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THEORY OF THE PARAMAGNETIC PHASE IN ITINERANT MAGNETISM
H. Capellmann
To cite this version:
H. Capellmann. THEORY OF THE PARAMAGNETIC PHASE IN ITINERANT MAGNETISM.
Journal de Physique Colloques, 1982, 43 (C7), pp.C7-351-C7-361. �10.1051/jphyscol:1982751�. �jpa-
00222358�
JOURNAL DE PHYSIQUE
page C7-35I Colloque C7, supplément au n°12, Tome 4'A, décembre 1982
THEORY OF THE PARAMAGNETIC PHASE IN ITINERANT MAGNETISM H. Capellmann
ILL Grenoble and Inst. Theor. Physik, TH Aaehen, F.R.G.
Résumé. - Une revue des théories modernes du magnétisme itinérant est présentée. Les fluctuations angulaires de la direction des spins dé- terminent la transition de phase vers l'état paramagnétique à haute température. La nature itinérante des électrons à l'origine du magné- tisme influence la phase paramagnétique, conduisant à des différences caractéristiques par rapport au cas du magnétisme localisé. Les fluc- tuations quantiques rapides qui résultent des sauts des électrons entre sites dominent le comportement à courte distance de la fonction de cor- rélations magnétiques et conduisent à un ordre magnétique à courte dis- tance très important, bien au-dessus de la température de transition T
c. Les conséquences pour la diffusion des neutrons sont discutées.
Abstract. - The modern theories of itinerant magnetism are reviewed.
Angle fluctuations of spin direction drive the phase transition to the paramagnetic phase at high T. The itinerant nature of the electrons responsible for the magnetism strongly influences the paramagnetic phase, leading to characteristic differences to localized magnetism.
Fast quantum fluctuations due to electronic hopping dominate the short range behaviour of the magnetic correlation functions and lead to very strong short range magnetic order far above the transition temperature T . The consequences on neutron scattering are discussed.
1. Introduction.- In the transition metals Cr, Mn, Fe, Co, Ni the electrons responsible for the magnetic properties and the ferromagne- tically (Fe, Co, Ni) or antiferromagnetically (Cr, Mn) ordered phases participate in the Fermi-surface. These electrons are itinerant, their wavefunctions are delocalized, therefore no truely localized magnetic moments exist. The delocalized nature of the wavefunctions is due to the possibility of the electrons to move around rather freely in the crystal (free hopping processes) .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982751
C7-352 JOURNAL DE PHYSIQUE
Let us establish the typical energy and timescales (I shall mainly dis- cuss the ferromagnetic transition metals and I shall take Fe as an example) :
The delocalized electronic wavefunctions are characterized by a band- structure Eks. The bandwidth W for the 3-d type electrons (which carry the magnetism) is of order Wz5eV. This corresponds to a typical hop- ping time tq (the time an electron needs to travel from one lattice site to a neighbouring one) of order tdyh/w (h = Planck's constant). As in any metal, the local charge density y i and the local spin density
A i -2 are not described by sharp quantum numbers even in the ground state.
Therefore even in the ground state the local charge and spin densities fluctuate(they are not sharp), we choose the word "quantum fluctuations"
to characterize this feature. t is the typical timescale for quantum fluctuations. We shall call this the "fast" time scale (of order 4 10-'~sec in Fe).
In the ground state (T = 0) the system is characterized by a band- structure in which the degeneracy of Ekqand Ek, has been lifted
d is the exchange splitting (of order 1.5 ev, 1 eV, .5 eV for Fe, Co,and Ni).
More spin up than spin down electrons are present in the ground state, resulting in a net polarization, but no localized magnetic moments exist, the local densities g,, zi2 are not sharp. Only after averaging over the "fast" quantum fluctuations, one obtains an "average moment per atom" or "atomic moment". (The quotation marks are to indicate that these "moments" are not localized but are the result of an averaging process) .
At finite temperatures anumber of different magnetic excitations may in principle occur and contribute to the phase transition from the magnetically ordered state at low temperatures to the paramagnetic
state at high temperatures: The "atomic moments" may change in ampli-
tude (longitudinal fluctuations) and in direction (transverse fluctua-
tions): These fluctuations occurring at finite temperatures are to be
distinguished from the quantum fluctuations (occurring even at T
=0
as well as at high T): The latter are fast and only an average over those gives a meaning to thel'atomic moments". These quantum averaged
"atomic moments" may themselves be time and space dependent, f luctua- ting s l o w l ~ (if compared to t
)on a time-scale given roughly by:
9
Em is a typical magnetic excitation energy (e.g. a spin wave energy), which is of order k B times the transition temperature Tc. This is a
"slow" time scale (typically sec for Fe) if compared to t q' The modern theories of itinerant magnetism [ I - 2 are based on trans- verse fluctuations driving the phase transition in the ferromagnetic transition metals, amplitude fluctuations costing too much energy to be of importance. In the antiferromagnetic metals Cr and Mn amplitude fluctuations are probably important, too.
2. The paramagnetic phase. - Although transverse fluctuations dominate in driving the phase transition - as is the case in localized magnetism, e.g. a Heisenberg model allows those type of fluctuations only, the lo- cal densities being sharp excluding amplitude fluctuations altogether -
drastic differences to localized magnetism are to be expected. The de- Localized nature of the itinerant electrons leads to unusual effects in the paramagnetic phase, in particular the short range part (up to 20 2
)of the magnetic correlation functions differ sharply from their counterparts in localized magnetism.
To demonstrate the important features I want to discuss the spin-den- sity - spindensity correlation function
2 .
where gi is the operator for the total spin-density at lattice site iti.
The Fouriertransform of C(R,t) is directly proportional to the neutron scattering function S (q, a )
where
J
2 a
f (q) is the formfactor and the sum is over all lattice sites.
JOURNAL DE PHYSIQUE
For l a r g e R t h e e q u a l t i m e c o r r e l a t i o n f u n c t i o n C ( r ) i n t h e paramagne- t i c p h a s e above Tc s h o u l d be of O r n s t e i n - Z e r n i k e form
f o r r l a r g e
where f i s t h e "thermodynamic" c o r r e l a t i o n l e n g t h . I n t h e language used f o r second o r d e r phase t r a n s i t i o n s , t h i s l a r g e r p a r t i s u n i v e r - s a l , i t s f u n c t i o n a l form b e i n g i n s e n s i t i v e t o much o f t h e m i c r o s c o p i c d e t a i l s , e . g . whether t h e e l e c t r o n s a r e i t i n e r a n t o r l o c a l i z e d . We have c a l l e d 5 t h e "thermodynamic" c o r r e l a t i o n l e n g t h t o d i s t i n g u i s h it from t h e o t h e r c h a r a c t e r i s t i c l e n g t h s i n t h e problem which we s h a l l d i s - c u s s below. 5 d i v e r g e s a t Tc i n much t h e same way a s i t d o e s i n l o c a - l i z e d magnetism.
The d r a s t i c d i f f e r e n c e s between i t i n e r a n t and l o c a l i z e d magnetism show up i n t h e s h o r t r a n g e b e h a v i o u r of t h e c o r r e l a t i o n f u n c t i o n . T h i s i s because t h e d e l o c a l i z e d n a t u r e of t h e i t i n e r a n t e l e c t r o n s i n t r o d u c e s two new c h a r a c t e r i s t i c l e n g t h s which have no c o u n t e r p a r t i n l o c a l i z e d
magnetism.
The two new Lengths a r e due t o t h e e x i s t e n c e of t h e F e r m i f u n c t i o n f ( E k s ) , which d e t e r m i n e s whether a n ( e x t e n d e d ) e l e c t r o n i c s t a t e of wave v e c t o r k and s p i n s i s o c c u p i e d o r n o t i n a m e t a l . The f i r s t new l e n g t h i s g i v e n by t h e r e c i p r o c a l of t h e Fermiwavevectork F:
T h i s l e n g t h i s r a t h e r s h o r t , t y p i c a l l y comparable t o t h e l a t t i c e con- s t a n t . I t d e t e r m i n e s t h e e l e c t r o n i c s t r u c t u r e on t h e v e r y s h o r t l e n g t h scale and t h e v e r y s h o r t t i m e s c a l e t . The v e r y f a s t e l e c t r o n hopping p r o c e s s e s (we c a l l e d them "quantum f l u c t u a t i o n s " above) a r e accompa- n i e d by a n i n s t a n t a n e o u s e x c h a n g e - c o r r e l a t i o n h o l e : A t e q u a l t i m e t h e P a u l i p r i n c i p l e w i l l e n s u r e t h a t no e l e c t r o n s of e q u a l s p i n w i l l be a t t h e same p l a c e ( t h e exchange h o l e ) , w h e r e a s e l e c t r o n s of o p p o s i t e s p i n may come c l o s e t o e a c h o t h e r . C o r r e l a t i o n e f f e c t s w i l l t e n d t o keep o p p o s i t e s p i n e l e c t r o n s a p a r t , t o o ( t h e c o r r e l a t i o n h o l e ) , b u t t h i s s e p a r a t i o n i s n o t complete. T h i s l e a d s t o a n e g a t i v e p o l a r i z a t i o n of t h e exchange c o r r e l a t i o n h o l e , which w i l l show up i n a n e g a t i v e c o n t r i - b u t i o n t o t h e c o r r e l a t i o n f u n c t i o n C ( R , t ) a t s h o r t d i s t a n c e s (of o r d e r lF) and a t v e r y s h o r t t i m e s ( o f o r d e r t c7 ) .
A second p u r e l y " i t i n e r a n t l e n g t h " ( h a v i n g no c o u n t e r p a r t i n l o c a l i z e d
magnetism) i s i n t r o d u c e d due t o t h e "smearing o u t " of t h e F e r m i f u n c t i o n
i n t h e v i c i n i t y of t h e F e m i e n e r g y EF a t f i n i t e t e m p e r a t u r e s . Within a
r e g i o n of w i d t h k B ~ around EF t h e F e r m i f u n c t i o n d r o p s from o n e t o z e r o which l e a d s t o a "momentum s p r e a d " gp
p.& k g i v e n by
vF i s t h e F e r m i v e l o c i t y . The "momentum s p r e a d " sp i s r e l a t e d t o a n u n c e r t a i n t y i n p o s i t i o n 6 x
T h i s s e r v e s a s a d e f i n i t i o n f o r a c h a r a c t e r i s t i c l e n g t h A x
=le ,
t h e " e l e c t r o n i c c o h e r e n c e l e n g t h "
T h i s is t h e l e n g t h o v e r which a n e l e c t r o n w i t h F e r m i - v e l o c i t y vF c a n
- 1 t r a v e l i n t h e t i m e i n t e r v a l tm = k( k g ~ ) .
Thermal d i s o r d e r c a n change t h e e l e c t r o n i c p r o p e r t i e s i n a d r a s t i c way o n l y o v e r d i s t a n c e s l a r g e r t h a n le. F o r s h o r t e r d i s t a n c e s t h e de-
l o c a l i z e d n a t u r e of t h e e l e c t r o n i c w a v e f u n c t i o n s e n s u r e s p h a s e cohe- r e n c e
(t o d e s t r o y t h i s p h a s e c o h e r e n c e o v e r d i s t a n c e s c o m p a r a b l e t o t h e l a t t i c e s p a c i n g o n e n e e d s e n e r g i e s o f t h e o r d e r o f t h e F e r m i e n e r g y E F ) . I n t h e v i c i n i t y o f Tc, t h e t y p i c a l t h e r m a l e n e r g i e s k a T c a r e much s m a l l e r ( b y one t o two o r d e r s o f m a g n i t u d e ) t h a n t h e F e r m i e n e r g y EF. T a k i n g i r o n a s a n example: k B ~ c i s of o r d e r . I e V , w h e r e a s EF i s of o r d e r s e v e r a l e V . T h e r e f o r e t h e " e l e c t r o n i c c o h e r e n c e l e n g t h "
l e w i l l be c o n s i d e r a b l y l a r g e r t h a n t h e l a t t i c e s p a c i n g a t t e m p e r a t u r e s c o m p a r a b l e t o T c ( a l e n g t h le of o r d e r 20 seems r e a s o n a b l e ) . " E l e c t r o - n i c p h a s e c o h e r e n c e " means t h a t e l e c t r o n i c w a v e p a c k e t s a r e t y p i c a l l y s p r e a d o v e r v o l u m e s o f t h e o r d e r of le 3 . Thermal e n e r g i e s a r e n o t a b l e t o l o c a l i z e e l e c t r o n s i n s m a l l e r volumes. A t t h i s p o i n t we w a n t t o s t r e s s t h e d i f f e r e n t p h y s i c a l o r i g i n and meaning of t h e " e l e c t r o n i c c o h e r e n c e l e n g t h " le a n d thel'thermodynamic c o r r e l a t i o n l e n g t h " f :
le d e t e r m i n e s o v e r which r e g i o n s s i n g l e ( e v e n non i n t e r a c t i n g ) e l e c t r o n s s p r e a d t h e i r w a v e f u n c t i o n s i n a p h a s e c o h e r e n t way g i v e n a c e r t a i n amount of t h e r m a l d i s o r d e r . One c a n t h i n k of t h e e l e c t r o n s b e i n g con- t a i n e d i n w a v e p a c k e t s o f d i m e n s i o n le.
T h e s e w a v e p a c k e t s of d i m e n s i o n le w i l l i n t e r a c t w i t h e a c h o t h e r . T h i s
i n t e r a c t i o n - i n t h e s y s t e m s we a r e d i s c u s s i n g h e r e - l e a d s tothemagne-
C7-356 JOURNAI, U E PHYSIQUE
t i c a l l y o r d e r e d s t a t e a t low t e m p e r a t u r e s . d e t e r m i n e s o v e r w h i c h r e - g i o n s d i f f e r e n t w a v e p a c k e t s a r e c o r r e l a t e d d u e t o t h e i r i n t e r a c t i o n : A l t h o u g h i n t h e v i c i n i t y o f t h e t r a n s i t i o n t e m p e r a t u r e l e , t h e d i m e n s i o n a t h e s i n g l e e l e c t r o n i c w a v e p a c k e t , d o e s n o t c h a n g e d r a s t i c a l l y ,
d i v e r g e s b e c a u s e t h e i n t e r a c t i o n b e t w e e n d i f f e r e n t w a v e p a c k e t s l e a d s t o i n f i n i t e r a n g e c o r r e l a t i o n s a t Tc.
Now l e t u s d i s c u s s how t h e t w o l e n g t h s lF a n d 1 w i l l a f f e c t t h e c o r r e - e.
rl a t i o n f u n c t i o n C ( R , t ) and i t s F o u r i e r t r a n s f o r m C ( q ,
h ) )i n t h e paramagne- t i c p h a s e a b o v e Tc. T a k i n g i r o n a s a n e x a m p l e , t h e e x p e r i m e n t a l l y a c - c e s s i b l e t h e r m a l e n e r g i e s k,<T a r e much smaller t h a t EF, t h e r e f o r e 1, w i l l b e much l a r g e r t h a n t h e l a t t i c e s p a c i n g a ( t y p i c a l l y le^: 208 a b o v e T i n F e ) . The l e n g t h lF ( t h e d i m e n s i o n o f t h e e x c h a n g e c o r r e l a -
C
t i o n h o l e ) w i l l b e c o m p a r a b l e t o a .
T h e r e a r e t w o t y p e s o f s p i n - f l u c t u a t i o n s w h i c h d e t e r m i n e t h e s p e c t r u m o f C ( R , t ) : The " f a s t " q u a n t u m f l u c t u a t i o n s w h i c h f o r m a c o n t i n u u m i n e n e r g y f t h e S t o n e r continuum: e x t e n d i n g a l l t h e way up t o e n e r g i e s o f t h e o r d e r o f t h e b a n d w i d t h ( ~ . ; 5 e V ) . T h e s e h i g h e n e r g y q u a n t u m f l u c t u a t i o n s
r e p r e s e n t t h e s t r u c t u r e and t h e d y n a m i c s o f t h e e x c h a n g e - c o r r e l a t i o n h o l e , t h e m a i n w e i g h t o c c u r s f o r w a v e v e c t o r s c o m p a r a b l e t o k F . I t i s i m p o r t a n t t o k e e p i n mind t h a t t h e h i g h e n e r g y quantum f l u c t u a t i o n s i n F e c a n n o t b e e x c i t e d t h e r m a l l y i n a n a p p r e c i a b l e way.
T h e s e c o n d t y p e o f low l y i n g e x c i t a t i o n s a r e s l o w t r a n s v e r s e f l u c t u a - t i o n s ( t h a t i s w h a t t h e s p i n waves a r e a t low t e m p e r a t u r e s ) . I n t h e p a r a m a g n e t i c p h a s e o n e c a n t h i n k o f t h e s e a s a m a c r o s c o p i c p o p u l a t i o n o f s p i n waves. T h i s m a c r o s c o p i c p o p u l a t i o n c a n t a k e p l a c e o n l y f o r w a v e l e n g t h s l a r g e r t h a n l e t w h e r e a s t h e d e l o c a l i z e d n a t u r e o f t h e elec- t r o n i c w a v e f u n c t i o n s s u p p r e s s e s t h e t h e r m a l p o p u l a t i o n o f s h o r t wave- l e n g t h f l u c t u a t i o n s .
T h i s r e s u l t s i n v e r y s t r o n g s h o r t r a n g e m a g n e t i c o r d e r i n t h e paramagne- t i c p h a s e , a f e a t u r e which i s t h e b a s i s f o r t h e modern t h e o r i e s o f i t i n e r a n t m a g n e t i s m [I , 2 ] . The s h o r t r a n g e o r d e r i s s t r o n g e n o u g h i n t h e p a r a m a g n e t i c p h a s e f o r s p i n w a v e s o f w a v e l e n g t h s s m a l l e r t h a n 1
e s t i l l t o p r o p a g a t e [ 3 ] . The p r o p a g a t i n g c h a r a c t e r o f t h e s p i n w a v e s a b o v e Tc f o r w a v e l e n g t h s up t o 2 0 8 i s r e p r e s e n t e d i n t h e s p e c t r u m o f
A,
C (q,
6 ) )by a p e a k a t f i n i t e
I,:f o r q - v a l u e s up t o q-l,Zn'< [ . F o r s m a l -
.
J C
l e r q v a l u e s C ( q , t . ) ) i s p e a k e d a r o u n d
;) =0 o n l y , t h e r e s p o n s e i s d i f -
f u s i v e . A c t u a l l y t h i s d i f f u s i v e p e a k a l s o e x i s t s f o r t h e l a r g e r q v a -
l u e s and i s d u e t o c o r r e l a t e d r e g i o n s moving a s a w h o l e . The p r o p a g a -
t i n g s p i n wave p e a k s a t f i n i t e W a r e w e l l s e p a r a t e d f r o m t h i s c e n t r a l
p e a k C 4 ] . A f u r t h e r r e m a r k a b l e b e h a v i o u r i s r e f l e c t e d i n t h e t o t a l
w e i g h t i n t e g r a t e d o v e r LJ :
T h i s q u a n t i t y i s m e a s u r e d i n p a r a m a g n e t i c N e u t r o n - s c a t t e r i n g . E x p e r i - m e n t a l l y a s w e l l a s t h e o r e t i c a l l y it i s i n t e r e s t i n g t o d i s c u s s C ( q ) f o r d i f f e r e n t fi , w h e r e fi i s t h e r a n g e i n e n e r g y o v e r w h i c h t h e i n t e g r a - t i o n h a s b e e n p e r f o r m e d .
I f t r u e l y a l l e n e r g i e s f r o m - & t o
+da r e i n t e g r a t e d o v e r t h e i n s t a n - t a n e o u s c o r r e l a t i o n f u n c t i o n i s o b t a i n e d :
I f t h e i n t e g r a t i o n i s o n l y o v e r a r a n g e f r o m - A u t o + A O , a n a p p r o - x i m a t e t i m e a v e r a g e o f t h e c o r r e l a t i o n f u n c t i o n i s o b t a i n e d [ 5 ] :
w h e r e 277
,
at-;, .
I f a l l t y p i c a l t i m e s c a l e s o f t h e s y s t e m a r e l o n g compared t o A t ( i . e . i f a l l t y p i c a l e n e r g i e s a r e s m a l l compared t o
% A G ) )t h e n ? (CJ, w
)w i l l b e p r a c t i c a l l y z e r o f o r > A , a n d cmAt w i l l b e i d e n t i c a l t o t h e e q u a l t i m e c o r r e l a t i o n f u n c t i o n . F o r t h i s t o b e t r u e t h e v a l u e s o f A W a r e v a s t l y d i f f e r e n t i n i t i n e r a n t s y s t e m s ( t h e r e
% ~ ( 3h a s t o b e o f t h e o r d e r o f t h e b a n d w i d t h , i . e . o f o r d e r 5eV f o r F e ) f r o m t h e v a - l u e s o f 6 0 i n l o c a l i z e d s y s t e m s . L e t u s d i s c u s s a l o c a l i z e d H e i s e n b e r g model f i r s t .
A l l r e l e v a n t e n e r g i e s i n t h e H e i s e n b e r g model a r e g i v e n by t h e e x c h a n g e c o n s t a n t s , t h e y d e t e r m i n e t h e s p i n wave s p e c t r u m a n d b T i s t y p i c a l -
B
Cl y o f t h e o r d e r o f t h e maximum s p i n wave e n e r g y . T h e r e f o r e a n i n t e g r a - t i o n up t o k g ~ ~ c o m p r i s e s a l m o s t a l l p o s s i b l e f l u c t u a t i o n s :
9
-
td,The r e s u l t i s p r a c t i c a l l y i d e n t i c a l t o t h e e q u a l t i m e c o r r e l a t i o n f u n c t i o n , w h i c h i s L o r e n t z i a n i n q ( i . e . O r n s t e i n - Z e r n i k e l i k e i n r ) . I n t e g r a t i n g C ( q ) o v e r t h e B r i l l o u i n z o n e o n e o b t a i n s :
T h i s r e s u l t i s i n d e p e n d e n t o f t e m p e r a t u r e . A t low t e m p e r a t u r e s ( T - r O )
C7-358 JOURNAL DE PHYSIQUE
2 .
t h e p a r t S 1s o b s e r v e d i n n e u t r o n s c a t t e r i n g a s t h e m a g n e t i c Bragg peak and t h e p a r t S i s a background c o n t r i b u t i o n , d i s t r i b u t e d o v e r a l l q ( d u e t o s p i n w a v e s ) . A t h i g h t e m p e r a t u r e s (T>>Tc) C ( q ) i s c o n s t a n t i n q , t h e e n t i r e c o n t r i b u t i o n S ( S + l ) b e i n g s p r e a d o v e r a l l q . C l o s e t o Tc 5 becomes l a r g e and C ( q ) p e a k s a r o u n d q = 0 . But e v e n a t Tc t h e r e a r e many F o u r i e r c o m p o n e n t s f o r l a r g e r q ( s a y of t h e o r d e r o f t h e z o n e b o u n d a r y ) , i n d i c a t i n g t h a t t h e a v e r a g e a n g l e between n e a r e s t n e i g h - b o u r s i s a l r e a d y q u i t e l a r g e a t T c ( o f t h e o r d e r of 80 d e g r e e s ) . T h i s i s i l l u s t r a t e d by t h e f a c t t h a t t h e q u a n t i t y q C ( q ) ( w h i c h 2 i s a mea- s u r e t o i n d i c a t e which q - v a l u e s c o n t r i b u t e s i g n i f i c a n t l y t o t h e F o u r i e r - s p e c t r u m , t h e f a c t o r q2 b e i n g a p h a s e s p a c e f a c t o r ) rises r a t h e r s t e e p - l y w i t h q o u t t o t h e zone boundary i n a H e i s e n b e r g model f o r a l l tem- p e r a t u r e i n t h e p a r a m a g n e t i c p h a s e .
T h i s s t o r y i s changed c o m p l e t e l y i n a n i t i n e r a n t s y s t e m . A t t e m p e r a - t u r e s above Tc b u t w e l l below t h e F e r m i t e m p e r a t u r e TF( ksTF i s o f o r d e r o f t h e b a n d w i d t h , t y p i c a l l y o n e t o two o r d e r s o f m a g n i t u d e l a r g e r t h a n k g ~ c ) t h e e l e c t r o n i c c o h e r e n c e l e n g t h le i s l a r g e compared t o t h e l a t t i c e s p a c i n g and t h e r e a r e v e r y few t h e r m a l l y e x c i t e d e x c i t a - t i o n s o f w a v e l e n g t h s s m a l l e r t h a n le.
T h e r e f o r e i n t h e i t i n e r a n t s y s t e m t h e q y a n t i t y
i s p e a k e d r a t h e r s h a r p l y a r o u n d t h e m a g n e t i c Bragg p e a k s , and t h e r e a r e p r a c t i c a l l y no F o u r i e r c o m p o n e n t s f o r q > - 7 2 7 . I n t h e i t i n e r a n t
1 ,
s y s t e m C t h ( q ) d r o p s r a t h e r q u i c k l y , v e r y much l i k e a G a u s s i a n ( a n d n o t l i k e a L o r e n t z i a n a s i n t h e H e i s e n b e r g m o d e l ) . The i n d e x t h i s t o i n d i c a t e t h a t t h e t h e r m a l l y e x c i t e d p a r t of t h e s p e c t r u m o n l y h a s b e e n i n t e g r a t e d o v e r , t h e r e f o r e t h e q u a n t i t y o b t a i n e d i s a n a v e r a g e o v e r t h e f a s t e r quantum f l u c t u a t i o n s . The q u a n t i t y q C t h ( q ) a s a f u n c t i o n o f q 2
- I
h a s a p e a k i n t h e v i c i n i t y o f 21T e, , a s a r e s u l t o f t h e a b s e n c e o f t h e r m a l l y e x c i t e d F o u r i e r c o m p o n e n t s f o r l a r g e r q . T h i s i n d i c a t e s t h a t t h e s h o r t r a n g e p r o p e r t i e s o v e r l e n g t h s c a l e s up t o 1 h a v e n o t c h a n g e d
e a p p r e c i a b l y from what t h e y w e r e a t T = 0.
I n t h e i t i n e r a n t s y s t e m a l s o t h e sum of Cth(q) o v e r t h e B r i l l o u i n - z o n e
i s much s m a l l e r t h a n t h e e q u i v a l e n t q u a n t i t y i n a H e i s e n b e r g model,
which would p r o d u c e t h e same o r d e r e d moment a t T
=0 .
I itinerant I:4senberg th
Taking Fe as an example, Ith is smaller by approximately a factor of five than what one gets if a Heisenberg model is used to describe the magnetic properties. According to equ. 14 . I is given by
t,
It is the local correlation function averaged over the timeinter-
- 1
val tm
=%, ((Z*T~) . Because of the fast electron hopping processes (quantum fluctuations) this differs considerably from the equal time correlation function. During the time tm electrons from within a re- gion of order l3 may have reached the site considered, (remember
e
le
=vF. tm). Therefore the quantity Ith is equivalent to a spatial average of the correlation function over a region characterized by the length let much larger than the lattice spacing, reducing the value of Ith from what it would have been if a Heisenberg model were applicable.
The fact that the local spin density is not sharp ( H ] f 0 ) not only influences the thermally excited part of the spectrum. Also the high frequency parts differ drastically in itinerant systems and lo- calized systems. Whereas the localized Heisenberg model has no excita- tion energies % W which are larger than k B ~ c , the itinerant system has a continuum of possible excitations over a range of the order of the bandwidth W. If these high energy fluctuations are integrated over, additional interesting structure is obtained in the itinerant system.
b/
-
tic structure of the exchange-correlation hole of a Fermi-liquid. This will be the case at low temperatures in the ordered phase as well as above Tc in the paramagnetic phase. As long as T is small compared to the Fermi temperature TF no appreciable changes will occur in the high frequency part of the spectrum.
The total intensity in the high frequency part of the spectrum
is typically much larger than the thermally excited intensity in the
itinerant system:
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The experimental observation of the short time part of the correlation function is difficult. Neutron scattering so far is restricted to energy transfers of the order of . I to . 2 eV, much snallerthan the band- width of Fe for example (which is several eV). Therefore the high fre- quency quantum fluctuations and their interesting magnetic structure could not be observed so far in iron.
However, other materials exist with a much smaller effective band- width. Typically these materials will have complicated crystal struc- tures of low symmetry. It is then possible that hybridization and gaps opened up at the Brillouin-zone boundary lead to very flat bands and the effective bandwidth might be comparable or smaller than available neutron energies. It is then possible to observe the quantum fluctua- tions in the ordered phase for T d O and above Tc. This has recently h e n demonstrated for the itinerant magnetic system MnSi L 6 1 .
3. Conclusion: - We have discussed the characteristic features of iti- nerant magnetism and their consequences on the spin-spin-correlation
r)
function C(q,L3). Drastic differences to localized magnetism exist.
a) At temperatures above Tc but low compared to TF the thermally excited fluctuations are restricted to wavelengths larger than the electronic coherence length le (of order 2 0 2 in Fe) . This leads to very strong short range order above T
c'
- I
