The electronic-structure and band-gaps in transition-metal compounds

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Journal of Magnetism and Magnetic Materials 54-57 (1986) 607-611 607

T H E E L E C T R O N I C S T R U C T U R E A N D B A N D G A P S I N T R A N S I T I O N M E T A L C O M P O U N D S

J. Z A A N E N , G . A . S A W A T Z K Y a n d J . W . A L L E N +

Physical Chemistry Department, Materials Science Center, Unwersity of Groningen, 9747 A G Groningen, The Netherlands

A many body theory recently introduced for describing the electronic structure of rare earth metals is applied to 3d transition metal compounds. The band gaps, their character and systematics, as well as various spectral distribution functions, are calculated and compared to recently obtained experimental data, for NiO and NiS.

The electronic structure of 3d transition metal compounds has been a controversial topic of discussion since 1937 when De Boer and Verwey [1] pointed out that the insulating behaviour of many of the compounds was inconsistent with the then available theory [2] for the electronic structure of solids. This problem was basically solved by Mott [3] and Hubbard [4] who showed that if the d - d C o u l o m b interaction is larger than the one electron dispersional part of the band width a correlation gap will result with the Fermi level in the gap. These ideas have formed the basis for the understanding of a large range of physical properties of T M compounds of which the most important for our discussions are: the band gaps, the optical and magnetic properties. F o r the Mn ~ Cu oxides and halides these properties are semi-quantitatively well described by as- suming a large d - d C o u l o m b interaction which prevents polarity fluctuations in the d band and stabilizes a ground state with an almost integral number of d electrons on each ion corresponding to that expected for an ionic compound. The deviation from ionicity is caused mainly by hybridization (covalency) with the anion p band which involves a charge transfer energy. F r o m the above follows that the local magnetic moment is given by that of an ion (d") placed in the appropriate point group symmetry which in turn is governed by the T a n a b e - S u g a n o [5] diagrams. Also the excitations in the d" manifold are given by these diagrams and corre- spond to sharp optical transitions [6] observed for energies below the band gap. These excitations should be described as Frenkel excitons [7]. Important to note is that in the above description the transitional symmetry and therefore the d band dispersion has not entered. In fact many of the properties can he described by consid- ering a TM ion as an " i m p u r i t y " in the solid.

The translational symmetry enters in the description of the collective magnetic properties, generally success- fully described by spin-only Hamiltonians with inter- atomic exchange interactions given by Andersons theory of superexchange [8] and the G o o d e n o u g h - K a n a m o r i rules [9]. These theories are also based on the assump- tion of large d d C o u l o m b interactions.

* XEROX Palo Alto Research Center, 3333 Coyote Hill Road, CA 94304, Palo Alto, USA.

The problems remaining concern mainly the nature of the band gap, the magnitude of the Coulomb interactions, and the nature of the states at the Fermi level of the metallic sulfides and some early T M oxides. In the numerous discussions on the nature of the band gap in insulators, N i O has played a central role as an example. The band gap could be of d d, O 2 p - T M 4 s , O 2 p - T M 3 d or T M 3 d - T M 4 s character as discussed by numerous authors [7,10 12]. Recent sophisticated local density functional band structure calculations obtain a d - d gap of 0.3 eV in N i O [13] although F e O and C o O are still metallic. That one obtains a gap at all is impressive but recent direct measurements [14,15] on N i O show that the band gap is an order of magnitude larger. Also there is considerable evidence that the band gap in Ni compounds is of a charge-transfer ( O 2 p - N i 3 d ) rather than d - d nature. This is suggested by Raman [16] experiments and also by the direct relation between the band gap and anion electronegativity [10]. The measured gaps are NiS (0 eV), NiI2 (1.7 eV) [17], NiBr 2 (3.2 eV) [17], N i O (4.3 eV) [14]. On the other hand the band gaps in the early T M c o m p o u n d s Ti, V are known to be of d - d character [7].

In this paper we present a new [1819] way of looking at the ground and excited states of T M compounds and arrive at a classification scheme which is consistent with available experimental data as well as known trends in the d d Coulomb interactions and anion electronegativ- ities.

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608 J. Z a a n e n el al. / Electronic structure in transition metal compounds

dispersional part of the b a n d width is small as in Ni dihalides [21] a n d N i O [13] ( = 0.5 eV).

We could of course also break the a n i o n translational s y m m e t r y e n d i n g up with a m u c h used cluster a p p r o x i m a t i o n [18]. However, the a n i o n p b a n d w i d t h s for the halides, oxides and sulfides are between 3 a n d 4 eV and therefore not negligible. O u r calculation shows that the cluster a p p r o a c h is valid only if the b a n d gaps are quite large ( > 2 eV).

For the calculation we use an A n d e r s o n i m p u r i t y H a m i h o n i a n

H - ~%,,c;,,,'j.,, + ~.%,d~)d,,, + ~. U((jlm ) d + d , d ] d,,,

nA m q / m + £ K,,l,,,( +' d,,,~k, , + {j,,,d,,,) '+ m/,H = Hh,,< + L/map + H hyhr.

The first term describes the " h o s t " b a n d structure which in o u r case involve the a n i o n p a n d cation 4s, p b a n d s . T h e 2nd a n d 3rd terms (tth,,p) describe the d-electronic s t r u c t u r e of a T M ion in the p o i n t group s y m m e t r y of the solid including crystal field splittings a n d in U the Slater integrals F °, F 2 a n d F 4. The 4th term describes the h y b r i d i z a t i o n interaction between the " i m p u r i t y " d states and the host b a n d s . In fig. 1 we show the various possible situations which could be e n c o u n t e r e d . The solid lines are s h o w n for H h'+' - 0 a n d the d o t t e d lines indicate what h a p p e n s to the redistribution of d weight u p o n switching on H hvh'. We point out that the states below E I represent the N 1 electron eigenstales (electron removal spectrum} a n d the states a b o v e E v represent the N + 1 electron eigenstates (electron a d d i t i o n spectrum}. The large splitting in the d d weight (-= U ) is due to the difference in the d ionization and affinity which, taking the H u n d s rule g r o u n d state term in each case, is given by U - / f ( d " ') + E ( d " + ') - 2 E ( d " ) . n-1 i n+ A d

I'll"

~ U e f f >l~, [d dn_l I n+l B .,e~ = U e f f ~ d i i i ~ - - - \ r ~_ E g a p - ~ , / ... :-~ II ' II n-i k d n+1 c d ~ = U e f f ~ q , j ,

'1

s

I.

i I D dn-~ <_~ ~ U e f f ~ d n+1 ,

, ,

i _ ,

I I I / ",.., ~gap I L ~ ~: . . . ~ ' ~J I E , dn~ 1 ~ U e f f ] :, d n÷l , i ~ , , i -~ I i i ~ - "> i " ~ E g a p ~ l EF

Fig. 1. An artists concept of the possible situations encountered for strongly correlated impurities in solids.

Defined in this way U c o n t a i n s b o t h the m o n o p o l e ( F ' ) a n d the higher multipole ( F 2. F a) Slater integrals of which F ° is strongly screened in the solid whereas F z a n d F 4 r e m a i n close to the free ion values. U therefore d e p e n d s on the d" g r o u n d state term. The c o n d u c t i v i t y gap is defined by

E~,p = E~'"id ( U - 1 ) + E~,'"id ( N + 1 ) 2 E,~'"id ( N ). T a k i n g the g r o u n d state energy (E~'4W(N)) for the N electron system as zero, Eg,p is simply the m i n i m u m energy required to remove a n electron plus that to add one electron. This is exactly what is m e a s u r e d by the onset of the p h o t o - and inverse p h o t o - e l e c t r o n spectra, respectively.

Fig. l a shows what h a p p e n s for a strongly correlated i m p u r i t y in a metallic host. Here we also show the various possible term splittings of the d" 1 (f,, ~) and d " + 1 (f,,. 1) states. These are observed in the beautiful p h o t o inverse p h o t o electron spectra of rare earth metals by Lang et al. [22]. Shown as d a s h e d lines are the results of switching on H hv6' causing a " v i r t u a l b o u n d state" b r o a d e n i n g and the possible f o r m a t i o n of a K o n d o r e s o n a n c e at the Fermi level as predicted theoretically [20] a n d observed by p h o t o a n d inverse p h o t o emission [23].

The situations we can e n c o u n t e r in T M c o m p o u n d s are shown in figs. l b , c with a large, a n i o n p-cation 4s, p, " h o s t " b a n d gap which is k n o w n to be 6 - 1 1 eV [11]. D e p e n d i n g on the size of U and the position of the d " i d,,~t states relative to the " h o s t " b a n d s the gap can be either a n i o n p - T M d , T M d T M d , T M d TM4s, or even a n i o n p - T M 4 s .

T h e most interesting cases for the late T M com- p o u n d s are shown in figs. l d a n d e. Here the a n i o n p b a n d lies in the d d correlation gap forming a charge t r a n s f e r gap (fig. ld). If the d" t state is close to or in the anion p b a n d (fig. le) a " ' b o u n d state" can be p u s h e d up, out of the b a n d by H h>h' with a strongly mixed character. If the charge transfer energy ( a n i o n p d " + ~ ) is also small this gap can close forming a metal b u t with a peak at the Fermi level very reminiscent of a K o n d o resonance. T h e zero gap case of fig. l d corre- s p o n d s to a p type metal (CuS) a n d the zero gap case of fig. l c is close to reality for NiS as s h o w n below.

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J. Zaanen et al. / Electronic structure in transition metal compoundY 609 n-I(UPS) n ( n e u t r o [ } n+l(BIS} U÷A I I

dO54--

U ! A A dnL -~-~ ~' , ', T~sn -1 ~L I ,~UY- E l l n - l ) U÷A

1

I

U+A d

n+IL

- ~

,,

A=IEd-EL

d

n+1

~r-',

T6n+1

dn ~ T6n E.(n+l}

Fig. 2. Diagram showing the various states and parameters used in the calculation.

the b a n d gap we also d o the calculation with one electron r e m o v e d (left h a n d side) and one electron a d d e d (right h a n d side) which again should be d o n e for every irreducible r e p r e s e n t a t i o n of the p o i n t group. The lowest energy states o f these relative to the ground state energy are a direct measure of the b a n d gap.

The p a r a m e t e r s used in the calculation are also indicated in fig. 2. F o r example for divalent Ni c o m p o u n d s the charge transfer energy ( A ) is given by

A = (E(dg_L) + E ( d S ) ) = ( d - - ( L '

m e a s u r e d from the center of the anion p b a n d (L is a ligand hole).

N o t e that the N - 1 particle states in fig. 2 have

b e e n d r a w n for U > A V = W / 2 w h e r e W i s the anion

p b a n d width, as suggested by Fujimori [18] for Ni c o m p o u n d s .

Switching on g hybr n o w mixes the various config- urations within each d i a g r a m resulting in a very com- plicated s p e c t r u m of discrete, b o u n d and shifted con- t i n u u m states. F o r the b a n d gap calculations we are only interested in the lowest energy state for each case which is in all cases shifted d o w n by g hybr as indicated in fig, 2 by 6 ~. In terms of our p a r a m e t e r s a n d the h y b r i d i z a t i o n shifts (8 ' ) the b a n d gap is

E s , p = A - W / 2 + 2 8 " - 6 ''+~ - 8 " 1 ( U ~ > A - W / 2 ) ,

Eg.p=U+28"

(~n+l--sn-'

(U<A--

W / Z ) , with W as the anion p b a n d width a n d with 6 " 1 always measured from the lowest energy state of the system with H hybr set to zero. We see immediately from this that the gap follows the charge transfer energy for U < A as observed in divalent Ni c o m p o u n d s .

N o t e that to obtain the generally used picture with the N - 1 particle states drawn below E v and the N + 1 particle states above E F we have to rotate the N - 1 particle spectrum of fig. 2 about the N particle ground state line. This requires knowledge of 8" and can cause some p r o b l e m s in d e t e r m i n i n g U directly from p h o t o - emission and inverse p h o t o e m i s s i o n experiments.

The details of the calculations, which follow proce- dures outlined by G u n n a r s s o n and S c h 0 n h a m m e r [20] will be published elsewhere [25]. As parameters we used W = 3 eV a n d V = 1 eV. The calculation has been d o n e for n = 10, 9, 8, 0, 1, 2, 3 c o r r e s p o n d i n g to Cu I ~, Cu 2 + Nl-~+, Sc ~+ Ti ~+, V ~+ but we find that the results are rather insensitive to n being d e t e r m i n e d more by U, A and V.

The energy gaps are shown in fig. 3 as a function of A and for various values of U. We notice that for U = 0 the gap is always zero. This will be the case in any calculation in which the exchange and correlation are replaced by an effective one particle potential. U p o n taking into account the translational symmetry of the T M ions a small gap may occur because of the d b a n d dispersional relations. We also see that for U large ( U > A) the gap is proportional to ,3 and goes to zero for A < W / 2 resulting in a metallic ground state. For U << A (right hand side) the gap is proportional to U as in a Mott H u b b a r d system. Next to these extreme cases there is an interesting region for U = ,3 ___ W / 2 . In this region the d" ~ state lies inside or close to the edges of the d " L continuum. The N - 1 particle ground state is then d e t e r m i n e d by a b o u n d state p u s h e d out of the d"_L c o n t i n u u m due to the hybridization with the d" discrete state. This b o u n d state will b r o a d e n into a narrow b a n d if we include the translational symmetry of

the T M ions and contains b o t h d " L and d" ~ character. I

It is i m p o r t a n t to note that this b o u n d state can occur for each IR s p a n n e d by the n - 1 d electrons and will therefore exhibit the same kind but s o m e w h a t reduced multiplet splittings. This was d e m o n s t r a t e d nicely in the

31-

-2 -1 U= y (9) / ! (B) .a~///.~;._._~¢ (s) ~ : i - - + .. . . . + 13) 4- ~ & - - Z X - - & - - &

~ 0

~ J o ~ O - - O - - O - - O - - O - -

,I

I I I I 1 I I I I I I

0 1 2 3 L 5 6 7 8 9 10

A

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610 J. Zaanen et al. / Electronic structure in transition metal compounds" ..Q >.- I--. 7 U.I l'-- Z

cluster calculations of F u j i m o r i et al. [18].

A clear d e m o n s t r a t i o n of the success of the a b o v e theory is not only the b a n d gap systematics but also its success in describing the p h o t o a n d inverse p h o t o e m i s - sion data which are a direct measure of the N - 1 a n d N + 1 particle spectral distributions. In fig, 4 we show two recently studied examples N i O [14] and the metallic NiS [26]. The a s s i g n m e n t scheme at the top is d o n e using a cluster theory a p p r o a c h for NiO. At the high p h o t o n and electron energies used the m e a s u r e d spectral weight is m a i n l y t h a t for d electron removal a n d ad- dition. Inverse p h o t o e m i s s i o n has also been d o n e at lower energies (10 eV) by H u f n e r et al. [15,17].

For N i O we see the gap of 4.3 eV a n d the j u s t discussed b o u n d state as the first p h o t o ionization peak. The arrows d r a w n between the N i O and NiS d a t a show the shifts of the structures seen due to a lower A a n d also a lower U. W e notice the large total d electron spectral weight spread of a b o u t 30 eV ( - 10 to + 18 eV) for N i O a n d 15 eV for NiS. It is this large spectral weight spread as well as the b a n d gaps a n d peaks positions which is a challenge to describe theoretically. The d electron p h o t o (inverse) emission s p e c t r u m is given by

lphot,,,,,,, 0 c I m ( ' v g l d + ( d ) z

l f i ( d +)dlq'~).

is the g r o u n d state wave function a n d is of the form kVg (3A 2~z ) = A IdS(3A 2g)) + '~' 0~/, Id9N (~A2g))

k + ~',8~A, ]d]°kk'(3A2~)),

1, k "

with the coefficients d e t e r m i n e d by the a b o v e men- tioned g r o u n d state calculation.

d9_L 2 d 7 d 8 / d 9

dlOL

)

PES

.,

BIS

Eph(eV)

1486 eV

;b.

i

~ . /

1 2 ~ . . j ' , i \ 4S 41o

_.

//

NiO"

• .~ ,-:: ,. : . ~ , ~ / / / t t /

J

1 4 8 6 _{J L _ _} [ i i i ~1 , i i i I i ~ , i I I , [ i [ - 2 0 - 1 0 0 1 0 2 0 E N E R G Y A B O V E EF ( e V )

Fig. 4. Photo and inverse photoemission spectra of NiO and NiS. The bar diagram shows the assignments according to a cluster calculation.

To indicate the spectral d i s t r i b u t i o n s of the N - 1 a n d N + 1 basis states of fig. 2 u p o n including H h>b~ we show in fig. 5a a n d c the imaginary parts of the G r e e n ' s functions (e.g. ~ a ( d S k l l / ( Z - H ) l d S k ) = GSL) entering the N + 1 electron spectrum. We see indeed the a p p e a r a n c e of a b o u n d state d e t e r m i n i n g the first ionization state in b o t h NiO a n d NiS. In N i O this state has substantial d 7 as well as dXL character. For NiS the lowest ionization state at the Fermi level is a mixture of the electronic c o n f i g u r a t i o n s d 7, dSL a n d d~L 2. This state together with the lowest affinity state of d 9 a n d d m L c h a r a c t e r a n d the g r o u n d state of d ~, d~L form a s h a r p structure at E~ which is very reminiscent of a K o n d o resonance. T h e exact n a t u r e of this state is NiS is very sensitive to the choice of A, V a n d U a n d therefore to the lattice p a r a m e t e r s which may indicate a m e c h a n i s m for the phase transitions observed in NiS [28].

The calculated spectra are shown in figs. 5b and d. T h e energy spread is the same as in figs. a and c but now we take into a c c o u n t the g r o u n d state wave functions in the a b o v e equations. We notice that a l t h o u g h the first ionization state in N i O has a lot of d~L c h a r a c t e r it still has a very large spectral weight for d electron removal c o n s i s t e n t with p h o t o n energy d e p e n - d e n t m e a s u r e m e n t s [29]. This is a direct result of the constructive interference for the lowest energy state be- tween the various ionization c h a n n e l s from the mixed g r o u n d state (i.e. d s --, d 7, dgL --, dS_L, d l ° L 2 - , d~L). This is a general tendency as p o i n t e d out by G u n n a r s o n a n d S c h 6 n h a m m e r [20]. The theoretical spectra were calculated for only one of the m a n y IR's for the n - 1 electron s p e c t r u m a n d therefore d o not show the splitting in the first ionization peaks observed e x p e r i m e n - tally.

W e nov,, finally come to a classification scheme b a s e d on the size a n d n a t u r e of the b a n d gaps s h o w n in fig. 4. We realize that ~ is roughly p r o p o r t i o n a l to the a n i o n electron negativity for a given TM. We also realize that U decreases in going from Cu to Ti com- p o u n d s from a b o u t 8 - 9 eV [30] to a b o u t 1 or 2 eV.

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J. Zaanen et al. / Electronic structure in transition metal compounds 611 , ', ~ ,~ i I "rl ,,~ ,, i ' i ' P B (i i I'1 G , ~ , , I , I II II I | i I _ _ ~ - i'~' i 0 10 E F -10 E F llO ENERGY (eV)

Fig. 5. A and C show the imaginary parts of the N - I and N + 1 particle Green's functions as defined in the text for NiO ( ~ = 6 , U = 8 , V = 2 ) and NiS ( A = 2 , U = 4 , V = 2 ) in both cases W = 4 eV. B and D are the corresponding photo- and inverse photoemission spectra for NiO and NiS respectively. A broadening of 1.4 eV (fwhm) was used in all cases.

gap is d e t e r m i n e d by the strongly mixed b o u n d state as discussed above. The regions marked C', C ' D a n d D describe metals. In D the Fermi level lies in the anion p b a n d resulting in hole c o n d u c t i o n as in m a n y Cu sulfides a n d selenides. The region C ' D is rather special because here the Fermi level is d e t e r m i n e d by the " b o u n d state" discussed above with a very high density of states which however is of strongly mixed character• We believe that NiS, CoS a n d p e r h a p s FeS belong to this region.

In our o p i n i o n the described calculation forms a g o o d basis for classification of T M c o m p o u n d s . The p a r a m e t e r s used can be d e t e r m i n e d by c o m p a r i n g calculated a n d experimental p h o t o and inverse p h o t o - emission as well as optical spectra. In future p a p e r s we s h o w that the same theory can also be used to describe core level spectroscopies. Aside from the extension of the calculation to include all irreducible representations of the point group the major challenge remaining is to

A Egap u

-2 -1 0 1 3 /, 5 (5 7 8 9 10 11 12

ZX

Fig. 6. A classification scheme based on the calculation described in the text. The various regions are explained in the text. This diagram is calculated for W = 3 and V = 1 eV.

find a way of i n c o r p o r a t i n g the T M translation symmetry. This is of u t m o s t i m p o r t a n c e for an u n d e r s t a n d i n g o f the collective magnetic a n d t r a n s p o r t properties. In this regard we believe that the described theory forms a basis for describing the n a t u r e of the quasi particles which could be used to i n c o r p o r a t e the T M translation symmetry.

O n e of the authors (G.A.S.) would like to thank the X E R O X C o r p o r a t i o n for their hospitality during his stay at which time the ideas for this work were con- ceived. This work was s u p p o r t e d in part by the N e t h e r - lands F o u n d a t i o n for Chemical Research (SON) with financial aid from the N e t h e r l a n d s Organization for the A d v a n c e m e n t of Pure Research (ZWO).

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