Correlations in d and f Electron Systems - American Chemical Society

Chapter 20

Correlations

in d and f Electron Systems G. Stollhoff and P. Fulde

Downloaded by CORNELL UNIV on September 1, 2016 | http://pubs.acs.org Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch020

Max-Planck-Institut für Festkörperforschung, D-7000 Stuttgart 80, Federal Republic of Germany

A discussion is given of electron corre lations in d- and f-electron systems. In the former case we concentrate on transi tion metals for which the correlated ground-state wave function can be calcu lated when a model Hamiltonian is used, i.e. a five-band Hubbard Hamiltonian. Various correlation effects are discussed. In f-electron systems a singlet groundstate forms due to the strong correlati ons. It is pointed out how quasiparticle excitations can be computed for Ce systems.

E l e c t r o n c o r r e l a t i o n s i n d - and f - e l e c t r o n systems are presently i n the center of considerable activity in theoretical solid-state physics. There are two d i s t i n c t classes o f materials which require a better understanding of electron correlations for their intriguing physical properties. To one c l a s s b e l o n g t h e r e c e n t l y discovered high-T superconducting materials (1) like La2- M Cu04 (M=Ba, S r ) , Y B a 2 C U 3 0 7 o r Bi2Sr2CaCU20g-^. The c h a r a c t e r i s t i c s t r u c t u r e o f t h e s e m a t e r i a l s i n v o l v e s layers of (Cu02) with approximately one h o l e per formula unit, depending on t h e degree o f doping or oxygen d e f i c i e n c y . From e x p e r i m e n t s i t i s known t h a t the electrons i n the layers are strongly correlated. F o r e x a m p l e , La2Cu04 i s n o t a m e t a l d e s p i t e o f t h e f a c t t h a t i t s h o u l d have a h a l f - f i l l e d band. Instead i t i s a Mott-Hubbard i n s u l a t o r . To t h e o t h e r c l a s s o f m a t e r i a l s belong t h e heavy- fermion systems (2-6). Examples are Ce compounds l i k e C e A l 3 , CeCU2Si2 o r U compounds like UPt3 o r UBei3« I n t h i s case the 4 f o r 5 f electrons are responsible f o r the unusual p h y s i c a l properties o f the systems. They a r e o n l y weakly c o u p l e d t o t h e r e m a i n i n g electrons and due to the strong on-site Coulomb c

x

x

5

n

0097-6156/89/0394-0279$06.00/0 ο 1989 American Chemical Society

Salahub and Zerner; The Challenge of d and f Electrons ACS Symposium Series; American Chemical Society: Washington, DC, 1989.


280

THE CHALLENGE OF d A N Df ELECTRONS

repulsions one is again dealing with strong electron correlations. The most s t u d i e d d e l e c t r o n systems a r e t h e transition metals. Here t h e o n - s i t e Coulomb i n t e r a c t i o n s and the d-band width are of similar size and therefore correlations are moderately strong. We shall concentrate on t h e s e systems and d e s c r i b e our present understanding of the groundstate and o f e f f e c t s caused by c o r r e l a t i o n s (7) . The l a t t e r include reductions of charge fluctuations, a build up of atomic magnetic moments due t o Hund's rule correlations, as w e l l as modifications i n the Stoner-Wohlfarth c r i t e r i o n for the onset of magnetic order. At present the correlated ground-state wave f u n c t i o n c a n be computed o n l y f o r a model Hamiltonian, for which we choose a five-band Hubbard system. This approach contrasts present day band-structure c a l c u l a t i o n s based on t h e local density (LDA) or spin density (LSD) approximation to the density-functional theory (8) . In LDA or LSD calculations one does not attempt to compute a ground-state wave function but instead calculates directly certain ground-state properties such as the density distribution or energy. Exchange and correlation are dealt w i t h by u s i n g r e s u l t s from the homogeneous electron gas. The advantages of that approach are i t s c o n c e p t u a l and c o m p u t a t i o n a l s i m p l i c i t y and a number o f o u t s t a n d i n g s u c c e s s e s . But the price one has to pay are uncertain approximations. F u r t h e r m o r e t h e r e i s no way o f d e t e r m i n i n g t h e many-body wave f u n c t i o n . In the second p a r t of the lecture we w a n t to demonstrate i n t h e s i m p l e s t p o s s i b l e way t h e p r o b l e m o f strongly correlated f-electrons. It is intimately connected with the Kondo lattice problem, or alternatively with the formation of a singlet state (5,6). Correlations

i n Transition Metals

We s t a r t from a model H a m i l t o n i a n which d e s c r i b e s the f i v e c a n o n i c a l d bands w i t h d i s p e r s i o n s c^fk^ (i/=l,..5) and i n c l u d e s o n - s i t e interactions Η = Σ //ok Ηχ(1)=

c ^ k j n ^ k )

\

Σ i j σσ

+ Σ H!(l) 1

[ Uijaiî(l)ajÎ.(l)aja.(l)a

(1)

i

0

(l)

+

1

+Jij[aia(l)aj^,(l)a +ai+(l)ai+.(l)a

j a

i

o

i(l)a

.(l)a

j a

j

o

(l)+

(l)]]


20. STOLLHOFF & FULDE

281

Correlations in d and f Electron Systems

The o n l y f r e e p a r a m e t e r i n t h e o n e - p a r t i c l e p a r t o f t h e H a m i l t o n i a n i s t h e t o t a l b a n d w i d t h W. The n ( k ) are the number operators for the Bloch states. The i o ( i ) ( i o ( D ) create (destroy) electrons w i t h s p i n a i n atomic-like d orbital i at site 1. The on-site i n t e r a c t i o n m a t r i x elements U^j have the form / / 0

a

a


u

i j

= U + 2J -

2Jij

(2)

where U and J are average Coulomb and exchange i n t e r a c t i o n c o n s t a n t s and J ^ j c o n t a i n s t h e anisotropies. We s h a l l a s s u m e t h a t J = 0 . 2 U a n d A J = 0 . 1 5 J w h e r e J a n d ΔJ a r e t h e i s o t r o p i c a n d a n i s o t r o p i c p a r t o f J i j / respectively. Further details can be found in Ref. (9,10). T h e H a r t r e e - F o c k (HF) g r o u n d s t a t e i s w r i t t e n as |φ > 0

c

=

ïï c ^ ( k ) I0> ki/σ

ki/

F

where the o p e r a t o r s denotes the vacuum.

c^(k)

(3)

= Σ i l

o %(k) refer to By e x p a n d i n g u

«!(//,k)

aia(l)

e

1

Bloch

states

and

^

ι0>

(4)

one c a n decompose ι > i n t o d i f f e r e n t configurations. In figure 1 two of them are shown with different interaction energies. One configuration (a) is favourable while the other (b) is unfavourable with respect t o t h e Coulomb r e p u l s i o n e n e r g y . One notices that in (b) the electron numbers at different sites f l u c t u a t e much more a r o u n d t h e c h o s e n a v e r a g e number o f 2.5 than i n (a). Electron correlations decrease the weight of unfavourable configurations which i s too large in the HF g r o u n d state. Stated differently, their e f f e c t i s to reduce charge f l u c t u a t i o n s at the different sites as compared with the ones contained in ΙΦ >. Electron correlations are introduced into the g r o u n d - s t a t e wave f u n c t i o n by t h e ansatz 0

0

i*o> = e

s

The o p e r a t o r S = - Σ i j l

ΙΦ >

(5)

0

S is uij

of the Oij(l)

form (6)


282

THE CHALLENGE OF d AND f ELECTRONS

with nit(l) Oij(l) =

ni|(l)

ni(l)

nj(l)

si(l)

sj(l)

(7)

The η ^ ( 1 ) , s i ( l ) a r e t h e e l e c t r o n number and s p i n o p e rators f o r the d i f f e r e n t s i t e s , i . e . η^ (1) = a ^ ( 1 ) a i ( 1 ) a n d ni=2_ η ^ . Furthermore, i n order to exclude s i n g l e p a r t i c l e e x c i t a t i o n s , contractions w i t h i n the operators O i ^ ( l ) a r e e x c l u d e d when e x p e c t a t i o n v a l u e s a r e e v a l u a ted. F o r more d e t a i l s see e . g . R e f . ( 9 ) . The parame t e r s n i j f o l l o w from m i n i m i z i n g the energy σ

σ

0


σ

n

c

9

)

η,ηη

The expectation values are evaluated within the R=0 a p p r o x i m a t i o n i n t r o d u c e d by F r i e d e l and coworkers (see e . g . R e f . (11)) i n w h i c h o n l y t e r m s w i t h 1=1» a r e kept. W i t h t h e s e a p p r o x i m a t i o n s c a l c u l a t i o n s become v e r y s i m p l e . I n t h e f o l l o w i n g a number o f r e s u l t s a r e presented. The r e d u c t i o n o f charge f l u c t u a t i o n s as a f u n c t i o n o f d band f i l l i n g i s shown i n f i g u r e . 2 f o r a b c c l a t t i c e and U/W = 0 . 5 . P l o t t e d i s t h e mean s q u a r e d e v i a t i o n o f the e l e c t r o n number

Δη

2

2

= < Ψ ΐη (1)ΐΨ > 0

0

-

0

2

where η ( 1 ) = Σι η ^ ( 1 ) . F o r c o m p a r i s o n we a l s o c o r r e s o p o n d i n g r e s u l t s f o r t h e HF g r o u n d - s t a t e . r u l e c o r r e l a t i o n s can be s t u d i e d by c o m p u t i n g S

2

(10)

0

show t h e Hund's

2

= 0

0


(11)


20.

STOLLHOFF & FULDE

Correlations in d and/Electron Systems

283

Figure 1. (a) "favourable" and (b) "unfavourable" c o n f i g u r a t i o n c o n t a i n e d i n t h e n o n m a g n e t i c HF g r o u n d state ΙΦ >. The c i r c l e s s y m b o l i z e atoms and t h e five segments the different d-orbitals. The d-electron occupancy p e r atom i s chosen t o be 2 . 5 . 0

Figure 2. Charge f l u c t u a t i o n s as f u n c t i o n o f d-band f i l l i n g n U . E l e c t r o n c o r r e l a t i o n s influence strongly the energy difference between nonmagnetic and magnetic states, l e a d i n g t o d r a s t i c changes o f the Stoner-Wohlfarth crit e r i o n for the onset of ferromagnetic order. The r e a s o n i s t h a t e l e c t r o n i c charge f l u c t u a t i o n s are smaller i n a ferromagnetically ordered than i n a nonmagnetic state. Therefore e l e c t r o n c o r r e l a t i o n s decrease the energy by a l a r g e r amount o f a s t a t e w h i c h i s n o n m a g n e t i c , t h a n o f a f e r r o m a g n e t i c s t a t e . F o r example, i n Fe t h e energy g a i n d u e t o f e r r o m a g n e t i c o r d e r i s 0 . 5 6 e V / a t o m when t h e HF a p p r o x i m a t i o n i s made a n d a r a t i o U / W = 0 . 4 4 i s assumed. When d e n s i t y and in addition spin correlations are i n c l u d e d , t h i s energy reduces t o 0.22 eV/atom and 0.15 e V / a t o m , r e s p e c t i v e l y (10) . I n t h e LDA t o t h e density functional theory, Hund's rule correlations are not taken into account, because t h e y are n o t p r e s e n t i n an unpolarized homogeneous electron gas from which the exchange-correlation potential i s taken. When t h e L S D approximation is applied instead, they are partially included. Spin correlations, however, modify the generalized Stoner parameter strongly ( 1 3 ) . The latter can be related to the exchange correlation energy E ( M ) f o r f i x e d m a g n e t i z a t i o n M by w r i t i n g


2

c

C

X C

Q

0

Mo E

X C

(M )

= E

0

x c

(0)

+

\

J dM I 2

x c

(M)

(13)

ο where I = I ( 0 ) i s the o r i g i n a l 4 d i s p l a y s the magnetic f i e l d parameter as obtained within mations for the case of Co. x c

Stoner parameter. Figure dependence o f the Stoner three different approxi I i i s the result of a


20.

STOLLHOFF & FULDE

Ό Downloaded by CORNELL UNIV on September 1, 2016 | http://pubs.acs.org Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch020

285

Correlations in d and f Electron Systems

5

10 2

F i g u r e 3. Atomic spin correlations S as f u n c t i o n o f d-band filling n. U/W=0.5. (Adapted from r e f . 9.) 0

1.6

ι—ι—ι—ι—ι—ι—ι—r

F i g u r e 4. S t o n e r parameter I(M) and l o s s o f k i n e t i c energy D(M) (dashed line) for Co as functions of magnetization. 1^ - f u l l c o r r e l a t i o n c a l c u l a t i o n , I2 - neglecting spin correlations, I3 - n e g l e c t i n g a l l c o r r e l a t i o n s , r e s u l t i n g from J ^ j . (Adapted from r e f .


286

THE CHALLENGE OF d A N Df ELECTRONS

complete correlation calculation. I2 is the Stoner p a r a m e t e r when s p i n c o r r e l a t i o n s a r e n e g l e c t e d . Since spin correlations play no role for fully magnetic s t a t e s , b o t h c u r v e s become e q u a l a t M =1.6. W h i l e I2 does not d i s p l a y s i z e a b l e magnetic f i e l d dependencies and compares in this respect with results of LSD c o m p u t a t i o n s , 1^ i n c r e a s e s b y 20% f r o m M=0 t o M . 13 finally is the curve obtained when a l l exchange contributions (~Jij) to the interaction part of the H a m i l t o n i a n ( E q . 1) a r e t r e a t e d i n H F a p p r o x i m a t i o n . I2 a n d I3 m a y b e c o n s i d e r e d a s l o w e r a n d u p p l e r l i m i t o f t h e d e f i c i e n c i e s o f L S D . A l t h o u g h a c h a n g e o f I b y 20% may s e e m s m a l l , i t h a s t h e e f f e c t o f c h a n g i n g t h e C u r i e temperature T by approximately a f a c t o r o f two, because T ~ ( I N ( O ) - l ) / * a n d I N ( 0 ) * 1, However, even t h a t is not s u f f i c i e n t i n order to b r i n g the l a r g e r c a l c u l a t e d values for T i n agreement w i t h experiments. This is due t o the fact that Stoner theory does not contain fluctuations of the order parameter. For improved calculations of T see e . g . Ref. (14). Another point of considerable importance i s the n o n l o c a l character of the exchange. The l a t t e r always favours non-uniform distributions of electrons (or holes) a m o n g t h e d i f f e r e n t d o r b i t a l s , e . g . e g a n d t2g orbitals, when t h e system is cubic. Direct Coulomb interactions as well as correlations favour uniform occupations of the different atomic orbitals and therefore counteract the effect of nonlocality of the exchange. Despite this, the anisotropies caused by exchange are important, i n p a r t i c u l a r f o r b u l k N i (10) as w e l l as f o r i t s s u r f a c e (15). Finally it is of interest to compute spin c o r r e l a t i o n s between n e i g h b o r i n g s i t e s , i.e. m a x

m

a

x


c

1

c

c

c

2

S (o)

= , o

(14)

0

where δ denotes a n e a r e s t neighbor o f s i t e 0. IΨ> is c a l c u l a t e d by s t a r t i n g from the nonmagnetic S C F - s t a t e as b e f o r e , b u t b y i n c l u d i n g i n S ( s e e E q . 6) a l s o operators of the form O i j ( 1 , l + a ) = £ > i ( 1 ) S j (1+©) . The effect of these operators i s that additional ferromagnetic corre lations betweeen electrons on neighboring sites are b u i l t i n t o I V » > , e x c e p t f o r b a n d f i l l i n g s c l o s e t o ncf. The i n d e x 1 s t a n d s for l i g a n d o r b i t a l a n d we a s s u m e t h a t i t i s r a t h e r extended s o t h a t we m a y n e g l e c t C o u l o m b r e p u l s i o n s w i t h i n that orbital. The H a m i l t o n i a n t h e n r e a d s Η = ε,Σ σ

1σ1

χ

+ σ

c

f Σ φ σ

+ V Σ (φσ+ΐσ^σ) σ

σ

+

Un£nf * τ

(15) + with η = f f and U v e r y l a r g e . We w a n t t o d i s c u s s t h e s o l u t i o n s o f t h e e i g e n v a l u e problem f o r two e l e c t r o n s . F i r s t we s e t V = 0 . In that case, because ci>tf the ground s t a t e i s a quartet w i t h e n e r g y E = c i + C f , i . e . one e l e c t r o n i s i n t h e f o r b i t a l and the o t h e r i s i n the 1 o r b i t a l . The e x c i t e d s t a t e i s a s i n g l e t with E = 2 c i , i . e . both electrons are i n the 1 orbital. W h e n VfO i s t a k e n i n t o a c c o u n t , the grounds t a t e quartet s p l i t s i n t o a low l y i n g s i n g l e t f

σ

a

a

Q

s

72

•*o> =

(φΐ-φΐ)

U-(V/Ac)2)

I0>

- ^

φ|ΐΟ> (16)

with

energy

l*cl>

=

2

Ε =εχ+ε£-2ν /Δε 0

(1-(V/Ac)2)

and a

φ|ΐΟ>+

triplet

j2 ( φ ί - φ τ ) Ό > (17)

Φΐ

,0>

'*c2> = ? ε 3 > = ΦΪ»0> w i t h energy Et=ci+Cf. We h a v e s e t ε χ - ε £ = Δ ε 4 ) . The f o r m a t i o n o f t h e s i n g l e t ιΨ> w i t h triplet state of e x c i t a t i o n energy Ε =2ν characteristic feature of strongly correlated systems. When t h e f o r b i t a l i s e m b e d d e d i c o n d u c t i o n e l e c t r o n s , t h e energy g a i n due t o f o r m a t i o n becomes , ψ

0

β χ

ΔΕ= Dexp[

2N(Ô)V^

2

(see figure an e x c i t e d /Δε, is a f-electron n a sea of the singlet

]

2

(

1

8

)

i n s t e a d o f ΔΕ=2ν /Δε, as i n the case o f two o r b i t a l s . H e r e D i s t h e c o n d u c t i o n - e l e c t r o n band w i d t h and 2N(0) is their density of states. We h a v e set the Fermi energy equal to zero. The energy gain is usually identified with a characteristic temperature kgTi^E,



288

THE CHALLENGE OF d AND

f ELECTRONS

t h e Kondo t e m p e r a t u r e . The low l y i n g e x c i t a t i o n s l e a d t o h e a v y - f e r m i o n b e h a v i o u r when t h e i o n s w i t h f e l e c t r o n form a l a t t i c e . The above calculation suggests that a singlet f o r m a t i o n due to strong correlations with a triplet e x c i t e d s t a t e s h o u l d be f o u n d i n a p p r o p r i a t e m o l e c u l e s . The effect r e q u i r e s an e v e n t o t a l number o f v a l e n c e electrons. I n o r d e r t o d e t e c t i t one s h o u l d s e a r c h e.g. f o r m o l e c u l e s c o n t a i n i n g Ce, w h i c h a r e d i a m a g n e t i c , b u t w h i c h show a f - e l e c t r o n c o u n t c l o s e t o 1, when p h o t o emission experiments are performed. The f o r m a t i o n o f a s i n g l e t s t a t e due t o s t r o n g c o r r e l a t i o n s i m p l i e s a l s o a new k i n d o f e l e c t r o n - p h o n o n coupling. The e n e r g y g a i n Δ Ε due t o s i n g l e t f o r m a t i o n d e p e n d s on t h e h y b r i d i z a t i o n V, w h i c h i n t u r n d e p e n d s on p r e s s u r e Ρ o r volume Ω · In p a r t i c u l a r i n a s o l i d t h i s dependence ΔΕ(V) is very strong (see Eg. (18)), r e s u l t i n g i n a s t r o n g e l e c t r o n phonon c o u p l i n g . Its s t r e n g t h c a n be c h a r a c t e r i z e d by an e l e c t r o n i c G r u n e i s e n parameter

= η

-

din T din Ω

K ±y

< '

M e a s u r e d v a l u e s o f n a r e a s l a r g e a s 100-200 i n h e a v y f e r m i o n systems (17). One important problem, which i s presently under i n t e n s e i n v e s t i g a t i o n s i s t h a t of the Fermi s u r f a c e of s t r o n g l y c o r r e l a t e d f - e l e c t r o n systems. I t was a s u r prise, at least to the present authors, that the m e a s u r e d F e r m i s u r f a c e o f t h e h e a v y - f e r m i o n s y s t e m UPt3 (18) i s v e r y much i n a c c o r d w i t h t h e one computed w i t h i n LDA ( 1 9 ) . T h e r e i s no a p r i o r i r e a s o n why t h e t o p o l o g y o f t h e F e r m i s u r f a c e s h o u l d come o u t c o r r e c t l y when e l e c t r o n c o r r e l a t i o n s a r e s t r o n g and a LDA i s made. But f o r UPt3 i t d o e s come o u t s u r p r i s i n g l y w e l l , a l t h o u g h t h e m e a s u r e d e f f e c t i v e masses a r e o f f b y a f a c t o r o f o r d e r 20 a s compared w i t h t h e c a l c u l a t e d o n e s . Detailed i n v e s t i g a t i o n s h a v e shown (20) t h a t t h e g o o d a g r e e m e n t in the case o f UPt3 i s due to a large spin-orbit s p l i t t i n g and a c r y s t a l - f i e l d (CEF) s p l i t t i n g , w h i c h i s much l e s s t h a n kgT^, i . e . t h e e n e r g y g a i n due t o s i n g l e t formation. In that case, the theory becomes a one-parameter ( w h i c h i s t h e e f f e c t i v e mass) t h e o r y , and t h e t o p o l o g y o f t h e F e r m i s u r f a c e due t o the heavy q u a s i p a r t i c l e s i s c o m p l e t e l y d e t e r m i n e d by t h e g e o m e t r y of the unit c e l l . I n c a s e s i n w h i c h t h e CEF s p l i t t i n g i s l a r g e r t h a n kgTR, one e x p e c t s d i f f e r e n c e s b e t w e e n t h e m e a s u r e d F e r m i s u r f a c e and t h e one w h i c h f o l l o w s from a p p l y i n g t h e LDA. I n o r d e r t o improve t h e computation o f t h e F e r m i s u r f a c e one c a n p r o c e e d a s f o l l o w s , at l e a s t f o r Ce compounds. One a p p l i e s t h e LDA t o t h e d e n s i t y f u n c t i o n a l theory f o r a l l e l e c t r o n s , except the f-electrons. The potential a c t i n g on t h e l a t t e r i s


20. STOLLHOFF & FULDE

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Correlations in d andf Electron Systems

d e s c r i b e d by an energy dependent phase s h i f t $1=3(ε), for which a simple, phenomenological ansatz is made. Only these channels w i t h i n t h e 1=3 m a n i f o l d o b t a i n a phase s h i f t d i f f e r e n t from z e r o , w h i c h have t h e symmetry Γ o f the c r y s t a l - f i e l d ground state. The l a t t e r is usually known from inelastic neutron scattering experiments. T h e s l o p e

Correlations in d and f Electron Systems - American Chemical Society

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