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>
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Submitted on 1 Jan 1982
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JOURNAL DE PHYSIQUE
Colloque
C7, supplément
au n°12,
Tome 4'A, décembre
page
1982
C7-35I
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 déterminent la transition de phase vers l'état paramagnétique à haute
température. La nature itinérante des électrons à l'origine du magnétisme influence la phase paramagnétique, conduisant à des différences
caractéristiques par rapport au cas du magnétisme localisé. Les fluctuations quantiques rapides qui résultent des sauts des électrons entre
sites dominent le comportement à courte distance de la fonction de corrélations magnétiques et conduisent à un ordre magnétique à courte distance 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 ferromagnetically (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 discuss the ferromagnetic transition metals and I shall take Fe as an
example) :
The delocalized electronic wavefunctions are characterized by a bandstructure Eks. The bandwidth W for the 3-d type electrons (which carry
the magnetism) is of order Wz5eV. This corresponds to a typical hopping 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
-2
A i 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
4
fluctuations. We shall call this the "fast" time scale (of order
10-'~sec in Fe).
In the ground state (T = 0) the system is characterized by a bandstructure 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
are not sharp. Only after averaging
exist, the local densities g,,
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)
zi2
.
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 amplitude (longitudinal fluctuations) and in direction (transverse fluctuations): 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'atomicmoments". These quantum averaged
"atomic moments" may themselves be time and space dependent, fluctuating 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 kB times the transition temperature Tc. This is a
sec for Fe) if compared to t
"slow" time scale (typically
q'
The modern theories of itinerant magnetism [ I - 2 are based on transverse 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 local densities being sharp excluding amplitude fluctuations altogether drastic differences to localized magnetism are to be expected. The deLocalized 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-density - 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
f
2
a
(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 paramagnet i c phase above Tc should be of Ornstein-Zernike form
f o r r large
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 being i n s e n s i t i v e t o much of t h e microscopic
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
called
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
5
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.
d i v e r g e s a t Tc i n much t h e same way a s i t does i n local i z e d magnetism.
5
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 behaviour 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
magnetism.
counterpart i n localized
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 Fermifunction
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 occupied 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 cons 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 accompan 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 time 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 each 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
opposite spin electrons apart, too ( t h e c o r r e l a t i o n h o l e ) , but 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
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 )
lF)
c7
.
A second p u r e l y " i t i n e r a n t l e n g t h " (having 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 Fermifunction
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 one 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 "
uncertainty i n position 6 x
sp i s
r e l a t e d t o an
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 coherence length"
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
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 ~-)1
.
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. For s h o r t e r d i s t a n c e s t h e del 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 coher 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 comparable t o
t h e l a t t i c e s p a c i n g one n e e d s e n e r g i e s of t h e o r d e r of t h e Fermienergy
E F ) . I n t h e v i c i n i t y of 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 (by one t o two o r d e r s of magnitude) t h a n t h e Fermienergy
EF. Taking i r o n a s a n example: k B ~ ci s of o r d e r
I e V , whereas 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 "
le
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
comparable 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 coherence" means t h a t e l e c t r o n i c wavepackets a r e t y p i c a l l y
3
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
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 and 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 (even 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 cont a i n e d i n wavepackets of dimension le.
T h e s e wavepackets of dimension 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
interaction
-
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-
JOURNAI, UE PHYSIQUE
C7-356
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 .
determines overwhichre-
g i o n s d i f f e r e n t wavepackets a r e c o r r e l a t e d due t o t h e i r i n t e r a c t i o n :
Although 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 temperature l e , t h e dimension
a t h e s i n g l e e l e c t r o n i c wavepacket, does n o t change d r a s t i c a l l y ,
d i v e r g e s because t h e i n t e r a c t i o n between d i f f e r e n t wavepackets 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. r
l 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.
Taking i r o n a s a n example, t h e e x p e r i m e n t a l l y ack,<T a r e much smaller t h a t EF, t h e r e f o r e 1,
c e s s i b l e thermal energies
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
above T
C
( t y p i c a l l y le^:208
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 -
t i o n h o l e ) w i l l be comparable t o a .
T h e r e a r e two t y p e s of s p i n - f l u c t u a t i o n s which d e t e r m i n e t h e spectrum
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 of t h e band w i d t h ( ~ . ; 5 e V ) . These h i g h e n e r g y quantum 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 main weight o c c u r s f o r wavevectors comparable t o k F . I t is
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
paramagnetic phase one c a n t h i n k of t h e s e a s a macroscopic p o p u l a t i o n
o f s p i n waves.
T h i s macroscopic population can t a k e p l a c e only 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 elect 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 wavelength fluctuations.
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 enough i n
t h e p a r a m a g n e t i c phase f o r s p i n waves of w a v e l e n g t h s s m a l l e r t h a n 1
s t i l l t o propagate [ 3 ]
. The
e
p r o p a g a t i n g c h a r a c t e r of t h e s p i n waves
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 spectrum of
A,
C (q,6 ) ) by a p e a k a t f i n i t e
I,:
f o r q-values
up t o q-l,Zn'<
J
.
ler q v a l u e s C(q,t.)) i s peaked around
);
= 0 only,
[C .
For smal-
the response is 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 peak a l s o e x i s t s f o r t h e l a r g e r q val 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
peak C 4 ]
.A
f u r t h e r remarkable behaviour 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 is measured i n paramagnetic N e u t r o n - s c a t t e r i n g .
Experi-
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 , where fi i s t h e r a n g e i n e n e r g y o v e r which t h e i n t e g r a t i o n has been performed.
I f t r u e l y a l l e n e r g i e s from - & t o
+d
a r e integrated over the instan-
taneous c o r r e l a t i o n function is obtained:
I f t h e i n t e g r a t i o n is o n l y o v e r a r a n g e from - A u t o + A O , a n approximate 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 is o b t a i n e d [ 5 ]
:
where
at-;, 277 .
,
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
, a n d cmAt
w i l l be i d e n t i c a l to
be practically zero f o r
>A
t h e equal t i m e c o r r e l a t i o n function. For t h i s t o be t r u e t h e values of
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 % ~ ( h3a s t o b e
o f t h e o r d e r of 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 ) from t h e v a l u e s of 6 0 i n l o c a l i z e d systems. L e t u s d i s c u s s a l o c a l i z e d Heisenberg
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 C
l y of t h e o r d e r of 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
9
k g c ~o m p~r i s e s
almost a l l possible fluctuations:
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
the equal t i m e correlation
f u n c t i o n , which i s L o r e n t z i a n i n q (i.e. Ornstein-Zernike 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 zone one 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 of 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 .
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
the part S
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 ) being spread over a l l q. Close t o
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
Tc
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
5
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
2
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 i s a meas 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 temp e r a t u r e i n t h e paramagnetic phase.
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
is of
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
than
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 system 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
7
2
In the itinerant
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 >
system Cth
1,
.
( 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 over, t h e r e f o r e t h e q u a n t i t y obtained i s an average over t h e
2
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
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,
-I
,
a s a r e s u l t of t h e absence of
thermally e x c i t e d Fouriercomponents 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)
over t h e Brillouin-zone
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 .
Iitinerant
th
I:4senberg
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 Because of the fast electron hopping processes
val tm = %, ((Z*T~)
(quantum fluctuations) this differs considerably from the equal time
correlation function. During the time tm electrons from within a remay have reached the site considered, (remember
gion of order l3
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 localized systems. Whereas the localized Heisenberg model has no excitation 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:
JOURNAL DE PHYSIQUE
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 bandwidth of Fe for example (which is several eV). Therefore the high frequency quantum fluctuations and their interesting magnetic structure
could not be observed so far in iron.
However, other materials exist with a much smaller effective bandwidth. Typically these materials will have complicated crystal structures 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 fluctuations in the ordered phase for T d O and above Tc. This has recently
h e n demonstrated for the itinerant magnetic system MnSi 6
L 1.
3. Conclusion: - We have discussed the characteristic features of itinerant magnetism and their consequences on the spin-spin-correlation
function C(q,L3). Drastic differences to localized magnetism exist.
r)
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 Tc'
b) Even above Tc propagating spin waves for wavevectors q, 2.T
exist.
-I
&A
C) The total intensity Ith of thermally excited fluctuations for
Tc < T < < TF in an itinerant system is much smaller than in a Jocalized Heisenberg model giving the same ordered moment at T =O.
d) The itinerant system has very fast quantum fluctuations at low
T as well as above Tc. These high frequency fluctuations form a
25 W. They peak for q-values of the
continuum and extend up to
order of kF , indicative of the magnetic structure of the exchange-correlation hole. For T << TF the total intensity lit of
qu
the quantum fluctuations is much larger than the low frequency
thermally excited part Ith.
References
1
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BROWN P . J . ,
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