o o ( T ) + m ' ( r , ( m ' ( r , T ) ) = 0.
(5) T), (6)
V0(T ) is the actual volume while ~2o is an arbitrary reference volume, which in the following will be taken as the (hypothetical) volume of the nonmagnetic state at zero temperature in ease of a ferromagnetic system. 60¢= (f~¢ - I~0)/~2 0 is a critical volume allowing for a magnetic volume collapse. Spin and volume fluctuations will be described by (m'Z(r, T))4= 0 and (60'2(r, T ) ) * 0. At zero temperature the magnetic moment per atomic volume ~ a is m 0 with I~a = 4~rr3s/3 ( = a3o/2 for bcc and = a3o/4 for fcc, respectively). K denotes the bulk modulus. We would like to emphasize that a microscopic description of metallic magnetism in the frame of the extended multiband Hubbard model, which includes electronic s-, p- and d-electron and lattice degrees of freedom, contains too many degrees of freedom to be really tractable within the path-integral formulation. The phenomenological theory has serious drawbacks but it contains a small number of parameters and allows one in a transparent way, to describe the mutual influence of spin and volume fluctuations. Total energy calculations with the help of the fixed spin-moment method [11-13] have shown that the volume dependence of a series of metallic systems must be described by two seperate branches. A nonmagnetic branch with a minimum at low volume lies energetically close to a ferro- or antiferro-magnetic branch with minimum energy at a larger volume. With increase in temperature, pressure or by alloying, both branches can intersect at a critical volume, where the magnetic moment becomes unstable and the system can, without energy change, either occupy the high-spin or the low-spin state. It seems that all unusual properties of
165
metallic systems with a tendency towards a volume instability can be discussed on the basis of this picture. The present phenomenological theory exactly reproduces this kind of behaviour and allows a systematic classification of metallic systems. A necessary condition of itinerant ferromagnetism is that, at zero magnetic field and in equilibrium, the magnetic part of eq. (4) has negative energy. At zero temperature the equilibrium conditions
O~'~'~'/O~ = 60{ r + 3y60 + 43602 + 2a'm 2 } - am 2,
0=
(7)
0 = O,~/Om
= 2m
(a(60 c- 6o) + a'~o2 + 2bin 2 + 3cm 4}, (8)
lead for ~ = 0 to
60°=
K+2a' ~ / ( x + 2 a ' ) 2 am 2 6~, 436~, 2 + 3~-
amg
+ ~-a--r,
b m~ = - ~-~ 4-
(9) b2 ~cc2
• 2
a(60c - 60°) + a 60° (10) 3c '
which shows that 60¢ plays a fundamental role. If 60¢ is too large, only mo = 0 is possible while for 60c << 6°0, a stable magnetic solution is expected. Now, results for total energy curves fix the Landau coefficient and always yield positive values for a('k Furthermore, since the coefficient a ' of the magnetoelastic coupling term is much smaller than a, the critical volume 60¢ is given by H = 8.~/Om = 0 for m-~ 0, i.e. 60~ is a well-defined quantity for each metallic system and corresponds to the perpendicular intersection of the H = 0 line and m = 0 axis in the volume-versus-moment plane. Hence, at 60cthe system can spontaneously polarize without any change in volume and energy. The behaviour of the total energy as function of the various parameters can be discussed in detail if eqs. (9, 10) are inserted into eq. (4). From this discussion we conclude that for (60c - 600) < 0 there is a stable ferromagnetic solution with a secondorder phase transition, while for (~0~- O~o)> 0 and b < 0 a first-order phase transition together wlth'a
166
M. Schr&er etal. / Magnetism and magnetic volume collapse
magnetic volume collapse can be expected. In particular, we find that for t% = too the binding surface E ( m o, Vo) has always two close-in-energy lying local minima, one at m o = 0 and one at m 0 ~ 0, which define locally stable low- and highspin states. The condition that both minima have equal energy is E = 0 (and toe = too). For t% = too the critical volume is given by toe-- - 2 a b / ( 3 c x ) . Hence, for two close-in-energy lying low- and high-spin states to exist, the fourth-order coefficient has to bc negative. At zero temperature the coefficient b is given in terms of derivatives of the density of states at the Fermi level according to b tx 3(p~//p0) 2 - ffo'//po Thus, b will be negative if the Fermi level lies near a minimum in the density-of-states curve. At finite temperatures spin fluctuations will lead to large renormalization effects. Yet these zero temperature considerations remain qualitatively correct. In the present work the necessary condition of metallic magnetism is (toe - too) < 0 which s!i~bt!y modifies the usual Stoner condition for itinerant ferromagnetism. The later follows from eq. (1) as al = Xo ~(T < a, where the unenhanced susceptibility is obtained from H = O.~/~m = ( 1 / g l L B ) A a m = ( a 1 - a ) m + aam 3 + aSm 5 + . . . for a = 0 . We have assumed that the molecular field and first expansion coefficient have a linear and quadratic volume dependence which gives rise to the analytical structure of the m 2 coefficients. Moreover, any temperature dependence of the Stoner-like form ¢ ( [ k B T / E F ] 2) has been oiifitted. Figures 1 - 3 show the binding surfaces for bcc and fcc Fe and FeaPt which were obtained with the help of eq. (4), whereby the coefficients were fixed with the help of local spin-density calculations for total energies E ( m o , 11"o). We would like to emphasize that, with the help of the few parameters in eq. (4) (which are listed in table 1), it is possible to reproduce quite accu-ately the results of 500 band-structure calculations which are neeessary to construct a single binding surface. We now discuss very briefly the most important features of our zero-temperature results, since the three fundamentally different types of surfaces are well suited for a description of the different kinds of metallic magnetism. The binding surface of bcc Fe is typical for
z u.l o
V
°~1 zuJ
~'C
~
~
2
.........
~
:
0
~
~
2./-*5 2.55 2.65 WIGNER-SEITZ RADIUS [a.u.]
m
Fig. 1. Total energy curves E(mo, Vo) for bee Fe at i mRy intervals. Upper inset: volumedependenceof the energy. Lower in~et: temperature variation of spontaneous magnetization, mean square amplitudes of spin fluctuations and relative volume change.
E O "5
I--
z ILl O (.3 I-LU
z
o
2.5 2.6 2.7 WIGNER-SEITZ RADIUS [o.u.] Fig. 2. Total energycurves E(me,V0) for foe Fe at 1 mRy intervals. Note that the H = 0 and P = 0 line d o not intersect. Hence, there is no local minimum at finite polarization.
M. Sclwi~ter et aL / Magnetism and magnetic volume collapse
75
167
80VOLUME [a.u?] 85
90
Fig. 3. Total energy curves E(mo, Vo) for Fe3Pt at 0.5 mRy intervals. Black dots: Temperature evolution of the system from zero temperature up to the Curie temperature, left: without anharmonic con'Lributions, right: with ~ a r m o n i c contributions. Upper inset: Energy as function of volume. Lower inset: Temperature variation of spontaneous magnetization, mean square amplitudes of spin fluctuations and relative: volume change.
s t a b l e m e t a l l i c f e r r o m a g n e t i s m . It h a s a p r o n o u n c e d local m i n i m u m a t the e q u i l i b r i u m posit i o n ( h i g h - s p i n state) a n d a s a d d l e p o i n t o n the m = 0 axis ( l o w - s p i n state) w h i c h lies 23 m R y ( = 3630 K ) a b o v e t h e g r o u n d s t a t e energy. H e n c e , b c c F e is a stable f e r r o m a g n e t i c s y s t e m w i t h n o p o s s i b i l i t y f o r a h i g h - s p i n to l o w - s p i n t r a n s i t i o n a n d n o t e n d e n c y t o w a r d s a v o l u m e i n s t a b i l i t y in t h e to r a n g e c o n s i d e r e d . T h e c o e f f i c i e n t s b a n d c in the c a s e o f b c c F e o f t h e m 4 a n d m 6 t e r m s are b o t h p o s i t i v e w h i l e a ( t o c - t o ) is n e g a t i v e w h i c h will l e a d to a s e c o n d - o r d e r p h a s e transition.
T h e b i n d i n g s u r f a c e o f fcc F e is o f p a r t i c u l a r interest. First, t h e m i n i m u m e n e r g y c o r r e s p o n d i n g t o a l o c a l m i n i m u m is o n t h e m = 0 axis ( l o w - s p i n state). T h u s , fcc F e is p a r a m a g n e t i c at T = 0. H o w e v e r , t h e r e is a clear t e n d e n c y o f the s y s t e m t o d e v e l o p a f u r t h e r local m i n i m u m at h i g h e r energy with larger volume and moment (high-spin state). S e c o n d , for l a r g e r v o l u m e s t h e i n c r e a s e in magnetization without any apparent volume c h a n g e o f f e r s t h e p o s s i b i l i t y for m e t a m a g n e f i c beh a v i o u r : I n a small i n t e r v a l o f r w s , b o t h n o n m a g n e t i c a n d f e r r o m a g n e t i c p h a s e s are locally stable.
Table 1 List of Landau coefficients which were used for the calculation of total energy surfaces. 8 and a ' in eq. (4) are set equal to zero. (F) and (P) denote ferromagnetic and paramagnetic ground states, respectively. ~2o is given in units of au 3 and stands for the atomic volume in the (hypothetical) nonmagnetic state. The ground state volume V0(0) is also #oven in units of au 3 per atom. All o~er units follow from the prescription that energies are measured in mRy and magnetic moments in/L B per atom
bcc Fe fcc Fe bcc Co fcc Co Fe3Pt Fe3Ni
(F) (P) (F) (F) (F) (F)
•0
v
~0(0)
V~(0)
'~c
a
b
c
gm
qm
1214 1180 1401 1363 1465 1174
- 393.7 - 154.3 -
70.3 68.6 70.7 69.5 77.1 69.5
76.2 68.6 74.5 72.5 82.4 73.5
-0.365 0.280 - 0.256 0.048 0.057 -0.035
20.30 26.05 32.30 28.10 29.95 29.65
0.77 - 1.06 0.90 - 4.20 - 0.91 0.41
0.04 0.09 0.26 1.14 0.20 0.05
3.19 1.4 1.4
10 8 8
168
M. Schri~ter et aL / Magnetism and magnetic volume collapse
Hence, the idea of high-spin to low-spin transitions without large changes in energy seems not to be farfetched. Actually, there is experimental evidence for such transitions in artificially stabilized fcc Fe at larger lattice spacings in a copper matrix [14]. In contrast to bcc Fe the coefficient b in the case of fcc Fe is negative while a(toc - t o o ) is positive. By making the coefficient b more negative and thereby enhancing the tendency towards a first-order phase transition we can stabilize the local minimum corresponding to the high-spin state. The total energy change while changing b is negligibly small. The binding surface of Fe3P exhibits fea~,ures of bcc Fe (stable high-spin state) and of fcc Fe (locally stable low-spin state). While b is still negative, toc is very close to the equilibrium volume. Therefore, this system should be on the verge of a large volume collapse. This tendency has been confirmed by measuring the Curie temperature and the hyperfine field under pressure [15]. The total energy surface has two close-in-energy-lying minima with an energy barrier of ouly 1 mRy (--- 158 K), as can be seen ;,n the upper inset of fig. 3. Since T~=500 K, it is tempting to assume that the low-spin, low-volume, 'state is thermodynamically accessible before reaching the Curie temperature provided that the energy bar-
ofcc Fe (NM), 27 10
.A
p
:s
~
rier does not increase with T. Actually, there is experiraental evidence for high-spin, highvolume-low-spin, low-volume transitions in Fe3Pt at finite temperatures [16]. Fe3Pt and Fe3Ni have slightly different binding surfaces. Low- and high-spin states of FeaNi still have comparable energies but the low-spin state does not any longer correspond to a local minimum on the binding surface. Fe3Pt, Fe3Ni and fcc Fe can be considered as INVAR-like systems. Their thermodynamic behaviour is governed by the thermal evolution of high- and low-spin states. All systems are close to a magnetic volume collapse, as is also visible from the phase diagram in fig. 4. We would like to add that a series of intermetallic compounds undergo first-order phase transitions associated with a magnetic volume collapse (for example, YMn 2). But, at present, we are unable to describe such systems since we do not have enough input data to fix the Landau coefficients. Similar expansions as in this work have often been used, especially in connection with the INVAR effect [17]. For work which is closely related to our work, see refs. [18-20]. For an extension of the expansion to the antiferromagnetic case by including the staggered magnetization with wave vector Q, see ref. [21]. New in this work is, that we use an extended and more appropriate expansion and that zero temperature band structure calculations serve as input to fix unequivocally the Landau coefficients. At finite temperatures we essentially follow the mode-mode coupling formulation of Murata and Doniach [22] and construct so a classical field theory for metallic systems which tend to volume instabilities. Before presenting finite temperature resuJts we briefly discuss on the microscopic origin of low-spin to high-spin transitions and the related INVAR effect.
~fee co 3. The origin of low-spin to high-spin transitions
~
bccFe~
Fig. 4. Phase diagram for metallic systems. (P), (F0 and (F2) denote the paramagnetic, first and second-order phase transition regions, respectively.
Low-spin to high-spin transitions are known to occur in many d4-d 7 transition-metal complexes, for example, as a function of the ligand field strength [23]. In fig. 5 the dS-dectron configura-
M. Schri~ter et al. / Magnetism and m~Enatir ~oluma ~,qlao~e eg
eg +,+ I-A --~- --~ -f' ,2g
A 1 ~' I~ ~
t29
A < Ac
A > Ac
HIGH SPIN
LOW SPIN
Fig. 5. d s electron configuration in a weak (left) and strong (right)ligandfield.
tion (either Fe 3+ or Mn 2+) in a weak and strong ligand field of octahedral sylrLmetry is schematically shown. A weak ligand field acts as small perturbation and occupation essentially follows Hund's rule giving rise to a high-spin state. In a strong field the tendency to minimize the crystal field energy acts against intraatomic correlative interactions which, for large enough fields, leads to a low-spin state. For each configuration between d 4 and d 7, there is ~ critical value A for which the ground states of different multiplicity cross. Theoretically, a low-spin to high-spin transition could be achieved in one and the same cornplex by enlarging the distance between ligands and the central transition-metal ion. The low-spin to high-spin transition in transition metals and compounds resembles the corresponding transition in inorganic complexes. However, there are two distinct features which are different. In fig. 6 we have plotted the volume dependence of 1-'12 and F25, energy levels (which
0 m -0.1
t / F12
( " - ~ ._._...~-. ~....o_e-----"" . . . . . . . "~"" EF t~ - 0 . 2 ~ F2s. ~ ~1 a: ~ ~-1 ~"-0.3 "'0" . . . . . . . . . . . . + -o-~ I.kl ,o== ....... LOW SPIN ' HIGH SPIN -t] -OA ~ ~ 95 100 4.05 VOLUME [a.u.3] --
Fig. 6. Low- and high-spin region of face-centered manganese. Open circles are band structure data taken from ref. [2]. The
lines are guides to the eyes.
169
correspond to the two-fold and three-fold degencrate eg and t~g levels in inorganic complexes). The points in fig. 6 are taken from band-structure calculations for fcc manganese [2] while the lines noted.are guides to the eyes. Two points have to be First, the energy splitting between Fl2 and F25, is not caused by crystal field (which is very small in transition metals) but is due to the difference in kinetic energies of electrons in overlapping dx2_~.2, d3_.2_,2 and d,y, dx:, d~,. orbitals. It is essentially Fdd~ which is negative and which lowers the energy of I25, with respect to Fi2. Second, the low-spin to high-spin transition in fcc Mn is only possible, because the Fermi level lies halfway between F25, and F12. Just before the transition this gives rise to a sharp edge in the spin-down density of states duc to flat occupied bands near F and L, and a sharp peak in the spin-up density of states due to flat unoccupied bands between F and L. When this spin-down and spin-up structures in the density of states simultaneously cross the Fermi level, causing simultaneous filling of spin-down and depletion of spin-up states, the moment will rise discontinuously from zero to a large value. We think that this observation of Fuster et al. [2] is correct. This simultanuous depletion and filling is only possible if the density of states just below and above the Fermi level is large enough and very similar. Since band structures of transition metal compounds are all similar, this condition can only be fulfilled if the FermJ level lies in between F25, and F12. This condition is indeed fulfilled for all transition-metal systems which are close to a low-spin to high-spin transition and it is not fulfilled in magnetically stable systems. If the total energy of the high-spin state is lower than the energy of the low-spin state, then the transition will stop when the gain in exchange energy will cancel the increase in elastic energy due to the volume expansion. No total energies and no binding surface were calculated for fcc Mn, so one does not know how close in energy are the low- and high-spin states. Figur~ 6 shows that, at the transition, spin-up F25, and spin-down F~2 levels cross ~nd that the level splitting b e t w e e n spin-down F2s, and Ft2 levels is now smaller than
170
M. Sehr6ter et al. / Magnetism and magnetic volume collapse
between F2s, and F12 in the low-spin state. Probably, the net amount of polarization in the highspin state is caused by occupying the more localized d levels according to Hund's rules. An interesting question is in how far these transitions are of any importance for the INVARlike systems. Since there is no real microscopic theory of the INVAR effect, we would like to make a brief comment based on our phenomenological theory and on the observations which can be drawn from fig. 6. Zero or small thermal expansion of an itinerant magnetic system is surprising. No volume expansion means that there is practically no shift of electronic energies with increase in temperature. One thinkable mechanism could be a pinning of levels near the chemical potential. Such a pinning could occur just after the transition to the high-spin state. Local volume fluctuations try to shift, let us say, the spin-up F25, and spin-down I'12 levels with respect to the artificially fixed Fermi level in fig. 6. Fight after the transition this costs energy since spin-up electrons have to be transferred to regions of spin-down electrons and vice versa, giving rise to spin fluctuations which tend to suppress the high-momentum state. For a situation as in fig. 6 volume and spin fluctuations will not cooperate but rather hinder each other and their contribution to the thermal expansion might be small or cancel. This could result in a pinning of the spin-down Fi2 and spin-up I'25, levels slightly below and above the chemical potential, respectively. This means that volume and spin fluctuations would not lead to an increase in volume and the origin of the INVAR effect would be of dynamical nature. Our phenomenological model is too crude to deal with such aspects. We obtain zero thermal expansion if the decrease in volume due to decreasing magnetization is compensated by the increase in volume because of spin and volume fluctuations. This is possible if parameters are so that the energies of low- and high-spin states remain close to each other with increasing temperature. This is a kind of static pinning effect within the phenomenological model. To sum up, a necessary condition for the occurrence of low-spin to high-spin transitions and
probably also for the appearence of the INVAR effect is that the Fermi level must lie between F25, and 1"12, i.e., the Fermi level lies near a minimum in the density-of-states curve. Since electronic data serve as input, we tentatively conclude that for an appropriate choice of Landau coefficients we can at least mimic certain aspects of such transitions.
4. A classical field theory for metallic systems The starting point is the free energy functional, #-=-~ 1
Inf~mdto e -aJeim' ,1
(11)
An exact evaluation of ~ is not possible, in spite of the fact that m and ~0 are classical fields. But similar to many particle quantum systems, a best one-particle ersatz problem can be formulated by making use of
+ Ydo= ~
o)o, a ak l m~k l 2 + bk l tok l 2 ,,
(12)
k
where -'~0 is the corresponding molecular-field problem to be solved with variational parameters a~k (a = x, y, z) and b k, and where the Fourier transform of the fields has been defined as r e ( r ) = ink=0 + ~ ' eik'mk, k
(13)
t o ( r ) = tok=0 + E ' eikrtok" k
The k = 0 modes correspond to homogeneous volume and magnetization (which is taken in the z-direction). With m,~k = xa, + iyak, o~k = u k + io k, functional integration is replaced by ] d x 0 du01rlk.o du k dvk(l-I ~ dx~k d y ~ ) . Due to the molecular-field ansatz, the higher order terms in eq. (4) can be truncated (the final form of ~ is rather lengthy and will be omitted). The Gaussian functional integrals are easily carried out, leading, for example, to the for'owing
171
M. Schr8ter et al. / Magnetism and magnetic volume collapse
expressions for the mean square amplitudes of spin and volume fluctuations
tO0= --t,~1-4- ~G--tw2,
(18)
(m~)o+ (m2)o +
tml ~ - t, ~ 4-
(19)
= ~
1 ~,
1 a~'
1 E' 1, k b/,
(14)
(15)
(w2)°=~
where the variational parameters are given by O~/'~a~k= O~/Obk=O. These expressions show that the fluctuation amplitudes will increase monotonously with temperature and will not saturate, which is also obvious from functional integrals for expectation values if the fields are scaled with ft. This is a serious drawback of the Gaussian approximation. However, we expect that the thermodynamics near T~ is well described by the classical field functional. With respect to the phase transition, the gradient terms are of crucial importance since they allow a description of the magnetic phase transition as a spin-fluctuation-driven instability of the ordered state. In principal, the coefficients of the gradient terms could also be obtained from band-structure calculations for inhomogeneous systems. Because such input data do not exist, we have additional parameters. Since coefficients of gradient terms measure the effective range of forces, they usually can be related to the transition temperature. The gradient terms contribute to the free energy in the form,
gm
k2
E ,k2
2 or mo=O,
tin2 = ~C ( a ( t o c - tOo)+ a'(~Oo2 +
,o, = ~ ( ~ + 2o'(m~0 +0+ 0)),
(21)
'°2-- - ~ ( m ~ 4-0, (22)
while the mean square amplitudes are determined by
q,{
al
2,~2g~. 1 - x~ arctan x~ / '2fl(m~)o,
__ , ( m ~ ) o ,
2,(~2)o,
=+_, x~=
q~&
(23)
~-~ to.
The variational fields are of the form, .~ = a ( ~ o - too) + a'(too~ + <'~>o)
(16)
which shows that a momentum cut-off is needed to avoid ultraviolet di.vergencies. The cut-off parameter can be related to the critical number of modes which will destroy the ordered state. We would like to mention that the approximation scheme (Gaussian approximation and momentum cut-off) used here gives rise to spurious effects like simulations of first-order phase transitions. The self-consistency relations for homogeneous magnetization and volume follow from ao~/Omo = 0~'/Oto0 = 0, leading to
m0= --lml + ~ - - t . ,
5(m~)o+ <-,~)o,
(17)
+ b( 6m2 + 6(m:2)o + ( m 2 ) o ) + 3c(5 m4 + 30m2(m~)o +6m2o( m2 )o + 15 (m~)oz + 2(m~ )o2 + 6(mz2)o(mZz )o), (24) ~ l = ~ ( ~ - too) + ~'(o,o~ + (~;')o)
+ 2~,(m2 + (m~)o + 2(,.~)o) + 34,,,~ + 6,,,~(..:)o 4- 4m~o(,,,~)o + 3(m2) 2 + 6(m~ )20+ 4(mZ=)o(rn 2 )o), (25)
M. SchrSter et al. / Magnetism and magnetic volume collapse
172
/~,0 = ½K + 3),~0o +
a'(m2o + (m2)o +
For brevity, contributions oc 8 have been neglected. The Curie temperature follows f r o m / z z =/~ .L = 0, leading to (m2)c=
q,,,
1
21
4 3
z
\/
-
4~r2flcg,,, = "2-f-c[ - b + ~b2 - 3-Cac ] '
,,<
(27) ac = a((o c - ~0)c + a'((°o2 + ((°2)0)c,
which shows that T~ 0c gin~q,, (which m a y serve to fix the additional parameters). The number of critical modes is given by 3q3/6"~ 2. A simple discussion of the formula for the spontaneous magnetization shows that the spin-fluctuationdriven phase transition should be of second order for tm2(Zc) = 0 while it is of first order for t2t(T¢) = t,,2(T~) leading to the following condition for a first-order phase transition, 5B2/7 < A < B 2, P
2
A = (a(~o c - (o0) + a ((o o + (~O2)o))/3c,
(29)
B = b/3c. Actually, the magnetization curve has always a nose-like feature at T~ resembling a first-order magnetic transition. This is not in contradiction to the condition for a magnetic phase transition within the classical field theory, which reads , ~ ( r c ) = ~,j_ ( r J = 0 ,
o
(28)
(30)
since t% and # 1 decrease monotonously from T = 0 to T = T¢ as is shown in fig. 7 for the case of Fe3Pt. It seems that this simulation of first-order transitions is a c o m m o n defect in the spin fluctuation theory based on the Gaussian approximation, where the spontaneous magnetization is determined by a~'/Orn o = 0. It also seems that the unified theory of Moriya et al. [24] carries the same defect [25]. There is no simple prescription to get around this difficulty since the Gaussian approximation violates the fluctuation dissipation theorem [26]. In the present case, the nose-like
200
oo¢
600TtKSO0, 1000
Fig. 7. Temperature dependence of the variational fields /t: and/~ ± for the case of Fe3Pt. The condition for the ferromagnetic phase transition is /~z(T~)=/L x (Tc) = 0. However, note that there is a discontinuity at the higher temperature Tc, which is due to the Gaussian approximation.
feature of m o ( T ~ T~) only disappears for qm ~ oo (hence, for T~ ~ 0). T h e condition for a true first-order phase transition is t,2,1(T~) = t,,2(Tc).
(31)
In order to fulfill this condition, t.,t(T) and t,n2(T ) have to cross once while t~,t(T ) and t,.2(T) have to cross twice. Figure 8 shows that this condition is nearly fulfilled for the case of Fe3Pt.
6
i
i
i
i
t,
.
0
.
.
.
.
.
200
400
600
T[K]
.
.
.
.
800
.
.
.
.
1000
Fig. 8. Temperature variation of the functions defined in eqs. (17)-(22) (see text for explanation).
173
M. SchriJter et al. / Magnetism and magnetic volume collapse
It is also visible how strongly spin fluctuations renormalize the Landau coefficients in eqs. (19) and (20). On the scale of fig. 8 the variation of t,0] and l~2 is less pronounced. The phase diagram for metallic systems which results from eq. (29) is shown in fig. 4. There is a small region of real first-order phase transitions. Even if none of the considered systems behaves discontinuously at T~ it is obvious that all but bcc Fe are close to the parameter space of first-order phase transitions.
5. T h e r m o d y n a m i c s
The entropy is easily evaluated with the help of
5a= - ~"BT/o,<,.l>o.<,~>o = - ( a -~' ~° l ] (m:2>o,(m~)o.(W2)o"
(32)
The final form of the free energy in Gaussian approximation is then ~ = d°- TSP,
(33)
K 2 + V,Oo 3 + 8(~Oo 4 _ 3(tO2>o) + a(,oc - ,~o)m~ e = ~'~o
× [ln 2flgmq~2(1 + x~_) - 2] + _~_/~± (,n~)o +
q3 12~ 2
×[ln2flg'~q2(l+x:) -{] + _~_l~,o(,n2,.,)o 3q3
12,rr2
q3 l.
12,rr2 J
(35)
This clearly shows that the free energy resembles, in parts, the free energy of a classical ideal gas (of spin fluctuations) with all the usual unphysical properties of an ideal gas for T---, 0. A second unphysical feature is connected with the increase of #- with T. The latter inconvenience could be avoided by using a temperature dependent cut-off parameter, q,,,tx if~3. Such a temperture dependent cut-off was also discussed by Murata [27]. For large cut-off parameters the corresponding derivatives of the free energy (entropy and specific heat) easily become negative. The specific heat follows from C= T(-~)
(36)
= Tk B ~ { ~ ( 1
a~kTaa~k)OT
+ a'(m~to~- (t~2>o((mz2)o +o)) + b(m:- 3(m~)02- 2(m~Z)o(m~)o
+ 1
- 2
leading to
+ 18(m2)o( m2 )o + 6(m 2 )2]
C=
- 30(m2) 3 - 18(rn~)02(m~ ) o - (m2)o 3q3 ×°2- 12(m~ )°3)+ ~
(34)
+
-• /~z(rn~)0 + 12~r'2q3'~--"
'
kB (303 + q 3 ) _ / 12~r2 -
- -
2, a/~: ~~
~m:/o
- ( m 2 ) o - ~ - -- (602>0~ ? .
q3 + 12~r-'--'~'
"fig"q" q3 [ln-~ ,rr z (1 + x-'2) - ~ ] 6a= - k B ( 121r2 [
bk aT
(38)
The general form of eq. (38) resembles the result for the specific heat obtained by Murata et al. [22] for the case that only spin fluctuations are retained. The consideration of volume fluctuations gives rise to the last term on the right-hand side of eq. (38). However, we would like to emphasize, that in spite of the additive nature of the different contributions in eq. (38) (which is due to the mean field-like ansatz eq. (12)), there is a strong interre-
174
M. Schr&er et al. / Magnetism and magnetic volume collapse
('~magneti,'- {~magnetic kB
q3,, 4,'ff 2 '
=
., /'f'1"
2 co o
(~VOLUt'4E
,
0
,~£
"~'--
0
m
. ,t" ~" f
|
'~',,,,, ,,
-:
"-G
i
0
200
. . . . .
i
400
- -
1
4
I
600 T [K] 800
12~r 2 "
•
] ~r
q2 ku
1000
Fig. 9. Temperature dependence of the different specific heat contributions as obtained from eq. (38) ior Fe3Pt.
lation between magnetic and volume contributions due to the self-consistency conditions. The result given in eq. (28) put in a different form has also been obtained by Wagner [19]. It is obvious from the analytical structure of eq. (38) that, at large enough temperatures, the, pecific heat necessarily will become negative for the case where the volume contribution is smaller than the magnetic contribution. Yet the behaviour of C near ~ is not unreasonable. By this we mean that the way in which the specific heat increases when approaching T~ from below and the m~gnituc?- of the discontinuity resemble results of more refined caculations. Since Fe3Pt is one of the most interesting INVAR systems, we have plotted in fig. 9 the different specific heat contributions for this system as obtained with the help of eq. (38), whereby the following dimensionless quantities were used,
Here, the indices refer to the different contributions in eq. (38). Since the volume variation below Tc is very small, the main contribution to the specific heat is of magnetic origin in this temperature range. For T > Tc the magnetic contribution always decreases with increase in temperature wtfile the volume contribution becomes larger. Thus, magnetic contributions and volume contributions partially compensate each other at high enough temperatures. This different behaviour for T > T~ is caused by the anharmonic term which leads to a negative contribution in eq. (26) and to a sign reversal in eq. (38). Details in the temperature range between T~ (temperature at which a second-order phase transition should occur due to the Landau condition) and Tc (temperature at which the pseudo-first-order phase transition occurs due to the Gaussian approximation) have been left out. Unfortunately, we cannot compare the different specific heat curves of fig. 9 with experiment, since detailed experimental specific heat data of ordered Fe72Ptzs do not exist. Other thermodynamic quantities can easily be calculated. For example, the relative volume change is given by
°h'(T) =
®{
(co2)°(T) + 3 -°~
+(m2>o(T)+<,n2>o(T)] 2~
a
[m2o+(,n:)o+(m~) °
9y""
2
,
(39)
which, in the case of a first-order phase transition, will lead to a discontinuous volume change of the order of
acoo(T ) =
a
2
-
).
(40)
The thermal expansion is given by CZto,a' = C~o,~, kB
1 (,~q~+ q3), 12~r2
1 atoo(T) a ( T ) = 1 + Coo(T ) aT '
(41)
M. SchrSter et aL / Magnetism and magnetic vole'me collapse
fixed-moment method. The classical gas of spin fluctuations can undergo a phase transition and, depending essentially on only two parameters, b and 0% metallic systems can be classified as magnetically stable with a second-order transition, as magnetically unstable with a first-order transition associated with a magnetic volume collapse, or as paramagnetic systems. Whereas stable magnetic systems possess only one local minimum on the binding surface, all systems being on the verge of a magnetic volume collapse have (at least) two minima which are close in energy. Two minima always exist if
8
Tc 6--. 6~ x-" o
calculated
4
IL
2 0000 0
o
-4
~
o
°o~ l
-2
~OQID
Tc
175
Fe72 PI2a • ordered o disordered
Do < l ~ - -
-6
V0(0 ) a n d b < O
(42)
-b
o
2b0
4b0 6bo
8b0
ooo
T [K] Fig. 10. Calculated thermal expansion of Fe3Pt as compared with experimentM results [30].
which leads to quantitative results being of correct order of magnitude. Figure 10 shows the calculated thermal expansion for Fe3Pt as compared to experiment. Besides, at very low temperatures the behaviour of a ( T ) is very reasonable. Further thermal properties, like compressibility, pressure dependence of T~, etc. can easily be obtained. In all calculated quantities the same unphysical features appear at very low and very high temperatures. At intermediate temperatures the mutual interplay of spin and lattice degrees of freedom can well be studied within the pr£~ent scheme. But the disadvantages due to the G~ass ian approximation and the neglect of quar~'~um mechanical aspects show that any improver~ent would be highly welcome.
6. Conclusions
We have shown that magnetovolume effects it~ metallic systems can be described with the help o~ a free energy functional of the Landau type, whereby the coefficients are fixed by zero temperature results for E ( m o, Vo) obtained with the
is fulfilled, where ~2~ is the volume at which the system can, without energy change, polarize and B 0 is the hypothetical nonmagnetic reference volume. The thermodyaamics of systems tending to a magnetovolume instability is closely related to the temperature evolution of the low- and high-spin states. Indeed, a two-states model coming close to the present picture was used to describe the a ~ 3' transition in iron [28]. The drawbacks of the present formulation have been critically discussed. They are mainly connected with the Gaussian approximation employed in tiffs work. An extension of the present formulation to avoid the Gaussian approximation is in progress. However, a systematic improvement is only possible within a quantum mechanical formulation. Especially, a microscopic theory of low-spin to high-spin transitions and of the INVAR effect has to be formulated. Most approaches, which can be found in the literature, are embedded in phenomenological models. The microscopic models which exist are mostly based on the single band or degenerate-band Hubbard model. Functional integration is performed in the static approximation, which is hard to overcome. Also, the incorporation of electronic details near the Fermi level would require tremendous computational efforts. For further references in this context we refer one to the work of KakehastJ [29] and for an overview of the present state of art of the spin fluctuation theory to ref. [6].
176
M. Schr6ter et al. / Magnetism and magnetic volume collapse
References [1] E.F. Wassermann, in: Ferromagnetic materials, vol. 5 (North-Holland, Amsterdam, 1989). [2] G~ Fuster, N.E. Brener, J. Callaway, J.L. Fry, Y.Z. Zhao and D.A. Papaconstantopoulos, Plays. Rev. B 38 (1988) 423. [3] D. Bagayoko and J. Callaway, Phys. Rev. B 28 (1983) 5419. [4] D.J. Kim, Phys. Reports 171 (1988) 129. [5] T. Moriya, Spin fluctuations in itinerant electron magnetism (Springer, Berlin, 1985). [6] Metallic magnetism, ed. H. Capellmann (Springer, Berlin, 1987). [7] P. Entel and M. Schr~ter, J. de Phys. 49 (1988) C8-293; Physica B 161 (1989) 160. [8] F. Jordan and P. Entel, Physica B 161 (1989) 121. [9] M. Shim~n, Rep. Prog. Phys. 44 (1981) 21. [10] W.A. Harrison, Electronic structure and the properties of solids (Freeman, San Francisco, CA, 1980). [11] J. Kiibler, Phys. Lett. A 81 (1981) 81. [12] V.L. Moruzzi, P.M. Marcus, K. Schwarz and P. Mohn, J. Magn. Magn. Mat. 54 (1986) 955. [13] V.L. Moruzzi, P.M. Marcus, K. Schwa~ and P. Mohn, Phys. Rev. B 34 (1986) 1784. [14] P.A. Montano, G.W. Femando, B.R. Cooper, E.R. Moog, H.M. N~k, S.D. Bader, Y.C. Lee, Y ~ Darici, H Min and J. Marcano, Phys. Rev. Lett. 59 (1987) 1041.
[15] M.M. Abd-Elmeguid and H. Micklitz, Phys. Rev. tl 40 (1989) 7395. [16] E. Kisker, E.F. Wassermann and C. Carbonc, Phys. Rev. Lett. 58 (1987) 1784. [17] T. Mori>a and K. Usami, Solid State Commun. 34 (1980) 95. [18] V.L. Moruzzi, Phys. Rev. Lett. 57 (1986) 2211. [19] D. Wagner, J. Phys.: Cond. Mat. 1 (1989) 4635. [20] P Mohn, K. Schwarz and D. Wagner, Physica B 161 (1989) 153. [21] M. Shimizu, J. Magn. Magn. Mat. 45 (1984) 144. [22] K.K. i'vlura~.aand S. Doniach, Phys. Rev. Lett. 29 (1972) 285. [23] C.J. Ballhausen, Ligand field theory (McGraw-Hill, New York, 1962). [241 T. Moriya and Y. Takahashi, J. Phys. Soc. Jpn. 45 (1978) 397. [25] S.N. Evangelou, Y. Hasegawa and D.M. Edwards, J. Phys. F 12 (1982) 2035. [26] S. Hirooka and M. Shimizu, J. Phys. F 18 (1988) L127. [27] K.K. Murata, Phys. key. B 12 (1975) 282. 1281 R.J. Weiss, Proc. Phys. Soc. 82 (1963) 281. [29] Y. Kakehashi, J. Phys. Soc. Japan 50 (1981) 1925. Y. Kakehashi and P. Fulde, Phys. Rev B 32 (1985) 1595. Y. Kakeliashi, Phys. Rev. B 38 (1988) 6928. [30] W. Stanun, Thesis, Duisburg (1988).
163
METALLIC MAGNETISM AND MAGNETIC VOLUME COLLAPSE M. SCHROTER, P. ENTEL and S.G. MISHRA t Theoretische Physik, Universitiit Duisburg and SFB 166, Postfach 10 15 03, D-4100 Duisburg 1, FRG Received 4 0 c t o b e i !989; in revised form 7 November 1989
We present a new description of possible magnetovolume instabilities in itinerant electron systems based on a generalized Cfinzburg-Landau expression. The Landau coefficients are input data and are obtained by reproducing binding surfaces (total energies as function of volume and the magnetic moment), which are the result of spin-polarized band structure calculations performed with the fixed-spin-moment(FSM) method. Finite temperature results are obtained by constructing a classical field theory. It is shown that essentially two Landau parameters (the fourth-order coefficient and the critical volume) may serve to classify itinerant magnetic systems as stable or potentially instable systems. In the first category the magnetic phase transition is of second order while, in the latter, the transition can be of first order, associated with a large magnetic volume collapse. We show that all INVAR systems are potentially instable.
1. Introduction
In metallic systems there is general interrelation between loss of magnetism and a decrease in the atomic volume. This tendency is very pronounced in transition-metal compounds and is responsible for magnetoelastic anomalies like the INVAR effect. For a recent review of the physical properties of INVAR systems and further references see ref. [1]. Of particular interest are hypothetical elemental systems such as bcc and fee manganese and fee iron for which the computed magnetic moment shows a discontinuous zero-moment to high-moment transition at a critical lattice spacing [2,3]. The close relation between magnetic and lattice properties in a metal was subject of intensive research during the last decade. For a recent theoretical review with special emphasis on the importance of the electron-phonon interaction in itinerant magnetic systems see ref. [4]. In this paper a new theory is presented which can account for many particularities of stable and unstable metallic systems. When trying to describe magnetism and volume changes simultaneously, one meets two obstacles. I Perraanent address: Institute of Physics, Bhnbaneshwar 752005, India.
The first is connected with the fact that neither the Heisenberg model of localized spins nor the Stoner theory of band magnetism is able to describe adequately the magnetic properties of transition metals. This is only possible in the frame of a unified theory of magnetism [5,6] (spin fluctuation theory) which in a quantitative respect is still in leading strings. On the other hand the coupling of spin to lattice degrees of freedom is a principle difficulty, since both degrees of freedom can be of equal relevance. In view ,~f these difficulties we use in this work a phenomenological model Harrd~tonian in form of a generalized Landau expansion which correctly reproduces total energy curves of transitionmetal compounds at zero temperature and which, in the frame of a classical field theory, leads to a general phase diagram for metallic systems. The drawbacks of the pheno.~enological theory at finite temperatures will be critically discussed. As stated above, the close connection of volume and magnetization changes is at the origin of low-spin to high-spin transitions, which can be viewed as a magnetovolume instability. We show that this instability can be discussed by starting from a Landau theory for metallic magnetism. The microscopic origin of such instabilities and
0304-8853/90/$03.50 © 1990 - El~vier Science Publishers B.V. (North-Holland)
M. SchrOter et al. / Magnetism and magnetic volume collapse
164
their relation to the INVAR effect is briefly discussed. Preliminary results were reported in ref. [71.
2. Ginzburg-Landau theory of metallic magnetism
Here, we briefly indicate how the generalized Landau expansion for transition metals can be derived from a microscopic quantum model. For simplicity, we first neglect lattice degrees of freedom. Then, it can be shown that the multiband hubbard model is suited for an evaluation of quasi-particle bands. An appropriate parametrization of the Hamiltonian in the Hartree-Fock approximation leads to renormalized on-site potentials for s, p and d electrons whose explicit form is given in ref. [8]. The solution of the corresponding self-consistency equations leads to realistic ~i:.~polarized band structures for transition metals, where transfer integrals are fixed at high-symmetry points and Uaa is chosen to give the correct amount of polarization. Shimizu showed that, on the basis of such a one-electron theory, the magnetic free energy can be written as [91
~[m I = ~
din' A(m', T)
-
½am2,
(1)
where za is the exchange splitting of the polarized bands and ct the coefficient of the molecular field. The exchange splitting can be expanded in a power g~ries of m by inverting a low-temperature expansion obtained from the relation for the number of spin-up and -down electrons and their corresponding chemical potentials (restricted microcanonical ensemble), i.e. +oo
.,,=f_
oo
d, o ( , ) / ( , - , , ,),
(2)
to Zi/gp. B = aim + a3 m3 + asm 5 + ... +O([kBT/EF]2), where the coefficients a~
leading
are given in terms of band density of states and its derivatives at the Fermi energy. It is generally bOieved that one can write down this expansion o~!.y for weak itinerant ferromagnets with small exchange splitting. However, in metallic ferromagnets we find that the expansion
is valid for any m, since the expansion parameter is, in principle, the ratio of magnetization density to the electron number density, m = m/(nPB), a quantity always less than unity since each electron can contribute, at most, one /xB to the itinerant magnetic moment. To give an example, we consider the free elect~on density of states and find, for the maglletic free energy, that =
-
2 +
+
nEF + ,#83,~ 8 + - . ' ,
(3)
a fast converging series. The only serious limitation is that this series expansion for metallic systems is, in principle, a low-temperature expansion and any use of it in the whole temperature range must be justified in the frame of a spin fluctuation theory. In the present work, this will be done by constructing a classical field theory which allows the evaluation of renormalized Landau coefficients. With respect to the spin-wave spectrum, only an expansion around k = 0 will be taken into account. It should be understood that m is, in principle, a vector and leads at finite temperatures to longitudinal and transverse fluctuations. Also a decomposition of m according to different electron orbitals has been left out in eqs. (1)-(3) for brevity. The coupling to the lattice is straightforward. In the original multiband Hubbard model the corresponding transfer integrals are taken as volume-dependent quantities. This is a well-defined procedure [10] and leads to volume-dependent Landau coefficients. We will neglect the dynamics of resulting electron-phonon interactions and only take into account the most essential static volume-dependencies. This procedure leads to the following phenomenological model Hamiltonian, where gradient terms have been added,
°~[ m, rO] = V o ~ fvo(r) d3r { g~( Vc° )2 + ½xr°2 + y~3 + 8~o4 + . . .
+ g , , E ( V , m # ) 2 + a ( % -- ~o)m 2 a#
+a'a~Zm2 + bm4 + cm6 + ... ~. ]
(4)
M. Schrrter et a L / Magnetism and magnetic volume colh~pse
The thermodynamic variables are magnetization and volume which are split into static and fluctuating parts
60(r, T ) = V ° ( T ) - ~2° V'(r, T) ~2o + j2o = 60o(T) + 60'(r, T),
( d ( r , r)> = o, re(r, T ) = t o o ( T ) + m ' ( r , ( m ' ( r , T ) ) = 0.
(5) T), (6)
V0(T ) is the actual volume while ~2o is an arbitrary reference volume, which in the following will be taken as the (hypothetical) volume of the nonmagnetic state at zero temperature in ease of a ferromagnetic system. 60¢= (f~¢ - I~0)/~2 0 is a critical volume allowing for a magnetic volume collapse. Spin and volume fluctuations will be described by (m'Z(r, T))4= 0 and (60'2(r, T ) ) * 0. At zero temperature the magnetic moment per atomic volume ~ a is m 0 with I~a = 4~rr3s/3 ( = a3o/2 for bcc and = a3o/4 for fcc, respectively). K denotes the bulk modulus. We would like to emphasize that a microscopic description of metallic magnetism in the frame of the extended multiband Hubbard model, which includes electronic s-, p- and d-electron and lattice degrees of freedom, contains too many degrees of freedom to be really tractable within the path-integral formulation. The phenomenological theory has serious drawbacks but it contains a small number of parameters and allows one in a transparent way, to describe the mutual influence of spin and volume fluctuations. Total energy calculations with the help of the fixed spin-moment method [11-13] have shown that the volume dependence of a series of metallic systems must be described by two seperate branches. A nonmagnetic branch with a minimum at low volume lies energetically close to a ferro- or antiferro-magnetic branch with minimum energy at a larger volume. With increase in temperature, pressure or by alloying, both branches can intersect at a critical volume, where the magnetic moment becomes unstable and the system can, without energy change, either occupy the high-spin or the low-spin state. It seems that all unusual properties of
165
metallic systems with a tendency towards a volume instability can be discussed on the basis of this picture. The present phenomenological theory exactly reproduces this kind of behaviour and allows a systematic classification of metallic systems. A necessary condition of itinerant ferromagnetism is that, at zero magnetic field and in equilibrium, the magnetic part of eq. (4) has negative energy. At zero temperature the equilibrium conditions
O~'~'~'/O~ = 60{ r + 3y60 + 43602 + 2a'm 2 } - am 2,
0=
(7)
0 = O,~/Om
= 2m
(a(60 c- 6o) + a'~o2 + 2bin 2 + 3cm 4}, (8)
lead for ~ = 0 to
60°=
K+2a' ~ / ( x + 2 a ' ) 2 am 2 6~, 436~, 2 + 3~-
amg
+ ~-a--r,
b m~ = - ~-~ 4-
(9) b2 ~cc2
• 2
a(60c - 60°) + a 60° (10) 3c '
which shows that 60¢ plays a fundamental role. If 60¢ is too large, only mo = 0 is possible while for 60c << 6°0, a stable magnetic solution is expected. Now, results for total energy curves fix the Landau coefficient and always yield positive values for a('k Furthermore, since the coefficient a ' of the magnetoelastic coupling term is much smaller than a, the critical volume 60¢ is given by H = 8.~/Om = 0 for m-~ 0, i.e. 60~ is a well-defined quantity for each metallic system and corresponds to the perpendicular intersection of the H = 0 line and m = 0 axis in the volume-versus-moment plane. Hence, at 60cthe system can spontaneously polarize without any change in volume and energy. The behaviour of the total energy as function of the various parameters can be discussed in detail if eqs. (9, 10) are inserted into eq. (4). From this discussion we conclude that for (60c - 600) < 0 there is a stable ferromagnetic solution with a secondorder phase transition, while for (~0~- O~o)> 0 and b < 0 a first-order phase transition together wlth'a
166
M. Schr&er etal. / Magnetism and magnetic volume collapse
magnetic volume collapse can be expected. In particular, we find that for t% = too the binding surface E ( m o, Vo) has always two close-in-energy lying local minima, one at m o = 0 and one at m 0 ~ 0, which define locally stable low- and highspin states. The condition that both minima have equal energy is E = 0 (and toe = too). For t% = too the critical volume is given by toe-- - 2 a b / ( 3 c x ) . Hence, for two close-in-energy lying low- and high-spin states to exist, the fourth-order coefficient has to bc negative. At zero temperature the coefficient b is given in terms of derivatives of the density of states at the Fermi level according to b tx 3(p~//p0) 2 - ffo'//po Thus, b will be negative if the Fermi level lies near a minimum in the density-of-states curve. At finite temperatures spin fluctuations will lead to large renormalization effects. Yet these zero temperature considerations remain qualitatively correct. In the present work the necessary condition of metallic magnetism is (toe - too) < 0 which s!i~bt!y modifies the usual Stoner condition for itinerant ferromagnetism. The later follows from eq. (1) as al = Xo ~(T < a, where the unenhanced susceptibility is obtained from H = O.~/~m = ( 1 / g l L B ) A a m = ( a 1 - a ) m + aam 3 + aSm 5 + . . . for a = 0 . We have assumed that the molecular field and first expansion coefficient have a linear and quadratic volume dependence which gives rise to the analytical structure of the m 2 coefficients. Moreover, any temperature dependence of the Stoner-like form ¢ ( [ k B T / E F ] 2) has been oiifitted. Figures 1 - 3 show the binding surfaces for bcc and fcc Fe and FeaPt which were obtained with the help of eq. (4), whereby the coefficients were fixed with the help of local spin-density calculations for total energies E ( m o , 11"o). We would like to emphasize that, with the help of the few parameters in eq. (4) (which are listed in table 1), it is possible to reproduce quite accu-ately the results of 500 band-structure calculations which are neeessary to construct a single binding surface. We now discuss very briefly the most important features of our zero-temperature results, since the three fundamentally different types of surfaces are well suited for a description of the different kinds of metallic magnetism. The binding surface of bcc Fe is typical for
z u.l o
V
°~1 zuJ
~'C
~
~
2
.........
~
:
0
~
~
2./-*5 2.55 2.65 WIGNER-SEITZ RADIUS [a.u.]
m
Fig. 1. Total energy curves E(mo, Vo) for bee Fe at i mRy intervals. Upper inset: volumedependenceof the energy. Lower in~et: temperature variation of spontaneous magnetization, mean square amplitudes of spin fluctuations and relative volume change.
E O "5
I--
z ILl O (.3 I-LU
z
o
2.5 2.6 2.7 WIGNER-SEITZ RADIUS [o.u.] Fig. 2. Total energycurves E(me,V0) for foe Fe at 1 mRy intervals. Note that the H = 0 and P = 0 line d o not intersect. Hence, there is no local minimum at finite polarization.
M. Sclwi~ter et aL / Magnetism and magnetic volume collapse
75
167
80VOLUME [a.u?] 85
90
Fig. 3. Total energy curves E(mo, Vo) for Fe3Pt at 0.5 mRy intervals. Black dots: Temperature evolution of the system from zero temperature up to the Curie temperature, left: without anharmonic con'Lributions, right: with ~ a r m o n i c contributions. Upper inset: Energy as function of volume. Lower inset: Temperature variation of spontaneous magnetization, mean square amplitudes of spin fluctuations and relative: volume change.
s t a b l e m e t a l l i c f e r r o m a g n e t i s m . It h a s a p r o n o u n c e d local m i n i m u m a t the e q u i l i b r i u m posit i o n ( h i g h - s p i n state) a n d a s a d d l e p o i n t o n the m = 0 axis ( l o w - s p i n state) w h i c h lies 23 m R y ( = 3630 K ) a b o v e t h e g r o u n d s t a t e energy. H e n c e , b c c F e is a stable f e r r o m a g n e t i c s y s t e m w i t h n o p o s s i b i l i t y f o r a h i g h - s p i n to l o w - s p i n t r a n s i t i o n a n d n o t e n d e n c y t o w a r d s a v o l u m e i n s t a b i l i t y in t h e to r a n g e c o n s i d e r e d . T h e c o e f f i c i e n t s b a n d c in the c a s e o f b c c F e o f t h e m 4 a n d m 6 t e r m s are b o t h p o s i t i v e w h i l e a ( t o c - t o ) is n e g a t i v e w h i c h will l e a d to a s e c o n d - o r d e r p h a s e transition.
T h e b i n d i n g s u r f a c e o f fcc F e is o f p a r t i c u l a r interest. First, t h e m i n i m u m e n e r g y c o r r e s p o n d i n g t o a l o c a l m i n i m u m is o n t h e m = 0 axis ( l o w - s p i n state). T h u s , fcc F e is p a r a m a g n e t i c at T = 0. H o w e v e r , t h e r e is a clear t e n d e n c y o f the s y s t e m t o d e v e l o p a f u r t h e r local m i n i m u m at h i g h e r energy with larger volume and moment (high-spin state). S e c o n d , for l a r g e r v o l u m e s t h e i n c r e a s e in magnetization without any apparent volume c h a n g e o f f e r s t h e p o s s i b i l i t y for m e t a m a g n e f i c beh a v i o u r : I n a small i n t e r v a l o f r w s , b o t h n o n m a g n e t i c a n d f e r r o m a g n e t i c p h a s e s are locally stable.
Table 1 List of Landau coefficients which were used for the calculation of total energy surfaces. 8 and a ' in eq. (4) are set equal to zero. (F) and (P) denote ferromagnetic and paramagnetic ground states, respectively. ~2o is given in units of au 3 and stands for the atomic volume in the (hypothetical) nonmagnetic state. The ground state volume V0(0) is also #oven in units of au 3 per atom. All o~er units follow from the prescription that energies are measured in mRy and magnetic moments in/L B per atom
bcc Fe fcc Fe bcc Co fcc Co Fe3Pt Fe3Ni
(F) (P) (F) (F) (F) (F)
•0
v
~0(0)
V~(0)
'~c
a
b
c
gm
qm
1214 1180 1401 1363 1465 1174
- 393.7 - 154.3 -
70.3 68.6 70.7 69.5 77.1 69.5
76.2 68.6 74.5 72.5 82.4 73.5
-0.365 0.280 - 0.256 0.048 0.057 -0.035
20.30 26.05 32.30 28.10 29.95 29.65
0.77 - 1.06 0.90 - 4.20 - 0.91 0.41
0.04 0.09 0.26 1.14 0.20 0.05
3.19 1.4 1.4
10 8 8
168
M. Schri~ter et aL / Magnetism and magnetic volume collapse
Hence, the idea of high-spin to low-spin transitions without large changes in energy seems not to be farfetched. Actually, there is experimental evidence for such transitions in artificially stabilized fcc Fe at larger lattice spacings in a copper matrix [14]. In contrast to bcc Fe the coefficient b in the case of fcc Fe is negative while a(toc - t o o ) is positive. By making the coefficient b more negative and thereby enhancing the tendency towards a first-order phase transition we can stabilize the local minimum corresponding to the high-spin state. The total energy change while changing b is negligibly small. The binding surface of Fe3P exhibits fea~,ures of bcc Fe (stable high-spin state) and of fcc Fe (locally stable low-spin state). While b is still negative, toc is very close to the equilibrium volume. Therefore, this system should be on the verge of a large volume collapse. This tendency has been confirmed by measuring the Curie temperature and the hyperfine field under pressure [15]. The total energy surface has two close-in-energy-lying minima with an energy barrier of ouly 1 mRy (--- 158 K), as can be seen ;,n the upper inset of fig. 3. Since T~=500 K, it is tempting to assume that the low-spin, low-volume, 'state is thermodynamically accessible before reaching the Curie temperature provided that the energy bar-
ofcc Fe (NM), 27 10
.A
p
:s
~
rier does not increase with T. Actually, there is experiraental evidence for high-spin, highvolume-low-spin, low-volume transitions in Fe3Pt at finite temperatures [16]. Fe3Pt and Fe3Ni have slightly different binding surfaces. Low- and high-spin states of FeaNi still have comparable energies but the low-spin state does not any longer correspond to a local minimum on the binding surface. Fe3Pt, Fe3Ni and fcc Fe can be considered as INVAR-like systems. Their thermodynamic behaviour is governed by the thermal evolution of high- and low-spin states. All systems are close to a magnetic volume collapse, as is also visible from the phase diagram in fig. 4. We would like to add that a series of intermetallic compounds undergo first-order phase transitions associated with a magnetic volume collapse (for example, YMn 2). But, at present, we are unable to describe such systems since we do not have enough input data to fix the Landau coefficients. Similar expansions as in this work have often been used, especially in connection with the INVAR effect [17]. For work which is closely related to our work, see refs. [18-20]. For an extension of the expansion to the antiferromagnetic case by including the staggered magnetization with wave vector Q, see ref. [21]. New in this work is, that we use an extended and more appropriate expansion and that zero temperature band structure calculations serve as input to fix unequivocally the Landau coefficients. At finite temperatures we essentially follow the mode-mode coupling formulation of Murata and Doniach [22] and construct so a classical field theory for metallic systems which tend to volume instabilities. Before presenting finite temperature resuJts we briefly discuss on the microscopic origin of low-spin to high-spin transitions and the related INVAR effect.
~fee co 3. The origin of low-spin to high-spin transitions
~
bccFe~
Fig. 4. Phase diagram for metallic systems. (P), (F0 and (F2) denote the paramagnetic, first and second-order phase transition regions, respectively.
Low-spin to high-spin transitions are known to occur in many d4-d 7 transition-metal complexes, for example, as a function of the ligand field strength [23]. In fig. 5 the dS-dectron configura-
M. Schri~ter et al. / Magnetism and m~Enatir ~oluma ~,qlao~e eg
eg +,+ I-A --~- --~ -f' ,2g
A 1 ~' I~ ~
t29
A < Ac
A > Ac
HIGH SPIN
LOW SPIN
Fig. 5. d s electron configuration in a weak (left) and strong (right)ligandfield.
tion (either Fe 3+ or Mn 2+) in a weak and strong ligand field of octahedral sylrLmetry is schematically shown. A weak ligand field acts as small perturbation and occupation essentially follows Hund's rule giving rise to a high-spin state. In a strong field the tendency to minimize the crystal field energy acts against intraatomic correlative interactions which, for large enough fields, leads to a low-spin state. For each configuration between d 4 and d 7, there is ~ critical value A for which the ground states of different multiplicity cross. Theoretically, a low-spin to high-spin transition could be achieved in one and the same cornplex by enlarging the distance between ligands and the central transition-metal ion. The low-spin to high-spin transition in transition metals and compounds resembles the corresponding transition in inorganic complexes. However, there are two distinct features which are different. In fig. 6 we have plotted the volume dependence of 1-'12 and F25, energy levels (which
0 m -0.1
t / F12
( " - ~ ._._...~-. ~....o_e-----"" . . . . . . . "~"" EF t~ - 0 . 2 ~ F2s. ~ ~1 a: ~ ~-1 ~"-0.3 "'0" . . . . . . . . . . . . + -o-~ I.kl ,o== ....... LOW SPIN ' HIGH SPIN -t] -OA ~ ~ 95 100 4.05 VOLUME [a.u.3] --
Fig. 6. Low- and high-spin region of face-centered manganese. Open circles are band structure data taken from ref. [2]. The
lines are guides to the eyes.
169
correspond to the two-fold and three-fold degencrate eg and t~g levels in inorganic complexes). The points in fig. 6 are taken from band-structure calculations for fcc manganese [2] while the lines noted.are guides to the eyes. Two points have to be First, the energy splitting between Fl2 and F25, is not caused by crystal field (which is very small in transition metals) but is due to the difference in kinetic energies of electrons in overlapping dx2_~.2, d3_.2_,2 and d,y, dx:, d~,. orbitals. It is essentially Fdd~ which is negative and which lowers the energy of I25, with respect to Fi2. Second, the low-spin to high-spin transition in fcc Mn is only possible, because the Fermi level lies halfway between F25, and F12. Just before the transition this gives rise to a sharp edge in the spin-down density of states duc to flat occupied bands near F and L, and a sharp peak in the spin-up density of states due to flat unoccupied bands between F and L. When this spin-down and spin-up structures in the density of states simultaneously cross the Fermi level, causing simultaneous filling of spin-down and depletion of spin-up states, the moment will rise discontinuously from zero to a large value. We think that this observation of Fuster et al. [2] is correct. This simultanuous depletion and filling is only possible if the density of states just below and above the Fermi level is large enough and very similar. Since band structures of transition metal compounds are all similar, this condition can only be fulfilled if the FermJ level lies in between F25, and F12. This condition is indeed fulfilled for all transition-metal systems which are close to a low-spin to high-spin transition and it is not fulfilled in magnetically stable systems. If the total energy of the high-spin state is lower than the energy of the low-spin state, then the transition will stop when the gain in exchange energy will cancel the increase in elastic energy due to the volume expansion. No total energies and no binding surface were calculated for fcc Mn, so one does not know how close in energy are the low- and high-spin states. Figur~ 6 shows that, at the transition, spin-up F25, and spin-down F~2 levels cross ~nd that the level splitting b e t w e e n spin-down F2s, and Ft2 levels is now smaller than
170
M. Sehr6ter et al. / Magnetism and magnetic volume collapse
between F2s, and F12 in the low-spin state. Probably, the net amount of polarization in the highspin state is caused by occupying the more localized d levels according to Hund's rules. An interesting question is in how far these transitions are of any importance for the INVARlike systems. Since there is no real microscopic theory of the INVAR effect, we would like to make a brief comment based on our phenomenological theory and on the observations which can be drawn from fig. 6. Zero or small thermal expansion of an itinerant magnetic system is surprising. No volume expansion means that there is practically no shift of electronic energies with increase in temperature. One thinkable mechanism could be a pinning of levels near the chemical potential. Such a pinning could occur just after the transition to the high-spin state. Local volume fluctuations try to shift, let us say, the spin-up F25, and spin-down I'12 levels with respect to the artificially fixed Fermi level in fig. 6. Fight after the transition this costs energy since spin-up electrons have to be transferred to regions of spin-down electrons and vice versa, giving rise to spin fluctuations which tend to suppress the high-momentum state. For a situation as in fig. 6 volume and spin fluctuations will not cooperate but rather hinder each other and their contribution to the thermal expansion might be small or cancel. This could result in a pinning of the spin-down Fi2 and spin-up I'25, levels slightly below and above the chemical potential, respectively. This means that volume and spin fluctuations would not lead to an increase in volume and the origin of the INVAR effect would be of dynamical nature. Our phenomenological model is too crude to deal with such aspects. We obtain zero thermal expansion if the decrease in volume due to decreasing magnetization is compensated by the increase in volume because of spin and volume fluctuations. This is possible if parameters are so that the energies of low- and high-spin states remain close to each other with increasing temperature. This is a kind of static pinning effect within the phenomenological model. To sum up, a necessary condition for the occurrence of low-spin to high-spin transitions and
probably also for the appearence of the INVAR effect is that the Fermi level must lie between F25, and 1"12, i.e., the Fermi level lies near a minimum in the density-of-states curve. Since electronic data serve as input, we tentatively conclude that for an appropriate choice of Landau coefficients we can at least mimic certain aspects of such transitions.
4. A classical field theory for metallic systems The starting point is the free energy functional, #-=-~ 1
Inf~mdto e -aJeim' ,1
(11)
An exact evaluation of ~ is not possible, in spite of the fact that m and ~0 are classical fields. But similar to many particle quantum systems, a best one-particle ersatz problem can be formulated by making use of
+ Ydo= ~
o)o, a ak l m~k l 2 + bk l tok l 2 ,,
(12)
k
where -'~0 is the corresponding molecular-field problem to be solved with variational parameters a~k (a = x, y, z) and b k, and where the Fourier transform of the fields has been defined as r e ( r ) = ink=0 + ~ ' eik'mk, k
(13)
t o ( r ) = tok=0 + E ' eikrtok" k
The k = 0 modes correspond to homogeneous volume and magnetization (which is taken in the z-direction). With m,~k = xa, + iyak, o~k = u k + io k, functional integration is replaced by ] d x 0 du01rlk.o du k dvk(l-I ~ dx~k d y ~ ) . Due to the molecular-field ansatz, the higher order terms in eq. (4) can be truncated (the final form of ~ is rather lengthy and will be omitted). The Gaussian functional integrals are easily carried out, leading, for example, to the for'owing
171
M. Schr8ter et al. / Magnetism and magnetic volume collapse
expressions for the mean square amplitudes of spin and volume fluctuations
tO0= --t,~1-4- ~G--tw2,
(18)
(m~)o+ (m2)o +
tml ~ - t, ~ 4-
(19)
= ~
1 ~,
1 a~'
1 E' 1, k b/,
(14)
(15)
(w2)°=~
where the variational parameters are given by O~/'~a~k= O~/Obk=O. These expressions show that the fluctuation amplitudes will increase monotonously with temperature and will not saturate, which is also obvious from functional integrals for expectation values if the fields are scaled with ft. This is a serious drawback of the Gaussian approximation. However, we expect that the thermodynamics near T~ is well described by the classical field functional. With respect to the phase transition, the gradient terms are of crucial importance since they allow a description of the magnetic phase transition as a spin-fluctuation-driven instability of the ordered state. In principal, the coefficients of the gradient terms could also be obtained from band-structure calculations for inhomogeneous systems. Because such input data do not exist, we have additional parameters. Since coefficients of gradient terms measure the effective range of forces, they usually can be related to the transition temperature. The gradient terms contribute to the free energy in the form,
gm
k2
E ,k2
2 or mo=O,
tin2 = ~C ( a ( t o c - tOo)+ a'(~Oo2 +
,o, = ~ ( ~ + 2o'(m~0 +
(21)
'°2-- - ~ ( m ~ 4-
while the mean square amplitudes are determined by
q,{
al
2,~2g~. 1 - x~ arctan x~ / '2fl(m~)o,
__ , ( m ~ ) o ,
2,(~2)o,
=+_, x~=
q~&
(23)
~-~ to.
The variational fields are of the form, .~ = a ( ~ o - too) + a'(too~ + <'~>o)
(16)
which shows that a momentum cut-off is needed to avoid ultraviolet di.vergencies. The cut-off parameter can be related to the critical number of modes which will destroy the ordered state. We would like to mention that the approximation scheme (Gaussian approximation and momentum cut-off) used here gives rise to spurious effects like simulations of first-order phase transitions. The self-consistency relations for homogeneous magnetization and volume follow from ao~/Omo = 0~'/Oto0 = 0, leading to
m0= --lml + ~ - - t . ,
5(m~)o+ <-,~)o,
(17)
+ b( 6m2 + 6(m:2)o + ( m 2 ) o ) + 3c(5 m4 + 30m2(m~)o +6m2o( m2 )o + 15 (m~)oz + 2(m~ )o2 + 6(mz2)o(mZz )o), (24) ~ l = ~ ( ~ - too) + ~'(o,o~ + (~;')o)
+ 2~,(m2 + (m~)o + 2(,.~)o) + 34,,,~ + 6,,,~(..:)o 4- 4m~o(,,,~)o + 3(m2) 2 + 6(m~ )20+ 4(mZ=)o(rn 2 )o), (25)
M. SchrSter et al. / Magnetism and magnetic volume collapse
172
/~,0 = ½K + 3),~0o +
a'(m2o + (m2)o +
For brevity, contributions oc 8 have been neglected. The Curie temperature follows f r o m / z z =/~ .L = 0, leading to (m2)c=
q,,,
1
21
4 3
z
\/
-
4~r2flcg,,, = "2-f-c[ - b + ~b2 - 3-Cac ] '
,,<
(27) ac = a((o c - ~0)c + a'((°o2 + ((°2)0)c,
which shows that T~ 0c gin~q,, (which m a y serve to fix the additional parameters). The number of critical modes is given by 3q3/6"~ 2. A simple discussion of the formula for the spontaneous magnetization shows that the spin-fluctuationdriven phase transition should be of second order for tm2(Zc) = 0 while it is of first order for t2t(T¢) = t,,2(T~) leading to the following condition for a first-order phase transition, 5B2/7 < A < B 2, P
2
A = (a(~o c - (o0) + a ((o o + (~O2)o))/3c,
(29)
B = b/3c. Actually, the magnetization curve has always a nose-like feature at T~ resembling a first-order magnetic transition. This is not in contradiction to the condition for a magnetic phase transition within the classical field theory, which reads , ~ ( r c ) = ~,j_ ( r J = 0 ,
o
(28)
(30)
since t% and # 1 decrease monotonously from T = 0 to T = T¢ as is shown in fig. 7 for the case of Fe3Pt. It seems that this simulation of first-order transitions is a c o m m o n defect in the spin fluctuation theory based on the Gaussian approximation, where the spontaneous magnetization is determined by a~'/Orn o = 0. It also seems that the unified theory of Moriya et al. [24] carries the same defect [25]. There is no simple prescription to get around this difficulty since the Gaussian approximation violates the fluctuation dissipation theorem [26]. In the present case, the nose-like
200
oo¢
600TtKSO0, 1000
Fig. 7. Temperature dependence of the variational fields /t: and/~ ± for the case of Fe3Pt. The condition for the ferromagnetic phase transition is /~z(T~)=/L x (Tc) = 0. However, note that there is a discontinuity at the higher temperature Tc, which is due to the Gaussian approximation.
feature of m o ( T ~ T~) only disappears for qm ~ oo (hence, for T~ ~ 0). T h e condition for a true first-order phase transition is t,2,1(T~) = t,,2(Tc).
(31)
In order to fulfill this condition, t.,t(T) and t,n2(T ) have to cross once while t~,t(T ) and t,.2(T) have to cross twice. Figure 8 shows that this condition is nearly fulfilled for the case of Fe3Pt.
6
i
i
i
i
t,
.
0
.
.
.
.
.
200
400
600
T[K]
.
.
.
.
800
.
.
.
.
1000
Fig. 8. Temperature variation of the functions defined in eqs. (17)-(22) (see text for explanation).
173
M. SchriJter et al. / Magnetism and magnetic volume collapse
It is also visible how strongly spin fluctuations renormalize the Landau coefficients in eqs. (19) and (20). On the scale of fig. 8 the variation of t,0] and l~2 is less pronounced. The phase diagram for metallic systems which results from eq. (29) is shown in fig. 4. There is a small region of real first-order phase transitions. Even if none of the considered systems behaves discontinuously at T~ it is obvious that all but bcc Fe are close to the parameter space of first-order phase transitions.
5. T h e r m o d y n a m i c s
The entropy is easily evaluated with the help of
5a= - ~"BT/
(32)
The final form of the free energy in Gaussian approximation is then ~ = d°- TSP,
(33)
K 2 + V,Oo 3 + 8(~Oo 4 _ 3(tO2>o) + a(,oc - ,~o)m~ e = ~'~o
× [ln 2flgmq~2(1 + x~_) - 2] + _~_/~± (,n~)o +
q3 12~ 2
×[ln2flg'~q2(l+x:) -{] + _~_l~,o(,n2,.,)o 3q3
12,rr2
q3 l.
12,rr2 J
(35)
This clearly shows that the free energy resembles, in parts, the free energy of a classical ideal gas (of spin fluctuations) with all the usual unphysical properties of an ideal gas for T---, 0. A second unphysical feature is connected with the increase of #- with T. The latter inconvenience could be avoided by using a temperature dependent cut-off parameter, q,,,tx if~3. Such a temperture dependent cut-off was also discussed by Murata [27]. For large cut-off parameters the corresponding derivatives of the free energy (entropy and specific heat) easily become negative. The specific heat follows from C= T(-~)
(36)
= Tk B ~ { ~ ( 1
a~kTaa~k)OT
+ a'(m~to~- (t~2>o((mz2)o +
+ 1
- 2
leading to
+ 18(m2)o( m2 )o + 6(m 2 )2]
C=
- 30(m2) 3 - 18(rn~)02(m~ ) o - (m2)o 3q3 ×
(34)
+
-• /~z(rn~)0 + 12~r'2q3'~--"
'
kB (303 + q 3 ) _ / 12~r2 -
- -
2, a/~: ~~
~m:/o
- ( m 2 ) o - ~ - -- (602>0~ ? .
q3 + 12~r-'--'~'
"fig"q" q3 [ln-~ ,rr z (1 + x-'2) - ~ ] 6a= - k B ( 121r2 [
bk aT
(38)
The general form of eq. (38) resembles the result for the specific heat obtained by Murata et al. [22] for the case that only spin fluctuations are retained. The consideration of volume fluctuations gives rise to the last term on the right-hand side of eq. (38). However, we would like to emphasize, that in spite of the additive nature of the different contributions in eq. (38) (which is due to the mean field-like ansatz eq. (12)), there is a strong interre-
174
M. Schr&er et al. / Magnetism and magnetic volume collapse
('~magneti,'- {~magnetic kB
q3,, 4,'ff 2 '
=
., /'f'1"
2 co o
(~VOLUt'4E
,
0
,~£
"~'--
0
m
. ,t" ~" f
|
'~',,,,, ,,
-:
"-G
i
0
200
. . . . .
i
400
- -
1
4
I
600 T [K] 800
12~r 2 "
•
] ~r
q2 ku
1000
Fig. 9. Temperature dependence of the different specific heat contributions as obtained from eq. (38) ior Fe3Pt.
lation between magnetic and volume contributions due to the self-consistency conditions. The result given in eq. (28) put in a different form has also been obtained by Wagner [19]. It is obvious from the analytical structure of eq. (38) that, at large enough temperatures, the, pecific heat necessarily will become negative for the case where the volume contribution is smaller than the magnetic contribution. Yet the behaviour of C near ~ is not unreasonable. By this we mean that the way in which the specific heat increases when approaching T~ from below and the m~gnituc?- of the discontinuity resemble results of more refined caculations. Since Fe3Pt is one of the most interesting INVAR systems, we have plotted in fig. 9 the different specific heat contributions for this system as obtained with the help of eq. (38), whereby the following dimensionless quantities were used,
Here, the indices refer to the different contributions in eq. (38). Since the volume variation below Tc is very small, the main contribution to the specific heat is of magnetic origin in this temperature range. For T > Tc the magnetic contribution always decreases with increase in temperature wtfile the volume contribution becomes larger. Thus, magnetic contributions and volume contributions partially compensate each other at high enough temperatures. This different behaviour for T > T~ is caused by the anharmonic term which leads to a negative contribution in eq. (26) and to a sign reversal in eq. (38). Details in the temperature range between T~ (temperature at which a second-order phase transition should occur due to the Landau condition) and Tc (temperature at which the pseudo-first-order phase transition occurs due to the Gaussian approximation) have been left out. Unfortunately, we cannot compare the different specific heat curves of fig. 9 with experiment, since detailed experimental specific heat data of ordered Fe72Ptzs do not exist. Other thermodynamic quantities can easily be calculated. For example, the relative volume change is given by
°h'(T) =
®{
(co2)°(T) + 3 -°~
+(m2>o(T)+<,n2>o(T)] 2~
a
[m2o+(,n:)o+(m~) °
9y""
2
,
(39)
which, in the case of a first-order phase transition, will lead to a discontinuous volume change of the order of
acoo(T ) =
a
2
-
).
(40)
The thermal expansion is given by CZto,a' = C~o,~, kB
1 (,~q~+ q3), 12~r2
1 atoo(T) a ( T ) = 1 + Coo(T ) aT '
(41)
M. SchrSter et aL / Magnetism and magnetic vole'me collapse
fixed-moment method. The classical gas of spin fluctuations can undergo a phase transition and, depending essentially on only two parameters, b and 0% metallic systems can be classified as magnetically stable with a second-order transition, as magnetically unstable with a first-order transition associated with a magnetic volume collapse, or as paramagnetic systems. Whereas stable magnetic systems possess only one local minimum on the binding surface, all systems being on the verge of a magnetic volume collapse have (at least) two minima which are close in energy. Two minima always exist if
8
Tc 6--. 6~ x-" o
calculated
4
IL
2 0000 0
o
-4
~
o
°o~ l
-2
~OQID
Tc
175
Fe72 PI2a • ordered o disordered
Do < l ~ - -
-6
V0(0 ) a n d b < O
(42)
-b
o
2b0
4b0 6bo
8b0
ooo
T [K] Fig. 10. Calculated thermal expansion of Fe3Pt as compared with experimentM results [30].
which leads to quantitative results being of correct order of magnitude. Figure 10 shows the calculated thermal expansion for Fe3Pt as compared to experiment. Besides, at very low temperatures the behaviour of a ( T ) is very reasonable. Further thermal properties, like compressibility, pressure dependence of T~, etc. can easily be obtained. In all calculated quantities the same unphysical features appear at very low and very high temperatures. At intermediate temperatures the mutual interplay of spin and lattice degrees of freedom can well be studied within the pr£~ent scheme. But the disadvantages due to the G~ass ian approximation and the neglect of quar~'~um mechanical aspects show that any improver~ent would be highly welcome.
6. Conclusions
We have shown that magnetovolume effects it~ metallic systems can be described with the help o~ a free energy functional of the Landau type, whereby the coefficients are fixed by zero temperature results for E ( m o, Vo) obtained with the
is fulfilled, where ~2~ is the volume at which the system can, without energy change, polarize and B 0 is the hypothetical nonmagnetic reference volume. The thermodyaamics of systems tending to a magnetovolume instability is closely related to the temperature evolution of the low- and high-spin states. Indeed, a two-states model coming close to the present picture was used to describe the a ~ 3' transition in iron [28]. The drawbacks of the present formulation have been critically discussed. They are mainly connected with the Gaussian approximation employed in tiffs work. An extension of the present formulation to avoid the Gaussian approximation is in progress. However, a systematic improvement is only possible within a quantum mechanical formulation. Especially, a microscopic theory of low-spin to high-spin transitions and of the INVAR effect has to be formulated. Most approaches, which can be found in the literature, are embedded in phenomenological models. The microscopic models which exist are mostly based on the single band or degenerate-band Hubbard model. Functional integration is performed in the static approximation, which is hard to overcome. Also, the incorporation of electronic details near the Fermi level would require tremendous computational efforts. For further references in this context we refer one to the work of KakehastJ [29] and for an overview of the present state of art of the spin fluctuation theory to ref. [6].
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