f Electron States in Condensed Matter - American Chemical Society

Chapter 17

f Electron States in Condensed Matter

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on April 18, 2016 | http://pubs.acs.org Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch017

Jaime Keller and Carmen de Teresa Division de Ciencias Básicas, Facultad de Química, Universidad Nacional Autónoma de México, Apartado 70-528, 04510 México D.F., Mexico

In condensed matter the d and f electron states give r i s e to a large variety of properties due to t h e i r p o s s i b i l i t y of being either atomic-like states or band-like states with small changes i n chemical com position, temperature, pressure, e t c . Both types of states are discussed both from the t h e o r e t i c a l (and computational) point of view and from the ob served spectroscopic, e l e c t r i c and magnetic properties point of view with s p e c i a l emphasis i n f electrons. Electron-electron c o r r e l a t i o n is shown to play a dominant r o l e i n these series of phenomena. Atomic-like f electron states i n condensed matter were f i r s t studied i n rare-earth and actinide metallic or non metallic com pounds. There the m u l t i p l i c i t y of the f states and related pro perties l i k e magnetic moment, Curie-Weiss s u s c e p t i b i l i t i e s and spectra (where the c r y s t a l f i e l d s p l i t t i n g i s measured) indicate that f o r most of the rare-earth series (RE) i t i s a good ap proximation indeed to consider those f electrons as atomic-like states. Then f o r the c a l c u l a t i o n of properties we can treat the f electrons i n those compounds within the same approximations as f o r the core electrons and assume that the i n t e r a c t i o n between f electrons i n d i f f e r e n t s i t e s i s carried through the conduction or the valence electrons. The magnetic i n t e r a c t i o n between the ions i n the magnetic metals f o r example, can then be considered as carried by the con duction electrons i n the well known Rudermann-Kittel-KasuyaYoshida (JJ i n t e r a c t i o n . The physical o r i g i n of t h i s i n t e r a c t i o n i s a point l i k e p o l a r i z a t i o n of the conduction electrons (CE), at the atomic s i t e s , by the magnetic moments of the f electrons, r e s u l t i n g i n an o s c i l l a t i o n of the spin density of the CE. The point l i k e approximation i s useful because the maxima of the f wave functions are found well inside the atomic core, i n r a d i i smaller than 0.7 atomic u n i t s . This p o l a r i z a t i o n i s carried from ion to i o n by the generated p o l a r i z a t i o n o s c i l l a t i o n of the con duction electron spins, which has a wave length λ = 2π/ε£/ ( e = ζ

£

0097-6156/89/0394-0246$06.00/0 o 1989 American Chemical Society

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

f


17.

KELLER & DE TERESA


247

Fermi energy r e l a t i v e to the bottom of the CB). The Fermi l e v e l wave lengths are i n general incommensurable with the c r y s t a l ' s interatomic or interplanar distances; as a r e s u l t , the magnetic ground states of the rare-earth intermetallics and pure metals have complicated s p a t i a l d i s t r i b u t i o n s , ferromagnetic, a n t i f e r r o magnetic, h e l i c a l , etc. There are, nevertheless, non metallic or metallic elements and compounds containing rare-earths or actinides, where the f electrons are c l e a r l y not properly described as l o c a l i z e d atomicl i k e states. A wide range of special properties are found as sociated with phenomena l i k e e l e c t r o n i c a l l y induced phase tran sitions, va&ch£« f ] u^iturations, mixed valency (MV), disappearance of Kondo r e s i s t i v i t y and heavy electron behaviour. As we w i l l see below, there i s a common o r i g i n to a l l these properties which arise from the f electrons forming f bands which are highly hybri dized with the s, ρ and d valence or conduction electrons, pre senting new types of c o r r e l a t i o n between the f electrons them selves, between the f electrons and the conduction electrons and between f electrons i n d i f f e r e n t atomic states. As a result we find a whole series of new electronic ground states. A further complication arises at high temperatures, because i t i s usually the case that the peculiar electronic ground state, which gave r i s e to the special properties, i s destroyed and the materials behave again as i f the f electrons were l o c a l i z e d , atomic-like states. The main reason for t h i s electronic t r a n s i t i o n i s that the correlations are weakened by the thermal disorder and that the entropy AS of the l o c a l i z e d , disordered, atomic l i k e magnetic moments adds a term AF = -AS T to the free energy F. At low temperatures, there are at least two ways i n which the f electrons, and t h e i r magnetic moments, behave: either they order spontaneously i n ferromagnetic, antiferromagnetic or com plicated magnetic structures, or the f electrons can form a heavy fermion state, strongly correlated with the conduction electrons. Some of these materials change again their ground state, at temperatures very close to absolute zero, to a non conventional superconducting state. The f i r s t example of a heavy electron system CeAl3 , i s a good material to study the heavy fermion systems because i t pre sents an extreme case of these properties ( i t does not become superconducting at low temperatures). Then i t should be useful to study the basic properties of these materials. There i s a series of materials with very interesting pro perties showing large q u a l i t a t i v e and quantitative changes i n t h e i r electronic structure a r i s i n g from small composition, tem perature or pressure changes. Among them the rare earth and actinide mixed valence compounds are currently the subject of extense experimental and theoretical studies (2-5). Boppart and Wachter (3) have presented a very clear example of t h i s behaviour i n t h e i r studies of the moment formation i n TmSe-^_ Te under pres sure where both mixed valency, or intermediate valence (IV), and semiconductor to metal t r a n s i t i o n are found. The particular i n terest of t h i s case i s not only that these materials have been extensively characterized, but also because they show, from com parison of valences determined by two d i f f e r e n t experimental methods, that a unique picture which considers only one type of a

a

a

x

x

American Chemical Society Library 1155 16th St., N.W.

Salahub and Zerner; The Challenge of d and f Electrons Washington, D.C. Society: 20036Washington, DC, 1989. ACS Symposium Series; American Chemical

248

THE CHALLENGE OF d AND f ELECTRONS

electronic structure change, cannot, at the same time, explain the volume and the magnetic properties change as a function of composition or pressure. The substitution of Te by Se i s of course a "compositional" pressure, to be added to the external pressure inducing the valence i n s t a b i l i t i e s . The valence of a material ν =v - n | , the difference between the number ν of electrons outside the core and the number n | of l o c a l i z e d f electrons i s determined from the analysis of two d i f f e r e n t experimental properties, say the change i n the l a t t i c e constant and the change of the molar Curie constant (see F i g . 4 of Ref. (3) and i t has been found that they do not coincide f o r a given set of com positions and external conditions. In f a c t , a l i n e a r i n t e r p o l a t i o n of the l a t t i c e constant and the Curie constant between those of Tm3+ and Tm2+ give d i f f e r e n t valences. This can be understood i f one r e a l i z e s that the Curie constant r e f l e c t s more the o r b i t a l character, while the l a t t i c e constant r e f l e c t s also the r e a l space extension of charges. As has been pointed out by Schoenes (6), the degree of d e r e a l i z a t i o n of the f states can be correlated with the o s c i l l a t o r strength of the f •> d t r a n s i t i o n . Table 1 i n r e f . (7) gives a s e l e c t i o n of o p t i c a l l y determined f d oscil l a t o r strengths for various Ce and U compounds. t o m i c


t o m i c

I.

Real Space Wave Functions of f Electrons i n Condensed Matter.

The t o t a l wave function of a material can be written as a sum of fixed configuration ( f ) states which w i l l be mixed by the actual e l e c t r o n i c c o r r e l a t i o n . This procedure i s usually known as configuration i n t e r a c t i o n (CI), being a standard method i n many body techniques (8-10) where i t i s usual that each c o n f i guration i s represented by one determinant constructed from ap propriate single p a r t i c l e wave functions of the d i f f e r e n t angular momenta. Each single p a r t i c l e wave function ψ i s , on the other hand, constructed as a l i n e a r combination of (usually) an atomic l i k e and a series of p o l a r i z a t i o n wave functions. This LCAO pro cedure i s suitable to construct a given single p a r t i c l e wave function f o r the actual geometry of the material. The c a l c u l a t i o n of the single p a r t i c l e 4f-electron wave functions i n atomic problems shows that even i f these states are occupied only a f t e r the 5s, 5p and 6s o r b i t a l s , the charge d i s t r i bution of the 4f i s such that most of i t i s inside the sphere of maximum charge density of the 5s and 5p. This i s also the case f o r the 5f electrons, i n the actinides, r e l a t i v e to the 6s and 6p states. The physical reason f o r t h i s behaviour i s the dominant role played by the e f f e c t i v e f potential V£ 3(r), a r a d i a l potential well confining the f-wave function to a small region of space, see Figure 1. The resultant f-wave function should be appropriately c a l l e d atomic-like. When the rare earth or actinide atom i s i n condensed matter, the boundary conditions and the potential well changes d r a s t i c a l l y , mainly i n the outermost part of the Wigner-Seitz c e l l , through the superposition of the atomic potential wells, with a lower potential i n the i n t e r s t i t i a l region. The change i n the potential makes i t necessary to include an extra contribution ψ£ ι , i n the bonding region to construct an appro priate f wave function Ψ£ i n t h i s material as: n

=


17.

K E L L E R & DE TERESA \p£ «(ε^ψ^ + / l

- a^ ψ£ΐ)Α

J | Ψ | d T = ι, 2

£f


1.0

2^0


I

249

where i n general:

2

|ψ i dx = 1

(1)

£

RCa.u. ) 3^0 4^2

Figure 1. The r a d i a l potential well V£_g (r) for f-electrons i n s o l i d γ-cerium and the r e s u l t i n g bottom of the band f-wave func tion. Note the deviation from atomic character at the WignerSeitz radius giving r i s e to a wide f-band i n these materials. Correlation w i l l contract the f wave function to an atomic l i k e d i s t r i b u t i o n reducing the band character even to the extent of producing atomic-like behaviour. The a u x i l i a r y Ψί' wave function i s not of atomic character, on the contrary i t must have contributions outside the sphere where the 5s, 5p and 5d (or 6s, 6p and 6d for actinides) have t h e i r maxima. The ψ£» i s usually taken to be orthogonal to ψ£. The angular part i s of course the same as the 4f (or 5 f ) ; but i n molecular and s o l i d state calculations the t could also be termed " p o l a r i z a t i o n " wave functions. Because the two components ψ£ and\l^t are orthogonal, the normalization i n t e g r a l J

A ||(a i|; n

+ νΤ^ψ

f

£ )

)|°ι , E2 within an f band, that i s bonding states at the bottom of the band are less l o c a l i z e d than the antibonding states at the end of the band. I f the average eigenvalue ε , the center of the f-band, i s close or above the fermi energy tne set a w i l l decrease i t s value. Because the f-electrons eigen values are found to be highly occupation dependent, s h i f t i n g by 1.5 to 5 eV with a unit change i n occupation, ε ^ < ε^ χ and and a^ > a j ^ , thus the average amount of atomic l i k e character, per electron, i n the f l state should i n general be taken to be smaller than i n the f configuration. n

n

n

n


η

η+

n +

n

Equation 1 i s an example of the expansion of one type of o r b i t a l into a l i n e a r combination of other sets, or LCAO procedure. A discussion of the r e a l space wave functions for RE and actinides, showing the large differences with the atomic l i k e wave functions can be found, f o r example, i n Freeman's introductory paper to the Physics of Actinides and Related 4f Materials (11), (especially Figure 8 of that paper for γ - U). See also (12). In our analysis here we have concentrated on the rare earths. Here i t should be stressed that f o r condensed matter a con venient analysis of the wave functions i s made i n the form of c e l l u l a r o r b i t a l s , these are defined inside the Wigner-Seitz c e l l s around each atom, a band can be formed f o r each angular momentum beginning at the energy at which these wave functions have zero slope at the surfaces of the Wigner-Seitz c e l l s up to the energies where the wave functions have zero value at the same points. In a sense we are speaking of a l i n e a r combination of c e l l u l a r o r b i t a l s (LCCO) procedure. The r e a l space wave functions ψ£ are the solutions of a Schrodinger l i k e equation

when the Ε i s minimized subject to the constraint (5) where the MC wave function Ψ i s a l i n e a r combination of determinants of the φ · Then 1

δ(ίΠ{ψ*},{ψ}] - ε ( ) |ψ| "dx

1

(6)

d x - l ) )= 0 N

contains a set of equations, one of which, defining

,

(7)

1


17.

K E L L E R & DE TERESA


251

ff

Equation 7" reduces to the_ Hartree Fock and CI i f only one determi nant i s used and Ε Ε^|_{ψ*},{ψ} |. But i n highly correlated sys tems the use of Equation 7 i s mandatory because c o r r e l a t i o n largely changes the from the Hartree-Fock value, then |Φ. - Ψ ? """mv 1


II.

Γ

ι « f i ( r ) with f-^(r) »

0.0

i n a range of r .

Configuration Space Wave Functions.

In configuration space the wave function for a given η occupancy of the f band w i l l be | f > , usually a determinant with η single f-states. The actual wave function for a p a r t i c u l a r experimental s i t u a t i o n h, for example an i n i t i a l state h = i or a f i n a l state h = f, i s n

n

|h>=EC |f >

(8)

n h

t h i s being the so called configuration i n t e r a c t i o n (CI) or multiconfiguration (MC) schemes (see, for example (8-10), where the t o t a l number of f electrons n w i l l not, i n general, be an integer value. The configuration wave function |f > i s usually represented by a determinant for the " a c t i v e " space where only the ν valence e l e c trons contribute to one column each: η columns for the f electrons, v-n-1 columns for the d electrons and one column for the (s-p) conduction electrons for each RE atom η - nf. In t h i s case the amount of atomic-like f-character: f

n

atomic n^ = ZnC, f hn η v

n 2

2

a

, < n η — f

/\ (9) 0

£

t o m i c

There should be some additional contributions to n ^ in Equation 9 , from the cross terms i n which can be important, mostly i f several C are of the same order of magnitude and at least two of these terms have η > 0. For a given h the parameter n | i s a suitable quantity to measure the amount of very l o c a l i z e d , atomic-like f-character, and i t s r e l a t i o n to the properties of the system. atomic i unique because i t depends on the choice of and ψ£» , but a consistent scheme w i l l lead to avoid t h i s , well known, shortcoming of population analysis. The configuration i n t e r a c t i o n ( i n the MC scheme) wave function |h> i s needed i n the condensed matter problem because the reduced symmetry of the c r y s t a l and, mainly, the e f f e c t of the scattering wave boundary conditions for energies above the i n t e r s t i t i a l po t e n t i a l or the existence of bonding states between the anions and the heavy metal ions, w i l l allow several types of coupling or cor r e l a t i o n between the f l e v e l and the valence or conduction e l e c trons. This coupling, usually denoted by Δ , i s i n fact the sum of several contributions which are responsible for either MV, Kondo or other e f f e c t s . In general there w i l l be a dominant configuration n n

t o m : L C

n

s

n o t


252 η C

THE CHALLENGE OF d AND f ELECTRONS contribution

n+1

(°

r

C

(C Δ

n-P

n

~

1.0)

Ε

and

Ε

then a second with lower amplitude E

- ^ η + 1 "~ ίΐ ) » î}

b e i n

t h e

8

total

energy

computed

n

for a fixed f configuration. For example i n the d i f f e r e n t phases of the Ce problem the ground state i s usually described as i n a f configuration, but i f a core hole i s present or for the case of BIS spectroscopy the f configuration w i l l then take a larger weight. In the ground state of α-Ce the increased i n t e r a c t i o n between the f electrons and the conduction band w i l l bring i n the f° configuration because i n t h i s case the energy difference E ^ i c ^ a ^ g i - ^ f ^ ^ o s i s small; t h i s being one of the reasons for the complicated behaviour of Ce metal and Ce compounds. For these materials the CI method has been extensively used by Gunnarson and Schonhammer (13) (and re ferences therein). The c o e f f i c i e n t s Cjj w i l l i n general depend on the hybridization of the f electron wave functions which i s the o r i g i n of the forma tion of an f band. But, as already pointed out by Haldane (14), these materials tend to behave as "low density" or impurity l i k e systems at intermediate temperature. This consideration i s import ant because i n a material such as CeAl3 , at intermediate tempera tures, transport properties correspond to incoherent scattering by each RE atom at the maximum resonance scattering cross section (15), as expected from an impurity l i k e Kondo-resonance, and the coherence between the RE atoms, although fundamental for the under standing of the low temperature regime, can be introduced a poste riori. We can describe the process of pressure induced valence tran s i t i o n s (as i n F i g . 1 of Ref. (3)) as a three step phenomenon. The f i r s t , i n the pressure range Ρ

f Electron States in Condensed Matter - American Chemical Society

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