Chapter 15 New Frontiers: Highly-Correlated Electronic Behavior

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Chapter 15

NEW FRONTIERS: HIGHLY-CORRELATED ELECTRONIC BEHAVIOR R. F. WILLIS and S. D. KEVAN

1. INTRODUCTION Preceding chapters in this volume have explored the current status and reviewed various applications of angle-resolved photoemission. The emphasis has been on the solid state and what might loosely be termed "delocalized electronic phenomena within a (mostly) independent single-electron band approximation." That is, on phenomena associated largely with valence states describable by wide band (large kinetic energy) behavior. At the other extreme, there exists an equally voluminous literature on photoelectron spectroscopy of individual atoms and molecules in the gas phase (ref. I), with the emphasis on sharp energy levels and strongly localized states. Both fields have matured over the past fifty years to a level at which both the photoelectron spectra and the underlying physics is well understood. There exists, however, an intermediate regime in which the electronic behavior possesses the characteristics of both delocalized ("band-like") and localized ("bond-like") states. The properties of these "narrow band materials," as they have come to be called (ref. 2), are dominated by strong electron-electron interactions which force us to look beyond existing theories of electronic behavior based on the above extremes. We choose this to be our closing theme for two reasons. Firstly, photoelectron spectroscopy provides fundamental information on the electronic properties of solids, which is used to advance

our theoretical understanding. The physics of strongly correlated electronic behavior in narrow band solids represents a new frontier in our conceptual understanding. Indeed, the discovery of high temperature superconductivity in narrow-band transition metal oxide compounds is serving to underline severe limitations in the existing way we think about electronic transport mechanisms.

To quote P.W. Anderson in a recent review (ref. 3) of current thinking on this particular issue: "angle-resolved photoemission is, for this field, the experimental probe that tunneling spectroscopy played in unraveling the BCS mechanism responsible for conventional superconductivity in wide band materials." This brings us to our second reason: these new materials are pushing us beyond the instrumental limits of our existing experimental equipment. Higher resolution and increased intensity are demanded which will set new standards in the future.

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In the following, we will attempt to summarize the ideas and methods which have evolved over the past few years concerning the theory of highly-correlated electronic states. In this, our own thinking has been advanced enormously by the papers and discussions published over the past ten years, represented by two NATO Advanced Study Institute proceedings - Moment Formation in Solids (1983) (ref. 4) and Narrow Band Phenomena (1987) (ref. 2). Our theme is the extent to which the various narrow-band phenomena show up in photoelectron spectroscopy, with specific

examples chosen to emphasize particular models and principles. Finally, we review the prospects for future high-resolution photoemission measurements.

2. HIGHLY-CORRELATED ELECTRONIC STATES IN SOLIDS The search for a good theoretical description of the electronic structure and related physical properties of narrow band materials has led us to two different approaches to describing the problem (ref. 5). namely: ab initio theories without any adjustable parameters; and model Hamiltonian theories with empirically determined parameters. In the ab initio methods (ref. 6), the usual approximation is either to limit the number of interacting electrons by considering only a small cluster of atoms - "large molecule approach" - or to replace the effect of the electron-electron interactions by some effective potential, as is done in density functional "band theory." The cluster calculations neglect long-range screening and lattice polarization effects. As a consequence, there is a tendency to predict band gaps which are too large and, due to the neglect of translational symmetry, they cannot describe different kinds of magnetic ordering. They are more successful in accounting for local excitations involving atomic multiplets, crystal, and ligand field level-splitting effects.

Density functional calculations (ref. 7) have been successful in

describing the cohesive energy, ground state charge-density distributions, types of magnetic order, and the F e d surfaces of many solids. They have failed to predict temperature dependent properties, give band gaps which are too small, and fail particularly badly in the present context of narrow band solids by predicting metallic behavior for known insulators - NiO, CuO, etc., being classic examples. The model Hamiltonian approaches have sought to give a more qualitative description, using highly simplified model-dependent many-body interactions.

They have been successful in

accounting for the physical interactions responsible for correlation gaps, mechanisms underlying magnetic exchange and superexchange, the Kondo effect, and thermal dependence of these properties. An important development in recent years has been the realization that careful analysis

of Auger and photoemission spectral lineshapes can provide values for and physical insights into the model-dependent key parameters (ref. 8). Two models in particular underline much of our present thinking - the Hubbard Hamiltonian

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(ref. 9), which is usually applied to describe insulating systems, and coming from the opposite extreme, the Anderson Hamiltonian (ref. 10) used to describe metallic behavior. 2.1 The Hubbard Hamiltonian The Hubbard model (ref. 9) is based on a single s-band characterized by one-electron matrix elements,

qj, which describe electronic hopping between

sites i and j, and an on-site two-electron

Coulomb repulsion, U. When U>>tj, the half-filled s-band splits to form a "correlation gap" and insulating behavior. Mott had earlier argued (ref. 11) that the insulating behavior observed in many transition metal compounds was due to just such a mechanism: a large onsite d-d Coulomb interaction U suppressing charge polarity fluctuations. The electron-electron interaction energy U thus overcomes the kinetic energy, expressed in terms of the band width W. This Mott-Hubbard metal-to-insulator transition (ref. 12) was an important development in explaining why so many compounds (NiO, CuO, NiS, etc.) were insulating despite having partially filled Bloch bands. The Hubbard Hamiltonian expresses this competition between the lowering of the kinetic energy (by delocalization and band formation) and the Coulomb correlation energy U (localization):

in which the sums are over spin

(3

and wave vectors k or lattice sites i for a given band of index

P. Qo expresses the banding and U=EA+E, is the sum of the (negative) electron affinity and (positive) ionization potentials of a singly occupied ion screened by the polarizability of the solid. Hubbard (ref. 9) showed that such a model Hamiltonian yields an insulator for a half-filled band when U>W and a metal when UW, the strong localization leaves the electron spin as the only low-energy-scale degree of freedom. These latter materials are magnetic insulators and the low-energy magnetic fluctuations can be described by spin-only Heisenberg-like Hamiltonians, generally expressing various "t-J models" in terms of the spin-spin exchange coupling parameter J. Because of their insulating character, Anderson (ref. 13) took U>>W, the d-band width for a transition metal compound, and showed that the interatomic cation exchange interactions occurred through the anion ligands via valance charge fluctuations (virtual excitations) involving both U and charge-transfer (superexchange interaction mechanism). The Mott-Hubbard model, together with Anderson's theory of superexchange has provided a framework for explaining much of the systematics of the magnetic, optical, and electronic properties of the insulting transition metal and rare earth compounds.

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2.2 The Anderson Hamiltonian The Anderson model Hamiltonian is an attempt to express the problem of magnetic moment formation in solids (ref. 4). It draws on an idea first proposed by Friedel (ref. 14) that for 3d transition metal impurities embedded in a metallic host, the d-orbitals could still retain their localized character - and hence their magnetic moment - by forming virtual bound states, broadened and shifted in energy. Hund's rule would tend to split the minority and majority spin states such that a magnetic moment would be retained if this splitting was large compared with the width of the virtual "resonant" level. Anderson expressed this in the form:

As in the Hubbard model, all Coulomb interactions are neglected excepting that between two electrons in the impurity d (or f ) shell. The first term in this Hamiltonian describes the host metal valence band structure, E,,

while the second and third terms express the impurity atomic states,

Edm, and interactions U(ijPm), including multiplet structures.

The final term represents

hybridization of the impurity states, m, with valence states, k, of the host via the one-electron matrix elements, V,.

In the simplest case of a single localized d-orbital with onsite Coulomb

repulsion U hybridizing through V,

to a continuum band of width

W and Fermi energy EF, the

virtual bound state formed has a binding energy relative to EF denoted Ed (or Ef) and broadened by the hybridization interaction. For U >> Ed > pVh2, where p is the continuum density of states, the occupation of the virtual bound d-state is integer and a stable local moment is formed.

2.3 Metal-Insulator U vs A Phase Diauam The introduction of hybridization into the model produces a surprising richness of physical phenomena. The notion that all charge-transfer fluctuations are suppressed, due to high values of

U in the Mott-Hubbard picture, implies that all gaps are d-d type. However, the band gaps in many transition metal oxides, sulfides, etc. seem to reflect a dependence on the electronegativity of the anion. This underlines the importance of (a) hybridization of the d-orbitals with the anion ligand orbitals in these insulators, and (b) hole creation in the ligand valence band due to transfer of an electron into a localized, empty d-orbital. That is, there is another charge fluctuation energy which is different from U - namely an energy A corresponding to 4"

d?+'

L where

denotes

a hole in the anion valance band. A is called the "charge transfer energy" which can be defined as A = Ed + EL, in which case E(d"+') ~

- E(d")

=U

- Ed where E(d"-'),

E(d") are the total energies

of the localized d-orbital states (ref. 15). Even in the ionic case, the states

qn-'dy'

(i. j

correspond to different cation sites) will have a dispersional width (2w) because of translational

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symmetry and overlap with the anion valence states of dispersional width (W). Also, the excited states, d""

L will have a dispersional width of (W+w).

Dependent on the relative values of U/A

and the band widths (a,W), various types of band gapping can occur. A simple total energy diagram (ref. 16), showing the relationship between these parameters for an ionic transition metal compound for which U >> 0, U > A and A > W, is sketched in Fig. 1.

: h E

-

Lu m c 0

I-

Increasing Bandwidth

.

Fig. 1. Total energy diagram depicting the various charge fluctuations in an ionic transition metal compound. U is the two-electron Coulomb repulsion energy and A is the charge transfer energy due to hybridization between d and ligand orbitals L, broadened into bands of width 2w and W+w (adapted from Ref. 16). For U > A, the gap is of the charge transfer type with a magnitude, A

U

-+

infinity, we can get a metallic ground state if A < W/2. Since

-

W/2. So, even for

w << W, generally,

this is

typical of p-type metals such as CuS. For A > W/2, the size of the gap scales as the anion electronegativity for a given cation and crystal structure. For U < A we are in the Mott-Hubbard regime with a d-d gap for U > w and a d-band metal for U < w. The early transition metal oxides exhibit this behavior. NiS has a zero band gap. This approach has formed the basis for an elegant classification scheme (ref. 17) which provides a framework for bringing together a wide variety of behaviors observed in transition metal compounds. A "phase diagram" can be drawn, Fig. 2, which depicts various types of insulating and metallic properties in terms of the parameters U/r and

with T being defined

as the metal d-state to ligand p-state charge transfer matrix element in these compounds. If A > U,

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the lowest lying hole/electron excitations are d-d fluctuations and a Mom-Hubbard metal-insulating gap is formed with E,,

= U. Both the electron and hole charge carriers move within a single

-

narrow d-band (width 2 0 ) with a large effective mass. At the opposite extreme, A < U, the lowest lying excitations are of the charge transfer type and Egap

A (related to the electronegativity of

the anion). The holes are now "light" particles in the anion valence band (width W+O) while the electrons remain "heavy" in the narrower d-band. The system is a charge-transfer semiconductor. The line A = U separates these two regimes.

I

1

W -

2

A

c

T

Fig. 2. U/T vs AfI' phase diagram depiciting various types of insulating and metallic behavior in transition metal compounds (Ref. 16). The stoichiometric compounds, La2Cu0, and YBa2Cu306.5, appear to fall into this latter charge-transfer semiconducting class.

Doping occurs by replacement of La2+ with S?'

or,

alternatively, introducing interstitial oxygen 02-defects (ref. 3) which produce charge compensation via light holes of primarily 0 (2p) character. A highly simplified density-of-states schematic representing the electronic properties of the undoped parent materials of these high-temperature superconductors is shown in Fig. 3 (ref. 18). As in the simpler oxides (CuO, NiO), the uppermost Cu 3d9 band is split into two "Hubbard bands,'' depending on whether an electron is extracted, d9+d8, (by photoemission) or injected, d9+d",

(by inverse photoemission).

The important parameters are the d-d Coulomb repulsion energy U of the localized two-hole state (d9+U+d8+d1O) and the charge transfer energy A required to excite an electron from the oxygen

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p-band into the upper d-band, the gap depending on the band width W of the oxygen p-band to a first approximation. Again, the relative strengths of the dimensionless parameters U/W and &W serve to define a wide diversity of behavior, in this case dependent on the level of doping, ranging from metallic through semiconducting to insulating and magnetic behavior (ref. 3). We will return to this subject in a subsequent section.

t

9

cu d-d

8

9

Cu d-d

10

Fig. 3. Schematic of the density of states distribution in undoped high-temperature oxide superconductors, e.g., La2Cu0, and also, CuO, NiO. (Adapted from Ref. 18). The richness of behavior originates from the interplay between essentially an atomic effect (the Coulombic repulsion of two holes or two electrons on the same ion) and a solid state effect (hybridization and charge fluctuations dependent on the band widths, w and W). The important point in the context of photoelectron spectroscopy is that the measurement of both the occupied and unoccupied states energy distribution is necessary to determine U and A. An example of photoemission at two different photon energies (hu = 66 eV and 120 eV) and inverse photoemission @IS) at x-ray energies (hu = 1486 eV) from NiO is shown in Fig. 4 (ref. 15). The two spectroscopies provide different projections of the density of states features due to wavelength dependent cross-section effects. These results classify NiO as a charge transfer insulator with a

4.3 eV gap, and comparison with ionic cluster calculations suggest a d-d Coulombic repulsion U of between 7 and 9 eV (ref. 15). The charge transfer parameter A can also be incorporated into an Anderson-type Hamiltonian,

EQ. (2), in which the transition metal cation is treated as an impurity screened by hybridization with the delocalized anion L states. The problem reduces to one of highly localized and correlated

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states, embedded in a semiconducting medium in which the long-range polarization effects of the solid (dielectric response) on the atomic ionization and affinity levels are accounted for. The Anderson Hamiltonian extends to other interesting phenomena. A notable case is the Kondo resonance effect associated with the low-temperature resistance minimum anomaly observed in dilute alloy systems containing d and f atoms (ref. 19).

PES

BIS 1486 eV

hv (eV)

'i

1

t

Fig. 4.Photoemission and inverse photoemission (BIS) spectra from NiO which is classified as a charge transfer insulator with a 4.3 eV gap. Comparison with ionic cluster calculations suggests a d-d Coulomb interaction, U, of between 7 and 9 eV (Ref. 15). 2.4 The Kondo Hamiltonian Under circumstances in which the ground state is assumed to have n localized f (or d) electrons bound to a simple impurity atom, such that the Hubbard states separated by a large energy U, then for each virtual bound state

(r =

KP

(f"-' and f"")

are

E(P-') and E(f""+') much larger than the broadening of

IV,I2);

a localized moment exists in an otherwise

non-magnetic host. Schrieffer and Wolff (ref. 20) showed that if the impurity atom retains only its spin degrees of freedom (i.e., spin rather than charge fluctuations), then the Anderson Hamiltonian transforms into the Kondo Hamiltonian:

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with S the impurity spin operator. All cerium intermetallics appear to lie in the Kondo regime of the impurity Anderson model in which the local moment due to the singly occupied 4f' shell has an antiferromagnetic exchange interaction J=(2V2U)/EAE&J) with each of N conduction electrons with energies E,. The ground state of this single local moment embedded in a spin-polarized conduction electron host is a singlet state. The result is a narrow many-body resonance located just above EF, shown schematically in Fig. 5, which is associated with the quenching of the Ce 4f magnetic moment. The position, width and intensity of this "Kondo resonance peak" is determined by the magnitude of TK, the Kondo temperature at which a resistance minimum is observed, which can vary from a few degrees Kelvin to hundreds of degrees Kelvin. An example is shown in Fig. 6, using the combined techniques of synchrotron photoemission and inverse photoemission @IS) (rzf. 21). Resonant photoemission from the 4d core level was employed to increase the 4f cross-section near the 4d absorption edge, near 120 eV. The experimental resolution employed in this case was 400 meV (PES) and 600 meV (BIS). The Kondo resonance temperatures for the two cases shown are of the order, TK-1-1OK in the case of CeAl and TK-900K for CeNi2, Tine linear coefficient of the specific heat y is proportional to TK-l reflecting the 4f spin entropy. Thus, the "heavy fermion" behavior of

CeAl can be understood as reflecting very small TK but extremely large y-values. No Kondo resonance peak is observed as a consequence in the spectrum of CeAl taken at 300K, Fig. 6.

I

\

Kondo

Resonance

U

I

'

* Density of States

Fig. 5. Schematic of density of states spectral weight representing Kondo resonance effects in cerium intermetallics.

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Ce 41 SPECTRAL WEIGHT --PES/BIS -THEORY

Al

(x 2.5)

-10

0

10

ENERGY AEOVEEF (eV)

Fig. 6. Experimental and calculated resonant photoemission and bremsstrahlung isochromat spectroscopy (BE)of the Kondo resonance in CeNiz (TK= 900K) and CeAl (TK = 1-1OK) taken at 300K (Ref. 21). 2.5 Remarks

The above model Hamiltonian approaches have provided a great deal of insight into the evolution of atomic-like to band-like electronic structure in narrow band materials. They have also revealed a surprising richness of physical phenomena. Unfortunately, although they are simple in form, they are proving remarkably difficult to solve for specific situations. They express strongly oversimplified descriptions of both the electron-electron interaction, as well as the one-electron dispersion relations and hybridization. For example, the electron-electron interaction is treated as an on-site interaction with the assumption that all other interactions are automatically incorporated

into renormalizations of the appropriate parameters in the Hamiltonian. The impurity model of the Kondo resonance effect, Fig. 5, ignores the lattice aspects of the cerium intermetallics and ducks the question of the competing role of long range RKKY exchange interactions mediating impurity-impurity atom coupling effects (ref. 22). For these cerium intermetallics which fail to order magnetically, even at the lowest-temperatures measured, or become superconductors, it is now common to refer to them as "Kondo lattice solids" to distinguish this unique and puzzling property. The translational symmetry is expected to produce fine structure in the Kondo peak (as

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yet unresolved) due to narrow band dispersion effects. Indeed, it is speculated that the dispersion might be of the order of TK and produce some of the heavy fermion superconducting properties of the other actinides (such as UPt,). The picture we are left with is that despite the now strong body of spectroscopic evidence that very large Coulomb interactions are at work to stabilize the "atomic" behavior in these materials, the heavy particle transport properties are indicative of quasiparticles at E, which are highly sensitive to translational symmetry for which the crystal momentum remains a good quantum number. There is a growing appreciation (refs. 2-4) that this duality in behavior is responsible for the new and exciting ideas emerging from a more critical look at the Ferrni liquid theories as applied to these narrow band solids (ref. 3). For example, near ground state properties may satisfy Luttinger's theorem (ref. 23) over some scale of temperature and energy, the properties of this "Luttinger liquid" being a generalization of the Landau behavior of Fermi liquids in normal wide band metals. Above some critical temperature (e.g., TK in the Kondo case) bare or "undressed' atomic-like excitations at higher energy scales may dominate. Theoretically, it would be most satisfying to be able to calculate the electronic structure from

an ab initio calculation. A theoretical breakthrough to the problem of fluctuating valence in a more general Anderson Hamiltonian has been accomplished by Gunnarsson and Schonhammer (ref.

24) who have been able to calculate the spectral density of states distributions of d- and f-states resulting from an impurity model. Also, and surprisingly, density functional theory describes the Fermi surfaces of the heavy fermion systems, like UP$, very accurately (ref. 25). However, in general, the ab initio methods run into grave problems when it comes to describing excited states, temperature dependencies, band gaps, optical properties and phase transitions. The most likely future scenario will be one in which ab initio methods are used to calculate accurate ground state properties from which the appropriate parameters used in the model Hamiltonians can be derived. This can then, in turn, describe the properties of the excited states and their interactions. The resolution of spectral features indicative of this wide variety of phenomena has been one of the recent triumphs of photoelectron spectroscopy. Also the discovery of superconductivity with high critical temperatures in a range of oxides containing transition metal and rare earth ions (ref. 26) has provided an enormous impetus to this field. In the following section we review some of

the progress made with reference to selected examples of particular relevance in the present context.

3. HIGHLY-CORRELATED ELECTRONIC SYSTEMS 3.1 Strong-Correlation Effects in Chemisorption

The band width W is a function of the number of nearest-neighbor coordinating atoms and the

degree of interaction of their overlapping wave functions. Two-dimensional layers deposited onto single crystal substrates might be expected, therefore, to show these strong correlation effects - indeed, it is a surprise that they are not more prevalent than has been observed to date. The Cs/GaAs(l 10) has recently been shown to be an interesting example of a two-dimensional Schottky barrier system for which a delocalized one-electron description fails to describe the excitation spectra (ref. 27). The central problem is that one-electron theory predicts a partially filled gap, due to Cs(6s)-induced interface states, resulting in a higher density of states at the Fermi level (ref. 28). Experimentally, the surface is found to be insulating at all coverages up to a full monolayer of Cs (ref. 29). Another intriguing fact is that STM studies (ref. 30) show a tendency for the metal adatoms to form pairs, zigzag chains, and ordered clusters indicative of "surface molecule" rather than metallic behavior. Above one monolayer coverage, the surface becomes metallic. We note that in the case of Cs, the density of the bulk metal is near the Mott limit for

an insulator-metal transition (ref. 27). High-resolution (<30 meV) electron-energy-loss spectroscopy (ref. 27) has revealed spectral features which can be interpreted as transitions between localized electronic states located within the GaAs band gap region at coverages below one monolayer. These experimental results are summarized the Fig. 7 (ref. 27). A 0.5 monolayer coverage (-2.2~10'~Cs atoms cm-2) forms local structures on the GaAs(ll0) substrate with one electron per Cs site and a Cs-Cs nearest neighbor spacing of 6.9A, Fig. 7(a). As shown in the inset molecular orbital diagram, the absence of any spectral features in the bulk band gap region at this particular coverage is due to the fact that charge transfer (CT) excitations occur between adjacent adsorbate atoms which localize two electrons on any one site. The Coulomb repulsion energy U-1.4 eV. Single particle transitions between the two atomic states cost an energy E>1.4 eV which is greater than the band gap onset. The singlet ground state is thus a Mott insulator in this model. For coverages 9 > 0.5 ML, patches of higher density compressed phase coexist with this 0.5 ML structure, Fig. 7(b). The compression reduces the Coulombic repulsion, U*-O.4eV, with reduced electron localization, and the separation between the levels is also reduced, E* < Egap. This scheme accounts for the dependence of the experimental spectra on Cs coverage, charge transfer excitations becoming observable in the gap region when E*-1.0 eV and U*-0.4 eV. The Cs structures shown in Fig, 7 contain one unpaired electron per surface unit cell which would give a metallic half-filled band in a Bloch-Wilson delocalized states scheme (ref. 6). However, the absence of states at EF in PES and IPS (ref. 31) support a non-metallic surface. In IPS, Cs-induced peaks at -(EF+1.2eV) and at -(EF+0.7eV) have been observed (ref. 31) for submonolayer coverages corresponding to Fig. 7. These peaks can be assigned to negative-ion final states of the respective regimes, (a) and (b), Fig. 7. The single occupied state 4 . 4 e V above

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the valance-band edge, Fig. 7(a), defines EF. In PES (ref. 29), a lack of screening of the localized-ion final state shifts emission from this level at EF into the valence band continuum so that no state at E, in the gap is observed. Also, no sub-band excitations have been reported. With EF at -0.4 eV (ref. 29), a final-state shift of > 0.4 eV to higher binding energy, analogous to core-level shake-up (ref. 3 l), is a reasonable assumption.

(a) 112 ML

structure

(b) compressed phase

hci

Fig. 7. Sub-monolayer overlayer structures of Cs/GaAs(l 10) showing charge-transfer excitations between localized electronic states. E and E* are single particle excitation energies, and U and U are the Coulomb repulsion energies relative to the bandgap energy, Egap (From Ref. 27.)

3.2 Negative-U Behavior Similar arguments to the above, invoking electron localization and correlation effects, have been used to explain two peaks observed in scanning-tunneling-microscopy (STM) differential current versus differential bias-voltage spectra of AdGaAs(100) (ref. 30). In this case, the two "Hubbard peaks" are observed at E,-0.5 eV and E,-0.7 eV, referenced to the valence (V) and conduction (C) band edges, i.e., U*-1.7 eV. Current tunnels from the STM tip into the upper level and into the tip from the lower level. This suggests that U* and E* may be negative quantities in this case! The concept of a "negative-U system" is not new. Anderson (ref. 32) came across the idea when considering the effect of bond (or lattice) relaxation effects on the electronic properties of vacancies and other localized point defects in silicon. The key idea is that for strongly covalent bonds, the bond strength between two atoms increases with the number of electrons N, the covalent

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part of the binding energy increasing nonlinearly as N2. This adds a negative correlation to the Hubbard interaction energy such that:

u

- u,

-

u,

(4)

Strong cancellation between these two terms is already implicit in free molecules. For example, H, and H2+ have binding energies of 4.72 eV and 2.78 eV and interatomic distances R = 0.74 A

and 1.06

A,

respectively. This "bond relaxation" effect gives an experimental value for U =

0.8 eV which is much smaller than the "frozen bond" value, U, 2 e2/R = 18 eV (ref. 33). For defect bonds in semiconductors, both U, and U, are reduced from the free bond values by interaction with the rest of the solid: U, is reduced by a factor roughly equal to the dielectric constant E, and U, is reduced by the elastic restoring forces of the crystal acting on the bond. The net result is a much reduced U > 0, or even U < 0, dependent on the cancellation, Eq. (4). The implication of bond relaxation at surfaces, where the adatom is free to relax fully, is particularly important in metal-semiconductor chemisorption (ref. 33). According to classical arguments, the dielectric screening at a semiconductor surface is (E+1)/2 which reduces the "bare-ion" Coulomb repulsion energy U, by a factor 7 for GaAs (ref. 33). In contrast, U, will be maximum and close to the free bond value. This favors a net Coulomb energy which is negative,

U, < U, in Eq. (4). Allan and Lannoo (ref. 33) have argued this to be the explanation for the polarity dependence of the STM tunneling experiments on Au/GaAs(l 10) (ref. 30). The argument also extends to the Cs/GaAs(l 10) system (ref. 29), the observed excitations reflecting a negative rather than positive U which would invert the order of the singledtriplet levels discussed earlier, Fig. 7. It remains to be seen to what extent this interesting idea remains generally valid for other chemisorption systems. 3.3 High Temperature Superconducting Oxides The richness of physical phenomena inherent in narrow band solids has been underlined by the recent exciting discoveries of superconductivity in perovskite compounds extending up to 125K (ref. 34). The incredible range of electronic behavior, varying with only small changes in

stoichiometry, is exemplified by the generic diagram of the cuprates, Fig. 8 (ref. 3). The introduction of a small number of interstitial oxygen defects or substitutional metallic ions into the insulating parent cuprate (e.g., La&3rCaCu06 and La2SrCu206+6derived from La2Cu04) changes the electronic properties to a remarkable degree (ref. 3). The crystal structure of the cuprate superconductors can be viewed as stacks of two-dimensional CuOz conducting layers intercalated with charge reservoir layers which serve to control the number of charge carriers. A schematic diagram illustrating this "charge transfer

585

model" (ref. 35) is shown in Fig. 9. The amount of charge transferred depends on the specific structural details and the competition between charge transfer and oxidation or reduction of the metal atoms i n the intercalate layers. As the canier concentration in the CuOz layers changes, a transition occurs from an insulating antiferromagnet, with long-range spin ordering between magnetic moments on the Cu atoms, to a doped semiconducting state containing a random array of oxygen defects with only short range magnetic-order (and a "spin-glass'' ground state). Further doping produces a strongly correlated metallic state with a superconducting ground state and, finally, a normal paramagnetic metal (Fig. 8). The N6el temperature of LaCuO, is about 350K. The system exhibits both Hubbard and Anderson charge and spin density fluctuations. In terms

of the Hubbard-Anderson parameters and the charge-transfer phase diagram, Fig. 2 earlier, these systems are extremely sensitive to changing values of U, A and W.

Amorphous Semiconductor

Strongly Correlated Metal

Paramagnetic Metal

e,

L

J

+-'

21a,

a

E a,

I-

Doping Level ( I-loles per Cu 0, ) Fig. 8. Generic phase diagram of the cuprate superconductors. The doping level is relative to the insulating parent compound. (Adapted from B. Batlogg, P. W. Anderson and J. R. Schrieffer, Ref. 3). The best photoemission results available to date have been for Bi,Sr2CaCu,0,

which, after

cleaving in vacuum, produces clean and chemically stable surfaces. Photoemission studies of the high temperature superconductors have been reviewed by Lindberg, et. al. (ref. 36). The main result is that for the normal phase above T,, the spectra show energy bands E(k) dispersing through the Fermi level EF which c o n f m s some kind of Fermi surface (ref. 37).

Inverse

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photoemission (ref. 18) is also consistent with a Fermi level crossing. Fig. 10 shows a band of unoccupied states extending up to about 1.5 eV above EF,characterizing the "Fenni liquid" state. The unoccupied band width of these metallic states is related to the hole concentration in the 0 (2p) bands, Fig. 3. The peak width of 1.5 eV is compatible with a Fermi liquid containing x+l holes in the band given that the undoped material already has one hole in the Fenni surface (ref.

38). A strongly correlated, more localized system with only x holes created by doping would be expected to produce a much narrower band width of only a few tenths of an eV, given that the doping level corresponds to a few tenths of a hole in the CuO, layers. The Fermi surface projected onto the (001) cleavage plane of Bi2SrzCaCuz08 (Fig. 11) (ref. 39) also shows electron-phonon "nesting" qualities indicative of possible collective charge- and spin-density-wave fluctuations.

Planes

CUO, Chains

cu 0 I Chains

A : Conduction Layers

B : Charge Reservoir Layers Fig. 9. Generic structure diagram of cuprate superconductors. The number of carriers in the conduction layers (A) is controlled by charge transfer from the charge reservoir layers (B). (Adapted from J. D. Jorgensen, Ref. 3). A big question remains: How does the transition from the doped semiconducting to metallic state occur? A correlated system with a half-filled d-band is a semiconductor, Fig. 3, whereas an ordinary Fermi-liquid band structure is metallic. Adding holes creates a metal in both cases. From the Mott-Hubbard-Anderson picture we might look into the fate of the split-off upper "Hubbard band' (d9-d" charge fluctuation, Fig. 3). A spectral peak shows up at this energy in the 0 (1s) absorption edge and inverse photoemission specha of CuO (ref. 18) (see N O , Fig. 4). Also, it is

587

inferred in infrared absorption spectroscopy on the transition metal oxides as a charge transfer excitation across the 2p-3d hybridization gap (A, Fig. 3). Upon doping, this infrared transition quickly loses strength in the superconducting oxides and new excitations begin to appear at lower energy indicative of possible transitions into unoccupied Fermi liquid states. However, to date inverse photoemission has not been able to find unambiguous evidence for this upper Hubbard band in either the oxide superconductors or the parent, undoped compound. The cusp-shaped feature at 2.9 eV above EF (Fig. 10) is a possible candidate except that, for the Bi2Sr2CaCu20, compound considered here, Bi (6p) states are also located at this energy. I

I

I

I

I

I

I

I

I

I

I

Inverse photoemission Ei = 18 eV

Fermi liquid

Bi,Sr,Ca,Cu,O, - 2 - 1

0

I

I

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8

9

10

Energy ( eV relative to Ferrni level )

Fig. 10. Inverse photoemission spectrum of the high temperature superconductor Bi2Sr2CalCu208 (Ref. 38). What of the superconducting state?

In Fig. 12, we show the results of angle-averaged

high-resolution photoemission measurements for Bi2Sr2CaCu208 above and below T,, which summarizes the original work of Imer, el. af. (refs. 3, 40). The onset of a superconducting gap

is indicated in the lower temperature spectrum. The calculated curves are based on BCS theory (ref. 41) giving a best-fit reduced gap parameter 2E,/kBTc = 8k1.4 corresponding to a superconducting gap half-width, E, = 3 e 5 meV with T, = 88K. This value supports the view that these materials are strongly-coupled BCS superconductors with extremely short correlation lengths between Cooper pairs.

The experimental resolution, Fig. 12, is not quite sufficient, however, to settle important questions relating to the nature of the quasiparticle states. For example, a set of angle-resolved

spectra from the normal state above T,, Fig. 13 (ref. 42), show unusual changes in the spectral lineshape as the hole states momentum moves through EF. The spectral peak shape and linewidth provides clues to what Fermi liquid description is appropriate. broadening of the electronic states grows like

r-

For example, the lifetime

IE-E,I" with n

1 whereas conventional

Landau Fermi liquid theory predicts n 2 2. The sharp cusp-shaped feature which develops close to EF is very similar to that calculated for a two-dimensional "Luttinger liquid" theory of the normal state spectrum (ref. 43) - shown as the dashed curve, Fig. 13. Adding the experimental background intensity gives a fairly good fit to the experimental results.

Fig. 11. Fermi surface in the (001) plane of the Bi2Sr2CaCu208 system (from Krakauer and Picket, Ref 39). The experimental results, Figs. 12 and 13, are extremely sensitive to background intensity variations which can shift the peak positions and distort the overall lineshape analysis. This is clearly seen in the difference between the measured and calculated curves, Fig. 12, the experimental area between the high and low temperature curves appearing to violate the density-of-states sum rules. Higher k-resolution and AE-resolution with improved signal/noise characteristics is clearly desirable. 4. HIGHER RESOLUTION PHOTOELEmRON SPECTROSCOPY The case for higher-resolution (greater sensitivity) photoelectron spectroscopy from these narrow band systems is implicit in the above arguments. To date, however, meV resolving power has been limited to just a few measurements in the vacuum ultraviolet (VUV) photon energy range accessible either by laboratory rare-gas discharge lamps or by normal incidence grating

589

monochromators at synchrotron radiation light sources (refs. 21, 40, 42). Pioneering work in the field of photoemission from narrow band solids was f i s t reported in 1985 with regards to low temperature studies of low energy excitations in the a and y phases of metallic Ce layers evaporated onto a sapphire plate in ultrahigh vacuum (ref. 44). A detailed calculation based on the Anderson impurity model, and including the 4f' spin-orbit splitting, accounted perfectly for the fine scale spectral features observed. A total system resolution of 20 meV (FWHM) was necessary to do so. Similar high resolution measurements were subsequently reported for the low energy excitations in the high-T, superconductor, Bi2Sr2CaCu208, taken with the He11 line

(40.8 eV) and synchrotron radiation at hu = 22 eV (refs. 40,42). These measurements still remain the state-of-the-art. Much poorer resolution has been achieved in the photon energy range beyond

50 eV due to monochromator resolutions typically AE > 150 meV at hu = 122 eV.

BINDING ENERGY (elecwonwIIs1

Fig. 12. Angle-averaged photoemission above and below T, for Bi2Sr2CaCu208. Upper plots are experimental data; lower curves are theoretical. (Adapted from Refs. 3, 40) This has proven to be a real limitation, particularly as regards identifying spectral details and interpreting the electronic properties of the high

T, superconductors (ref. 3). For example,

returning to the question of identifying the upper Cu 3d9-3d" Hubbard band in the cuprates, Fig.

10, it is possible to enhance the cross-section of the Cu 3d-states relative to the 0 2p-states by going to these higher photon energies. For YlB~Cu3O,,there exists a peak at about 1.5 eV above E, but it appears to have oxygen p character from its high intensity at low photon energies

590

(ref. 45). This feature has been resolved by IPE but here, again, the best resolution achievable to date (AE

- 260 meV) (ref. 18) is not a significant improvement.

Also, and still on the subject of

the high-T, oxides, the apparent sum-rule violations, which come from analyzing changes in the background continuum as T is lowered through T, (Fig. 13), most likely arise (to some extent) from

stray

electron

emission

associated

with

surface

roughness.

Such

rough

specimen-surface-related problems are also likely to be the cause of the hotly disputed optical rotation effects which claim to support anyon statistics (ref. 46).

1,

0.5

a,

I

0.3

0.2

Oil

E'

I

BlNDlNG ENERGY leleclron voiid

Fig. 13. Angle-resolved photoemission spectra from Bi2Sr2CaCu208in the normal conducting state (Ref. 42). The calculated spectrum for the two-dimensional "Luttinger liquid" theory of the normal state is shown by the dashed cusp-like feature (Anderson, Ref. 3). Recent advances in synchrotron radiation photoelectron spectroscopy suggest that these problems will be resolved in the very near future. The first is the announcement of commercial monochromators combining excellent spectral resolution with electron energy analyzers which began to approach a total system resolution at the soft X-ray wavelengths comparable to the above

-

in the VUV (ref. 47). Fig. 14 is a plot of monochromator resolution AE (FWHM)versus photon energy, E, showing AE

E3R over a wide spectral energy range. With a single 1221-lines/mm

plane-grating ellipsoidal mirror configuration, this commercial monochromator (ref. 47) covers the

591

whole grazing-incidence VUV and soft X-ray photon energy-range from 38 eV to 2000 eV with

- 10.000 at hu = 60 eV. This translates into an optimum monochromator resolution of AE - 4 meV at hu = 40 eV, 17 meV at hu = 120 eV, and 300 meV a resolving power up to E/AE

at hu = 800 eV. This allows variable wavelength photoemission studies at an overall energy resolution previously only achieved with He11 radiation. Due to the strong enhancement of the 4f cross-section at the 4d-4f resonance, rare-earth intermetallics other than cerium, with 4d and even 5d shells, are now accessible to this level of high-resolution photoemission studies.

30

100 300 Photon Energy E ( e V )

1000

Fig. 14. Resolution, AE (FWHM)versus photon energy, E, of the SX700/II monochromator operated in first order (solid line) and second order (dashed line) of diffraction from a 1221 lines/mm plane grating. The straight lines represent the relation AE = E3D (Ref. 47). The next major step forward is to improve the accuracy of k-dependent measurements by reducing the slit width of the electron analyzer. However, this sacrifices flux. High-brightness synchrotron radiation sources using magnetic undulators and wigglers hold promise of replacing the lost flux to allow both high spatial and energy resolution measurements. Also, more highly focussed (and coherent) light beams on the sample, down to microns size levels (ref. 48), will greatly improve measurements from "inhomogeneous materials'' superconductors arid heavy-fennion intermetallics.

-

such as many high T,

592

ACKNOWLEDGEMENTS The authors wish to thank the National Science Foundation (NSF-DMR 8818884) and the DOE

(DOE/ER/45275) for support, and colleagues whose work is referenced here for permission to publish their data. We also thank Gary Mankey for help preparing the diagrams. REFERENCES

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34. J. F. Berdnorz and K. B. Mueller, Z. Phys. B, 64 (1986) 1 8 9 C. W. Chu, et. nl., Phys. Rev. Letts., 58 (1987) 405. 35. R. J. Cava, et. al., Physica C, 165 (1990) 419. 36. P.A.P. Lindberg, Z.-X. Shen, W. E. Spicer and I. Lindau, Surf. Sci. Reports, 11 (1990) 1-137. 37. J. C. Campuzano, et. al., Phys. Rev. Lett, 64 (1990) 2308. 38. W. Drube,F. J. Himpsel, G. V. Chandashekhar and M. W. Shafer, Phys. Rev. B, 39 (1989) 7328. 39. H. Krakauer and W.E. F’ickett, Phys. Rev. Lett., 60 (1988) 1665. 40. J. M. h e r , et. al., Phys. Rev. Lett., 62 (1989) 336. 41. J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev., 106 (1957) 162; ibid, 108 (1957) 1175. 42. C. G. Olson, et. al., Phys. Rev. B, 42 (1990) 381. 43. P. W. Anderson, Phys. Rev. B, 42 (1990) 2624. 44. F. Patthey, B. Delley, W. D. Schneider and Y. Baer, Phys. Rev. Lett., 55 (1985) 1518; ibid, 58 (1987) 2810. 45. J. A. Yarmoff, D. R. Clarke, W. Drube, U. 0. Karlsson, A. Taleb-Ibrahimi and F. J. Himpsel, Phys. Rev. B, 36 (1987) 3967. 46. S. Speilman, K. Fesler, C. B. Eom, T. H. Geballe, M. M. Fejer, A. Kapitulnik, Phys. Rev. Lett., 65 (1990) 123. 47. The Karl Zeiss Company’s SX700/II monochromator is one example, coupled with a Leybold-Heraeus EA-11 127.5mm mean-radius hemispherical analyzer, giving a system resolution currently 40 meV at hn = 122 eV. See articles by G. Kaindl, et. al., Synchrotron Radiation News, 3 (1990) 22; ibid, 4 (1991) 19. 48. B. T. Tonner, Synchrotron Radiation News, 4 (1991) 27.