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Interacting virtual state approach to the magnetism of transition metals-Cr and FeHr-
199 I N T E R A C T I N G V I R T U A L STATE A P P R O A C H TO T H E M A G N E T I S M OF T R A N S I T I O N M E T A L S - C r and F e H r Y. T E R A O K A and J. K A N A M O R I * College of General Education, University of Osaka Prefecture, Sakai 591 and *Department of Physics. Osaka University. Toyonaka 560, Japan A new theory based on the Anderson model of 3d virtual states is developed to discuss the magnetism of transition metals from the point of view unifying the concept of a localized moment with that of itinerant electrons. It explains quantitatively various features of experimental data on the spin density wave in Cr, and leads us to a new view of the mechanism of the formation of the SDW. It can elucidate also the mechanism underlying the transition from the antiferromagnetic state Io the ferromagnetic one in FeRh.
I. Introduction
We propose here a localized m o m e n t treatment of the magnetically ordered states of a transition metal which corresponds to an extension of the A l e x a n d e r - A n d e r s o n - M o r i y a theory of the exchange interaction between two transition element magnetic atoms [1]. We treat the inter-atomic electron transfer as a perturbation on the 3d virtual bound state of each atom in the one-electron approximation: we determine the 3d levels in the H a r t r e e - F o c k scheme to minimize the total energy including the perturbation energy of the electron transfer. The magnitude of the magnetic moments and the effective exchange integral of their interaction thus calculated depend on the relative orientation of the moments, reflecting the itinerant character of d electrons. After discussing the formalism in section 2, we present an application to the spin density wave (SDW) in Cr in section 3. It is generally believed that the SDW is generated by the nesting of the Fermi surfaces around the F and H points in reciprocal space. The c o m m o n l y used model which considers only the states near the nesting Fermi surfaces explicitly, however, seems to be inadequate for discussing the SDW quantitatively, since the calculated SDW amplitude is too small c o m p a r e d with the observed value unless one adds a contribution from a not well defined "electron r e s e r v o i r " [2]. This fact is consistent with Windsor's calculation of the unenhanced q-dependent susceptibility to which the nesting states prove to contribute only an eighth at m a x i m u m [3]. Our approach, on the other hand, yields good quantitative agreement with the data on the SDW first and third harmonics [4], the charge density wave (CDW) [5], the magnetic m o m e n t of the antiferromagnetic Physica 91B (1977) 199-204 © North-Holland
(AF) state at the S D W - t o - A F transition in C r Mn alloys [6], the volume change at the same transition [7] and the strain wave [8]; we shall not discuss the last two subjects in this paper. We can explain also the first order transition at Tn. A new interpretation of the mechanism underlying the SDW formation in Cr will be given as a conclusion. In section 4 we discuss the FeRh ordered alloy of the CsCI type which undergoes a transition from the AF to the ferromagnetic (F) state with increasing temperature. The Rh atom has no magnetic m o m e n t in the AF and acquires about I/.t B in the F, while the Fe atom has about 3/~R in both phases [9]. We present a microscopic theory of the AF-to-F transition and also the magnetization process of the AF state under an external field at low temperatures, calculating the electronic structure by use of the present approach. In section 5 we discuss briefly various elaborations of the approach, since we assume the simplest version of it in the preceding sections. 2. F o r m a l i s m
In this paper we assume the single orbital Anderson model with a constant virtual level half-width A to describe the unperturbed state. The perturbation expansion of the one-electron Green function to the second order of the interatomic electron transfer yields the local state density of electrons with spin (7 at the jth atom given by o
pj.(E) = p~.(E)
+ ~'. v; l
{Pi-~,(~) - -
-
o
Pit E)
----~
\ (Ei+l,, - E i , , £
200
with p'),, r e p r e s e n t i n g the I o r e n t z i a n c e n t e r e d at the H a r t r e e - F o c k a t o m i c level K,,.
O<,',,(~) = (Alrr)l{(e
li'i<,):
+ A:[.
(2)
Here e is the e n e r g y v a r i a b l e , V; is the a,~erag, e t r a n s f e r integral to the n e i g h b o r s specified by / and /-2,, is to be d e t e r m i n e d s e l f - c o n s i s t e n t l y to satisfy h),, : k] + Un, ,.
(3)
in ~:his E] r e p r e s e n t s the e n e r g y level in lhe a b s e n c e of the C o u l o m b i n t e r a c t i o n b e t w e e n e l e c t r o n s : % ,, is the a v e r a g e n u m b e r of elecIron., with spin cr which is o b t a i n e d by integrating p; ,Je) given by eq. (I) up to lhe F e r m i e n e r g y at a b s o h t t e zero. In the f o l l o w i n g calct,httion we take a c c o u n l of the t r a n s f e r to the n e a r e s t and next n e a r e s t n e i g h b o r s only. s p e c i f y i n g the c o r r e s p o n d i n g t r a n s f e r integrals by Vi and V:. r e s p e c l i v e l v . We a s s u m e V l / A = 0 . 3 2 and V:/A 0.12 u n l e s s o t h e r w i s e specified; the ratio V J V , is in good a g r e e m e n t with that o b t a i n e d from the a v e r a g e s of Pettifor's parameters of the tight b i n d i n g c a l c u l a t i o n of the bcc d band [10[. We a s s u m e that the e x p a n s i o n to s e c o n d o r d e r in the /,' values is q u a n t i t a t i v e l y m e a n i n g f u l in a bcc metal in the c a l c u l a t i o n of those quantitie,, which do not d e p e n d on the details of lhe e n e r g y s p e c t r u m . In fact the stale d e n s i t y of the n o n m a g n e t i c (NM) state calcuhtted by use of eq. (I) r e p r o d u c e s well the two peak s t r u c t u r e of the d b a n d in a bcc metal, as is s h o w n in fig. I.* I n t e g r a t i n g eq. (1) to the F e r m i e n e r g y E~. we o b t a i n s e l f - c o n s i s t e n t e q u a t i o n s for the hi,
values, e~ ix d e t e r m i n e d by the c o n d i t i o n thai the a v e r a g e n n m b e r of e l e c t r o n s per atom c a [ cttlated from the hi,, values should agree wilh a given value, N,r T h e s e l f - c o n s i s t e n t e q u a t i o n s are solved n u m e r i c a l l y b} an iterati~e mcthod. The a b o v e m e n t i o n e d N t and the (X,ulomb in tegral p a r a m e t e r I_!/&) are varied as p a r a m e t e r s to find the most a p p r o p r i a t e situation for the problem under consideration.
3. The spin density wave in (Tr We a s s u m e in the c a l c u l a t i o n of the S D W thal the p e r i o d in units o f the l a t t i c e c o n s l a n t ix ;il3 i n t e g e r p w i t h the w a v e v e c t o r p o i n t i n g in the I0011 d i r e c t i o n . F u r t h e r m o r e , the n o d e s o f the
e n v e l o p e of atomic m o m e n t s are a s s u m e d to he h a l f w a y b e t w e e n lwo a d j a c e n t (001) atomic p l a n e s , as is illustrated in fig. 25. S i n c e a t o m s on the same (001) plane are in the same state, we d e t e r m i n e in the s e l f - c o n s i s t e n t c a l c u l a t i o n n~, values of a t o m s for o n e - q u a r t e r of the period s p e c i f e d , m a k i n g use of the s y m m e t r y of the solution s h o w n in fig. 2 with r e s p e c t to the loop
I:ig. 2 A schematic representation ,af lhc envelope m3,J :tt,c,mic mc, mer'Hs in the 5,1)W '.tatL. with p 8. \ f r o , a i n d i c a t e the a t o m i c i l l o m e n l s of e a c h {0(1]) a t o m i c pl:mc
p(E)
~E
Fig. I. The density of stales calculated with eq. (iI for the l i o n r n a g n e t i c st;ate.
* In the ca>e of an fcc metal which we do nol discuss hc~c. il turns out t h a t w e need at leaM the third order term in V~. Note that such a tern] is absenl in the bcc ca,,c
of the e n v e l o p e . The e q u a t i o n s are ,~olved b~ iteration with arbitrarily c h o s e n initial values of the %,,: v a r i o u s c h o i c e s of the initial values 'arc a d o p t e d to see thai they c o n v e r g e to the same final solution after iteration. The iteration i,~ c o n t i n u e d until the difference b e t w e e n the input and output of each %, at a cycle b e c o m e s smaller than 10 ~ and the root m e a n s q u a r e of the fNote that in most cases we do not need an explicit value of 3. if other energ), p a r a m e t e r , , a r e m e a s u r e d in unit,, ~>f 3 :~We have adopted in some calculation', a model in v,hich the node i,, placed on the atomic phmc to confirm that the re,,ult doe,, not depend appreciahl~, on the node D>~ilion
201 difference over all ni~ values is smaller than 10 7"
The calculation of the AF, F and nonmagnetic (NM) states can be carried out in a much simpler way. Assuming different quantization axes of spin for atoms and modifying eq. (1) appropriately, we can obtain the helical (H) solution as well. When U is not too large, we can obtain the condition for the appearance of a magnetic solution of a given p with infinitesimal magnetic amplitude by expanding the self-consistent equations in powers of deviation of each nj,, from its value in the NM state, Na/lO. We confine ourselves here to the case Nd ~> 5, because this case yields the observed phase relation between the SDW and the charge density wave (CDW) [8]. Then we find that the magnetic solution is lower in energy than the NM for Nd larger than a critical value No; N~ turns out to decrease with increasing p, being smallest for the AF. Nc values for the SDW and H for the same p coincide with each other. N c decreases with increasing U. When U is sufficiently large, the magnetic solution with finite amplitude becomes energetically equal to the N M at a critical value of Ne when Ne is increased from N a - - 5 . This critical value which we call again N~ can be determined by energy comparison only. Thus the range of N~ is divided into the second order regime corresponding to larger N~ for smaller U and the first order regime corresponding to N~ close to 5 for larger U. Fig. 3 shows the calculation of the m a x i m u m amplitude of the SDW as well as that of the AF for a given value of U.
10-
05 L
SDW20 i~
~ HELIX 20
AF 00 t 515
,
~./l
....................
~5.20 Nd <
5.25
OO
:E/& --0.005
Fig. 3. The u p p e r part s h o w s the m a x i m u m a m p l i t u d e of the S D W w i t h p = 20 as well as the m a g n i t u d e of the a t o m i c m o m e n t in the A F a n d H s t a t e with p = 20 as a f u n c t i o n of Nd. U/A = 4.42 is a s s u m e d . The d o t s s h o w the m a x i m u m a m p l i t u d e of the first h a r m o n i c of the S D W . The l o w e r part s h o w s the e n e r g y of t h e s e s t a t e s m e a s u r e d f r o m that of the n o n m a g n e t i c state. T h e full line, for e x a m p l e , is the e n e r g y of the S D W w i t h p = 20.
There the choice of U is such that N~ values for large p are close to the boundary between the second order regime and the first order one, as can be seen from the steep rise of the amplitude with increasing Na at Na = N~. As is discussed below, the SDW can be of lowest energy for such a choice of U. We believe that the first order nature of the transition at the Ndel temperature in Cr is related to this first order appearance of the magnetic solution; we shall report a calculation at finite temperatures in the future. The first order appearance is due to the presence of the deep valley at the middle of the state density of the NM shown in fig. 1. As is well known, we can expect the first order transition when the Fermi level is near a minimum of the state density. Fig. 3 shows also the energies of the SDW, AF and H states plotted against Na, from which we can see that the SDW is of lowest energy in a limited range of N a with the amplitude around 0.4 P'B per atom. By varying U, we find that the SDW can be the ground state for U/A = 4.3 4.5; Nc values of the SDW and AF are around the boundary of the first order regime for such choices of U/A. The energy of the H state, on the other hand, remains always higher than that of the AF state. It is difficult within the adopted accuracy of numerical solution to determine precisely the period p of the SDW of lowest energy, though the p value of lowest energy seems to be around 20 with the tendency to decrease first and increase after passing a minimum with increasing Ne. An example of the envelope of the SDW whose energy is lower than the AF state is shown in fig. 4. When N a or U is increased beyond the range where the SDW is of lowest energy, the envelope b e c o m e s more rectangular, or in other words, approaches the antiferromagnetic state with periodic antiphase boundaries. The numerical solution obtained a b o v e is consistent with various experimental data. The magnitude of the magnetic m o m e n t in the AF state is shown to be almost equal to the amplitude of the first harmonic of the SDW for a given Ne except for Ne near N~ (see fig, 3). This result is in good agreement with the experimental data on C r - M n alloys which show the S D W - t o - A F transition [6], while the nesting model seems to fail to explain it [2]. Furthermore, the envelope of the SDW shown in fig. 4
202
-*~ J
"x X\
¢" / \
•
\
-
l/
••
z
N
\,%
s •
Fig. 4. The calculated envelope of lhe SI)W wilh p :20 for N , , - 5.209 with the maximum an'lplitude of 0.38 # , / a l o m i~ given by the full curve. The broken curve is lhc []r,.l harmonic of the SDW. The dot-dashed curve represents the as'>ocialed CDW on an arbilrary scale. The amplinlde of the first harmonic of the CDW is menlioned in lhe text. Nole thai electrons accumuhtle a! lhe loop of the SD~vV.
g i v e s the ratio of the a m p l i t u d e of the lhird h a r m o n i c to that of the first h a r m o n i c as 0.021 w h i c h is in g o o d a g r e e m e n t with the cxp e r ) m e n t a l v a l u e 0.0165_+0.05 141. T h e ~tss o c i a t e d C D W has an a m p l i t u d e of 0,()(t6g elect r o n s / a t o m w h i c h ix c o n s i s t e n t with a r e c e n l d i r e c t m e a s u r e m e n t [5]. A l s o lhe strain w a v e a m p l i t u d e e s t i m a t e d f r o m this C D W is in a g r e e m e n t with e x p e r i m e n t a l d a t a [4, gl. T h e m e c h a n i s m w h i c h l o w e r s the e n e r g y of the S D W r e l a t i v e to the A F is a c o m p r o m i s e b e t w e e n the t w o a n t ) f e r r o m a g n e t i c c o u p l i n g s of the n e a r e s t and n e x t n e a r e s t n e i g h b o r s w h i c h ix a c h i e v e d with the help o f the C D W f o r m a t i o n . N o t i n g that the c o u p l i n g of the next n e a r e s t n e i g h b o r s is s a c r i f i c e d in the A F , we m a y e x p e c l that the spatial m o d u l a t i o n o f the a t o m i c magnetic m o m e n t can l o w e r the S D W e n e r g y . In fact we find that by w l r y i n g V, the r a n g e of N,; w h e r e the S D W is l o w e r in e n e r g y than the A F t e n d s to d i s a p p e a r for s m a l l e r 17,,. On the o t h e r hand, the H state w h e r e one might e x p e c l a s i m i l a r compromise w i t h o u t C D W c a n n o t be lower in e n e r g y than the A F in the p r e s e n l c a l c u l a t i o n , w h i c h is c o n s i s t e n t with the wellk n o w n f a c t in the c a s e o f the H e i s e n b e r g m o d e l of the bcc lattice. In lhe S D W the a s s o c i a t e d C D W d e c r e a s e s the n u m b e r of e l e c t r o n s al n e a r l y n o n m a g n e t i c a t o m s a r o u n d the node and increases that at a t o m s a r o u n d the loop. T h e f a c t that we find the S D W g r o u n d state tit N<# n e a r the b o u n d a r y of the first o r d e r a p p e a r a n c e r e g i m e m e a n s that a slight d e c r e a s e of the electron n u m b e r a r o u n d the n o d e can r e d u c e the e n e r g y i n c r e a s e c a u s e d by the p r e s e n c e of n e a r l y n o n m a g n e t i c a t o m s , s i n c e a smaller n u m ber of e l e c t r o n s f a w ) r s the N M. A l s o lhe fact
lhal the S I ) W spatial m o d u h i l i o n can contain the higher h a r m o n i c s appropriately, c o n l r i b u l c s h> the e n e r g y lowering. A h o g e l h e r ~ve can un d e r s t a n d q u a l i t a t i v e l y the r e a s o n v~,hv' we find lhe S D W g r o u n d s t a l e in a r a t h e r limited range of the c h o i c e s of p a r a m e l e r s . If we :ire well inside the first o r d e r regime with large 17, for e x a m p l e , the spatial m o d u l a t i o n b e c o m e s difticult b e c a u s e of a well d e x e l o p e d a l o m i c magnetic m o m e n t in lhe AF. W h e n we take note of the v o l u m e c h a n g e :ll the S D W - t o - A F l r a n s i l i o n , we should c o n s i d e r lhe r e d i s t r i b u t i o n of e l e c l r o n s h e l w e e n the d s l a t e s and the s hand c a u s e d by a c h a n g e of lhe F e r m i e n e r g y . W c can ',how, h o w e v e r , lhal, il d o e s not m a k e an a p p r e c i a b l e c c m t r i b u l i o n Io the e n e r g y d i f f e r e n c e h e l w e e n the S D W and lhc A F d i s c u s s e d a b o v e . W c shall nol go inlo lhe d i s c u s s i o n of this s u b j e c l lind also thai of the r e l a t i o n b e t w e e n the p r e s e n l theor$ and lhe n e s t i n g m o d e l l h e o r y . W e tit) IlOt i n l e n d al p r e s e n t it) d e n y the p o s s i b i l i t y that lhe nesting of lh,3 F e r m i s u r f a c e p l a y s an i m p o r t a n l role in d e t e r m i n i n g the p e r i o d of the S D W . ()ur lheor,,. on the o t h e r hand, yields a q u a n t i t a t i v e l y g o o d d e s c r i p t i o n of the S D W and a r e a , , o n a b l e exp h m a t i o n for the facI lhal we find the S D W in ( ' r o n l y a m o n g the t r a n s i t i o n m e t a l s .
4. The AF-to-F transition in ordered FeRh alloy W e a s s u m e here the ( ' s ( ' l l y p e o r d e r i n g of ke and Rh a t o m s with equal c o n c e n t r a t i o n , deferring the c a s e of n o n s t o i c h i o m e t r y to future ptlblicalion. ()ur calculation shows within r e a s o n a b l e c h o i c e s of p a r a m e t e r s that Rh :lion1>. have no l o c a l i z e d m o m e n l in lhe A F Male. while Fe a t o m s h a v e a m o m e n l of a b o t l l -4 #,t~. It s h o w s thai the n u m b e r of e l e c t r o n s of Fe a l o m s ix d e c r e a s e d in F e R h to lhe c x l e n l lha! Ihc interaction belween Pc tilt)ms is a n t i f e r r o m a g n e t i c . T h e i n t e r a c t i o n b e t w e e n F'e and Rh a l o m s , on lhe o t h e r hand, f a v o r s lhe ferromagnetism. We discuss the magnelization process of lhe A F Male at absolute zero. in order to elucidate the situalion. Assuming lha[ the A F Male is the grotlnd stale, we apply an external field H to it. I'hen lhe sublattice magnetizations of Fe tllorns and the Rh magnetization take lhe configuration shown in fig. 5 ill the absence of anisolropy. We denole Fe atoms on the sublatlice 1 b ) Fel.
203
EFrFe '1' TMRh '(~e) !15
(Xl0 3)
EFe-Fe
0 ~i Rh
Fel
~
J
4.0
Fell
--~ ~ J
---
-
- . . . .
i1.0
Fig. 5. T h e c o n f i g u r a t i o n of the s u b l a t t i c e m a g n e t i z a t i o n s in the A F s t a t e of F e R h u n d e r a n e x t e r n a l field.
5 /.;-~
o.o Taking the spin quantization axis for each sublattice in the direction of its magnetization, we add the Z e e m a n energy - o ' H p . ~ c o s 8 and crH/xB with or = - 1 to E~ given by eq. (3) for Fe and Rh atoms, respectively; 8 is defined in fig. 5. n~ in eq. (3) is now the number of electrons with spin tr referred to the abovementioned quantization axis; we may assume nFeh, = nFell,~ and hence E F e l ~ = E F e l l ¢. We note that the electron transfer between tr spin states and - o - ones is allowed between FeI and F e l l and also between Fe and Rh. Calculating the total energy to the second order of the electron transfer, we determine the angle 8 by the torque balance condition, dE/d8 = 0, given by
111Rh
_ i-
-1.0
EFe_R h
-~.._
_
-
E~sin 8 E~
=
+ E [ xe-ah
Fe-Fe • ½(sin 8) + Ee~ sin 28 = 0,
- ( n F e . - nFe-,.)P.BH, 0
(5) 0
~Fe-Rh-- 8 V ] ~
(
--ex
. ERh~r -- E F e ¢
--
(4)
0
nRh°-- riFe°
0
nRh-°'--rife° ) ~
----EFe¢(6)
E Fe= 3v
N ( -- PFe~r(~F) 0
0
0
EFe_,r -- g F e . ] '
(7)
0 where nFe¢ is the quantity obtained by integrating poe to the Fermi energy. The first term in eq. (4) is the torque due to the external field; Eex for the second and third terms may be called the " e x c h a n g e energies" for F e - R h and F e - F e . The a b o v e expressions for Eex agree with those derived by Moriya [1], though our Ee~ varies with the change of the magnetic configuration with increasing H * . We note that the variation of Eex represents the deviation f r o m the Heisenberg model in which it is a constant. Fig. 6 shows examples of the calculated Eex vs. 8 relation; the corresponding magnetization processes are shown in fig. 7. We assume the same V~/A and VE/A as those for Cr in the calculation. * O u r E ~ is c a l c u l a t e d o n the b a s i s of the e l e c t r o n i c s t r u c ture w h i c h is d e t e r m i n e d s e l f - c o n s i s t e n t l y for a g i v e n H.
EFe-Rh Fig. 6. The "exchange energies" defined by eqs. (6) and (7) and the Rh atomic magnetic moment at the equilibrium angle 0 s a t i s f y i n g the t o r q u e b a l a n c e e q u a t i o n for a g i v e n v a l u e of H ; t h u s H v a r i e s w i t h & T h e full c u r v e s c o r r e s p o n d s to c a s e a in fig. 7 w i t h Nd = 7.218; the b r o k e n o n e to c a s e b w i t h Nd = 7.212; the d o t - d a s h e d o n e to c a s e c w i t h Nd = 7,206. U/A = 10 a n d (Ev, - ER,)/A = 0.054 are a s s u m e d for all cases.
The values of other p a r a m e t e r s are given in fig. 6. In fig. 7 we present three typical examples of the calculated magnetization curve at absolute zero, a, b and c. In case a the Rh m o m e n t and EF~ ~ shown in fig. 6 tend to saturate with decreasing 8, showing the localized m o m e n t behavior at the final state 8 - 0 . In this case the strength of EF~-Rh relative to that of EF~-Fe is small c o m p a r e d with other cases. As we increase the relative strength of E [ xe-Rh w e go towards case c via case b. In the present choice of p a r a m e t e r s this increase of the relative strength is brought in by a decrease in the average number of electrons per atom, Na (see ~.B/ATOM 2.0
"-:.--\\
F:ERRo. /
1.0 !,d
"
b /
/
o.o~o%1
o602
/
a.
/" /
obo3 ..H)~
Fig. 7. T h e m a g n e t i z a t i o n p r o c e s s in the A F s t a t e of F e R h at a b s o l u t e zero. T h e u n i t s of the e x t e r n a l field are s u c h t h a t ~BH/A = 0.001 c o r r e s p o n d s to H = 105 O e for A = 0.05 Ry. T h e c u r v e d is o n l y s c h e m a t i c .
204
fig. 6). For a fixed N,; smaller values of U/A a n d / o r E~:~- ERh also i n c r e a s e the relative strength of ~TFe ~ Rh , leading us t o w a r d s case c. It is highly i n t e r e s t i n g and d e s i r a b l e to d e t e r m i n e e x p e r i m e n t a l l y which case of the m a g n e t i z a t i o n c u r v e is realized in a given alloy. We can easily e x t e n d the a b o v e d i s c u s s i o n to finite t e m p e r a t u r e s within the o n e - e l e c t r o n app r o x i m a t i o n . Defining the c h a n g e e n e r g i e s by the t o r q u e b a l a n c e at finite t e m p e r a t u r e , we find that the relative strength of ~lIE;7.[:e , Rh i n c r e a s e s rapidly with i n c r e a s i n g t e m p e r a t u r e , leading us finally to case d in fig. 7, where the A F - t o - F t r a n s i t i o n takes place in the a b s e n c e of an external field. O u r c a l c u l a t i o n can r e p r o d u c e well the p h a s e d i a g r a m in the T - H plane [111. The m e c h a n i s m u n d e r l y i n g the t r a n s i t i o n is an increase of the m a g n e t i c p o l a r i z a b i l i t y of Rh atom and a d e c r e a s e of ,~., with i n c r e a s i n g ten> perature. In case b in fig. 7 the t r a n s i t i o n temp e r a t u r e is 3 1 6 K for A - 0 . 0 5 R y . Our calc u l a t i o n e x p l a i n s also the d e c r e a s e of the Fc a t o m i c m o m e n t by 0.1 /x u in the F phase compared to that in the A F p h a s e [91, the d i f f e r e n c e of the e l e c t r o n i c specific heat b e t w e e n the A F and the F phase and the v o l u m e c h a n g e associated with the t r a n s i t i o n . Details of the calc u l a t i o n will be r e p o r t e d in a f u t u r e p u b l i c a t i o n .
p s e u d o - G r e e n i a n f o r m a l i s m 112]. T h e p r e s e n c e of the s b a n d , or in o t h e r words the free electron b r a n c h , musl be taken into a c c o u n t in some p r o b l e m s , as was m e n t i o n e d at the end of seclion 3. Apart from these e l a b o r a t i o n s we can apply the p r e s e n t t h e o r y It) several p r o b l e m s where we need to u n i f y the c o n c e p t of localized mom e n t with that of itinerant e l e c t r o n s . A m o n g them the easiest one is the c a l c u l a t i o n of the spin wave stiffness in a f e r r o m a g n e t i c metal. We have made also an a p p l i c a t i o n to the s u r f a c e m a g n e t i s m of t r a n s i t i o n metals. We report on these p r o b l e m s as well as the details of the p r e s e n t s t u d y in future p u b l i c a t i o n . We t h a n k P r o f e s s o r discussions.
H. Miwa for v a l u a b l e
References I II I. Moriya, Progr. ]'hooF. Phs~. 33 11965) I s7 121 A. Shibatani, K, Motizuki :rod T. Nagamiya, Phy',. Re'.. 177 11969) 984.
A. Shibatani, J. Phys. Soc. Jap. 39 (1975) 8~1. 17;I C.G. Windsor, J. Phys. F. 2 11972) 742. [4] R. Pynn, W. Pres~, S.M. Shairo and S.A. Wermer, Phys. Rev, B13 (1976) 295. [5] Y. l'sunoda, private communication.
5. Concluding remarks We have p r e s e n t e d two a p p l i c a t i o n s of our " i n t e r a c t i n g virtual s t a t e s " a p p r o a c h in its simplest v e r s i o n . An o b v i o u s e l a b o r a t i o n of the c a l c u l a t i o n is to d i s t i n g u i s h b e t w e e n the de and d-/ orbitals t o g e t h e r with the i n c l u s i o n of the i n t e r - o r b i t a l e x c h a n g e i n t e r a c t i o n in the A n d e r son model. We have to i n t r o d u c e then three kinds of t r a n s f e r integrals, ddcr, ddvr and ddTr'. We can take into a c c o u n t also the higher order t e r m s in the p e r t u r b a t i o n e x p a n s i o n of the transfer, which we c a r r y out on the basis of the
[6] Y. Hamaguchi, E.(). Wollan and W.C. Koehler, Ph,,x Rev. 138 (1065) A7~7. 171 Y. Ishikawa, S. Hoshino and Y. Endoh, J. Phys. Soc .lap. 22 11967) 1221. N. Kunitomi, private communication. [8] Y. Tsunoda, M. Mori, N. Kunitomi. Y. I'eraoka and J. Kanamori, Solid Stale ('ommun. I4 (1974) 287 Y. Tsunoda, Y. Nakai and N. Kunitomi. Solid State (ionlrnun. 16 (1975) 443. 191 G. Shirane, R. Nathans and ('.W. ('hen, Ph~,s. Rex. 1~,4 {1964) A 1547. [10l D.G. Penifor, ,I. Phys. ('. 2 11969) 1051. [Ill B.K. Poromarev. JETP ~6 (197:{I 1(15. 1121 J. Kanamori. K. Terakura and K. Yamada, Progr. l'heor. Phys. 46S 11970) 221.
I. Introduction
We propose here a localized m o m e n t treatment of the magnetically ordered states of a transition metal which corresponds to an extension of the A l e x a n d e r - A n d e r s o n - M o r i y a theory of the exchange interaction between two transition element magnetic atoms [1]. We treat the inter-atomic electron transfer as a perturbation on the 3d virtual bound state of each atom in the one-electron approximation: we determine the 3d levels in the H a r t r e e - F o c k scheme to minimize the total energy including the perturbation energy of the electron transfer. The magnitude of the magnetic moments and the effective exchange integral of their interaction thus calculated depend on the relative orientation of the moments, reflecting the itinerant character of d electrons. After discussing the formalism in section 2, we present an application to the spin density wave (SDW) in Cr in section 3. It is generally believed that the SDW is generated by the nesting of the Fermi surfaces around the F and H points in reciprocal space. The c o m m o n l y used model which considers only the states near the nesting Fermi surfaces explicitly, however, seems to be inadequate for discussing the SDW quantitatively, since the calculated SDW amplitude is too small c o m p a r e d with the observed value unless one adds a contribution from a not well defined "electron r e s e r v o i r " [2]. This fact is consistent with Windsor's calculation of the unenhanced q-dependent susceptibility to which the nesting states prove to contribute only an eighth at m a x i m u m [3]. Our approach, on the other hand, yields good quantitative agreement with the data on the SDW first and third harmonics [4], the charge density wave (CDW) [5], the magnetic m o m e n t of the antiferromagnetic Physica 91B (1977) 199-204 © North-Holland
(AF) state at the S D W - t o - A F transition in C r Mn alloys [6], the volume change at the same transition [7] and the strain wave [8]; we shall not discuss the last two subjects in this paper. We can explain also the first order transition at Tn. A new interpretation of the mechanism underlying the SDW formation in Cr will be given as a conclusion. In section 4 we discuss the FeRh ordered alloy of the CsCI type which undergoes a transition from the AF to the ferromagnetic (F) state with increasing temperature. The Rh atom has no magnetic m o m e n t in the AF and acquires about I/.t B in the F, while the Fe atom has about 3/~R in both phases [9]. We present a microscopic theory of the AF-to-F transition and also the magnetization process of the AF state under an external field at low temperatures, calculating the electronic structure by use of the present approach. In section 5 we discuss briefly various elaborations of the approach, since we assume the simplest version of it in the preceding sections. 2. F o r m a l i s m
In this paper we assume the single orbital Anderson model with a constant virtual level half-width A to describe the unperturbed state. The perturbation expansion of the one-electron Green function to the second order of the interatomic electron transfer yields the local state density of electrons with spin (7 at the jth atom given by o
pj.(E) = p~.(E)
+ ~'. v; l
{Pi-~,(~) - -
-
o
Pit E)
----~
\ (Ei+l,, - E i , , £
200
with p'),, r e p r e s e n t i n g the I o r e n t z i a n c e n t e r e d at the H a r t r e e - F o c k a t o m i c level K,,.
O<,',,(~) = (Alrr)l{(e
li'i<,):
+ A:[.
(2)
Here e is the e n e r g y v a r i a b l e , V; is the a,~erag, e t r a n s f e r integral to the n e i g h b o r s specified by / and /-2,, is to be d e t e r m i n e d s e l f - c o n s i s t e n t l y to satisfy h),, : k] + Un, ,.
(3)
in ~:his E] r e p r e s e n t s the e n e r g y level in lhe a b s e n c e of the C o u l o m b i n t e r a c t i o n b e t w e e n e l e c t r o n s : % ,, is the a v e r a g e n u m b e r of elecIron., with spin cr which is o b t a i n e d by integrating p; ,Je) given by eq. (I) up to lhe F e r m i e n e r g y at a b s o h t t e zero. In the f o l l o w i n g calct,httion we take a c c o u n l of the t r a n s f e r to the n e a r e s t and next n e a r e s t n e i g h b o r s only. s p e c i f y i n g the c o r r e s p o n d i n g t r a n s f e r integrals by Vi and V:. r e s p e c l i v e l v . We a s s u m e V l / A = 0 . 3 2 and V:/A 0.12 u n l e s s o t h e r w i s e specified; the ratio V J V , is in good a g r e e m e n t with that o b t a i n e d from the a v e r a g e s of Pettifor's parameters of the tight b i n d i n g c a l c u l a t i o n of the bcc d band [10[. We a s s u m e that the e x p a n s i o n to s e c o n d o r d e r in the /,' values is q u a n t i t a t i v e l y m e a n i n g f u l in a bcc metal in the c a l c u l a t i o n of those quantitie,, which do not d e p e n d on the details of lhe e n e r g y s p e c t r u m . In fact the stale d e n s i t y of the n o n m a g n e t i c (NM) state calcuhtted by use of eq. (I) r e p r o d u c e s well the two peak s t r u c t u r e of the d b a n d in a bcc metal, as is s h o w n in fig. I.* I n t e g r a t i n g eq. (1) to the F e r m i e n e r g y E~. we o b t a i n s e l f - c o n s i s t e n t e q u a t i o n s for the hi,
values, e~ ix d e t e r m i n e d by the c o n d i t i o n thai the a v e r a g e n n m b e r of e l e c t r o n s per atom c a [ cttlated from the hi,, values should agree wilh a given value, N,r T h e s e l f - c o n s i s t e n t e q u a t i o n s are solved n u m e r i c a l l y b} an iterati~e mcthod. The a b o v e m e n t i o n e d N t and the (X,ulomb in tegral p a r a m e t e r I_!/&) are varied as p a r a m e t e r s to find the most a p p r o p r i a t e situation for the problem under consideration.
3. The spin density wave in (Tr We a s s u m e in the c a l c u l a t i o n of the S D W thal the p e r i o d in units o f the l a t t i c e c o n s l a n t ix ;il3 i n t e g e r p w i t h the w a v e v e c t o r p o i n t i n g in the I0011 d i r e c t i o n . F u r t h e r m o r e , the n o d e s o f the
e n v e l o p e of atomic m o m e n t s are a s s u m e d to he h a l f w a y b e t w e e n lwo a d j a c e n t (001) atomic p l a n e s , as is illustrated in fig. 25. S i n c e a t o m s on the same (001) plane are in the same state, we d e t e r m i n e in the s e l f - c o n s i s t e n t c a l c u l a t i o n n~, values of a t o m s for o n e - q u a r t e r of the period s p e c i f e d , m a k i n g use of the s y m m e t r y of the solution s h o w n in fig. 2 with r e s p e c t to the loop
I:ig. 2 A schematic representation ,af lhc envelope m3,J :tt,c,mic mc, mer'Hs in the 5,1)W '.tatL. with p 8. \ f r o , a i n d i c a t e the a t o m i c i l l o m e n l s of e a c h {0(1]) a t o m i c pl:mc
p(E)
~E
Fig. I. The density of stales calculated with eq. (iI for the l i o n r n a g n e t i c st;ate.
* In the ca>e of an fcc metal which we do nol discuss hc~c. il turns out t h a t w e need at leaM the third order term in V~. Note that such a tern] is absenl in the bcc ca,,c
of the e n v e l o p e . The e q u a t i o n s are ,~olved b~ iteration with arbitrarily c h o s e n initial values of the %,,: v a r i o u s c h o i c e s of the initial values 'arc a d o p t e d to see thai they c o n v e r g e to the same final solution after iteration. The iteration i,~ c o n t i n u e d until the difference b e t w e e n the input and output of each %, at a cycle b e c o m e s smaller than 10 ~ and the root m e a n s q u a r e of the fNote that in most cases we do not need an explicit value of 3. if other energ), p a r a m e t e r , , a r e m e a s u r e d in unit,, ~>f 3 :~We have adopted in some calculation', a model in v,hich the node i,, placed on the atomic phmc to confirm that the re,,ult doe,, not depend appreciahl~, on the node D>~ilion
201 difference over all ni~ values is smaller than 10 7"
The calculation of the AF, F and nonmagnetic (NM) states can be carried out in a much simpler way. Assuming different quantization axes of spin for atoms and modifying eq. (1) appropriately, we can obtain the helical (H) solution as well. When U is not too large, we can obtain the condition for the appearance of a magnetic solution of a given p with infinitesimal magnetic amplitude by expanding the self-consistent equations in powers of deviation of each nj,, from its value in the NM state, Na/lO. We confine ourselves here to the case Nd ~> 5, because this case yields the observed phase relation between the SDW and the charge density wave (CDW) [8]. Then we find that the magnetic solution is lower in energy than the NM for Nd larger than a critical value No; N~ turns out to decrease with increasing p, being smallest for the AF. Nc values for the SDW and H for the same p coincide with each other. N c decreases with increasing U. When U is sufficiently large, the magnetic solution with finite amplitude becomes energetically equal to the N M at a critical value of Ne when Ne is increased from N a - - 5 . This critical value which we call again N~ can be determined by energy comparison only. Thus the range of N~ is divided into the second order regime corresponding to larger N~ for smaller U and the first order regime corresponding to N~ close to 5 for larger U. Fig. 3 shows the calculation of the m a x i m u m amplitude of the SDW as well as that of the AF for a given value of U.
10-
05 L
SDW20 i~
~ HELIX 20
AF 00 t 515
,
~./l
....................
~5.20 Nd <
5.25
OO
:E/& --0.005
Fig. 3. The u p p e r part s h o w s the m a x i m u m a m p l i t u d e of the S D W w i t h p = 20 as well as the m a g n i t u d e of the a t o m i c m o m e n t in the A F a n d H s t a t e with p = 20 as a f u n c t i o n of Nd. U/A = 4.42 is a s s u m e d . The d o t s s h o w the m a x i m u m a m p l i t u d e of the first h a r m o n i c of the S D W . The l o w e r part s h o w s the e n e r g y of t h e s e s t a t e s m e a s u r e d f r o m that of the n o n m a g n e t i c state. T h e full line, for e x a m p l e , is the e n e r g y of the S D W w i t h p = 20.
There the choice of U is such that N~ values for large p are close to the boundary between the second order regime and the first order one, as can be seen from the steep rise of the amplitude with increasing Na at Na = N~. As is discussed below, the SDW can be of lowest energy for such a choice of U. We believe that the first order nature of the transition at the Ndel temperature in Cr is related to this first order appearance of the magnetic solution; we shall report a calculation at finite temperatures in the future. The first order appearance is due to the presence of the deep valley at the middle of the state density of the NM shown in fig. 1. As is well known, we can expect the first order transition when the Fermi level is near a minimum of the state density. Fig. 3 shows also the energies of the SDW, AF and H states plotted against Na, from which we can see that the SDW is of lowest energy in a limited range of N a with the amplitude around 0.4 P'B per atom. By varying U, we find that the SDW can be the ground state for U/A = 4.3 4.5; Nc values of the SDW and AF are around the boundary of the first order regime for such choices of U/A. The energy of the H state, on the other hand, remains always higher than that of the AF state. It is difficult within the adopted accuracy of numerical solution to determine precisely the period p of the SDW of lowest energy, though the p value of lowest energy seems to be around 20 with the tendency to decrease first and increase after passing a minimum with increasing Ne. An example of the envelope of the SDW whose energy is lower than the AF state is shown in fig. 4. When N a or U is increased beyond the range where the SDW is of lowest energy, the envelope b e c o m e s more rectangular, or in other words, approaches the antiferromagnetic state with periodic antiphase boundaries. The numerical solution obtained a b o v e is consistent with various experimental data. The magnitude of the magnetic m o m e n t in the AF state is shown to be almost equal to the amplitude of the first harmonic of the SDW for a given Ne except for Ne near N~ (see fig, 3). This result is in good agreement with the experimental data on C r - M n alloys which show the S D W - t o - A F transition [6], while the nesting model seems to fail to explain it [2]. Furthermore, the envelope of the SDW shown in fig. 4
202
-*~ J
"x X\
¢" / \
•
\
-
l/
••
z
N
\,%
s •
Fig. 4. The calculated envelope of lhe SI)W wilh p :20 for N , , - 5.209 with the maximum an'lplitude of 0.38 # , / a l o m i~ given by the full curve. The broken curve is lhc []r,.l harmonic of the SDW. The dot-dashed curve represents the as'>ocialed CDW on an arbilrary scale. The amplinlde of the first harmonic of the CDW is menlioned in lhe text. Nole thai electrons accumuhtle a! lhe loop of the SD~vV.
g i v e s the ratio of the a m p l i t u d e of the lhird h a r m o n i c to that of the first h a r m o n i c as 0.021 w h i c h is in g o o d a g r e e m e n t with the cxp e r ) m e n t a l v a l u e 0.0165_+0.05 141. T h e ~tss o c i a t e d C D W has an a m p l i t u d e of 0,()(t6g elect r o n s / a t o m w h i c h ix c o n s i s t e n t with a r e c e n l d i r e c t m e a s u r e m e n t [5]. A l s o lhe strain w a v e a m p l i t u d e e s t i m a t e d f r o m this C D W is in a g r e e m e n t with e x p e r i m e n t a l d a t a [4, gl. T h e m e c h a n i s m w h i c h l o w e r s the e n e r g y of the S D W r e l a t i v e to the A F is a c o m p r o m i s e b e t w e e n the t w o a n t ) f e r r o m a g n e t i c c o u p l i n g s of the n e a r e s t and n e x t n e a r e s t n e i g h b o r s w h i c h ix a c h i e v e d with the help o f the C D W f o r m a t i o n . N o t i n g that the c o u p l i n g of the next n e a r e s t n e i g h b o r s is s a c r i f i c e d in the A F , we m a y e x p e c l that the spatial m o d u l a t i o n o f the a t o m i c magnetic m o m e n t can l o w e r the S D W e n e r g y . In fact we find that by w l r y i n g V, the r a n g e of N,; w h e r e the S D W is l o w e r in e n e r g y than the A F t e n d s to d i s a p p e a r for s m a l l e r 17,,. On the o t h e r hand, the H state w h e r e one might e x p e c l a s i m i l a r compromise w i t h o u t C D W c a n n o t be lower in e n e r g y than the A F in the p r e s e n l c a l c u l a t i o n , w h i c h is c o n s i s t e n t with the wellk n o w n f a c t in the c a s e o f the H e i s e n b e r g m o d e l of the bcc lattice. In lhe S D W the a s s o c i a t e d C D W d e c r e a s e s the n u m b e r of e l e c t r o n s al n e a r l y n o n m a g n e t i c a t o m s a r o u n d the node and increases that at a t o m s a r o u n d the loop. T h e f a c t that we find the S D W g r o u n d state tit N<# n e a r the b o u n d a r y of the first o r d e r a p p e a r a n c e r e g i m e m e a n s that a slight d e c r e a s e of the electron n u m b e r a r o u n d the n o d e can r e d u c e the e n e r g y i n c r e a s e c a u s e d by the p r e s e n c e of n e a r l y n o n m a g n e t i c a t o m s , s i n c e a smaller n u m ber of e l e c t r o n s f a w ) r s the N M. A l s o lhe fact
lhal the S I ) W spatial m o d u h i l i o n can contain the higher h a r m o n i c s appropriately, c o n l r i b u l c s h> the e n e r g y lowering. A h o g e l h e r ~ve can un d e r s t a n d q u a l i t a t i v e l y the r e a s o n v~,hv' we find lhe S D W g r o u n d s t a l e in a r a t h e r limited range of the c h o i c e s of p a r a m e l e r s . If we :ire well inside the first o r d e r regime with large 17, for e x a m p l e , the spatial m o d u l a t i o n b e c o m e s difticult b e c a u s e of a well d e x e l o p e d a l o m i c magnetic m o m e n t in lhe AF. W h e n we take note of the v o l u m e c h a n g e :ll the S D W - t o - A F l r a n s i l i o n , we should c o n s i d e r lhe r e d i s t r i b u t i o n of e l e c l r o n s h e l w e e n the d s l a t e s and the s hand c a u s e d by a c h a n g e of lhe F e r m i e n e r g y . W c can ',how, h o w e v e r , lhal, il d o e s not m a k e an a p p r e c i a b l e c c m t r i b u l i o n Io the e n e r g y d i f f e r e n c e h e l w e e n the S D W and lhc A F d i s c u s s e d a b o v e . W c shall nol go inlo lhe d i s c u s s i o n of this s u b j e c l lind also thai of the r e l a t i o n b e t w e e n the p r e s e n l theor$ and lhe n e s t i n g m o d e l l h e o r y . W e tit) IlOt i n l e n d al p r e s e n t it) d e n y the p o s s i b i l i t y that lhe nesting of lh,3 F e r m i s u r f a c e p l a y s an i m p o r t a n l role in d e t e r m i n i n g the p e r i o d of the S D W . ()ur lheor,,. on the o t h e r hand, yields a q u a n t i t a t i v e l y g o o d d e s c r i p t i o n of the S D W and a r e a , , o n a b l e exp h m a t i o n for the facI lhal we find the S D W in ( ' r o n l y a m o n g the t r a n s i t i o n m e t a l s .
4. The AF-to-F transition in ordered FeRh alloy W e a s s u m e here the ( ' s ( ' l l y p e o r d e r i n g of ke and Rh a t o m s with equal c o n c e n t r a t i o n , deferring the c a s e of n o n s t o i c h i o m e t r y to future ptlblicalion. ()ur calculation shows within r e a s o n a b l e c h o i c e s of p a r a m e t e r s that Rh :lion1>. have no l o c a l i z e d m o m e n l in lhe A F Male. while Fe a t o m s h a v e a m o m e n l of a b o t l l -4 #,t~. It s h o w s thai the n u m b e r of e l e c t r o n s of Fe a l o m s ix d e c r e a s e d in F e R h to lhe c x l e n l lha! Ihc interaction belween Pc tilt)ms is a n t i f e r r o m a g n e t i c . T h e i n t e r a c t i o n b e t w e e n F'e and Rh a l o m s , on lhe o t h e r hand, f a v o r s lhe ferromagnetism. We discuss the magnelization process of lhe A F Male at absolute zero. in order to elucidate the situalion. Assuming lha[ the A F Male is the grotlnd stale, we apply an external field H to it. I'hen lhe sublattice magnetizations of Fe tllorns and the Rh magnetization take lhe configuration shown in fig. 5 ill the absence of anisolropy. We denole Fe atoms on the sublatlice 1 b ) Fel.
203
EFrFe '1' TMRh '(~e) !15
(Xl0 3)
EFe-Fe
0 ~i Rh
Fel
~
J
4.0
Fell
--~ ~ J
---
-
- . . . .
i1.0
Fig. 5. T h e c o n f i g u r a t i o n of the s u b l a t t i c e m a g n e t i z a t i o n s in the A F s t a t e of F e R h u n d e r a n e x t e r n a l field.
5 /.;-~
o.o Taking the spin quantization axis for each sublattice in the direction of its magnetization, we add the Z e e m a n energy - o ' H p . ~ c o s 8 and crH/xB with or = - 1 to E~ given by eq. (3) for Fe and Rh atoms, respectively; 8 is defined in fig. 5. n~ in eq. (3) is now the number of electrons with spin tr referred to the abovementioned quantization axis; we may assume nFeh, = nFell,~ and hence E F e l ~ = E F e l l ¢. We note that the electron transfer between tr spin states and - o - ones is allowed between FeI and F e l l and also between Fe and Rh. Calculating the total energy to the second order of the electron transfer, we determine the angle 8 by the torque balance condition, dE/d8 = 0, given by
111Rh
_ i-
-1.0
EFe_R h
-~.._
_
-
E~sin 8 E~
=
+ E [ xe-ah
Fe-Fe • ½(sin 8) + Ee~ sin 28 = 0,
- ( n F e . - nFe-,.)P.BH, 0
(5) 0
~Fe-Rh-- 8 V ] ~
(
--ex
. ERh~r -- E F e ¢
--
(4)
0
nRh°-- riFe°
0
nRh-°'--rife° ) ~
----EFe¢(6)
E Fe= 3v
N ( -- PFe~r(~F) 0
0
0
EFe_,r -- g F e . ] '
(7)
0 where nFe¢ is the quantity obtained by integrating poe to the Fermi energy. The first term in eq. (4) is the torque due to the external field; Eex for the second and third terms may be called the " e x c h a n g e energies" for F e - R h and F e - F e . The a b o v e expressions for Eex agree with those derived by Moriya [1], though our Ee~ varies with the change of the magnetic configuration with increasing H * . We note that the variation of Eex represents the deviation f r o m the Heisenberg model in which it is a constant. Fig. 6 shows examples of the calculated Eex vs. 8 relation; the corresponding magnetization processes are shown in fig. 7. We assume the same V~/A and VE/A as those for Cr in the calculation. * O u r E ~ is c a l c u l a t e d o n the b a s i s of the e l e c t r o n i c s t r u c ture w h i c h is d e t e r m i n e d s e l f - c o n s i s t e n t l y for a g i v e n H.
EFe-Rh Fig. 6. The "exchange energies" defined by eqs. (6) and (7) and the Rh atomic magnetic moment at the equilibrium angle 0 s a t i s f y i n g the t o r q u e b a l a n c e e q u a t i o n for a g i v e n v a l u e of H ; t h u s H v a r i e s w i t h & T h e full c u r v e s c o r r e s p o n d s to c a s e a in fig. 7 w i t h Nd = 7.218; the b r o k e n o n e to c a s e b w i t h Nd = 7.212; the d o t - d a s h e d o n e to c a s e c w i t h Nd = 7,206. U/A = 10 a n d (Ev, - ER,)/A = 0.054 are a s s u m e d for all cases.
The values of other p a r a m e t e r s are given in fig. 6. In fig. 7 we present three typical examples of the calculated magnetization curve at absolute zero, a, b and c. In case a the Rh m o m e n t and EF~ ~ shown in fig. 6 tend to saturate with decreasing 8, showing the localized m o m e n t behavior at the final state 8 - 0 . In this case the strength of EF~-Rh relative to that of EF~-Fe is small c o m p a r e d with other cases. As we increase the relative strength of E [ xe-Rh w e go towards case c via case b. In the present choice of p a r a m e t e r s this increase of the relative strength is brought in by a decrease in the average number of electrons per atom, Na (see ~.B/ATOM 2.0
"-:.--\\
F:ERRo. /
1.0 !,d
"
b /
/
o.o~o%1
o602
/
a.
/" /
obo3 ..H)~
Fig. 7. T h e m a g n e t i z a t i o n p r o c e s s in the A F s t a t e of F e R h at a b s o l u t e zero. T h e u n i t s of the e x t e r n a l field are s u c h t h a t ~BH/A = 0.001 c o r r e s p o n d s to H = 105 O e for A = 0.05 Ry. T h e c u r v e d is o n l y s c h e m a t i c .
204
fig. 6). For a fixed N,; smaller values of U/A a n d / o r E~:~- ERh also i n c r e a s e the relative strength of ~TFe ~ Rh , leading us t o w a r d s case c. It is highly i n t e r e s t i n g and d e s i r a b l e to d e t e r m i n e e x p e r i m e n t a l l y which case of the m a g n e t i z a t i o n c u r v e is realized in a given alloy. We can easily e x t e n d the a b o v e d i s c u s s i o n to finite t e m p e r a t u r e s within the o n e - e l e c t r o n app r o x i m a t i o n . Defining the c h a n g e e n e r g i e s by the t o r q u e b a l a n c e at finite t e m p e r a t u r e , we find that the relative strength of ~lIE;7.[:e , Rh i n c r e a s e s rapidly with i n c r e a s i n g t e m p e r a t u r e , leading us finally to case d in fig. 7, where the A F - t o - F t r a n s i t i o n takes place in the a b s e n c e of an external field. O u r c a l c u l a t i o n can r e p r o d u c e well the p h a s e d i a g r a m in the T - H plane [111. The m e c h a n i s m u n d e r l y i n g the t r a n s i t i o n is an increase of the m a g n e t i c p o l a r i z a b i l i t y of Rh atom and a d e c r e a s e of ,~., with i n c r e a s i n g ten> perature. In case b in fig. 7 the t r a n s i t i o n temp e r a t u r e is 3 1 6 K for A - 0 . 0 5 R y . Our calc u l a t i o n e x p l a i n s also the d e c r e a s e of the Fc a t o m i c m o m e n t by 0.1 /x u in the F phase compared to that in the A F p h a s e [91, the d i f f e r e n c e of the e l e c t r o n i c specific heat b e t w e e n the A F and the F phase and the v o l u m e c h a n g e associated with the t r a n s i t i o n . Details of the calc u l a t i o n will be r e p o r t e d in a f u t u r e p u b l i c a t i o n .
p s e u d o - G r e e n i a n f o r m a l i s m 112]. T h e p r e s e n c e of the s b a n d , or in o t h e r words the free electron b r a n c h , musl be taken into a c c o u n t in some p r o b l e m s , as was m e n t i o n e d at the end of seclion 3. Apart from these e l a b o r a t i o n s we can apply the p r e s e n t t h e o r y It) several p r o b l e m s where we need to u n i f y the c o n c e p t of localized mom e n t with that of itinerant e l e c t r o n s . A m o n g them the easiest one is the c a l c u l a t i o n of the spin wave stiffness in a f e r r o m a g n e t i c metal. We have made also an a p p l i c a t i o n to the s u r f a c e m a g n e t i s m of t r a n s i t i o n metals. We report on these p r o b l e m s as well as the details of the p r e s e n t s t u d y in future p u b l i c a t i o n . We t h a n k P r o f e s s o r discussions.
H. Miwa for v a l u a b l e
References I II I. Moriya, Progr. ]'hooF. Phs~. 33 11965) I s7 121 A. Shibatani, K, Motizuki :rod T. Nagamiya, Phy',. Re'.. 177 11969) 984.
A. Shibatani, J. Phys. Soc. Jap. 39 (1975) 8~1. 17;I C.G. Windsor, J. Phys. F. 2 11972) 742. [4] R. Pynn, W. Pres~, S.M. Shairo and S.A. Wermer, Phys. Rev, B13 (1976) 295. [5] Y. l'sunoda, private communication.
5. Concluding remarks We have p r e s e n t e d two a p p l i c a t i o n s of our " i n t e r a c t i n g virtual s t a t e s " a p p r o a c h in its simplest v e r s i o n . An o b v i o u s e l a b o r a t i o n of the c a l c u l a t i o n is to d i s t i n g u i s h b e t w e e n the de and d-/ orbitals t o g e t h e r with the i n c l u s i o n of the i n t e r - o r b i t a l e x c h a n g e i n t e r a c t i o n in the A n d e r son model. We have to i n t r o d u c e then three kinds of t r a n s f e r integrals, ddcr, ddvr and ddTr'. We can take into a c c o u n t also the higher order t e r m s in the p e r t u r b a t i o n e x p a n s i o n of the transfer, which we c a r r y out on the basis of the
[6] Y. Hamaguchi, E.(). Wollan and W.C. Koehler, Ph,,x Rev. 138 (1065) A7~7. 171 Y. Ishikawa, S. Hoshino and Y. Endoh, J. Phys. Soc .lap. 22 11967) 1221. N. Kunitomi, private communication. [8] Y. Tsunoda, M. Mori, N. Kunitomi. Y. I'eraoka and J. Kanamori, Solid Stale ('ommun. I4 (1974) 287 Y. Tsunoda, Y. Nakai and N. Kunitomi. Solid State (ionlrnun. 16 (1975) 443. 191 G. Shirane, R. Nathans and ('.W. ('hen, Ph~,s. Rex. 1~,4 {1964) A 1547. [10l D.G. Penifor, ,I. Phys. ('. 2 11969) 1051. [Ill B.K. Poromarev. JETP ~6 (197:{I 1(15. 1121 J. Kanamori. K. Terakura and K. Yamada, Progr. l'heor. Phys. 46S 11970) 221.