Journal of Structural Chemistry, Pot 40, No. L 1999

STRONG ELECTRON

C O R R E L A T I O N E F F E C T S IN

X-RAY A N D P H O T O E L E C T R O N HIGH-TEMPERATURE

SPECTRA OF

SUPERCONDUCTORS UDC 537.531:539.26:539.184

P. V. Avramov and S. G. Ovchinnikov

This review is a detailed analysis of the literature (161 references) dealing with experimental and theoretical investigations of the electronic structure of high-temperature superconductors. Special attention is paid to the sudden perturbation model, which was effectively used to describe and interpret a series of X-ray and photoelectron experiments. INTRODUCTION At present, one can state with confidence that the discovery of high-temperature superconductivity (HTSC) gave a strong impetus to the development of many fields of experimental and theoretical physics. This influence was most conspicuous in high-resolution photoelectron spectroscopy and strong electron correlation theory. The first attempts to understand the peculiarities of the electronic structure of high-temperature superconductors and correlate them to the nature of HTSC were made immediately after the discovery of these compounds [1]. Soon it was realized that experimental X-ray and photoelectron data may not be interpreted directly and unambiguously. This is due to the complexity of the atomic and band structures of HTSC materials and to strong electron correlation effects in the spectra [2-4]. Currently, there is a wealth of published data on the electronic structure of HTSC oxides obtained by both experimental and theoretical methods. All these investigations were concerned with separate physical aspects of these compounds; there were no attempts to correlate the specific features of HTSC oxides to spectroscopic experiments, primarily, to X-ray and photoelectron studies. Curiously enough, the major challenge in this field is comparison between theory and spectroscopic experiment which would allow one to judge the adequacy of theoretical models for electronic structure description of HTSC. The available publications in this field present a wealth of information on the electronic structure of these compounds, the magnetic and dielectric properties of their precursors and related compounds, strong electron correlation effects, the form of the Fermi surface, the nature of the in-gap states arising in the course of doping and the effect of this process on the electronic structure, the energetics of the superconducting gap, the structure of the occupied and vacant bands, and some other properties. This information in its most part was obtained experimentally: by X-ray photoelectron, photoelectron, X-ray, X-ray emission, X-ray absorption, and optical spectroscopy. Thus the angle-resolved photoelectron spectroscopy (ARPES) data made it possible to determine the form of the Fermi surface for various copper oxide superconductors and related compounds with CuO 2 planes analogous to the planes in La2CuO4. In the metallic phase, low- and high-doped regions were isolated. The high-doped regions obey the Luttinger theorem and have a large Fermi surface which agrees with band structure calculations. However, strong electron correlation (SEC) effects lead to formation of a fiat band of the dispersion law of electrons in the vicinity of the point X = (~r, 0) and to a narrowing of the energy gap in the vicinity of t F compared to band structure calculations. In the region of low-doped metallic compositions, the non-Fermi fluid properties are pronounced; the Luttinger theorem is violated, and the photoemission spectra contain shadow bands. All these results indicate that the electronic structure L. V. Kirenskii Institute of Physics, Siberian Branch, Russian Academy of Sciences (Krasnoyarsk). Translated from Zhumal Struktumoi Khimii, Vol. 40, No. 1, pp. 131-183, January-February, 1999. Original article submitted September 30, 1997. 108

0022-4766/99/4001-0108522.00 o1999 Kluwer Academic/Plenum Publishers

of occupied states in this region of compositions is formed in conditions of strong local antiferromagnetic fluctuations, which give rise to a pseudogap and shadow bands; they also lead to the non-Fermi fluid behavior of the mass operator in the vicinity of the Fermi level and to violation of the Luttinger theorem. It was noted that ARPES gave the In'st results permitting one to trace the evolution of the photoelectron spectrum of Bi-2212 due to variation of hole concentration, but the situation in the vicinity of the dielectric-metal concentration transition is not clear. Photoelectron spectroscopy of the core states gave information about more general characteristics of electronic structure - the oxidation states of copper and the occupancies of many-electron configurations arising from SEC effects. Thus a comparison between the Cu2p X-ray photoelectron spectra (XPS) and model calculations shows that the occupancy of the Cu3d 9 configuration in the ground state is up to 50%. In a completely doped system of LaSrCuO 4 type, the main peaks in the high-energy X-ray and X-ray photoelectron spectra of the inner levels Culs and Cu2p are formed by the Cudl~ configurations. X-Ray absorption spectra (X.AS) confirm that SEC is largely responsible for the specific electronic structure of HTSC, in particular, for the electronic structure of vacant electronic states. XAS studies revealed the two-particle contributions generated by doping and showed that one dopant acts on two copper centers to give vacant Cudzz electronic states. In addition to experimental studies, one-electron calculations were accomplished to determine the electronic structure of the key compounds, but they were not generally accompanied by theoretical modeling of the available spectroscopic experiments. From the very start it became clear that in the absence of theoretical modeling the majority of experimental spectra may not be interpreted in an unambiguous and straightforward way. The reason again lies in strong electron correlation effects. Theoretical modeling was effectively applied for some cases, primarily for dielectric phases. On the other hand, model studies of many-electron effects in HTSC (Anderson's model, versions of the Hubbard model, t - J model, etc.) helped to understand the general role of SEC effects in these compounds. However, such approaches preclude theoretical modeling of one-electron excitation processes; as a result, in most cases, theory may not be directly compared with experimental data, as the spectrum of two-hole post-threshold Cup states in HTSC obtained by N. Kosugi's group. As a matter of fact, all previous investigations gave no more than a fragmentary view of the electronic structure of HTSC; some effects were studied only with one-electron approaches, whereas others, with many-electron model approaches. No realistic model has been developed which would allow one to describe from a single standpoint large amounts of experimental data by including both one- and many-electron effects (for example, X-ray absorption spectra and various photoelectron spectra). Here we present a review of experimental data [core level X-ray photoelectron spectra, photoelectron spectra (primarily, angle-resolved photoelectron spectra), inverse photoemission spectra, K a core level X-ray emission spectra, and X-ray absorption spectra] along with the results of theoretical model and ab initio calculations. The literature data are analyzed from a single standpoint using the sudden perturbation model, which is treated here in detail. In the framework of this model, we investigate the mechanisms of formation of the above spectra, including both one- and many-electron components. 1. BASIC DATA ON THE ELECTRONIC STRUCTURE OF HIGH-TEMPERATURE SUPERCONDUCTORS 1.1. Experimental Procedures for Electronic Structure Investigations

The experimental information about the electronic structure of solids is currently obtained by spectroscopic methods: X-ray photoelectron, photoelectron, inverse photoemission, core and valence level X-ray emission, X-ray absorption, and optical spectroscopy. X-Ray photoelectron spectra (XPS) are obtained by irradiating a sample by monochromated X-ray quantum beam with an energy equal to or higher than the ionization energy of the core levels. The energy of photoelectrons excited from certain core levels is measured in the course of experiment. Measuring the chemical shift on the energy scale relative to the spectrum of an element, one obtains information about the oxidation state of the element in the compound. The many-electron processes during electronic structure excitation are examined by analyzing the satellite structure of the spectrum. 109

Photoelectron spectrum (PES) is formed by electrons excited from the valence band. This technique, which is analogous to the X-ray photoelectron method, uses either hard or soft (of the order of several dozens electron-volts) X-radiation. Photoelectron spectra are essentially more complex than X-ray photoelectron spectra, as they convey information about the structure of the valence band, which is determined by both one- and many-electron effects. Inverse photoemission spectroscopy (IPES) studies vacant electronic states. The experimental procedure involves irradiating the sample with a monochromated beam of free electrons. These electrons are inelastically scattered on the vacant electronic states localized in the surface layers and emit braking radiation, whose energy equals the difference between the energies of exciting radiation and vacant state interacting with the electron and intensity is proportional to the density of vacant states for the given energy. X-ray emission spectra are formed by electron transitions from the core or valence levels to the core vacancy which has a lower energy and was previously formed by exciting X-radiation. In the course of experiment, the intensity and energy of X-ray quanta emitted in these processes are measured. The emission spectra formed by transitions from one core orbital to another carry information about the charged state of the atom. The position of the peaks of the core level X-ray emission spectra on the energy scale is sensitive to the state of other atoms through the valence shell of the atom and insensitive to the Madelung potentials. The same information may be extracted by analyzing the satellite structure of these spectra. Valence band X-ray emission spectra convey information about the local densities of states of the valence band subject to dipole selection rules. X-ray absorption fine structure (XAFS) is the most informative method of investigating vacant electronic states. These spectra are formed by inelastic scattering of exciting X-radiation on the absorbing atom and convey information on the local densities of vacant states before or after the ionization threshold in conformity with dipole selection rules. During the experiment, the intensity of X-ray absorption is measured as a function of the energy of exciting radiation. Analysis of these spectra gives both structural information (positions of form resonances on the energy scale) and electronic structure information (pre-threshold and short-range fine structure); the spectra are formed with participation of both one- and many-electron mechanisms. Optical spectra result from electronic transitions between the filled and vacant states under the action of electromagnetic radiation at optical wavelengths and are associated with the combined density of states of the valence band and conduction band. 1.2. Photoelectron Spectra of Superconductors The Cu2p XPS spectra, which are relatively easily measurable in the course of experiment, were studied most frequently [3, 7-15]. An analysis of these spectra is important for obtaining information about the valence and charged state of copper in these compounds. Figure 1 presents typical Cu2p XPS of CuO, La2_xSrxCuO 4 (x = 0, 0.15), and Cu20 [3, 7]. In contrast to the spectrum of Cu20, the spectrum of any of the other two compounds has two intense features assigned to the transitions to the 3d 1~ and 3d 9 states. The complex form of the 3d 9 peak arises from the Coulomb interaction between the 2p and 3d vacancies in the final state, which leads to an almost rectangular form of this peak for "good" samples (prepared under certain conditions). During sample degradation (annealing, exposure with time), the 3d 9 peak at first becomes asymmetric and then starts to lose intensity. The satellite structure of Cu2p XPS was widely used for estimating charge distribution in the ground state [16, 17]. Most estimations made with Anderson's model [18-24] using experimental data show that in CuO and La2_xSrxCuO 4 the content of the 3d 9 configuration is up to 50-70%, and the contribution from 3d 8 is insignificant. Several publications gave an erroneous interpretation of Cu2p XPS; in most of these papers it was concluded that Cu(3 + ) ions make a considerable contribution to the chemical bond. The most interesting (from theoretical viewpoint) spectra seem to be the copper 3 + spectra of NaCuO 2 [25] (Fig. 2), whose local structural parameters for the environment of copper (in particular, the CuO4 cluster) are similar to those of HTSC. Two major peculiarities distinguishing the Cu2p XPS of NaCuO 2 from the spectra of La2CuO 4 and CuO are as follows: considerable shift of the main and satellite maxima to the short-wave region and the emergence of an additional long-wave satellite with an energy of 937.5 eV. Investigating the valence band photoelectron spectra and inverse photoemission spectra provides information 110

z

~176

/. !

ICuzO

/

I

I

I

915

".. I

930

Binding energy, eV Fig. 1. Cu2p3/2 XPS [3, 7] of CuO, La1.85Sro.15CuO4, La2CuO4, and Cu20.

# I 948

I

Binding energy, eV

I

928

Fig. 2. Cu2P3/2 XPS [25] of CuO (a) (half-width F = 3.25 eV), La2CuO 4 (b) ( F = 3.30 eV), YBa2Cu30 7 (c) ( F = 3.20 eV), and NaCuO 2 (d) ( F = 1.60 eV).

about the density and nature of occupied and vacant electronic states. Interpreting these spectra is impossible without qualitative quantum chemical Fermi calculations, except the simplest cases. Figure 3 shows the typical photoelectron and inverse photoemission spectrum of Lal.8Sr0.2CuO4 [26]. Analogous spectra are characteristic of the nondoped oxide La2CuO 4 [27]. As can be seen, the intensity of both spectra at the Fermi level is either small (Lal.sSro.2CuO4 [26]) or nearly vanishes (La2CuO 4 [27]). Based on these facts one can make inferences about the density of states at the Fermi level. The splitting between the experimental PES of the La2CuO 4 surface is so insignificant that may in principle be interpreted as the absence of a dielectric gap. Precision photoemission spectroscopy is also effectively employed to measure the superconducting gap in the superconductive state [28-30]. The results of these works indicate a superconducting gap of approximately 24 meV for all compounds under study. -.A different approach to PES analysis was adopted in [31, 32]. The photoelectron and resonance photoemission spectra of copper in CuO and HTSC show a pronounced satellite structure, which was assigned by the authors to the 111

PES

BA

hv=40.8 eV I

-12

I

-8

IPES

~

hv=9.5eV

/

I

I

I

I

-4 EF=O 4 8 Binding ene~y, eV

I

12

I

16

Fig. 3. Experimental [26] photoelectron spectrum (PES) measured at hv = 40.8 eV and inverse photoemission spectrum (IPES) measured at hv = 9.5 eV of Lal.sSr0.2CuO 4. Both spectra are normalized to the

intensity at the Fermi level (see insert 10x 10).

3d s final states. Five 3d s states with different energies and hence with different Hubbard correlation energies were isolated. Comparing the experimental photoelectron spectra with theoretical one-electron calculations (e.g., [3, 13, 31, 32], etc.) proved to be rather fruitful. The spectra coincided in shape, due to which the one-electron calculations by both cluster and band structure methods gave a qualitative description of the overall structure of the valence band. It was shown that the top of the valence band consists of Cud and Op electrons. On the other hand, serious discrepancies between one-electron theory and experiment were noted. It was shown that 1) the fundamental peaks in experimental photoelectron spectra are shifted by 1-2 eV relative to the theoretical ones; 2) a number of satellite structures not described by one-electron theory are present in the experimental spectra; 3) it should be emphasized that all one-electron calculations gave the metallic ground state. These discrepancies unambiguously indicate that valence electrons in HTSC are strongly correlated. Other points under discussion were the adequacy of Fermi surface descriptions in one-electron theory and strong electron correlation effects on the form of the Fermi surface. As reported for YBa2Cu30 7 [33, 34] and Bi2Sr2CaCu208+ x [34, 35], the experimental Fermi surface is in good agreement with one-electron band structure calculations, indicating [34] the Fermi-fluid behavior of electrons on the Fermi surface in the normal state. However, as maintained in [7], this does not prove that many-body effects are completely absent. Thus several structures with heavy Fermions are known at present that have a Fermi surface which corresponds to one-electron theory, although electron masses in them may be as high as 2000 [361. Moreover, the agreement between the calculated and experimental band structures is much worse at higher binding energies [371. 1.3. Angle-Resolved Photoelectron Spectra of HTSC Recently, the most important and interesting data about the structure of occupied electronic states were obtained by angle-resolved photoelectron spectroscopy (ARPES). It is generally believed that the photoelectron spectrum is proportional to the following one-electron spectrum functions: 1

2

The function A _ defines the photoemission spectrum (PES), and the function A+ defines the inverse photoemission 112

spectrum (IPES). Thus it is suggested that the matrix element of the optical dipole transition is simply constant, which is generally not true. A more complete description of PES formation demands using two-particle correlation functions defining the creation and annihilation of electron-hole pairs by interaction with light. Thus for ARPES, the probability of light interaction with the electron shell of a substance, causing a transition of the valence electron ct~a to the photoelectron state ap+ with an energy ep, equals

Ifp,o))=~a IM(p,q,a)le-** f ~d t

+ + Cp-qo). e io)t (Cp_q(t) apo(t) ape

Here the transition occurs by the absorption of a photon with a frequency w, wave vector q ~ 0, and polarization a; M(p, q, a) is the matrix element of the dipole transition. If we neglect the top part describing the interaction of photoelectrons with valence electrons, the two-particle correlator is divided into a product of one-particle correlators" ' l(p, o)) = Z I M(p, q, a ,)12~ , ~o, 2dt~ eff~

Cv_q,a> =

G

Here WA is the work function. The matrix element is a function of the wave vector and polarization and takes into account selection rules. The two-dimensional wave vector p / / f o r the photoelectron recorded by ARPES is P// = [ 2 m ( ~

E n,v N - 1 + E g ) ] 1/2 sinO,

where 0 is the polar emission angle. Structure of the top of the valence band in SrzCuO2Cl 2. The structure of the top of the valence band in Sr2CuO2CI 2 has attracted attention because this compound has a K2NiF4 type structure with CuO 2 layers separated by double SrCI layers and is an analog of the tetragonal structure of La2CuO4. The electric and magnetic properties of Sr2CuO2CI 2 and La2CuO 4 are also analogous: according to the band structure picture, both dielectrics must be metals with a half-occupied band and both are antiferromagnets. For single crystals of Sr2CuOzCI2, TN = 256 K. The orthorhombic phase was not found in Sr2CuO2CI 2 when the temperature was lowered to T = 10 K. An important difference between the two compounds lies in the fact that Sr2CuO2CI 2 may not be doped [38]; it is believed that the CuO 2 planes in this compound do not contain charge carriers. At the same time, its isostructural analog, Sr2CuO2F2+y, at large values o f y is a superconductor with Tc = 46 K [39]. The dispersion law of holes at the top of the valence band was experimentally investigated by ARPES on single crystals of the nondoped dielectric Sr2CuO2C12 and superconductor Bi2Sr2CaCuO2+y [40] for the I ' M direction (Fig. 4). At the temperature of spectrum recording (T = 350 K) the samples are above the N6el point, in the region of short-range antiferromagnetic order with the correlation length ~AFM = 250/~ [41]. For fast and local measurements of photoelectron spectra, this means that the spectra are sensitive to antiferromagnetic order effects. According to two-magnon Raman scattering data [42], the exchange interaction between the nearest neighbors is J = 125• meV. ARPES spectra were obtained using synchrotron radiation with a resolution of 75 meV in energy and (1/20)~ in wave vectors k x, ky. The lattice parameter is taken to be unity. The_characteristic points of the Brillouin zone for the square lattice are denoted by X = ~(1, 0), Y = ~(0, 1), M = ~(1, 1), M = ~(1/2, 1/2). As can be seen in Fig. 4a, the top of the band is reached at the point M; the energy reckoned from the Fermi level is E(M) = -0.8 eV. The intensity of the peak shows a nonmonotonous behavior, being maximal in the vicinity of M and zero in the vicinity o f / ' a n d M. The total band width is W = 280• meV. The ARPES data of Sr2CuO2C12 were compared with those of the metallic composition Bi2Sr2CaCuO2+y [43] (Fig. 4b); the peaks in the spectrum of the latter are narrower and vanish after crossing the Fermi level near the point ~(0.45, 0.45). Below the Fermi level the dispersion law resembles the corresponding segment of a dielectric. Thus in the region of k from ~(0.27, 0.27) to ~(0.45, 0.45), dispersion is 270• meV in Bi-2212 and 240• meV in Sr2CuO2CI 2. This coincidence hints that the hard band model is applicable; doping simply shifts the Fermi level inside the valence band. As will be shown below, this simple behavior takes place only for the I ' M direction of the Brillouin zone. 113

[I, 0o,,,I "

03

- 1,1 b

"~------------"~--0.91.0.91 ~

k--0.73.0.73 ~,-0.64. 0.64

o.6. o.o L

_

~-0.55.0.55

0.55.0.55

~- 0.5.0.5

. v

:z:"

,~.0.45.0.45

~.0.41.0.41 ~'--0.36.0.36

~ "~

o.3z o.32 r ~'---0.27. 0.27 ~-O.la. O.la ~-- O. 0 ililifil[Itlt]llit[

-1.5

iilliil

1.0 -0.5 0 -1.0 -0.5 Energy relative to EF, eV

lliltl

0

0.5

Fig. 4. ARPES data for Sr2CuO2Cl 2 (a) and Bi2Sr2CaCu2Os+y (b) from [40]. The two-dimensional Brillouin zone is shown at the top; the full circles correspond to the occupied states, the size of the circle shows wave vector resolution.

Figure 5 gives photoelectron spectra for the F X and XY directions. The absence of peaks in the I " X direction contradicts the hard band picture, because doped metallic compositions are known to have a fiat band with E F in the vicinity of (~r, 0) [43, 44]. This prompts that the states near the point X appear with doping, significantly changing the form of the Fermi surface of metallic compositions compared to the hard band model. The overall picture of the dispersion law of electrons at the top of the valence band of Sr2CuO2CI2 is shown in Fig. 6. The calculations in the framework of the t - J model [45, 46] yield the bandwidth W = 2.2 ] in a wide range of the t/J parameters. The values of W and J given above are used to fred W/J = 2.2_+0.5, which is consistent with the t - J model. For other directions in the BriUouin zone, there is no agreement with the t - J model. In [47], the ARPES spectra of Sr2CuO2CI2 single crystals were measured at different polarizations of incident photons. For the [1, 1] direction, the dispersion law of electrons at the top of the valence band was obtained the same as in [40]. For the [1, 0] direction, ARPES were found to depend on optical polarization (Fig. 5b) so that the electronic states that are odd with respect to reflections in the plane of the normal and [t, 0] direction do not show any peak on the spectrum function curve. At the same time, the even states show a distinct quasiparticle peak with dispersion of 0.2 eV. The difference between the nonpolarized [40] and polarized [47] spectra demonstrates an important role of matrix elements of dipole transition for the form of the photoelectron spectrum line. It is useful to compare the ARPES data with the data of optical and electron energy loss (EELS) spectroscopy. According to optical absorption spectra, the charge transfer excitation energy with q = 0 for Sr2CuO2C12 is 2 eV [48]. In an analogous publication which appeared later [49], the absorption edge proved to be 1.42 eV at T--,0. Angle-resolved EELS spectra were reported in [50]. In the region of small angular vectors q~0, EELS defines the same excitations as in the case of optical spectroscopy. At q ;e 0, the imaginary part of inverse dielectric permittivity is measured ( d I / d ~ ~ q - 2 I m [-1/e (q, w)]); therefore, for q # 0, the AREELS method permits one to study the dispersion of excitations, including that of forbidden transitions. For Sr2CuO2CI 2 single crystals, the EELS spectra

114

I

;

S,SS

~%

~,

1,0

/

17

",s

- I1,1

0,1

18 Kinetic enegry, eV

b

a )4, 0

0 1.0, 0

0.1

).9, 0

0.2 0.3

).8, 0

e-

e=

).7, 0

0.5 0.6 0.7 0.8 0.9 0

).3, 0 ).1,0 3.0 -1.0

a:._=

0.4

).6, 0 ).5, 0 ).4, 0

-1.5

III .r=. v v

-0.5

-1.5

-1.0

-0.5

Energy relative to E F, eV

Fig. 5. Photoemission spectra [40] for Sr2CuO2Cl2 in the directions (0, 0 ) - ( ~ , 0) (a) and ( 0 , ~ ) - ( ~ , 0 ) (b). The dashed line in Fig. 5a indicates the maximal possible energy for any peaks in the given direction. Figure 5c shows the spectrum in the (0, 0 ) - ( ~ , 0) direction for polarized light; even (with respect to reflections) electronic states - full circles; odd states - open circles.

IJ.

uj - 0 . 8 -

jJ~%% .p.s

-

'%+

0

-r - 1 . 0 c UJ

-1.2(g/2, ~/ I

(o,o)

I

I

(~,~)

I

(~,o)

I

I

(o,o) (~, o)

I

(o,l~)

Reciprocal lattice vector k

Fig. 6. Comparison of the experimentally determined dispersion law for Sr2CuO2CI2 (circles) with calculated data for the t - J model (solid line) [40].

115

l

r

=lo. -Jz

.

J 0

i tl

ill

2

Ji

4

)i

IIIIII

6

I II

---..-..

i: II

q I I I I I I I

II

il

8 0 2 4 6 8 Energy loss, eV Fig. 7. Electron energy loss spectra for Sr2CuO2CI2 in [100] ( f Ar) (a) and [110] (FM) (b) directions [49].

measured in the I"X and F M directions are presented in Fig. 7a, b [50]. It can be seen that the optically allowed transitions have maximal dispersion in the F M direction with a bandwidth of about 1.5 eV. In the F X direction, the dispersion of the allowed transitions decreases, and a forbidden transition with an energy of 4.5 eV appears. As noted in [50], the peak intensities are redistributed from allowed to forbidden transition, the integrated intensity remaining constant. For the F M direction, the intensity of excitations depends weakly on the wave vector. Large (1.5 eV) dispersion for allowed transitions is difficult to explain in terms of transitions from the valence band to the conduction band, since ARPES (see above) gives the bandwidth of 0.3 eV for the valence band and suggests the same order of magnitude for the conduction band. Therefore, the authors of [50] developed a simple model of an exciton of a small radius, which is a pair of an electron on a copper atom and a hole on the nearest oxygen atom with zero spin; the pair can move through the antiferromagnetic lattice without violating the antiferromagnetic order. As a result, the bandwidth of the exciton proves to be greater than that of the Fermion, which is a hole whose motion is greatly hindered by antiferromagnetic correlations. An adequate description of such processes requires a more detailed calculation of two-particle spectra including both electron-hole continuum and bound states. Shadow zones and short-range antiferromagnetie order in Bi2Sr2CaCu20$+ x. The existence of strong antiferromagnetic fluctuations not only in the dielectric region of weak doping but also in the region of superconducting compositions is well known from neutron magnetic scattering data for La2_xSrxCuO 4 [51] and YBa2Cu3OT_y [52]. The influence of these fluctuations on the electronic structure was widely discussed in recent years. Important data on the electronic structure of Bi2Sr2CaCu2Os+x directly showing the influence of magnetic fluctuations on the Fermi surface were obtained by the ARPES technique [53]. The standard scheme of an ARPES experiment suggests giving the direction of the vector k = (kx, ky) and measuring the spectra along few highly symmetric directions in the Brillouin zone. In contrast to this, in [53] the range of energies AE = 10 meV in the vicinity of the Fermi level was limited, and the authors constructed a map of the intensity of emitted photoelectrons in this region for the whole Brillouin zone using 6000 discrete values of k with an angular resolution better than 0.1~ This gave a cross section of the Fermi surface by the (kx, ky) plane (Fig. 8). One can dearly see two types of line: very intense lines centered around the X and Y points of the Brillouin zone and weakly intense lines around the center of the zone and the angular points of M type. Figure 8c shows that the weak lines are derived from the strong ones by shifting the lines by the wave vectors F X or FY; this indicates a new spacing, 2x2. A low-energy electron diffraction (LEED) study did not reveal any rearrangement of atomic structure of the surface of the single crystal; therefore, the doubling of the spacing is associated with magnetic correlations. Since photoelectron excitation is a local and swift process, its characteristic spatial and temporal scales may be small compared to the length and time of spin fluctuation correlation; then the 2 x 2 superstructure behaves as a quasistatic structure, and weak lines may well be attributed to narrowing of the Brillouin zone. 116

r

. , , ~ Y.

r'~,,., z '~

Fig. 8. Map of intensity of photoelectrons excited from Bi2Sr2CaCu208+ x in the energy region of 10 meV around the Fermi energy. The logarithmic scale intensifies the weak features. The outer circle corresponds to the radiation angle of 90* (a); simplified scheme (a) indicating the points of the Brillouin zone and intense (solid lines) and weak (dashed lines) lines (b); scheme of extended Brillouin zones showing that, when shifted by the vector F X or I"Y, the system of intense lines forms a system of weak lines (c); Fermi surface of Bi2Sr2CaCu20 2 as calculated by the FLAPW method [38] (d) [53].

These states were predicted theoretically [54] and called shadow states. Considering the effect of spin fluctuations on the electronic structure, the authors of [54] defined the form of spin susceptibility from phenomenological analysis and showed that, for strong antiferromagnetic correlations, the transitions (relation between the vectors k and k' such that Ik' - k - Q I < 1/~, where Q = (~, ~) and ~ is the length of correlations) lead to shadow zones. Based on the calculations [54], however, one can assess that the correlation length ~ must be sufficiently large to ensure the observation of the corresponding satellites in ARPES spectra (~ = 20a, where a is the Cu-Cu distance). This interpretation of the data of [53] raised doubt [55], because neutron scattering data showed that in the region of superconducting compositions Bi-2212 ~ is only several a. However, subsequent calculations with microscopic models by the exact diagonalization and Monte Carlo quantum methods [56] and perturbation theory calculations [57] showed that shadow zones can appear as satellites on the spectrum function curve at a small length ~ = 2.5a, which agrees with neutron scattering data. These works will be treated in detail below. The authors of [53] also noted that their data about the Fermi surface, for example, k F in the (1, 0) and (0, 1) directions and the curvature of the surface near k F in these directions, agree with the results of band structure calculations [58]. At the same time, the experiment failed to reveal the small pockets centered around the points M = (:r/2, :r/2). Conversely, the shadow zones found experimentally are missing in the band structure calculations. This is quite explicable, because the one-electron band structure calculations do not include the dynamic renormalization of electrons by spin fluctuations giving rise to shadow zones. Variation of band structure with doping in Bi-2212 systems. An ARPES investigation of changes in the band structure caused by changes in hole concentration was recently accomplished on two groups of Bi-2212 samples [59]. The first group was Bi2Sr2CaCu208+ x single crystals, in which the concentration of carriers was changed by annealing in air at T = 600"C to prepare overdoped samples with TC = 85 K and in argon at T = 550~ to prepare underdoped samples with T C = 67 K. The samples are referred to as overdoped and underdoped compared to the optimal concentration of holes giving the maximal TC. 117

The second group of samples involved thin single crystal films with the composition Bi2Sr2Cal_xDyxx Cu2Os+x obtained by molecular beam epitaxy. The concentration of holes in these films is controlled by the quantity of Dy [66]. The samples with x = 1, 10, and 17.5% Dy had TC = 85, 65, and 25 K, respectively; T C was measured as the null resistance point on the curves of the temperature dependences of resistance. For x = 50% Dy, the films were dielectric. The spectra were measured in vacuum at T = 110 K. For both groups of samples, ARPES are given in Fig. 9. The data in Figs. 9al and 9a2 are completely identical to the ARPES data for the typical Bi-2212 samples with a slightly excessive concentration of holes; the Fermi level intersects the F M line around 45% its length and around 25% X M length [61]. The spectra in Fig. 9d for the dielectric sample are characterized by rather broad peaks (probably, because of scattering on Dy atoms); nevertheless, the dispersion of the centers of these peaks and that of the peaks of Sr2CuOeCI2 [40] (Fig. 4a) are very much alike. The data for the samples with 10% Dy (Fig. 9bl, b2) show an intermediate behavior between the two limiting cases: an optimally doped system and a dielectric system. The value of k F along the F M line did not markedly change compared to the 1% Dy sample, although one can note broadening of peaks and their decreased intensity in the vicinity of e F. At the same time, the behavior near the point X (Fig. 962) radically changes. For the 10% Dy sample, the peaks in the vicinity of X are broadened and shifted toward larger binding energies so that none of them crosses eF in the X M direction. This absence of crossings for one line in k space might be explained by impurity effects, namely, by a shift caused by additional scattering on the Dy impurities. However, the insignificant changes in ARPES spectra for the 10 and 1% Dy samples preclude this explanation and indicate that the electronic structure is modified with doping. The second group of samples (Fig. 9cl, c2) confirms this conclusion. The underdoped sample (TC = 67 K) in the F M direction has ARPES analogous to the spectra of the films with 10% Dy (TC = 65 K); the Fermi level is crossed around 45% length from the center of the Brillouin zone. For the F X and X M directions, the spectra 6c2 and 662 are also very similar, although the two types of doping are structurally diverse. Substitution of Ca by Dy occurs between the CuO 2 planes, and elimination of oxygen during annealing predominantly occurs from the BiO layers. Nevertheless, according to ARPES, the changes in the electronic structure are identical. The dispersion laws of electrons obtained by ARPES are shown in Fig. 10 for both groups of samples. For all samples remaining superconductors, the dispersion law in the [1, 1] direction and the value of/c F change slightly. In the dielectric region of compositions, the dispersion law in the (0, 0)-(~r/2, 3r/2) direction still resembles the corresponding region of the dispersion law for metallic samples, although the bandwidth decreases. The dispersion law is absolutely different in the vicinity of X. On passing from optimally doped to underdoped samples, the cross section of the Fermi surface on the X M fine vanishes. This leads to great changes in the Fermi surface (Fig. 11). The optimally doped samples with TC = 85 K show large cross sections of the Fermi surfaces, which agree with the Luttinger theorem [62]. If the hard band model worked, the decrease in hole concentration would lead to a decrease in the area of the cross section, which would nevertheless retain its form (as shown by the dashed line in Fig. 11). The crossings of the Fermi surface by the X M line would then be preserved, but this conflicts with the experiment, showing that the energy gap opens on the Fermi surface along the X M line. The authors of [59] propose three possible interpretations of their results. The first interpretation is coupling of quasipartides without pair coherence above Tc. This might open the gap without a coherent superconductive state. Coupling of quasiparticles at temperatures above TC was discussed in terms of Ginsburg-Landau generalized theory for d-type superconductivity [63-65]. Coupling of d-spinons [66, 67] in terms of the RVB concept [68] was also discussed for temperatures above TC. For both classes of pairing, the photoelectron energy shift near the point X is explained as the emergence of a gap with dxe_y2 symmetry, which is maximal for X and zero along the F M line. The amplitude of the gap may be estimated from the 20-30 meV shift of binding energy for the point X. Another interpretation concerns superstructure formation due to antiferromagnetic ordering or a rearrangement of atomic order. In this case, the points F and M become equivalent in the new Brillouin zone, and shadow zones appear on the side of M. The mixing of the shadow and initial zones opens the gap in the vicinity of X, as in spin density wave theory [69]. The order is not necessarily long-range, as discussed in the previous section. The Fermi surface cross section by the shadow zone in this case is shown by the dashed line in Fig. 11. The resulting small hole pocket is centered around the point M. The third interpretation is also based on d-coupling of spinons, but SU(2) symmetry is preserved on doping [70]. This approach also suggests formation of a hole pocket.

118

~"

(~'~,)

Jo (0'0)

o~, ~

) tuoJj eoue~,s!C]

-.

~

.

111

.=_ "~ ~ r

~e.f

-

~

&

H

o o

o

o

L

o

9u n "leJ ' ~ j s u e ~ u l

119

0

olb 0.4 I ~ (0, O)

I I I I I I I I I I I (n/2, ~/2) ( I0) Reciprocal lattice vector k

(o, o)

Fig. 10. Dispersion laws for samples with a variable concentration of holes for the Bi2Sr2Cal_xDyxCu208+ x film (a) and Bi2Sr2CaCu2Os+y crystals (b). Notation for different concentrations: a) 1% Dy, T c = 85 K - full oval; 10% Dy, T c = 65 K -- rhomb; 17.5% Dy, T c = 25 K - rectangle; 50% Dy, dielectric - triangle; b) sample annealed in air, T c = 85 K - oval; sample annealed in argon, T c = 67 K - rhomb [59].

Comparing the results of [59] and [53] and the results of spectrum calculations of quasiparticles in systems with strong correlations [45, 71, 72, 54, 56, 57], we conclude that the spectrum partly becomes dielectric due to spin fluctuations, since shadow zones were directly observed for Bi-2212 in [53]. In this case, reconstruction of the Fermi surface with a decreased concentration of holes involves alteration of the surface topology. The ARPES data for Sr2CuO2C12and Bi2Sr2CaCu2Os+y taken from [40] (Fig. 4) also agree with this conclusion. A comparison of the dispersion laws in Figs. 4 and 10 leads us to draw the following conclusion concerning the electronic structure transformation of the CuO 2 plane due to doping: in a nondoped sample, the top of the valence band is reached at a point M = (~t/2, ~/2). As is known from optical and photoelectron spectroscopy data for dielectric compositions, doping gives rise to in-gap states as deep admixture levels [73-77]. Details of the electronic structure modification in the vicinity of the dielectric-metal transition due to doping are currently not clear. For Bi-2212 samples with Tc = 25 K, i.e., in a metallic underdoped phase, new states appear near the X point which form the flat zone. It is conceivable that these states appeared from the in-gap states of the dielectric phase, which naturally explains their nondispersion nature. As the concentration of holes increases to optimal, these states form a band having a saddle singularity in the vicinity of the X point and a maximum at the point M = (~, ~). Band structure of SrzRuO 4. The role of CuO 2 layers remains a key problem for understanding the mechanism of superconductivity. Can other laminated perovskites not containing copper be superconductors? The recent discovery of the superconductors Sr2RuO 4 with TC = 0.93 K [78] make it possible to draw an interesting comparison with copper oxides. Sr2RuO 4 crystals have a body-centered tetragonal lattice of K2NiF4 type with lattice parameters a = b = 3.8694, c = 12.764/~ at room temperature (X-ray diffraction data [79]). The striking differences between Sr2RuO 4 and laminated cuprates are as follows: a) even nondoped samples are superconductors, in contrast to the isostructural La2CuO 4 and Nd2CuO4; b) the Ru 4 + (4d 4) ion in the low-spin state has spin S = 1, whereas Cu 2 + (3d 9) has S = 1/2; c) in Sr2RuO4, t h e p - d hybridization with oxygen involves the tzg (dxy, dxz, and dyz) orbitals; in copper oxides, this is the eg (dx2_y2) orbital. 120

(o,o)

(=, o - Tc=85 K ~ - Tc=67 K /

90

/

I

(0, ~)

(=,~) Reciprocal laffice vector k

Fig. 11. Fermi surface cross sections for Bi-2212 superconductors with different hole concentrations [59]. Gap opening is shown on the (at, 0 ) - (at, at) line when the concentration of hole decreases. The trine dashed line denotes the expected Fermi surface cross section in the rigid band model. Full symbols - overdoping; open symbols - underdopiug.

kx, k 1 -0.3-0.2-0.1 0

>=

0

0.1

0.2

0.3

-5

E -10 LU - 1 5 -20 m

19 t--

-10

F,

,,~

,F

-15

uJ -20 -25 -30 -0.3 - . -. . .2 0.3 Reciprocal lance vector k shifted by -0.814, ,~-1

Fig. 12. Flat bands in the vicinity of the point (at, 0) for Sr2RuO 4 according to ARPES data [80]. Notation for the Brillouin zone points in [80] is inverse compared to ours; here X = (at, at), M = (at, 0).

The ARPES spectra of Sr2RuO4 were measured at 10 K in vacuum with an energy resolution of 22 meV and angular resolution of 1", which corresponds to AK = 0.06/~-1 [80]. As in the case of cuprates, a saddle point was found on the curve of the dispersion law near the point (at, 0) (Fig. 12), leading to Van Hove's singularity. Figure 13 shows two-dimensional Fermi surface cross sections measured in [80]. The results obtained in [80] 121

F

~

1.6 s

I

Y |

1.2 ; .

(

o.8

,, 0

,'F, L , , , , 0.4"'

F

0.8

v1.2

1.6

~

F

Reciprocal lattice vector kx, A-~ Fig. 13. Fermi surface of Sr2RuO 4. Open circles - ARPES data, solid lines - band structure calculations [80].

were compared with the data of LDA calculations [81]; it was noted [80] that good agreement with theory may be achieved by using a 0.77 meV shift in band structure calculations. According to [81], the three zones observed on the Fermi surface are predominantly dxz-, dyz-, and dxy-like. The two Fermi surfaces centered around the point (~r,~) are hole-like, and the surface around the point (0, 0) is electron-like. The Van Hove's singularity lies 0.17 meV below eF. Previous ARPES experiments revealed saddle points at K = (~, 0) and Van Hove's singularities for many cuprates (Table 1): YBa2Cu307_y [82], YBa2Cu40 8 [82, 83], Bi-2201 [84], Bi-2212 [43], and Nd1.85Ce0.tsCuO 4 (NCCO) [85]. As can be seen from Table 1, the n-type system NCCO is distinguished by the form of the dependence p(T) - T 2 [86] and by the remote position of Van Hove's singularity from eF. For cuprates and Sr2RuO 4, Van Hove's maximum is close to the Fermi level. In many theories, the proximity of e F to Van Hove's singularity was regarded as a reason for the high values of T c in cuprates ([87, 88] and reviews [89-91]). However, the low Tc in Bi-2201 and Sr2RuO 4 cast doubt on this scenario. Probably, the TC of these two compounds were high due to Van Hove's singularity but were lowered by the action of some other factors. For Sr2RuO 4 this may be proximity to the ferromagnetic state with strong ferromagnetic fluctuations suppressing Tc [92]. Anyhow, the case of Bi-2201 and Sr2RuO4 suggests a more complex relationship between the high values of T C and the proximity of Van Hove's singularities to the Fermi level than proposed by the scenarios of [87, 88].

TABLE 1. Superconductive Transition Temperature, Temperature Dependence of Electric Resistance in the Normal Phase, and Position of Van Hove's Singularity Measured Relative to the Fermi Energy (from [80]) Compound YBaECU307_y YBa2Cu408 Bi2(Sro.93Pro.o3)2CuO6 Bi2Sr2CaCu208 +y Ndl.85Ceo.15CuO4 Sr2RuO4

122

Tc, K

ab(T)

92 82

Linear

10 83 25 0.93

The same >)

Quadratic Linear [58]

Evil, meV < 10 19

<30 <30 350 17

[821 [82, 83]

[841 [43] I851 [801

1.4. X-Ray Absorption Fine Structure of HTSC

The CuK spectra of HTSC are the best studied XAFS spectra of these compounds. They are formed by X-ray excitation of the copper ls electron to the vacant Cup orbitals. Copper oxides have no vacant Cup states up to the ionization threshold; hence, the CuK spectra of HTSC and related oxides have no intense pre-threshold lines. Historically [1], the CuK absorption spectra were predominantly employed to determine the oxidation state of copper in HTSC. Thus it was erroneously concluded that Cu(III) is greatly involved in formation of the electronic structure of La2CuO 4. More recently, the nature of the main short-range XAFS peaks was explained based on one-electron calculations [93-96]. The features determined by photoelectron scattering on the nearest atoms [1, 93] and on the following coordination spheres [94-96] were revealed. In [95], the experimental spectrum was compared with the result of the one-electron extended cluster calculations of the polarized CuK spectra; it was inferred that the peak lying 7 eV higher than the fundamental maximum in the z-polarized spectrum is many-electron. However, no attempts, even if qualitative, were made in [95] to assess the validity of this hypothesis. The CuK spectra are well reproducible and hence are quite reliable to be used to obtain difference spectra for different conditions and variations of composition. The authors of the experimental work [97] effectively employed this property of CuK spectra to obtain for the fh'st time the spectra of two-hole Cup states of La2CuO 4 and YBa2Cu30 7. The sig~nificance of this work should be emphasized specially, since this approach made it possible to reveal the spectra corresponding to the included electronic states of doped centers. The CuL 3 spectrum is another spectrum of HTSC well studied experimentally. The mechanism of formation of CuL 3 XAFS of HTSC involves X-ray quantum excitation of the Cu2p core level and electron transition to the d-type bound state or to the s- or d-orbital at positive energies with subsequent escape from the system. Since the photoelectron wave function belongs to the mother system, it becomes possible to calculate the matrix elements of X-ray transitions in the one-electron approximation. Previously, the mechanism of formation of CuL 3 X-ray absorption spectra of HTSC ceramics was studied by the multiscattering method in the one-electron approximation (e.g., [2, 96]) or by using many-electron calculations based on Anderson's model [98]. The first approach does not allow one to describe strong correlation effects, which play a significant role in the mechanism of formation of these spectra; the second approach strongly diffuses the absorption picture, reducing the whole spectrum to a whiteline, precluding investigations of the complex post-threshold region. Currently there is lot of experimental evidence [98, 99] indicating that in copper-containing HTSC oxides the mechanisms of formation of CuL2,3 spectra considerably differ from the simple one-electron picture based on the crystal field model. The differences primarily involve electron transitions to the states missing in this model - so-called nondiagram transitions. These transitions are formed either by various strong correlation effects or by photoelectron scattering on possible potential barriers induced by the neighboring atoms and the chemical bond in the compounds. These differences were reported in [99] for the CuL 3 spectrum of YBa2Cu3OT_ 3 (Fig. 14). The fundamental peak A of the whiteline was assigned [99] to the Cu2p 6 3d 9--,Cu2p5 3d 1~ transition (for any 6); peak B was attributed to the Cu2p6 3d 9L~Cu2p5 3dl~ transition formed with participation of strong correlation effects (for ~ = 0.07-0.30), which agrees well with the results of the theoretical work [100] involving the Cu2p XAFS of YBa2CU3OT_~ using Anderson's model. When ~ increases (the concentration of electron vacancies decreases), the intensity of peak B falls to 0 and peak C appears instead of it at a distance of 2.8 eV. Based on the energy level of the fundamental peak in the CuL 3 spectrum of Cu20 [101], peak C was assigned to copper ions in the oxidation state 1 +. This interpretation of peak C certainly deserves consideration, but the mechanism of formation of the CuL 3 spectrum of monovalent copper was not interpreted in [99] nor in [101]. Moreover, the absolute oscillator strengths of absorption for the Cu2p orbital were not given in [101] for YBa2Cu3OT_ ~ and Cu20; this does not permit one to correctly analyze the CuL 3 spectrum of the complex mixed system YBa2Cu3OT_ ~ formed by the superposition of various types of copper ions in the oxidation states 1 +, 2 +, and 3 +. Thus one has an impression that even inclusion of many-electron effects in consideration may not explain all features of the CuL 3 spectra of copper-containing HTSC. According to [100], a nondoped system (with one electron vacancy per formula unit) has only one 2p6dxgz_y2.-,2pSd10 X-ray transition, although the initial state has two d 9 type and two dl~ type configurations due to the hybridization of the vacant states. In doped systems with more than one vacancy per formula unit, many-electron effects are much more important due to the appearance of Cud 8, Cud 9L, and cudl~ contributions. This leads to 123

A ,1, C u - Lm-XAS

~

6 =0.07

~,,

"~

0.29 ,,'t

928

I

..

I

932

I

I

936

I

Photon energy, eV

Fig. 14. Experimental CuL 3 X-ray absorption (XAS) spectra [99] of YBa2Cu307_ ~ for different 6.

significant differences between the CuL 3 spectra of included electron states of doped compounds and the spectra of nondoped compounds, which involve shake-up satellites on the whiteline [2, 100, 101]. 1.5. Core Level X.Ray Spectra The core level X-ray emission spectra are characterized by good reproducibility, which manifests itself in the electronic structure studies of solids. The reason for this lies in the fact that the effective depth of escape of X-ray quanta is several hundreds angstr6ms and considerably exceeds the depth of escape of electrons used in X-ray photoelectron spectroscopy. It is also significant that the energy of the inner emission line depends only on the state of the atom. The available literature presents ample data on CuKa X-ray spectra (see, e.g., [102, 27, 103]) (transition from the Cu2p orbital to the Culs orbital previously ionized by an X-ray quantum). Most theoretical and experimental papers investigating these spectra interpret the spectral features in terms of Anderson's model or Hubbard's double-band model. Generally they describe only the mechanisms of spectrum formation for the one-hole configuration; the contribution of the two-hole configuration for which correlation effects are very important is not included. The shift of the fundamental maximum of CuKa spectra during a transition from nonsuperconductive to superconductive state of HTSC oxides, including La2_xSrxCuO4, was studied in [104, 105]. It was shown that only in YBa2Cu30 7 is the fundamental CuKa maximum shifted by 0.35 eV due to a change in the leading configuration. The theoretical spectra of one-hole systems are described in detail in [104, 105]. It was shown that the spectrum has a weakly intense satellite reflecting the density of Cud 9 configurations whose energy level is 0.4 eV higher than that of the fundamental maximum reflecting the density of cudl~ states in the ls and 2t9 hole configurations. As shown in [105], the shift of the CuKa energy level may not be measured using Larson's model without isolating satellite structure contributions. Thus the spectra may not be analyzed without invoking many-electron approaches. 1.6. Optical Spectra of H T S C C o m p o u n d s

In the optical spectra of dielectric La2CuO4, Nd2CuO4, and YBa2Cu30 6 (see, e.g., [106, 107]), the absorption threshold lies at 1.5-1.75 eV and was assigned to the charge transfer gap showing itself in angle-resolved photoelectron spectra. 124

The spectrum function above the gap decreases with doping, the threshold is shifted to the higher-frequency region, and new spectral features arise in the medium and far infrared regions. Also, very intense low-frequency features appear, which indicate that the gap doses during the doping. The density increases quicker than would be expected based on the free electron model. This suggests that the density of states above the gap is transferred to the low-frequency region during doping; in terms of strong electron correlations this is interpreted as a transfer of states from the highest to lowest Hubbard band. Thus optical experiments indicate that there are strong electron correlations in these compounds. 1.7. One-Electron Calculations of the Electronic Structure of Cuprates

In the first attempts to describe the electronic structure of copper-containing HTSC materials by ab initio cluster and one-electron band structure methods, the results of calculations were generally compared with photoelectron and X-ray emission spectra (e.g., [108, 1-9, 26, 110-113]). These calculations gave virtually the same picture of the electronic structure of these compounds (Fig. 15, [114, 27]) formed by the Cu3d and O2t9 orbitals. Limitations on their applicability for HTSC became evident at once: 1. These one-electron calculations gave a zero magnetic moment on copper, whereas, experimentally, all nondoped HTSC are antiferromagnets with a magnetic moment on the copper ion/, ~ 0.5/zB; HTSC in general do not possess long-range antiferromagnetic order but show strong spin fluctuations [115]. 2. The experimental photoelectron spectra are shifted by approximately 1-2 eV downfield compared to the results of band structure calculations for both La2_xSrxCuO 4 and YBa2Cu307_ & 3. The calculations predicted the metallic character of the ground state of nondoped oxides of LazCuO 4 or YBa2Cu30 6 type, but in experiment they are insulators. 4. In the framework of one-electron calculations, it is impossible to interpret the oxygen and copper core level X-ray photoelectron and X-ray spectra because of their complex satellite structure. 5. Comparisons between theory and experiment showed that the simple one-electron approach may not be used to describe a series of singularities in the copper absorption X-ray spectra. In more recent works [116-118], an attempt was made to "modernize" the one-electron approach by introducing a correction to the vacant state potential. This correction is actually analogous to the Ua parameter introduced in Hubbard's model, which will be treated below. This approximation was effectivelyused to qualitatively describe the forbidden gap, the shift of the photoelectron spectra down on the energy scale, and the magnetic moment of copper atoms in the ground state of nondoped oxides such as La2CuO4, CaCuO2, Sr2CuO2CI2, and YBa2Cu306. The nature of the electronic states of the top of the valence band and the bottom of the conduction band was also described correctly and coincided with the most reliable experimental and theoretical data. Regretfully, no modeling was performed in these works for X-ray or photoelectron

X2 .y2 3 Z2 "r2~\

xy ,z=-

Cu

\

Y///A \ ~ ~ / / / / / ~ , ~ ,,,-- P ~(2 )

~

0

Fig, 15. Scheme of the electronic states of the Cu3d and O2p orbitals of the CuO 2 plane in terms of crystal field theory and interpretation in terms of LCAO calculations by LDA methods [27, 114]. The hatched regions denote occupied states; or(*) and ~(*) symbolize the (anti)bonding pdo states. 125

spectra; indirect comparison between the calculated and experimental data does not allow one to draw the final conclusion about the adequacy of this approach. 1.8. Inclusion of Strong Electron Correlations

As mentioned above, the one-electron approach failed to describe the electronic structure of nondoped cuprates and their physical and spectral properties caused by strong correlation effects. Two methods were previously used to take these effects into account: model calculations using Hubbard's or Anderson's models and ab initio approaches as configuration interaction (CI) or self-consistent field multiconfiguration interaction (MC SCF) methods. Model calculation is the most easily perceptible and simple method of including strong electron correlations. The simplest model is the tight binding model, describing the electronic structure of the CuO 2 plane and induding only the atomic orbitals of the CuO 4 cluster: two occupied Opx,r orbitals and one half-occupied Cudxz_y2 orbital. In this case, the model Hamiltonian is i, o

i, o

(i,j)o

(i,j)o

Here summation is done over the indices of atoms in the cluster; (i,j) symbolizes summation over the indices of the nearest neighbors; o is the spin index. This model takes into account three bands containing five electrons, but in practice the model is recorded as a hole representation in which the Cu3dl~ state is considered a vacuum state. In the case of one vacancy, the whole electronic structure of the CuO 2 plane is reduced to only one Hubbard band Cu3d 9202p6. In this model, d + and p + are standard and denote the hole creation (annihilation) operators on the copper and oxygen d or p orbitals of the CuO 2 plane. The charge transfer gap A is determined by the difference between the energies of the copper and oxygen p and d states (A - ep - ed) and is positive in the hole representation. The transfer integrals t ~ and t/~ are the parameters of the system, which are determined either from experimental data or from ab initio estimates. The signs of these parameters depend on the symmetry of the system and the values are much less than A (t~Jd,t~j << A). The model does not describe one of the basic properties of strongly localized copper d orbitals, namely, their strong Coulomb interaction included in Emery's model, which is a three-band analog of the Hubbard one-band model: i

i

(i,j)

Here ndo = dio + dio and n p = PioPio + are the densities of the Cu3d and O2p holes, respectively; Ud and Up are the parameters of the Hubbard interaction on the same orbitals of copper and oxygen; Upd describes the copper--oxygen interaction. In the hole representation, these values are positive and correspond to repulsion; Ud dominates when the electronic structure is formed, and this suppresses the Cu3dg-,cu3d 8 transition. The limiting case, when all the three parameters of the Hubbard interaction are zero (Ua = Up = Upa = 0), corresponds to one-electron calculations in the local density functional approximation, when the top a* band (Fig. 15) is doubly degenerate (in terms of this model) and hence half-occupied. An increase in the Hubbard repulsion parameter Ua removes the degeneracy of the a* band, forming the lowest filled (LHB) and upper vacant (UHB) Hubbard bands. In the case of Ua < A, the electronic structure corresponds to the Mott-Hubbard insulator, for which the highest filled band is a Cud type band; when Ud > A, we have a charge transfer insulator. In the latter case, the lowest Hubbard band is at a lower energy level than the oxygen subband; this leads to a charge transfer when the energy of electronic excitation from the oxygen sublattice to the copper centers is minimal. The Hubbard three-band model offers a simple and natural explanation to magnetism on the individual copper centers in nondoped cuprates. Indeed, if the copper d band is split into two Hubbard bands corresponding to d 9~d 8 and dl~ 9 excitations, then the number of the other electrons per formula is even and the insulating nature of the compounds is quite understandable. Since the d 9 configuration corresponds to the magnetic ion, the presence of magnetism is quite explicable [119]. The long-range antiferromaguetic order for such compounds is determined by the spin superexchange between

126

the copper centers with one vacancy; it is defined using the unitary transformation of the three-band model leading to Heisenberg's two-dimensional model:

H = Jcc ~ (Si Sj - 1/4 ndnd), (i,j)

where Jcc is the exchange coupling constant, and Si is the spin operator on the copper center:

Jcc = (4t;a/A)(1/Ua + 2/(2A + Up)). The lower experimental estimate for this constant is Jcc ~- 0.15 eV. Another successful application of Hubbard type models for nondoped HTSC is qualitative description of the CuKa and Cu2p XPS spectra of these compounds [31, 120-122, 123, 124]. However, most authors described the mechanisms of spectrum formation for the one-hole configuration and neglected two-hole configurations, for which correlation effects are very important. For a two-hole configuration, this problem was solved on one structural unit of HTSC in [125]. For a charge transfer insulator, it would seem that the extra hole must settle on the oxygen subband, which comes right after the upper Hubbard band. This viewpoint, however, was disproved in [125]. It was shown that the covalent mixing of the copper and oxygen atomic states (in band structure theory [Fig. 15] this corresponds to the :r-subband of essentially oxygen nature) forms the triplet and singlet (ZRS - Zhang-Rice singlet) states Cu3d 902p5 due to the Hubbard splitting of the occupied states in the one-electron :r-subband. According to the data of [125], the singlet is the highest occupied state (in electron representation) and is the first to be completed by the extra vacancy formed by doping. In terms of the Hubbard one-band model this means that both the Zhang-Rice singlet and the Cu3d 1~ vacuum state are singly degenerate and behave like the upper and lower Hubbard bands. Due to this, they may be described in terms of the half-occupied effective Hubbard model: nn

M

=

-,

nnn

Cjo + H q - , ' (i,j)

(C,o+ (i, la/a

+ Hq +

U n,,ni,, i

where nia = c+ cia is the electron density with the spin or, and U --- A. In addition to the transition integral between the nearest neighbors t (430 meV), there is a transition integral with the centers of the next coordination sphere (t' = - 7 0 meV). In a single CuO 4 cluster, the only oxygen state mixing with the copper d state is the totally symmetric combination P~a -- 1/2 p+. The other three oxygen states PNBao are nonbonding. In the half-occupied state we have a vacancy

2z

+

with S = - 1 / 2 for the cluster. Addition of a hole (which corresponds to doping) leads to the problem of two holes on the four states interacting with the copper states. This gives rise to five configurations (according to the number of basis wave functions): Cud 8 O2p 6 corresponding 4 corresponding to the IP~tP~j,) state, cudgo2p 5 corresponding to the singlet to the Idjtd)~ + + ) state, Cudl~ IS) = (IP~td)~) + + + Id~P~j,)) / r andtriplet IT) = (IP~,td~) - I d ~ P ~ ) ) / v~ states, and Cud902pS corresponding to the states formed by the nonbonding oxygen states of I P ~ t dj~ ) type. Diagonalization of the Hamiltonian in this model is trivial; in the case of a charge transfer insulator, the ground state is the Zhang-Rice singlet, whereas the triplet lies 2-4 eV higher than the singlet and is unimportant in the low-temperature physics of the phenomenon. The Zhang-Rice singlet in this model is represented as an effective spinless hole moving on the two-dimensional lattice of copper spins in the subspace of doubly occupied states. The whiteline (transition from the Cu2p core orbitals to the bound vacant Cud states) of the CuL 3 absorption spectra was modeled in [98] for both one- and two-hole configurations of the CuO4 cluster in the p - d three-band model. It was shown that doping leads to a marked intensity in the z-polarized spectrum of the CuO 4 cluster. A comparison between [98] and [126] shows that there is good agreement between theory and experiment, confirming the pronounced increase in the density of Cudz2 states as a result of doping. With all advance in the understanding of the electronic structure of HTSC by using Hubbard type many-electron 127

models, it was not explained why the optical and primarily photoelectron spectra are adequately described by band structure theory since it would seem that the split Hubbard bands must be unambiguously revealed in spectroscopic experiments [119]. The many-configuration ab initio (MC SCF or CI) calculations is another approach to include strong electron correlations. These calculations of oxide systems with Cu(II) primarily demonstrate strong localization of the electrons of the top of the valence band (e.g., [127, 128]), confirming the applicability of Hubbard type models. The most detailed paper [128] showed that ionization of both core and valence orbitals is accompanied by a strong screening effect, leading to many-electron shake down satellites in the photoelectron spectra due to the charge transfer from the occupied O2p orbitals to the vacant Cu3dcr orbitals. Among the papers devoted to ab initio CI investigations of doped copper [formally Cu(III)] oxide systems, it is worthwhile to mention [129]. The electronic structure of the compounds La-Sr-Cu-O and N d - C e - C u - O was calculated by the multiconfiguration variational method in the duster approximation. The clusters CuO6, CuO4, Cu2Oll, and Cu20 7 were chosen as objects of calculation. It was shown that the ground state of the hole-doped cluster CuO 6 changes from 1Alg to 3Blg when the internuclear distance copper-apical oxygen changes at a nearly superconducting concentration of holes. The ground state of the electron-doped cluster CuO 4 in these calculations is of 3Big symmetry, and the doping electron has moved to the Cu4s orbital. The authors managed to correctly describe the antiferromagnetie ordering in Cu2Oll and Cu207; it was shown that doping destroys antiferromagnetism in both p- (Cu2On) and n(Cu207) type systems, although the mechanisms of these processes differ. 1.9. General Concept of the Electronic Structure of HTSC Oxides

Thus we currently have the following concept of the electronic structure of HTSC ensuing from the above data obtained by various experimental and theoretical methods [130]. Hubbard repulsion rifts the degeneracy from the upper half-occupied one-electron band or* (Figs. 15, 16a), splitting it into the lower and upper Hubbard bands depending on the ratio between the parameters tpp, tpcl, Ua, and A (Fig. 16b, c). Thus, according to Zaanen-Sawatzky-Allen's classification [131], there are three variants of electronic structure (Fig. 16): a) d type metal for Ua = 0. This case was discussed in the section devoted to one-electron calculations; b) Mott-Hubbard insulator for tpp, tpd < Ua << A; c) charge transfer insulator for tpp, tpd < A < Ud. The experimental resonance photoelectron spectroscopy data, allowing determination of the local O2p and Cu3d states, indicate that the electronic structure of HTSC corresponds to a charge transfer insulator. The measurements

a

NB I

b LHB UHB

C

LHB

E:p

i

Ed

E

UHB

Fig. 16. Zaanen-Sawatzky-Allen's classification scheme [1311 (a-d) of the one-particle spectra of transition metal compounds: a - metal, b Mott-Hubbard insulator, c - charge transfer insulator (CTI). Diagram d shows CTI with Zhang-Rice singlet-triplet spfitting. Full regions occupied states, (N)[AIB - (non)[anti]bonding states, L(U)HB lower(upper)Hubbard bands, ZRS - Zhang-Rice singlet, T - triplet, E c r - renormalized gap with a charge transfer, E - energy. 128

[132-134] showed that the Ud parameter of the Hubbard three-band model for La2_xSrxCuO4, YBa2Cu307_ a, and Nd2_xCexCuO4 is much larger than A. The experimental data were compared with the results of cluster calculations, and Ud was estimated at 7.3-10.5 eV. Hubbard repulsion acts analogously on the one-electron x-subsystem (containing four electrons per unit cell), splitting it into the triplet state containing three electrons and the singlet state with one electron. According to [125], the latter state is the highest-energy occupied state for nondoped HTSC and is an analog of UHB in terms of the Hubbard model, whereas the triplet is at a lower energy level and corresponds to LHB. Figure 16d depicts the electronic structure of a nondoped center calculated with the Hubbard model, which was treated above. The ZRS peak corresponds to the Zhang-Rice singlet, which, according to [125], is the ground state of a two-hole configuration. 2. SCHEME FOR CALCULATING THE SPECTRA OF HTSC 2.1. Many-Electron Model of the CuO2 Plane

In the many-electron approach, the Hamiltonian of the many-band p - d model describing the valent state of copper and oxygen may be recorded in the hole representation [135-137]:

H = Hd + np + npp + Hpd,

H d = E Hd(r), r

Ha(r) = ~ [(~,U-/*)d a+o d r a o +

(1/2)Ua

o lira - o ] + E (Vd rlrl + drw' di2a' ~ n~2' - :a drl~ lira + dr2~ 00'

lip = E Hp(i), i 0

0"

O'

+

+

Hp(i) = E [(ePa -- tz)PiaaPiaa + + (1~2)Up nia n~aa] + E (Vp nil hi2 - Jp PiloPila' Pi2a' Pi2a), aa

r

Hpd = E Hpd (i, r), i, r

+ H C + Uaa nr2 o liia o' _ jaa d~2a + drAa'Piao' Hpd(i'r) = ~a E (TaaPiaodrAo + + Piao), ~0'

M , ,=E E

(i,j) aflo

(ta#Piaopi#a + HC). +

(1)

Here epa and ed2 are the one-particle energies of the p and d hole orbitals a and A, respectively; Up, Ud are Hubbard correlations; Vp, Vd are the matrix elements of Coulomb repulsion on the same and different orbitals of oxygen and copper; Jp, Jd are the Hund exchange integrals on the oxygen and copper atoms; Taa and tAa are the matrix elements of the p - d and p - p transitions between the nearest neighbors; VAa and J ~ are the matrix elements of the Coulomb and exchange interactions between the copper-oxygen neighbors; /z is the chemical potential estimated by a self-consistent procedure and lying inside the dielectric gap for a nondoped system. The quality of the results obtained with this model will evidently depend on the basis used; therefore, it is necessary to take into consideration at least the dx2_yZ and dz2 orbitals of copper and the Px and py orbitals of all oxygen atoms. The energy of the dx2_y2 orbital was chosen equal to ca, whereas the energy of the dz2 orbital was ed + Ad. The energy of the Pry orbitals was chosen equal to el,. The first two terms in (1) describe the intraatomic interactions including the Hubbard correlations Up and Ud, interorbital Coulomb interactions, and Hund exchange. The last two terms in Eq. (1) correspond to the interatomic p - p and p - d transitions and Coulomb interaction. The parameters of Hamiltonian (1) are considered as empirical;

129

they were determined by comparing the electronic structure of the ground state of La2CuO 4 with optical and magnetic data [138]: Vp = 3 eV and V d = 4.5 eV, Jp = Jd = 0.5 eV, Taa = 1.5 eV and taa = 0.2 eV, V2a = 0.6 eV and Jaa = 0.2 eV, ed = 0, Ad = 1.5 eV, and ep = 2 eV. The dependence of the results on the choice of Up and Ud will be discussed below; Up and U d are considered infinitely large, unless stated otherwise. 2.2. Ground State of the CuO4 Cluster

The top of the valence band in hole representation expressed in terms of the many-electron method consists of quasiparticles [139] with an energy f/s -- E0(2, S) - E0(1), where E0(1 ) and E0(2, S) arc the energies of the ground states of the cluster including one- and two-particle sectors of Hilbcrt space. The state S = 1/2 of the energy E0(1 ) is spin degenerate, whereas Eo(2, S) may be a singlct (S -- 0) or triplet (S -- 1). Dispersion in the system fls~f~s(k ) results from intercluster transitions, and the dispersion laws differ between S. For this reason, the X-ray spectra must differ between systems with different spins. Using the t h r e e - b a n d p - d model in the case of T ~ << A, Up, Ua ( T ~ is the parameter of t h e p - d transition; A = ea -- ed, the energy of charge transfer), for the effective exchange integral/cu-o [130], JCu-O -- 87"2" (1/(A + Up) -I- 1/(Ud - A)). Clearly, for infinite Up, Ud, the exchange integral/Cu-O is 0, and the singlet and triplet are degenerate. The finite values of Up and Ud lead to large Jcu-o. For the typical parameters Ud = 10 eV, Up = 6 eV, T2a = 1-1.5 eV, and A = 2-3 eV, JCu--O = 2 eV. Thus a correct estimation of the effective exchange integral demands that finite parameters of Coulomb repulsion Up and Ud be taken for calculation. For electronic structure calculations of strongly correlated copper oxide systems, the many-band p - d model is more realistic than the three-band one. Evidently, the basis functions of the triplet are not changed by the inclusion 9 . + + . of the finite values of+ U, and Ud. For the smglet level, three +new+ states are added: dxz_vz ~dxe_vz ~ [0) with the energy + r . . ,,,. .,,, . 2edx2 ,2 + Ua ; dz 2, t dz2, ~ 10} with the energy 2Ca,2 + lid ; Pi, tPi,, [0) with the energy 2ep + Up (z = 1, 2, 3, 4). The many-'band p - d model is transformed into the three-band model for Aa = edzz -- edx2_y2 "-, oo, where Ad is the splitting of the dz2 and dx2_yZ levels in the crystal field. For the CuO4 cluster with two vacancies, the complete two-body basis with finite U a and Up contains 72 states. Since the spin is a quantum number, the Hamiltonian matrix in this basis is partitioned into four units: one unit corresponding to the singlet state (21x21) and three triplet-state units (6x6, 8x 8, and l x 1). Exact diagonalization of these matrices affords a set of two-body molecular states and their energies explicitly including strong electron correlations. Note that the proper states in a two-hole cluster are always a mixture of d 8, d 9L_.,and dl~ configurations. The two-hole state in the CuO 4 cell may be singlet (Zhang-Rice singlet [1251) or triplet [1401. The singlet (eS) and triplet (eT) energies are calculated by exact diagonalization of the CuO 4 duster. For minor variations of parameters, a crossover may appear between the singlet and triplet, and the splitting parameter Ae = e T - . e S changes sign. In our case, for A =ep - ed = 2 eV, the two-hole ground state is a triplet; for A = 1.5 eV, this is a singlet. Figure 17 [141] presents the results of the calculation using the m a n y - b a n d p - d model for U d = 12 eV, Up = 8 eV, Txa = 1.5 eV, and A = 3 eV (curve 1). This curve corresponds to the minimal set of parameters conventionally termed the "three-band model + dz2 orbital," since all parameters not belonging to the three-band model were set to be 0. This set unambiguously indicates the influence of the dz2 orbital when its energy decreases to realistic values. When Ad decreases, the influence of the interorbital Coulomb interaction increases, as calculated for curve 2, for which we set that I,"d = 4.5 eV. The set of parameters for curve 3 contains all the parameters used for a complete calculation of the CuO 2 layer.

130

As eV _

9

xl

1

I

I

I

I

I

l

I

I

I

I

0 2 4 6 8 d, eV Fig. 17. Plot of the energy Ae = e T - e S of splitting between the two-hole triplet and singlet states in the CuO4 cluster vs the crystal field parameter Ad = ed~2 -- ed~2_y2. The parameters of the model (in eV) are: 1 - Ud = 12, Up = 8, A 3, Taa = 1.5, all others are 0; 2 - analogously to (1), except that Vd = 4.5; 3 -- Ud = 12, Up = 8, A = 2, T2a = 1.5, taft = 0.2, It'd = 4.5, Vp = 3, Vpd = 0.6, Jp = Jd = 0.5, Jpd = 0.2.

As can be seen, when the energies of the dxz_y2 and dz2 orbitals approach each other, the exchange splitting Ae decreases. The virtual transitions to the nondegenerate orbital states lead to antiferromagnetic exchange and stabilize the singlet, whereas the virtual transitions to the degenerate states lead to ferromagnetic exchange and stabilize the triplet state; this takes place when the density of states increases proportionally to the degeneracy on the Fermi surface (Stoner's criterion). Thus a transition from three-band to many-band p - d model leads to a decrease in Ae from 2-4 eV in the former case to 0.1-0.5 eV in the latter. Introduction of other small parameters to the many-band model can lead to inversion of the singlet and triplet states. For example, these may be parameters of oxygen-oxygen transitions t2a [140], interatomic Coulomb and exchange p - d integrals Vpd and Jpd, or inclusion of apical oxygens.

2.3. Sudden Perturbation Model for X-Ray Absorption Spectra X-Ray absorption regarded in this section is a one-electron and one-photon process. An electron system absorbs an X-ray quantum, and an electron from any core orbital is transferred to the highly excited state. Other processes, which occur simultaneously or after this (for example, X-ray emission or Auger processes) differ heavily in the energies of transitions, which helps to easily distinguish them, and in the nature of emitted or absorbed particles. In X-ray spectroscopy, it is traditionally accepted that, when an X-ray quantum induces a transition of an electron system from the ground state (with a wave function tlJ0 and energy Eo) to the final quasistationary highly excited state with an X-ray hole on the core orbital [wave function qJ*, energy E*, and lifetime estimated from the experimental widths of X-ray lines (AE = F ~ 1 eV) and the ratio of Heisenberg uncertainty r* = At > h / A E -~ h -- 10 -15 s)], the system goes through a certain "nonrelaxed" highly excited transition state (W~ and E~); in this state, a hole has already formed on the core orbital but the remainder of the system has not yet adapted itself to the hole (is yet nonrelaxed in terms of X-ray spectroscopy) (Fig. 18). This assumption is generally called a frozen orbital approximation (FOA). This scheme suggests that the time of valence shell rearrangement r r is greater than the time of the action of the perturbation or, which is the same in the given process, the time of electron escape from the core shell r e = a/v = a / ~ , where a is the effective path of the electron before it quits the system, which is approximately 2 au, v is the escape velocity, and ho~ is the energy of the absorbed quantum. Let us verify this. At any instant, the nuclear electronic system must satisfy the Schr6dinger equation, namely, the stationary equation for the ground (HoW0 =E0qJ0) and final (H*qj* =E'W*) states and the nonstationary equation [ih (Og2 900 = Htg2 t] for the nonstationary (with respect to time) electron escape from the system, where qJ t and n t define the wave function and the Hamiltonian of the transient system, respectively, and t is time. 131

a

Ground state valence band

Nonrelaxed =

valence band

Relaxed valence band f

t'- h(0

FOA /

/

Core orbital Ground state

Vo, Eo

Relaxed highly exited X-ray state q', E*

Nonrelaxed

highly exited X-ray state q,'~', E~'

Schematic X-ray quantum excitation of an electron system: a - ground state of the system before absorption of an X-ray quantum hco; b - final state in the frozen orbital approximation (FOA) def'med by the wave function W~ and energy E~ with a hole on the core level, a photoelectron in continuum, and a nonrelaxed valence band; c - final quasistationary state with electron system adjusted to the X-ray hole, det'med by the wave function W* and energy E*. Fig. 18.

As mentioned above, the physics of the X-ray experiment is such that the system qJ~ must go over to ~* in a radiationless way, i.e., without emitting or absorbing any particles (photons or electrons). Evidently, E 0 ;~ E~ ;~ E*, and qJ~ is not a solution of the Hamiltonian H0,/-F, o r H t. Hence it appears that there is no state defined by qJ~ or E~. Under the action of an X-ray quantum, the ground stationary state (~0 and E0) is transferred to the quasistationary state, which is the final state for the process under consideration (lifetime C), via the transition state qJ t(t)E*(t) (lifetime equals the time r e of core electron escape from the system); hence, the time of rearrangement of the electronic system r r = r e. The change in the electronic shell energies induced by the relaxation to the X-ray hole is of an order of 1-10 eV, whereas the total excitation energy of the system h co - - 1000-5000 eV, i.e., the valence shell relaxation energy in X-ray processes is a small perturbation of the system. Indeed, suppose the electronic system possesses certain inertia and fails to rearrange before electron escape from the system, i.e., by the moment when the action of the perturbation has ceased. Then any changes in the electronic shells must be accompanied by energy absorption or emission, but these are absent in the phenomena under analysis by definition. Nevertheless, it should be borne in mind that the interaction between the electronic system and the X-ray quantum is an extremely swift process. For such processes [142, 143],

a / v << hey, where a is the effective path of an electron before it quits the system, v is its velocity, and ev is the valence shell energy. According to [142, 143], this expression is transformed into 2V~v/h co << 1. The latter completely satisfies the applicability conditions for the sudden perturbation model [144] and the Born approximation [145] (first approximation of perturbation theory) in scattering theory. The essence of the sudden perturbation model is an instant change of the Hamiltonian of the system under the action of perturbation, with the probability of transition for large perturbation defined as I = [(WoIV*)I 2 132

Thus we see that the final state in the process of X-ray absorption is a completely relaxed quasistationary state with an X-ray hole on the core level and that the rearrangement time of the electronic system exactly equals the time of escape of the excited core electron from the system. The sudden perturbation model is applicable on condition that the electron escape time r e is much smaller than the lifetime 3" of the highly excited X-ray state (r e << 3*). This is confirmed by the comparison between the scales of the total excitation energy and the energy of valence shell rearrangement (%/ha) = 10-3-10-4). For copper oxide HTSC, the necessity of including the many-electron states in X-ray and photoelectron investigations arises due to the strong Coulomb interaction between the core hole and the Cu3d electrons. Let us consider the form of the absorption spectrum in the sudden perturbation approximation caused by the formation of the ls or 2p core hole, which is better known as Larson's model in one-electron calculations [146-148]. The Coulomb interaction of holes on the Cu3d and core orbitals in the final X-ray state is defined by adding to (1) the term

I4~, d

=

V~, d ~,

a ~+o d , ~ n c ,

(2)

r~ ~ o t

where nc = ~ , nco is the operator of the number of vacancies on the core orbital. The interaction of the vacant electronic cy

states with the 2p and ls X-ray core vacancies is described by the Coulomb matrix elements VcP,d = 7.5 eV and V,c,s d = 7 eV, respectively. It is noteworthy that final states without holes (2p-,d 1~ dl~ dlOLs(e), dl~ or ls--,dl~ dlOLL transitions), with one hole (2p-,d9s(e), d 9Ls(e), d 9 or ls--,d 9, d gL transitions), or with two holes (2p--,dSs(e) and Is-*d 8) on copper can appear in the course of formation of X-ray absorption spectra. The many-electron wave function of the system before hole formation on the core orbital of copper may be recorded as .n , ~d) g'in = vc win, O,

(3)

where ~on is the wave function of the core electron; n is the occupation number of the core orbital; ~Pm, (pa)0 is the wave function of the ground state of the system of copper and oxygen valence electrons with an energy E~,~ defined by Hamiltonian (1) provided that n d + n p = n h = const, where nd and np are hole concentrations in the copper d states and oxygen p states; n h is the number of holes in the cell, which is 1 or 2 depending on the extent of doping. The wave function of the system in the final state with a constant number of vacancies in the valence shell (except whiteline formation in the CuL 3 spectrum in which the number of Cu3d vacancies decreases by 1) and one photoelectron in the continuum of s, p, or d nature well after the ionization threshold may be recorded as w~m) = ,pn-1 ~l g,~,rod),

(4a)

where ~ot is the wave function of the photoelectron in the l state to which the photoelectron has passed after excitation, with the energy el ; ~0~,m d) is the function of the ruth state of a system o f p and d electrons in the final state with the energy E,~m. Here the index m runs through all possible states of Hamiltonian H + Hc, d calculated for the state with a vacancy on the core orbital. The energies of the initial and final states are, respectively, Ein=nec

+ ~.(pa) ~'in, O, E.f,m = (n - 1)e c + e l + E~,~m,

(5a)

where E~,m d) is the energy of the ruth state of the highly excited final state. With strong electron correlation and whiteline formation neglected, the energy of the absorbed X-ray quantum is h o~ = el - ec ;

with strong electron correlation included, the energy of the same quantum is 133

htO=el-e

c -I- A E m ,

(6a)

AEm=E~,md)-Ei(t~n,~O0 .

Thus the energy of the absorbed quantum may be represented in the one-electron approximation as hco0 = hoJ - AEm. X-ray absorption is described by the Hamiltonian k

:'%+tk,

where/(0,k) = [ @ c [ e r [7'k)[ 2 is the one-electron dipole matrix element, c + is the hole creation operator on the core orbital, and lk is the hole annihilation operator in the valence shell or continuum. Thus in the case of the constant number of d holes (transitions to the s or p orbitals), the intensity of the X-ray transition is Im(hW ) = [(g,in[erlg'(m))[ 2 = [(~Oc(ec)[er[~ol(el))l 2 I ( g , ~ ( E ~ )

I~md)(E~md)))[2.

(7a)

In the absence of a strong Coulomb interaction between the core hole and the valent vacancies (Vc, d = 0), the states ~ ( E ~ ) a n d

g,~,md)(E~,0 d)) are orthogonal; the last term l ( c ' m ) ( A E m ) = [ ( ~ ( E ~ )

(7a)

this

equals

Jm, 0" In

case,

the

probability of transition

is

determined by the

Ig,f,fP2(E~/',md)))[2in matrix

element

/(0,/)(hw0 ) =

@c(ec) lerl~ot(et)) alone, which is calculated by the one-electron method. Because of the Coulomb interaction (2), the states of the valence p and d electrons (holes) before and after photoionization are nonorthogonal. Therefore, the absorption spectrum has contributions not only from the ground state but also from various excited levels of the final state. The mechanism of whiteline formation in the CuL2,3 spectrum is more complex because the number of holes in the d shell decreases. The wave function of this process may be represented as u~(m) = ~on-l'~a~, ~m(n h - 1),

(4b)

where g'~,~m(nh -- 1) denotes the many-electron wave function from the (nh -- 1)st sector of Hilbert space. In this case, the energy of the final state is Ef, m = (n - 1)ec + E~,~m(nh -- 1),

(5b)

and the energy of the absorbed X-ray quantum is (6b)

hw = -ec + AEm.

The one-electron energies e d of the d orbitals of the initial and final states are then included in the many-electron energies E~n,"d and E~,m ). For processes of this kind, they were taken from one-electron X a calculations and are approximately - 2 to - 3 eV. If there is no strong Coulomb interaction (the whole electronic system does not relax to the creation of an X-ray hole), the energies of the initial and final d shells may be represented as Ei ~ = ne c + ( 1 0 - nh)e d and E ? = (n - 1)e c + (10 - n h + 1)ed, respectively. The energy of the one-electron transition in this case is e?

-

e ~ = h o , o ---

-

.

Thus for the given case, we can write the following expression: hto 0 = h t o - A E m. The intensity of the whiteline (when the number of d holes decreases by 1) may be represented as lm(ht~ = I(g'inlerl@m))12 = I(~~176

(pd) (pa') E '~,d,nh d) 2 [(~Pin, 0(E(pd) in, o) ldr2a[g'~m( -- 1)) [2.

As in (7a), we denote the one-electron part by ](c'm)(AEm) and the many-electron part by/(~ 134

(7b)

Snmming over all possible many-electron transitions, we transform Eqs. (7a) and (7b) into the equation defining the whole spectrum,

I(ha)) = X Ira(ha)) = m

Thus the total absorption spectrum

X ](0,1)(ha ) _

AEm)[(c,m)(AEm)"

(8)

ra

I(ha))

consists of a set of one-electron spectra in which the intensity of

the fundamental line is proportional to the many-electron factor/(c, m)(~Em) from (7a), (To) with m = 0. It also consists of satellites, separated from the fundamental line by the energy &Era = E~,m d) - E ~

and having an intensity determined

by the many-electron factors (7a), (To) with m # 0. Thus the X-ray absorption spectrum of a strongly correlated electronic system is a discrete convolution of two spectra. One is a discrete spectrum/(c, m)(l~.m) of transitions inside the system o f p and d electrons (holes). The other is the one-electron spectrum/(0, 0( h co _ &E(mh)) of transitions from the Is and 2p core orbitals to the vacant electronic states before and after the ionization threshold according to dipole selection rules and shifted on the energy scale due to the Coulomb rearrangement of the valence electron system.

2.4. One-Electron Model of X-Ray Absorption Spectrum Currently, the self-consistent fieldXa scattered wave (SCF-Xa--SW) method [150] is the most consistent method of calculating the X-ray absorption spectra of nd metals near the ionization threshold using the one-electron approach. Here the electronic structure of CuO610-, CuO 9 - (La2_xSrxCuO4), CuO58-, and CuO 7- (YBa~Cu307_~) clusters corresponding to the formal states of copper 2 + and 3 + is calculated using the X a - O M E G A program complex [151]; the electronic wave functions and the intensities of one-electron X-ray transitions in a dipole approximation over the whole energy range was calculated using the Xa-CONTINUOUS program [152]. The parameters of the clusters were chosen according to internuclear spacings [153, 154] and are given in Table 2. In the SCF-Xa-SW method, the so-called MT approximation for the form of the potential constructed from the nuclear and electronic charges is used along with the traditional Slater approximation for the exchange correlation term Vex = -6a((3/8Jr)p(r)) 1/3 (where a is the Slater constant of exchange interaction chosen for each type of atom according to [155], and p(r) is the total electron density at the point r). The whole space of the molecule or cluster is

TABLE 2. Parameters of Clusters in One-Electron Calculations Selected from th~ Structural Data of [153, 154]

Oxide YBaCuO

YBaCuO

LaCuO

LaSrCuO

Ref.

[1541

[1541

[1531

[1531

Cluster

Point group

CuO 8 -

CuO~-

CuO~ ~

CuO~-

D4v

D4v

Cu-O distance, au

a parameter [1551

Radius of spheres, au

3.674

Cu = 0.707

Cu = 2.59

4.344

O = 0.744 III = 0.744d II = 0.735

0 1 = 2.48 0 2 = 3.40 III = 6.58

3.674

Cu = 0.707

Cu = 2.59

4.344

O = 0.744 III = 0.7a.a.A. II = 0.735

0 1 = 2.48 0 2 = 3.40 III = 6.58

3.579

Cu = 0.707

Cu = 2.14

4.588

O = 0.744 III = 0.7A.A.A. II = 0.735

01 = 2.05 0 2 = 2.45 I I I = 7.04

3.579

Cu = 0.707

Cu = 2.14

4.588

O = 0.744 III = 0.74a.a. II = 0.735

01 = 2.05 0 2 = 2.45 III = 7.04 135

divided into regions of three types: atomic spheres (lst type) centered by the nuclei of atoms with the radii chosen by Norman's procedure [156], interspheric region (2nd type), and outer sphere (3rd type) whose center and radius are chosen such that the interspheric region be minimal and touch the atomic spheres. Inside the 1st and 3rd types, the potential is spherically averaged and chosen constant in the 2nd sphere. We think this is a rather coarse approximation but it permits one to use a very wide basis (up to l = 12) in the framework of the scattered wave technique and to calculate electronic wave functions and hence the intensities of X-ray transitions both before and after the ionization threshold.

2.5. Polarization Dependence of X-Ray Absorption Spectra In polarized X-ray absorption spectra, the dipole selection rules are defined by the integral over three angular harmonics:

[(Ylcmc(r) [Ylrmr(r) [Yllm/)(r) ]2,

(9)

where Ylcmc(r) describes the initial core 2p state; ~rmr(r ) defines the polarized photon field; Yl/ml(r) is the angular harmonic of the final vacant state of s or d type. These integrals are nonzero only in the case of me + m r + mf = 0 [144]. Then, according to [2], the z-polarized CuL 3 spectrum will be expressed as

Iz = 2/312,o + 12,1,

(lOa)

Ixy = I2, 2 + 1/612,0+ 1/212,1 .

(10b)

and the xy-polarized spectrum will be

The form of the final spectra allowing for the density of vacant one-electron states and the extent of doping in La2_xSrxCuO 4 as well as strong correlation effects in doped and nondoped cells was synthesized by adding the contours of the one-electron spectra according to formulas (8), (10a), and (10b), taking into account relation (9). The xy-polarized CuL 3 spectrum of nondoped La2CuO 4 was formed by the one-electron transition from the Cud 9 configuration to the only possible configuration Cud 10 with an intensity 0.3428 and energy 2.03 eV using the one-electron contours of the spectrum components calculated for the CuO 1~ cluster by the SCF-Xa-SW method. The polarized spectra of the cells with two vacancies in the triplet ground state are more complex; thus thexy component is formed by transitions from the two-hole ground state to the four configurations of the final state with a vacancy on the Cu2p core orbital from the one-hole sector with weights 0.0560, 0.2241, 0.0037, and 0.0285 and the z component with weights 0.2238, 0.0000, 0.0148, and 0.0000. The energies of these four configurations are 1.9405, 2.1424, 9.8151, and 10.3131 eV, respectively. The form of the spectrum components was calculated by the SCF-Xa-SW method for the CuO69- cluster modeling the state of copper 3 + in the one-electron approximation. The integrated intensity of the polarized pre-threshold lines depends only on the occupation of the corresponding vacant d orbitals (x2 - y2 or z) in the initial state according to selection rules for &m (10a) and (10b), and the number of many-electron transitions depends on the number of configurations in the final state. Our model neglects the orthorhombic distortion of the CuO 2 plane. This leads to the absence of a whiteline in the z component in the spectra of the nondoped compound because of the absence of the dz2 orbital contribution to the initial state, whereas the presence of this line in the spectra of the doped compound is described as the mixing of the d 8(dx2_y2 + dzz) and d 9L(dz2) states with weights (0.38) 2 and "(-0.46) 2, respectively. The CuL 3 spectra of the singlet state were synthesized by the same procedure. The intensities of transitions to the final configurations were 0.222, 0.001, 0.042, and 0.000 in the xy-polarized spectrum and 0.000, 0.005, 0.000, and 0.002 in the z-polarized spectrum, respectively; the energies of configurations were 2.139, 2.280, 10.363, and 10.956 eV, respectively. As can be seen, in the singlet state the density of bound vacant dz2 states is nearly 0. The form of the CuK spectra was synthesized analogously by formulas (Ta) and (8). Thus the spectrum of nondoped La2CuO 4 was formed from the one-electron spectrum of the CUdl0L configuration (weight 0.765, energy 2.7 eV) and the one-electron spectrum of CUd 9 (weight 0.235, energy 10.6 eV). The spectrum of LaSrCuO 4 with a two-hole singlet ground state was also formed from the spectra of two configurations: CudX~ (weight 0.849, energy 2.3 eV) and Cud 9L_ (weight 0.144, energy 12.1 eV). The spectrum of LaSrCuO 4 with a two-hole triplet ground state 136

was formed from the spectra of three configurations: cudl~ (weight 0.630, energy 3.425 eV), Cud 9L_ (x2 - y2; weight 0.151, energy 11.7 eV), and c u d g L (z2) + CUd s (weight 0.219, energy 16.5 eV). We emphasize that the weights and energies of many-electron configurations cited here were calculated using the parameters of Hamiltonian (1) obtained previously [138]; these parameters were not regarded as fitting parameters and were not varied here. 2.6. Small Concentration Approximation (Independent Center Model)

In a doped crystal of La2_xSrxCuO4, there are one- and two-hole cells. The spectra of these partially doped superconductors were constructed in an approximation that the highly correlated two-hole states generated by the doping atoms do not interact with each other because of their small concentration. Thus the weights of the one- and two-hole components in the spectra of La2_xSrxCuO4 (x = 0.2) were taken in accordance with the extent of doping. For example, the spectrum of Lal.sSro.2CuO4 was formed from the spectrum of La2CuO 4 with a weight of 0.8 and the singlet or triplet spectrum of LaSrCuO4 with weights 0.2. The half-widths of the Lorentzian and Gaussian broadening of the CuKa and Cu2p XPS were chosen to be 0.3 eV. For modeling the total Cu2p XPS and CuKa spectra, the spin-orbit splitting of the Cu2p orbital was taken to be 20 eV and the statistical weights of the CU2pl/2 and Cu2P3/2 states were chosen to be 1/3 and 2/3 in accordance with the statistics of the L - S bond. 3. STRONG ELECTRON CORRELATION EFFECTS IN X-RAY AND X-RAY PHOTOELECTRON SPECTRA 3.1. Cu2p XPS Spectra of La2CuO4 Type Compounds As noted in Sect. 1, the multiplet structure of the spectra of copper oxides in the Cud 9 configuration is well defined in terms of Anderson's many-electron theory and was described in detail (for example, in [125]). Yet we would like to recall the typical experimental Cu2p XPS spectra of Cu20, CuO, La2CuO4, La1.85Sr0A5CuO 4 (Fig. 1) [3, 7], and NaCuO 2 (Fig. 2) [25]. Figure 19 presents the Cu2p XPS spectra of the one-hole (a) and two-hole (b) configurations calculated with an e x t e n d e d p - d model. According to the results of calculations, the fundamental maxima in the two spectra correspond to Cu 1+. Thus in the one-hole configuration (Fig. 19), the occupancy of Cudl~ is (0.91) 2, and the occupancy of Cud 9 is 0.422. In the two-hole configuration, the occupancy of Cudl~ is even slightly increased, (0.92) 2, the weight of Cud 8 is negligible, (0.05) 2, and the weights of two Cud9L configurations are (0.37) 2 and (0.11) 2.

sotA

XPS Cu2p La2CuO 4

4o .=_ 9o

0 2

t 6

A

1 10

1 14

l 18

8o XPS Cu2p LaSrCu04

o

Z

40 0

I

I

A A I

6 10 14 Binding energy, eV

I

18

Fig. 19. Theoretical Cu2p XPS of La2CuO 4 (a) and LaSrCuO 4 (b) neglecting the spin-orbit splitting of the Cu2p orbital. 137

t/) t-

40-

.c_ . - -

t~ 0

7

]~ Cu2pl/2

CU2pz/2

200

I

-19

-14

I

-9

I

I

I

I

-4 1 6 11 Binding energy, eV

I

16

Fig. 20. Theoretical Cu2p XPS of Lal.SSr0.2CuO4 constructed including the spin-orbit splitting of the Cu2p orbital (A~ = 20 eV) in the independent center approximation.

Addition of a hole gives rise to one more short-wave satellite maximum at 18 eV, which is associated with the transitions to two c u d g L configurations with weights (0.12) 2 and (0.81) 2 and Cud 8 configuration with weight (0.56) 2. Thus these calculated data qualitatively support a certain growth of the short-wave part of the 3d 9 peak for doped La2CuO4 and the appearance of an additional satellite in the spectrum of NaCuO 2. The greater differences in the position of the peaks on the energy scale for NaCuO z are clearly due to the fact that the Cud 8 configuration is absent in this compound [157] and the fundamental maximum mainly arises from CUd 9L; hence copper is always bivalent in this compound. The overestimated (by about 3 eV) energy splitting in La2CuO 4 seems to be due to the rough determination of the parameters of the p-d model. Consideration of spin-orbit splitting of the 2p core level and of the doping effect on the spectra in the independent center approximation showed (Fig. 20) that in the spectrum of Lal.sSro.2CuO4 the fundamental peak reflecting the occupancy of the CUdl~ configuration must have an indistinct (in proportion to the extent of doping) asymmetric short-wave structure (peak at - 1 7 eV) associated with the energy splitting between the cudl~ and cudl~ configurations of the clusters with the formation oxidation states 2 + and 3 +. A comparison of our results with the experimental spectrum [3] (Figs. 1 and 2) supports this viewpoint. The results mainly coincide with the data of [125] for Cu2p XPS, the only difference being the high-energy satellite separated from the fundamental line by 14 eV and missing in [125]; in our opinion, the reason for this is the fact that we carry out exact diagonalization of the Hamiltonian including all two-particle states, whereas the authors of [125] perform numerical diagonalization in a given basis, which is more limited. 3.2. CuKa Spectra of La2CuO4 Type C o m p o u n d s As mentioned in Sect. 1, the theoretical spectra of one-hole systems were described in detail in [104, 105]. The spectrum has a weak satellite reflecting the density of Cud 9 configurations and lying on the energy scale 0.4 eV

"=•'

80-

"0

N

40-

m

z0

0 -2

I

-1

I

0

I

1

I

2

Binding energy, eV

Fig. 21. Theoretical CuKa spectrum of the two-hole configuration corresponding to LaSrCuO4. 138

"•c

40-

._r "o

m 20-

~ P3/2 Cu

N

2pl/a

LEO o

Z

-2

I

3

I

I

I '~

8 13 18 Binding energy, eV

k

I

23

Fig. 22. Theoretical CuKa spectrum of Lal.8Sr0.2CuO 4 including spin-orbit splitting of the Cu2p orbital (Ae = 20 eV) in the independent center approximation.

higher than the main maximum reflecting the density of the Cud:~ state in the ls and 2p hole configurations. As shown in [105], the shift of the energy level of CuKa may not be measured using Larson's model without separating out the contributions of the satellite structure. Addition of a hole per cluster in our case changes the main maximum (Fig. 21), which now reflects the density of the Cudl~ configurations of the ground, Is intermediate, and 2/7 fmal hole states. The weakly intense satellite lying 0.4 eV higher than the main maximum now reflects the density of the Cud 9L_.configurations, and the appearing short-wave intense satellite with an energy of 1 eV is formed from two Cud 9/. configurations and one Cud 8 configuration with weights (0.56) 2 + (0.57) 2. Inclusion of spin-orbit splitting of the Cu2p core orbital and the superposition of the two-hole and one-hole spectra in the independent center model indicates (Fig. 22) that doping brings about asymmetry and a weakly intense short-wave shoulder on the main peak. 33. Basic Energy Levels of the Initial and Final X-Ray States of the CuO4 Cluster

The weights of the d 9 and d:~ configurations of the nondoped CuO 4 cell in La2CuO 4 are 69 and 31%, respectively. Thus the copper ion in this cell is mainly in the classical oxidation state Cu 2 +. The doping does not alter the picture; the leading configuration of the CuO 4 cell of the completely doped compound LaSrCuO 4 is d 9/_. with a weight of 57% (36% dx2_y2 and 21% dz2); the weight of dl~ is 28%, and that of d 8 is 14%. The creation of a ls or 2p core hole leads to a dramatic rearrangement in the electronic structure of both doped and nondoped cells; the weights of the d 9 and d:~ configurations of the nondoped CuO 4 cell of La2CuO 4 are 18 and 82%, respectively. Thus the oxidation state of copper changes and is 1 + in these cases. The same picture is observed for doped CuO 4 cells, for which the weight of dl~ becomes 85%, whereas the weight of d 9L_ is 15% (14% dx2_y2 and 1% dz2). The weight of d 8 becomes negligible (0.3%). 4. UNDERESTIMATED FORBIDDEN GAP IN THE PHOTOELECTRON SPECTRA OF La2CuO 4 4.1. Representation of Photoelectron and Inverse Photoemission Spectra in the Sudden Perturbation Approximation

In the sudden perturbation approximation (Sect. 2), the wave function of the initial state participating in the formation of the photoelectron spectrum may be written as

0,,0(n ~Oin = ~_k, v Win, Ok k),,

(11)

~pf__~h Wf, , Cod). m ~ n k -+- I' )"

(12)

and the wave function of the final state as

(1,,0O(nk) is the wave function of the ground state of the system of copper and oxygen valence electrons with the Here ~P/n, 139

energy E~,dd, which are defined by Hamiltonian (1) and by the occupation number of the copper d shell in the ground state nh, which is 9 for both the photoelectron and inverse photoemission spectrum; ~ov k is the one-electron wave function after the ionization threshold; k is the occupation number of this state before the electronic system interacts with the excitation radiation, which is 0 for the photoelectron spectrum and 1 for the inverse photoemission spectrum; ~p~,md)(nh--+l)is the wave function of the ruth state of the system ofp and d electrons in the final state with the energy

E~,ma)(nh +_-1), the index rn running through all possible states of Hamiltonian (1) with the number of carriers altered during the experiment; (nh_+l) is the occupation number of the copper d orbital after the external action, which is 8 for the photoelectron spectrum and 10 for the inverse photoemission spectrum; h is the occupation number of the post-threshold state after the external action, which is 1 for the photoelectron spectrum and 0 for the inverse photoemission spectrum. The formation of the photoelectron spectrum is described by the Hamiltonian

HPES X-ray =

(13a)

E rk d+lk, k

where rk = I (9'i t er J~'v) ]2 is the one-electron dipole matrix element of electron transition from the valence one-electron orbital to the region of the continuous spectrum; d + is the hole creation operator in the valence shell; lk is the hole annihilation operator in the continuous region. The formation of the inverse photoemission spectrum is described by the Hamiltonian H X-ray IPES = E

k

rk dt~

(13b)

where rk, d, and 1+ have the same sense as in formula (13a). Then the probability of an electron transition will be

[ProES=

I(~intert~o~m))t2= I(~oilerl~v)l 2 [(~ddo(nh)td+[~p~,md)(nh-

1))[ 2

(14a)

for the photoelectron spectrum and /~ES__ i(~,inlerl~,}m))12= [(~,iierl~,v)12 [(~,~,ado(nh)ldl~,~m(nh + 1))12

(lab)

for the inverse photoemission spectrum. As shown in Sect. 2, the one-electron contour defined by the term I($oiler I~'v) 12 may be calculated by any ab initio quantum chemical method. The energies of the initial and final states are

Ein = kev + e~,~O0, El, m = kev 4- E~V,md)(nh_.+l),

(15)

and the energies of transitions are

110) = (h -- k)e v 4- A E m ,

AErn = E~,md)(r/h+l) -- E/~n,d~0.

(16)

In the case of missing correlations, the one-electron term of the energy for PES and IPES differs only in sign and corresponds to either absorption (PES) or emission (IPES) of an electromagnetic quantum. The relative shift of the one-electron contours of these spectra on the single energy scale will only depend on the values of AE PEs (- sign) and AEIm PES ( + sign) determined by formula (16). Recall that the initial state of La2CuO 4 is the same for both spectra - Cud 9 but the final states differ: Cud 8 for PES and Cud 1~ for IPES. 4.2. Shift of Photoelectron Spectrum Curves of HTSC Recall (Sect. 1) that PES and IPES are most often used for analyzing the density of states near the Fermi level. Figure 3 shows the typical photoelectron spectrum and inverse photoemission spectrum of Lal.sSr0.2CuO 4 [26]. 140

The nondoped oxide LR2CuO4 has analogous spectra [27]. As can be seen, the intensity of both spectra at the Fermi level is small (Lal.sSro.2CuO4 [26]) or nearly zero (La2CuO4 [271). Based on these facts, one can make inferences about the density of states at the Fermi level. However, the splitting between the experimental photoelectron spectra of the La2CuO 4 surface is so weakly pronounced that it can in principle be interpreted as the absence of a dielectric gap. In our many-electron calculations, the energies are -2.03 eV for the Cud 9 one-hole ground state, -0.93 eV for the Cud s state, and zero for the Cud 1~ state (by definition). Thus, according to (16), the shake-down process shifts the PES contour by about 1 eV down the energy scale, in good agreement with the results of a series of known papers (see, for example, [114, 108]); the IPES contour is shifted by 2 eV. For CuO, which is related to La2CuO4, the gap is 1.4 eV [158]. The IPES of the La2CuO 4 surface is shifted by 1 eV down relative to PES, due to which the gap between them is narrowed. This narrowing is also determined by strong electron correlation effects. 5. STRONG ELECTRON CORRELATION EFFECTS ON THE FORM OF CuK X-RAY ABSORPTION SPECTRA OF Laz_xSrxCuO 4 5.1. Background

As mentioned in Sect. 1, the role of strong electron correlations in the mechanisms of XAS formation has not been sufficiently studied. Meanwhile, there is direct experimental evidence that strong electron correlation effects participate in the mechanism of formation of the CuK X-ray absorption spectrum of La2CuO 4 [159]. This mechanism was studied in detail with various versions of the one-electron ab initio multiscattering method [93-96]. These investigations adequately described all features except peak C lying 7 eV higher than the main peak. When the cluster size was increased to 50-60 atoms [94-96], this peak was obtained only with xy-polarization, whereas the experiment also shows this structure in the z-polarized spectrum [159]. Addition of a vacancy per structural unit during doping considerably complicates the picture due to the appearance of contributions from the Cud 8, Cud 9L, and Cudl~ configurations. This leads to drastically complicated CuK absorption spectra of HTSC of the inclusion states [97]. To study strong correlation effects on the form of CuK X-ray absorption spectra, we use exact diagonalization of the Hamiltonian of the many-band p - d model for the CuO 4 cluster in the framework of the sudden perturbation model described in Sect. 2. The matrix elements of the ls~p(e) X-ray transitions were calculated for the CuO 1~ and CuO69- clusters by the SCF-Xa-SW ab initio method. The resulting spectra were synthesized by using the lineshape obtained by the one-electron method, and the statistical weights and configuration energies were calculated in the many-band p - d model. The spectrum of completely doped LaSrCuO 4 was calculated for both the singlet and triplet two-hole states. 5.2. Discussion of Results

Figure 23 shows the experimental [97] and theoretical one-electron CuK spectra of La2CuO 4. The synthesized spectrum including many-electron effects is in fair agreement with the experimental spectrum in the energy levels and relative intensities of peaks. Probably, the only exception is the long-wave part of the spectrum in the region of peak A. Previously, it was noted that the specific relative intensity and energy level of peak A in the CuO~~ cluster are only due to the small size of the cluster [94-96]. As shown by the calculation, the fundamental line in the CuK spectrum corresponds to the dl~ configuration (weight 0.882, peaks O, A, B, D, E); the only intense short-wave shake-up satellite [peak C in the experimental and B' in the theoretical spectrum (Fig. 23)] at 7.8 eV relative to the main line reflects the d 9 configuration (peaks O', A', B', D', E'). Thus the experimental peak C must be compared with the theoretical peak B'; this peak reflects not only photoelectron scattering on the neighboring atoms, as follows from the literature [94-96], but also the contribution of the Cud 9 configuration in La2CuO 4. The form of the experimental CuK spectrum of the inclusion states of LaSrCuO 4 ("trivalent copper") is much more complex [97] (Fig. 24). The experimental [97] (curve 1) and theoretical CuK spectra of LaSrCuO 4 are compared

141

q

=.

o.11i .

.

.

.

,

/

Jl l +0"

I

I

l

0

I

I

I

20

I

I

E, eV

40

Fig. 23. Experimental [97] (1) and theoretical (2) CuK spectra including many-electron effects and theoretical one-electron absorption spectrum (3) of La2CuO 4. Peaks O, A, B, D, and E refer to the fundamental line (dl~ configuration); O', A', B', D', and E', to the shake-up satellite with an energy of 7.8 eV relative to the fundamental line (d 9 configuration).

BA'

,

O' B; A' .

-~C

I

B" D D"

I

I

I

I

I

I

I

10 30 50 E, eV Fig. 24. Experimental [97] (1) and theoretical CuK X-ray absorption spectra of two-hole open states with a triplet (2) and singlet (3) ground states. Peaks O, A, B, D, and E refer to the fundamental line (d~~ state); O', A', B', D', and E', to the first satellite (dgL(x2 --"-~_) state); O", A", B", D", and E", to the second sateUite (d 9L(z2) and d s states).

with the singlet (curve 3) and triplet (curve 2) ground states; it follows that the triplet is the two-hole ground state of LaSrCuO 4 in the doped system La2_xSrxCuO 4. Its main line (peaks O, A, B, D, and E) belongs to the dl~ configuration (weight 0.912) with minor admixtures of d 9L (x2 - y2, weight 0.392) and d 9L (z2, weight 0.122). The first satellite (peaks O', A ', B', D', and E') refers to d 9/_. (xz -yZ, weight 0.902) with a slight admixture of dlOLL (weight 0.392). The second satellite (peaks 0", A", B", D", and E " ) reflects the density of the dgL (z2, weight 0.812) and d 8 (weight 0.572) states with a minor admixture of d 9/_. (xz -y2, weight 0.122). The indices under the experimental spectrum denote our peak assignments to the corresponding configurations. The slight difference between peaks A' and B' in relative intensities and position on the energy scale, in our opinion, is primarily due to the overestimated relative intensity of peak A in the one-electron calculation, naturally entailing a certain distortion of the form of the final spectrum. 142

B+A' ~B'(d a) ~B'(d 91

z

I

0

I

I

20

I

I

40

I

I

60 E, eV

Fig. 25. Experimental [97] CuK absorption spectrum of Lal.85Sro.15CuO4 (1) and theoretical CuK spectrum of Lal.sSro.2CuO4 synthesized using the triplet ground state of the two-hole open states (2). Peak B'(d 9) belongs to the d 9 configuration of the initial system; peak B(d 8), to the dl~ configuration of the open states with two electron vacancies per formula unit.

The model spectrum of the doped compound Lal.sSro.2CuO4 (Fig. 25) synthesized using the spectrum of the triplet state does not differ from the spectrum of the parent compound La2CuO 4. An exception is probably peak B(d s) corresponding to the dl~ configuration of the inclusion states of the completely doped compound LaSrCuO4 and separated from the main maximum B(d 9) belonging to dl~ of the starting compound by 2 eV. Summing up, this description of doping effects assumes that the doping changes the concentration of the holes but not the form of the Hamiltonian or the values of its parameters. Strictly saying, this is not so. These effects are mixed with fluctuation type effects of crystal field parameters near the Sr ion. Variation of the charged state of ions affects the position of the latter in the lattice, leading to the observed deformation of octahedra and to the structural transition from ortho to tetragonal phase when the concentration of Sr increases. A detailed comparison with a high-resolution experimental spectrum will probably demand taking these factors into account. Our approach, however, considers only the basic change in the electronic structure due to doping, namely, the appearance of in-gap states due to variation of the occupancies of two-hole levels. These states are well known for all doped dielectric copper oxides, and experimentally their appearance is the major change in the spectra at small concentrations. As x increases, these states form a metallic band, which seems to be responsible for superconductivity. 6. NONDIAGRAM TRANSITIONS IN CuL 3 POLARIZED X-RAY ABSORPTION SPECTRA OF HTSC CERAMICS 6.1. Strong Electron Correlation Effects on the CuL3 X-Ray Absorption Spectra of La2-xSrxCuO 4 The procedure for calculating the CuL 3 spectra is described in detail in Sect. 2. Recall that, as shown in [100], there is only one X-ray transition, 2p6dx~_y2-*2p5d 10, in La2_xSrxCuO 4 where x = 0 (with one electron vacancy per formula unit), although in the initial state there are two d 9 type and two dl~ type configurations due to hybridization of vacant states. When x > 0, the role of many-electron effects increases due to the contributions from Cud 8, Cud 9L, and cudl~ This leads to significant discrepancies between the CuL 3 spectra of the inclusion states of doped compounds and the spectra of nondoped compounds, namely, to shake-up satellites [2, 100, 101] on the whiteline. The experimental xy-polarized CuL 3 spectra of LazCuO 4 and Lal.92Sro.08CuO4 [2] are shown in Fig. 26a; the theoretical xy-polarized spectra of La2CuO 4, LaSrCuO 4, and La1.92Sro.08CuO4 are depicted in Fig. 26b for the case where the triplet is the two-hole ground state; the theoretical xy-polarized spectra of La2CuO4, LaSrCuO4, and La1.92Sro.08CuO4 for the singlet as the two-hole ground state are presented in Fig. 26c. 143

a

e-

Ellab

20-

~§

15-

::

10-

~ .~ .~ l ,.,

_=

5a) 0 o 2.0-

i:5

.

~

.b

.

+ Lal.92Sro.oeCU04

*:l:

~,~t_~ ~

O,

La2CuO 4

~*++ i

.

928

j

-+-

"~ I 936 Energy, e V

I 932

,=.

I 940

b

Ellab 9 La2CuO 4 .

.

. La+.~2Sro.osCu04 9 LaSrCu04

.0,+ 9 *+

9

9

.*+. ". ****§ ; era _=

Triplet

...

4- 9 09

.~.e4.* 9

i

I -1

1

~ 9 ~en~.

I 3

I 5

I 7

I 9

c 9

Ellab

9

. La2CuO 4 9 Lal.92Sro.osCu04 9 LaSrCu04

: i ?***\ ~ ~+9 **+ . 9: . ' ~

I -1

Singlet

**:.~+:1: . 4 . ~

I 1

I 3

....

I 5

9 ::

I 7

I 9

Energy, e V

Fig. 26. Experimental xy-polarized CuL 3 X-ray absorption spectra of La2CuO 4 and Lal.92Sro.08CuO 4 [2] (a), theoretical xy-polarized spectra of La2CuO 4, LaSrCuO 4, and Lat.92Sro.08CuO4 in the case of triplet as the two-hole ground state (b), and theoretical xy-polarized spectra of La2CuO 4, LaSrCuO 4, and Lal.92Sro.08CuO4 in the case of singlet as the two-hole ground state (c).

Figure 27 shows the experimental z-polarized CuL 3 spectra of La2CuO 4 and Lal.92Sr0.osCuO4 [2] (a), the theoretical z-polarized spectra of La2CuO 4, LaSrCuO 4, and La1.92Sr0.08CuO4 for the case of a triplet as the two-hole ground state (b), and the theoretical z-polarized spectra of these three compounds when the two-hole ground state is the singlet (c). The xy-polarized experimental and theoretical spectra of the nondoped compound La2CuO 4 (Fig. 26) have no nondiagram lines (electron transitions not described in the first approximation in crystal field theory) before the 144

_

EIIc * La2Cu04

+4. 44"

9~

2" § §

2-

* La 1.gzsro.oaCu04

e-

, I , @ O '1" r

4.4' @ 0

"'i:

1-

4"@

+~"@

%6

...,%"

0

4,+ @4.@ @ "~4-++ 4. @@4" 04"@+@@@+ @@

".@i~.@....

$L-~ 9 9

I

I

I

2.0~: 1.0/5 0 928

932 936 Energy, eV

b

EIIc 9 La2Cu04 9 Lal.gzSro.osCU04 9 LaSrCuOL

~._ 9

:

" ~, Tdplet

: e~ r

I

940

~

+++ ;

-1

I

I

I

I

3

5

7

9

-=

c

Singlet 6 m6Lt6 4~6e6066~ 9A ~9 Ae6A e~6A aAm ~ . ~

I

-1

I

1

I

3

I

5

I

7

6 ~ . 6 ~6

I

g

Energy, eV 27. Experimental z-polarized CuL 3 X-ray absorption spectra of La2CuO4 and Lal.92Sro.08CuO4 [2] (a), theoretical z-polarized spectra of La2CuO4, LaSrCuO4, and Lal.92Sro.08CuO4 in the case of the triplet as the two-hole ground state (b), and theoretical z-polarized spectra of La2CuO4, LaSrCuO4, and Lal.92Sro.08CuO4 in the case of the singlet as the two-hole ground state (c). Fig.

ionization threshold. On the z-polarized experimental spectrum (Fig. 27), however, one can see a sufficiently intense whiteline, which is absent on the theoretical spectrum. This difference arises from the fact that the orthorhombic distortion of the CuO 2 plane was not taken into account; as shown in [160], this distortion leads to this effect. The lineshape in the post-threshold part of both z- andxy-polarized spectra, as noted previously in the one-electron approach [96], is described adequately. The sharp increase in the intensity of the whiteline in the z-polarized spectrum (Fig. 27) distinguishes the experimental spectra of the doped compound from the spectra of the nondoped one. The absence of a whiteline in the theoretical z-polarized spectrum of the doped compound in the singlet ground state and its presence in the triplet state (Fig. 27) suggests that the triplet is the ground state in cells with two electron vacancies. In this case, the whiteline is formed by a transition from the ground state to the final state with a Cu2p vacancy and considerable occupation of the dz2 states in the orbitals d 8 (dx2y 2+dzZ) and d 9L (dz2) (weights (0.38) 2 and (-0.46) 2, respectively) with transition energy 1.94 eV and intensity 0.2238. The intensity of the transition of the whiteline in the z-polarized spectrum for the second (in energy) configuration is zero. Our model adequately defines the low-intensity long-wave satellite (xy-polarization) formed by the first (in energy) configuration of the final state with a 2p core vacancy (intensity of transition 0.0560) and fixed in the experimental spectrum (Fig. 27) at an energy level 0.4 eV below the whiteline [2]. The whiteline in the xy-polarized spectrum in the 145

triplet state is formed by the second (in energy) configuration with transition energy 2.14 eV and intensity 0.2241. The intensities of transitions to subsequent high-energy configurations with transition energies 9.82 and 10.31 eV in these spectra are nearly zero; therefore, the whitelines assigned to these transitions do not show themselves in the post-threshold regions of these spectra. The post-threshold spectra are mainly formed by superposition of the lines of the first two configurations. The small energy splitting between the lines and the large (versus the whiteline) half-widths of the features lead to an insignificant diffusion of the spectra in the positive region. In contrast to [126], our calculations show a pronounced intensity of s states in the post-threshold region (peak energy about 8 eV), which is primarily attributed to the duster effect. The weak shoulder at about 2 eV in the theoretical spectra for both polarizations is one-electron and will be discussed below. 62. Nondiagram Transitions in the CuL 3 Spectrum of YBazCu307-r As shown in Sect. 1 (Fig. 14), in the CuL 3 spectrum of YBa2Cu307_ ~ of prime interest is the short-wave satellite separated from the whiteline by 2.8 eV at 6 close to 1. In this case (6 = 1), the system has no two-hole highly correlated electronic states. Therefore, the electronic structure of YBa2Cu307_ ~ was modeled only using the one-electron cluster approach SCF-Xa-SW for the CuO 8 - and CuO 7- clusters. The spectrum of the CuO 8 - cluster corresponds to the oxidation state 2 + of the copper ion of the second type in the oxygen-deficient compound YBa2Cu306; the spectrum of the CuO 7- cluster corresponds to the oxidation state 3 + of the copper ion of the second type observed in Y B a 2 C u 3 0 7 . The formal oxidation state of the copper ions of the second type in the latter case is 2.5 +. However, we took into account the well known fact [97] that one doping center in HTSC acts on two copper ions and considered that the CuO 7- duster defines the electronic structure of copper at ~ -~ 0. Figure 28 gives the theoretical CuL 3 spectra of these clusters. The corresponding experimental spectra (Fig. 14) as interpreted in [99] are treated in Sect. 1. As can be seen, the spectrum of each cluster has an intense short-wave shoulder, which is separated from the whiteline by 2.8 eV and is analogous to peak C in the experimental spectra of YBa2Cu3OT_ ~ (Fig. 14) with large 6. As in experiment, a transition from one to two vacancies per formula unit (which is equivalent to a decrease in the duster charge by unity while the neighboring atoms of copper remain the same or to CuO58--*CuO 7-) leads to a sharp decrease in the intensity of this peak. As 6 decreases (the concentration of electron vacancies increases), the intensity of peak C decreases, and the peak is replaced by an intense satellite B, which was attributed [99] to strong correlation

b

a

e--

_=

i I

CuO~~

I

I

-4

I

I

0

-4

I

I

0

I

I

4

I

I

4

CuO~-

I

I

8

I

I

8

I

I

-4

I

I

-4

I

I

0

I

I

0

I

I

4

I

I

4

I

I

8

I

l

I

8

Energy, eV Fig. 28. CuL 3 one-electron X-ray absorption spectra of clusters: CuO 1~ and CuO58- (a), C u O 9 - and C u O 7- (b). 146

f, Ry

f, Ry 7.55-

a

-7.55-

b

-7.559 CuO~ ~

9 CuO~-

9 CuO~"

9 CuO7-

Fig. 29. One-electron potentials of the Cud states of the clusters: CuO61~ and CuO 9 - (a), CuO 8 - and CuO 7- (b). R - distance to the copper nucleus; f - potential.

effects. The emergence of peak C on passing to oxygen-deficient systems, in our opinion, may be explained by analyzing the one-electron Cud potentials. As can be seen (Fig. 29), the potentials of the Cud states of the CuO 1~ and CuO 8- clusters corresponding to the oxidation state of copper 2 + have a hump in the positive region analogously to the f-electron potential in rare earth elements [161]. The presence of this barrier in the potential leads to the emergence of quasistationary electronic states in the region of positive energies, which, in turn, appear in absorption spectra as highly intense giant resonance nondiagram spectral lines in lanthanides [162]. However, as opposed to the f-states, the Cud electron potential is sensitive to the atomic environment and chemical bonding. Thus the results of calculations for the copper 2 + clusters CuO 10- and CuO 8- were compared with those for the copper 3 + clusters CuO69- and CuO 7- (this is equivalent to a transition from one to two vacancies per formula unit); the positive barrier of the potential (Fig. 29) decreases, the decrease being much more dramatic for the CuO 7- cluster, drastically diminishing the intensity of the threshold satellite. The relatively minor change in the barrier height in the CuO69- cluster is thought to be the reason for the fact that, when the extent of doping x in La2_xSrxCuO 4 decreases, the structure analogous to peak C in the yttrium ceramics is not revealed in the post-threshold region, contrary to the case of YBa2Cu307_t~. Thus one cannot rule out that the CuL 3 spectra of HTSC with large concentrations of Cu 2 + may have lines that are due to the positive barrier in the Cud potentials, which generates the high density of the Cud quasistationary states near the ionization threshold. When the oxidation state of copper increases, the barrier decreases and so does the intensity of the peak reflecting the density of the quasistationary states in the region of positive energies. CONCLUSIONS Thus we see that a sufficiently adequate description of the electronic structure and its changes due to doping is missing, although a tremendous success has been attained in the electronic structure studies of nondoped nonsuperconducting oxides by both one- and many-electron theories and by indirectly comparing the results with spectroscopic and magnetic experiments. The theoretical data obtained by different methods often conflict with each other; the authors of most works did not make direct comparisons between the theoretical and available spectroscopic data. This is primarily due to the fact that a realistic theoretical model for describing most experiments is missing. A series of basic spectral characteristics of the compounds were described from a single standpoint using the many-electron theory of X-ray and X-ray photoelectron spectra constructed from the sudden perturbation model. In this theory, the X-ray and electronic spectra may be represented as a product (convolution) of the spectra of one-electron 147

transitions to vacant orbitals before and after the ionization threshold and many-electron transitions inside the system of valence electrons obtained, for example, using the SCF-Xa-SW method and the many-band many-electron p - d model. Thus strong correlation effects on the Cu2p XPS and CuKa X-ray spectra of La2_xSrxCuO 4 type compounds were studied by the exact diagonalization method of the CuO4 cluster in the many-band many-electron p - d model. It was shown that the main peak in the Cu2p XPS of LaSrCuO 4 corresponds to the Cudl~ configuration. Doping La2CuO 4 gives rise to a high-energy shoulder on the main maximum due to the splitting of the Cudl~ and Cudl~ configurations and distorts the form of the high-energy satellite. The main maximum of the CuKa spectrum has contributions from the Cudl~ two-hole configurations in the intermediate and final Is and 2p hole states. Doping also leads to one more intense short-wave satellite in the CuKa spectrum. The photoelectron spectrum studies showed that the forbidden gap obtained by the combined analysis of the experimental photoelectron and inverse photoemission spectra in a single energy scale for the surface of La2CuO 4 is underestimated by 1 eV. The electronic and satellite structure of the spectra of La2CuO 4 was calculated using the three-band p - d model and the sudden perturbation approximation. It was shown that the shake-down processes shift the one-electron contour of the final two-hole configuration of PES by 1 eV down the energy scale. The one-electron contour of the Cud 10 final configuration of IPES is shifted by 2 eV. This leads to an underestimated energy level splitting between the filled and vacant bands. The strong correlation effects on the CuK and CuL2, 3 X-ray absorption spectra of La2_xSrxCuO 4 (x = 0, 0.2, 1) were investigated in the sudden perturbation approximation. In this model, the main peak of the CuK spectrum for x = 0 was assigned to the Cudl~ configuration and the only satellite was attributed to the Cud 9 configuration. A comparison between the theoretical and experimental contours shows that the two-hole ground state in the CuO 4 cell is a triplet; this disproved the traditional viewpoint that only the Zhang-Rice singlet determines the nature of two holes in the unit cell. In the spectra of inclusion two-hole states, one can also observe other high-energy satellites reflecting the density of the Cud 9L_ and Cud 8 configurations. Similar conclusions were drawn in the studies of the CuL2,3 polarized spectra of completely doped cells of LaSrCuO 4 (x = 1). One of the peaks in the CuL2. 3 spectrum lying 2.8 eV above the threshold was assigned to transitions to the Cu3d quasistationary states in the region of positive energies due to the presence of a high barrier in their Hartree-Fock potential; this made the peak sensitive to the atomic environment and the charged state of copper in such compounds. This work was supported by the Scientific Council on HTSC (project 95027 under the State Program "High-Temperature Superconductivity"). REFERENCES .

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