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INTRODUCTION. Auger electron spectroscopy (AES) and inverse. AES (iAES) are of particular importance in developing the electronic theory of ...
Physics of the Solid State, Vol. 46, No. 9, 2004, pp. 1583–1590. Translated from Fizika Tverdogo Tela, Vol. 46, No. 9, 2004, pp. 1537–1543. Original Russian Text Copyright © 2004 by Moliver.

METALS AND SUPERCONDUCTORS

Auger-Spectroscopic Appearance of Electron Correlation at the Fermi Surface of Graphite S. S. Moliver Ul’yanovsk State University, ul. L’va Tolstogo 42, Ul’yanovsk, 432700 Russia e-mail: [email protected] Received October 8, 2003

Abstract—It is shown that, in Auger-electron spectra of three-dimensional semimetal graphite and two-dimensional graphite (a zero band-gap semiconductor), an energy gap should be observed between the thresholds (edges) of the forward and inverse processes (threshold gap). In the one-electron approximation, this gap is zero, since the threshold for the Auger spectrum of the forward process is the minimum hole energy in the valence band, while the threshold for the spectrum of the inverse process is the minimum energy of conduction electrons. Inclusion of the electron correlation at the Fermi surface within the quantum-chemical approximation of a single open electron shell for multiplet structures of the restricted Hartree–Fock method makes it possible to determine the threshold gap as 1.5 eV for a 48-atom cyclic model of three-dimensional graphite and as 2.0 eV for a 24-atom model of two-dimensional graphite. The threshold gap does not contain the Fermi energy, in con1 trast to the Auger spectrum thresholds, where --- (4.0 eV – εF) for the forward Auger spectrum (holes) and 2 1 --- (−1.1 eV + εF) for the inverse spectrum (conduction electrons), the sum of which gives this gap. The results 2 of calculations for the forward Auger spectra of three-dimensional graphite (including the conclusion that electron correlation of holes in the top valence bands is weak in the Auger process) are shown to agree with the experimental data. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION Auger electron spectroscopy (AES) and inverse AES (iAES) are of particular importance in developing the electronic theory of three-dimensional (3D) semimetal graphite, since these methods carry information on the electron correlation. In the one-electron approximation, the Auger electron spectrum describes the process that yields two holes in the valence band below the Fermi level εF. In other words, one valence electron transfers from the level εv 1 to the level εcore of a preliminarily generated deep core hole (the state close to the atomic orbital C1s in the case of graphite) and another valence electron transfers from the level εv 2 to a free state with kinetic energy εkin. In the same approximation, the inverse Auger spectrum describes the process resulting in the formation of two conduction electrons. More specifically, a probe-beam electron with kinetic energy εkin is captured into the vacant conduction level εc1 and another electron transfers from the core level to another conduction level εc2. We note that the abbreviation iAES is not conventional; it was introduced to associate various electron spectroscopy versions with the inverse Auger process, e.g., appearance-potential spectroscopy (APS). In what follows, we are interested only in systems with metal filling and spectrum edges (thresholds) caused by the lowest excitations of holes (AES) and

conduction electrons (iAES) at the Fermi surface. In the one-electron approximation, the spectra are double convolutions of the density of states with a matrix element of the Coulomb interaction of two final particles. The energy dependence of the matrix element can be found within the one-electron approximation, whereas the many-electron approximation radically changes the transition probability and includes correlation corrections of the following types: (i) The interaction of valence electrons with a deep hole. This component has almost no effect on the shape of the spectra and reduces to a constant, which is included in the empirical parameter εcore = 284.35 ± 0.05 eV [1]. This parameter is directly measured as the threshold for internal photoelectron emission in the xray absorption spectrum. (ii) The “direct” correlation associated with the interaction of two final particles, i.e., holes (AES) or conduction electrons (iAES). In the case of an insulator (semiconductor) with delocalized carriers, this correlation is small and is considered using the representation of large-radius excitons. In particular, this correction contains the static permittivity and, in effect, takes into account the “indirect” correlation of all the crystal electrons. The opposite extreme case of strong localization (e.g., narrow 3d bands of transition metals) under conditions of completely filled bands is also convenient for analytical consideration in the Hubbard approximation.

1063-7834/04/4609-1583$26.00 © 2004 MAIK “Nauka/Interperiodica”

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(iii) The indirect correlation associated with all valence electrons. It is impossible to strictly distinguish between this and the previous contribution. However, in the case where we have metal filling of bands and spectrum edges associated with shallow delocalized excitations near the Fermi level, exactly this type of correlation becomes dominant. In this study, we perform numerical simulation of this correlation. The kinetic energy of primary (in forward AES) or secondary electrons (in iAES) is involved in the binding energy, 1 AES ε v = --- ( ε core – ε kin – ε F ), 2 1 iAES ε c = --- ( ε kin – ε core + ε F ), 2

(1)

where all the energy parameters are positive: εF is measured downwards from the zero potential, at which εkin = 0; i.e., the Fermi energy is numerically equal to the work function. The particle energies (in the approximation of independent electrons) are also taken positive: εv and εc are measured downwards and upwards from the Fermi level, respectively. The first Auger spectrum in (1) was recently measured for 3D semimetal graphite under the same conditions as a photoelectron emission spectrum (PES) (a single sample in the same chamber without loss of vacuum and a single detector of electrons) [1]. Although each measured electron spectrum coincided with x-ray [2, 3] and Auger electron [4] spectra previously obtained for graphite, accurate fixing of the contact potential with the detector [1] allowed the authors to conclude that the critical points of the AES and PES spectra coincide (double differentiation was used, since the spectra strongly differed in shape). Since the photoemission spectrum is one-electron and its features coincide with the high-symmetry points of the band structure εv (k) = 2.3, 4.8, 7.8 eV, …, their coincidence with critical points (1) was interpreted as being due to weakness of the direct correlation of final holes from the upper valence bands in the Auger process. However, the shapes of the AES and PES spectrum edges differ significantly over the range from the Fermi level to the first critical point [1]. The difference in the spectrum shapes is evident, and differentiation reveals an additional critical point in the Auger electron spectrum. This point cannot be interpreted as belonging to a band. Referencing the Auger electron spectra to the Fermi level is always characterized by an error of no less than 0.1 eV. Therefore, it is reasonable to say that the electron correlation of the states at the Fermi surface (including the direct correlation of two final particles at this surface) is the main effect that controls the position of the Auger spectrum threshold of semimetal graphite. Such an interpretation is also applicable to the

threshold for the inverse Auger spectrum, for which no experimental data have yet been obtained. We note that, for correlation with a core hole [see item (i)], the edge shape of the AES spectrum of graphite is affected by the Auger process with an initial state consisting of an exciton formed by a core hole and a valence electron. This contribution to the resulting Auger spectrum was studied using convolution of the one-electron density of graphite valence bands with a model delta-shaped density of states of the exciton under the assumption that the exciton binding energy is small and is identical to the Auger spectrum threshold [4]. In fact, there is a satellite of the basic Auger process whose threshold is shifted by the binding energy of the core-hole exciton. The spectrum narrows since the exciton band is substituted for one of the valence densities of states in the convolution integral for the probability of the Auger process. The contribution of the exciton satellite can be experimentally separated, since the lifetime of the free core hole is significantly longer than the lifetime of an excitonic hole. The influence of the electron correlation on the spectrum threshold energies for this satellite is the same as in the case of the basic Auger process. The approach that is developed in this paper is also completely applicable to 2D graphite (zero band-gap semiconductor), i.e., to a system with the Fermi surface degenerated into a point. The thresholds of the Auger spectra calculated in this work and their comparison with experimental data for 2D (plane) and 3D hexagonal graphite show that these parameters can be important experimental characteristics of the electronic properties of cluster carbon materials. 2. AUGER PROCESSES AND QUANTUM-CHEMICAL MODELS Quantum-chemical calculation yields the total energy of a crystal model consisting of a certain number of atomic cores and valence electrons (or bare nuclei and all electrons). Let the crystal model contain N valence electrons. The total energy E(N) of the ground state is at least a self-consistent Hartree–Fock approximation, which is more or less accurate depending on the calculation method (semiempirical or nonempirical, ab initio method for calculating the matrix elements of the electron–electron interaction). The total energy can also account for the electron correlation if the configuration interaction or perturbation theory is applied to the self-consistent many-electron wave function [5]. The final states of the forward and inverse Auger processes correspond to the energies E(N ± 2), which can be calculated by changing the number of model electrons by two. In this case, the problem of a nonzero total charge of the model arises. For example, if the model is a building block of an infinite crystal and the calculation method takes into account only the valence PHYSICS OF THE SOLID STATE

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electrons, then a small compensating charge should be placed on the cores. Let ∆(N ± 1) be the correction to the total model energy associated with the difference between the electron charge of the model containing only valence electrons and the charge of a separated crystal volume involved in the Auger process. This correction is introduced with the following purpose. The correction itself can be determined in quantum-chemical calculations, whereas direct calculations of E(N ± 1) can be meaningless, e.g., because of the odd total spin of the model. In the forward Auger process, the initial energy is E(N) – ∆(N – 1) + εcore and the final energy is E(N – 2) + εkin. The charge correction takes into account that the charge of the separated crystal volume changes by +1 in the Auger process, while the charge of valence electrons of the model changes by +2. Substituting these energies into Eq. (1), we obtain a theoretical formula for the absolute position of the AES spectrum edge, 1 ε v = --- [ E ( N – 2 ) – E ( N ) + ∆ ( N – 1 ) – ε F ]. 2

(2)

In the inverse Auger process, the initial energy is E(N) + εkin and the final energy is E(N + 2) – ∆(N + 1) + εcore; thus, we obtain a theoretical formula for the absolute position of the iAES spectrum edge, 1 ε c = --- [ E ( N + 2 ) – E ( N ) – ∆ ( N + 1 ) + ε F ]. 2

(3)

The charge corrections make up a fixed fraction of the energy of electrostatic interaction of cores with each other and with all electrons; both corrections are independent of the electron correlation and are approximately equal. Therefore, summing Eqs. (2) and (3), we obtain 1 iAES AES U F = ε v + ε c = --- ( ε kin – ε kin ) 2 1 = --- [ E ( N + 2 ) + E ( N – 2 ) – 2E ( N ) ]. 2

(4)

In a system with metal filling, the energies of holes and conduction electrons are very close to the Fermi energy in the one-electron approximation. Hence, the energy gap between the thresholds of the forward and inverse Auger spectra (see Eq. (4)) is the characteristic energy of the electron correlation and can be measured experimentally. To this end, it is necessary not only to measure the forward and inverse Auger spectra but also to determine their threshold energy involved in relation (4), which is a complex problem. Its solution is facilitated by the fact that one does not have to determine the absolute position of the Fermi level but rather maintain a constant contact potential between the sample and the electron spectrometer. PHYSICS OF THE SOLID STATE

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Finally, if E(N ± 2) are taken to be energies of excited (rather than ground) states selected in a certain way, then Eqs. (2) and (3) can be used to determine not only thresholds but also specific features of Auger spectra. 3. OPEN-SHELL METHOD FOR 2D AND 3D GRAPHITE Although the above conclusion relates to an arbitrary many-electron system with an open shell, an adequate quantum-chemical model for calculating the total energies can be constructed only in a few cases. Due to the fact that the Fermi surface of semimetal graphite encloses a small phase volume, the electronic structure of graphite can be adequately described by a cyclic model based on an extended unit cell (EUC) of the crystal, which allows one to calculate the wave functions at several high-symmetry points of the Brillouin zone, including points close to the Fermi surface, e.g., inside electron pockets. In 2D graphite, the Fermi surface degenerates into a single high-symmetry point of the Brillouin zone (zero band-gap semiconductor); therefore, the model of a single open shell is adequate for 2D graphite. This model and the method for calculating its electronic properties [6] are characterized by the following features: (i) The crystal is modeled by a quasi-molecular EUC characterized by the D3h point symmetry and by the k set {Γ + 6T + 3M(Q) + 2K(P)} and consists of two crystal planes, with 24 carbon atoms in each. The model of 2D graphite (D6h symmetry) consists of a single plane; however, its k set is much the same. Therefore, all group-theoretic constructions remain valid due to simple doubling of the irreducible representations. The prefix “quasi” in the model title means that periodic boundary conditions are imposed on molecular orbitals (MOs); i.e., the entire set of Bloch states of the crystal is reduced to the k set indicated above (widely used alternative designations of the high-symmetry points of the Brillouin zone of hexagonal graphite are given in parentheses). Thus, the number of valence electrons in the models of the initial states of Auger processes is N = 192 and 96 in the cases of 3D and 2D graphite, respectively [see Eqs. (2)–(4)]. (ii) The open electron shell of such a quasi-molecular EUC models the Fermi surface of an infinite 3D crystal according to the following scheme. The graphite band states nearest to the Fermi surface are π-type, i.e., are composed of pz atomic orbitals (AOs). In turn, at the point K, which is at the center of the main electron pocket of the Fermi surface, the upper band π state of the model is doubly degenerate and corresponds to the bottom of the partially filled conduction band and the valence band top of the crystal. As indicated in the k set of the model, the Brillouin zone of graphite contains two points K, which cannot be transferred to one another by translation through a reciprocal lattice vec-

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tor; hereafter, they are designated as K' and K". The open shell formed by MOs with these quasimomenta includes four degenerate MOs grouped into two subshells with different wave vectors. Each subshell is a spatial MO doublet, which transforms according to the 2D representation E' or E" of the D3h point group. Three spinless configurations of the type (K')m(K'')n – m = (2K)n with an even number of electrons n = 2, 4, 6 can be formed from the MOs of the open shell. All the above also relates to the model of the “Fermi surface” of 2D graphite, which degenerated into the points K; only the designations of irreducible representations should be changed in the group-theoretic formulas. The tables given in [6] list all terms of every configuration, i.e., spectroscopic combinations of the Slater determinants, whose group-theoretic selection and designations correspond to (a) a certain wave vector, Γ or K, which is determined by summing the wave vectors of n electrons settled over four MOs of the open shell; (b) irreducible representations of the D3h point group (which is the point group of hexagonal semimetal graphite and of a 48-atom EUC); and (c) the type of filling of the open-shell MOs: χ is the pairing type (the open shell contains only doubly filled MOs), ψ is the exchange type (the open shell contains only singly filled MOs), and ϕ is the mixed type. For example, according to a self-consistent calculation, the model ground state with energy E(N) at the beginning of the Auger process is represented by a term with configuration (K')2(K'')2, whose wave function is composed of four Slater determinants differing in the filling of four MOs of the open shell [see Eq. (5)]. In fact, the group-theoretic analysis of the multiplet structure of the open electron shell of the cyclic model of semimetal graphite [6] is the basis of the proposed theoretical description of Auger processes; the remainder relates to calculation of the electronic properties. (iii) The construction of the model of semimetal graphite described above predetermined the method for calculating its properties: the restricted open-shell Hartree–Fock–Roothaan method. Although the abbreviation ROHF (restricted open-shell Hartree–Fock) is used for this method as in quantum chemistry [5], an open shell with any degeneracy (orbital and/or with respect to the wave vector), rather than the simplest version with half-filling of the upper MO as in conventional software packages, is meant in the case under consideration. Such an extended interpretation of the ROHF necessitated the development of the McWeeney projection method by using electron density matrices to calculate Fock matrices and by introducing a system of open-shell coefficients [6] and a symmetrization procedure (in the presence of subshells in crystal models). As a result of these modifications, the self-consistency cycle takes into account the electron correlation associated with the open shell (the calculated terms involve several determinants). The high spatial symmetry of the model is also used in full (for example, when selecting

molecular orbitals for an initial approximation and for calculating transition energies with allowance for the selection rules). (iv) The other features of the calculation are methodical and are associated with the AO basis (semiempirical Slater AOs C2s2p), parametrization of matrix elements of the electron–electron interaction [intermediate neglect of differential overlap (INDO)], techniques for summing over an infinite lattice, etc. The restrictions used on the basis size and Coulomb integrals are associated exclusively with the potential of the computer equipment employed and the experience gained in the study of solid-state carbon systems (diamond, its surface, and its structural defects; fullerene, its crystals and chemical compounds). We emphasize that the cyclic model and its electronic structure representation within the approximation of one open shell (ROHF) are not related to semiempirical calculations and are quite feasible in ab initio quantum chemistry. For this reason, details of the geometrical parametrization of graphite and its properties are hereafter omitted. 4. CALCULATION OF AUGER PROCESS THRESHOLDS It should be emphasized once again that the abovestated method for calculating the Auger process thresholds can be feasible at any level of quantum chemistry with MOs (ab initio, semiempirical, tight-binding approximation). However, since numerical results are considered below, we first of all discuss the quantumchemical formalism of the software developed by the author. These data were published in parts (semiempirical implementations of the conventional Hartree– Fock–Roothaan method always differ in some details associated with the class of problems to be solved): the Fock matrix construction for the self-consistency cycle [7]; implementation of the self-consistent field (∆SCF) method for electron excitations [8]; the procedure of orthogonal transformation of MOs of a degenerate electron shell, which is necessary to relate these MOs to equivalent wave vectors, i.e., subshells [9]; and the calculation features associated with the cyclic model of the crystal, which are important in estimating charge corrections in the Auger process models described by Eqs. (2) and (3) [9]. To demonstrate the quality of the semiempirical parametrization, Fig. 1 shows the MO energy spectra obtained using self-consistent ROHF calculations of the diagonal Slater sum of the configuration (2K)4(Γ + K) of two cyclic models of the initial Auger-process states: (i) 2D graphite (graphene), a zero band-gap semiconductor whose Fermi surface collapses into the point K of the Brillouin zone (a 24-atom EUC), and (ii) 3D hexagonal graphite, a semimetal with a small Fermi surface (a 48-atom EUC). PHYSICS OF THE SOLID STATE

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' ( Γ ) = αβαβ --------------K i'K i'K ''j K ''j E ( N ) = A 1χ (5) 4 i = 1, 2 j = 1, 2

∑ ∑

= –5657.65 eV ( –2828.53 eV ), 1 E ( N – 2 ) = --- [ A '1ψ ( Γ ) + A '2ψ ( Γ ) + 2E ψ' ( Γ ) ] 4 = –5658.67 eV ( –2829.02 eV ),

(6)

1 E ( N + 2 ) = --- [ A '1ϕ ( Γ ) + A '2ϕ ( Γ ) + 2E ϕ' ( Γ ) ] 4 = –5653.65 eV ( –2823.96 eV ),

(7)

where only the states of the open shell are given, α and β are the spin basis functions, and the numerical values for the 24-atom model of 2D graphite are parenthesized. Spectroscopic combinations of the Slater determinants for the terms entering in diagonal Slater sums (6) and (7) and their open-shell coefficients are given by 1 ' ( Γ ) = ------- ( ψ 11 + ψ 22 ) ( αβ – βα ), A 1ψ 4

1

24-atom EUC valence band, eV

–7 –9 –11 –13 –15 –17 –19 –21

M(Q)

Γ

T

K(P)

(b) 1

–7 –9 –11 –13 –15 –17 –19 –21

M(Q)

Γ

T

K(P)

Fig. 1. Spectra of molecular orbitals of cyclic models of (a) 2D and (b) 3D graphite. Closed squares and solid lines are σ bands, and open squares and dashed lines are π bands; numeral 1 indicates the open-shell states.

Analogous terms of the mixed-filling type (with subscript ϕ) entering into Eq. (7) have other open-shell coefficients and can be determined using the same formulas with the substitutions

1 A '2ψ ( Γ ) = ------- ( ψ 12 – ψ 21 ) ( αβ – βα ), 4 1 E ψ' ( Γ ) = ------- [ ( ψ 11 – ψ 22 ) ± ( ψ 12 + ψ 21 ) ] ( αβ – βα ), 8 ψ ij = K i'K ''j

ψ

ϕ,

( αβ – βα )

αβαβ ( αβ – βα ),

ϕ ij = ( K i'K i') ( K i''K ''j )K i'K ''j . Formally, these terms should be split by combinations of three non-Hartree–Fock Coulomb integrals, which

and are tabulated in [6]. PHYSICS OF THE SOLID STATE

(a)

48-atom EUC valence band, eV

The optimum basis interatomic distances are d = 2.90aB = 1.53 Å (experimental value, 1.42 Å) and c = 6.79aB = 3.59 Å (3.35 Å). Both spectra confirm the reliability of the calculation by showing small splitting (doubling) of the graphene bands as the interplane interaction in 3D graphite is included, except for the π bands at the Brillouin zone center Γ [10]. The orbital energy of the open shell is designated by diamonds (1) in Fig. 1; the dash-dotted line shows that the conduction band begins at this energy. This energy does not obey the Koopmans’ theorem [7]; therefore, in contrast to the case of orbital energies of closed and virtual shells, this theorem cannot be applied to estimate the energy of one-electron excitations, in particular, to determine the Fermi level. By decreasing or increasing the number of electrons of the EUC by two, we obtain quantum-chemical models of the final states of Auger processes, namely, diagonal Slater sums of the configurations (2K)2(Γ + K) and (2K)6(Γ + K) with open-shell coefficients [6]. As a result, the following terms or Slater sums of almost degenerate terms turned out to be ground states:

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24-atom EUC total energy, eV

–2820

final states of Auger processes. For convenience, in the tables from [6], the terms are designated by different symbols [see (a–c) in item (ii) of Section 3]. Applying definition (4) to energies (5)–(7), we immediately obtain a numerical estimate of the observed effective electron correlation at the Fermi surface of 3D (2D in parentheses) graphite,

(a) 2D graphite

–2822

–2824 2εc –2826 UF –2828 2εv –2830

48-atom EUC total energy, eV

–5650

N–2

N

N+2

(b) 3D graphite

–5652

–5654 2εc UF

–5656

2εv N–2

(8)

We estimate Auger process thresholds (2) and (3) by determining the charge corrections. According to their definition, we need to estimate the change in the total energy of the model caused by a change in the total charge of cores at a fixed state of the system of valence electrons. The total energy terms containing the charge correction are involved in any quantum-chemical calculation. In the model we proposed, this term is the “Madelung” energy EM defined as the energy of atomic cores in their field (repulsion) and in the self-consistent field of the valence electrons (attraction). This total energy component should be calculated to control convergence of lattice sums [11]; specific implementations [9] are not of principle importance. Since this part of the total energy is proportional to the squared charge of the atomic core (all cores of the model are identical) and since an identical extra charge is placed on each core (to retain the entire model neutrality) as the number of valence electrons changes by i, we have the simple proportionality i 2 M E ( N ± i ) ∼  1 ± ---- .  N

–5658

–5660

U F = 1.5 eV ( 2.0 eV ).

N

N+2

Fig. 2. Determination of the Auger process thresholds for cyclic models of (a) 2D and (b) 3D graphite. The terms of the three multiplets are separated by vertical lines and grouped in column pairs. The left-hand and right-hand columns contain the wave vectors Γ and K, respectively. Squares, triangles (symbols are paired in the case of double degeneracy), and circles designate the terms belonging to the representations A', E', and (E', i), respectively. Closed symbols show the pairing type of filling χ. The notations allow one to determine the term type using the tables from [6].

involve all four MOs of the open shell [6]. In fact, all these integrals strictly vanish. It is not necessary to perform a group-theoretic analysis, since direct calculation with a symmetrization procedure yields zero values for all three integrals at all values of the parameters (which is another argument in favor of a maximum complete consideration of the symmetry in quantum-chemical calculations). Figure 2 shows the self-consistently calculated complete multiplet structures of the models of the initial and

It follows from this relation (and it is confirmed by calculations) that the Madelung energies for the sequence of states (6), (5), and (7) with a gradually increasing number of electrons form a uniform series. Hence, simple averaging is sufficient to estimate the charge corrections (in electronvolts): 5.047 (4.697), 1 M M ∆( N ± 1) ≈ ± --- [ E ( N ± 2) – E ( N ) ] = 2 5.052 (4.798) (the values for 2D graphite are in parentheses). The error in linearity is of the same order of smallness as the difference in the determined charge corrections. Graphic subtraction for determining the Auger process thresholds, carried out at a Fermi energy of 4.6 eV [2] (the work function of semimetal 3D graphite), is shown in Fig. 2: the level E(N) is shifted by ±[∆(N ± 1) – εF]. Half differences between the obtained levels and the final-state energies E(N ± 2) yield the sought thresholds according to Eqs. (2) and (3). For 3D graphite (Fig. 2b), we get 1 ε v = --- ( 4.0 eV – ε F ) = – 0.3 eV, 2 1 ε c = --- ( –1.1 eV + ε F ) = +1.8 eV. 2 PHYSICS OF THE SOLID STATE

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These formulas correspond to the performed calculation, within which εF cannot be accurately determined; these numerical estimations are carried out with an experimental value of 4.6 eV for the Fermi energy [2]. The same value was also substituted into the formulas for the thresholds for 2D graphite (Fig. 2a), 1 ε v = --- ( 4.2 eV – ε F ) = – 0.2 eV, 2 1 ε c = --- ( –0.2 eV + ε F ) = +2.2 eV, 2

(10)

although the Fermi energy in graphene apparently differs from that in 3D graphite. We note that the indicated numerical estimates of the thresholds and of the effective correlation depend only weakly on the conditions for summing the exchange matrix elements, in contrast to other model properties (upper levels of the valence band, the elastic moduli, the multiplet levels of the open shell in the ground state) [9]. Moreover, the determined values of the Auger process thresholds are size-consistent; i.e., they do not contain a proportional dependence on the model size, as seen from Fig. 2, where the same energy range (10 eV) is shown for the 24- and 48-atom models. This independence shows that the calculation procedure is reliable and that the results reflect the real properties of open-shell electrons rather than artifacts depending on quantum-chemical parameters. The first of the thresholds (9) determined for holes agrees with the conclusion (based on experimental data [1]) that the correlation correction to the forward Auger process is small; at least, this correction is of the order of the experimental error in determining the Fermi energy. There is also other experimental verification. The negative effective correlation εv can be compared to the negative critical energy of the forward Auger spectrum [1]. Its exact value is not given in [1], since this spectral region cannot be explained using the oneelectron approximation and other explanations (exciton satellite of the initial core hole, correlation of final holes) were not considered. This value was determined within a unified technique for all critical points and, judging from the presented spectra, is approximately −0.8 eV. We can see that the electron correlation at the Fermi surface of graphite should most significantly manifest itself in the inverse Auger process. The sum of the calculated thresholds does not contain the uncertainty associated with the calculated values of the charge correction and the Fermi energy in Eq. (9); therefore, it is the value (8) of the observed quantity (4) that is the main numerical prediction in this study. 5. CONCLUSIONS The quantum-chemical method for calculating the thresholds of Auger electron spectra proposed in this paper is intended to analyze the effects of the electron PHYSICS OF THE SOLID STATE

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correlation of final particles generated at the Fermi surface of semimetal graphite, namely, two holes in the forward Auger process or two conduction electrons in the inverse process. Since it is oriented toward a specific electron system, i.e., is conceptually a materialsscience approach, this method loses in generality in comparison with analytical methods based on approximations, such as the Hubbard model [12–15]. However, this approach has certain advantages. (i) No special approximations are required to describe the “direct” correlation of final particles in the Auger process. Within a quantum-chemical model, this Coulomb interaction is necessarily included in the selfconsistent Hartree–Fock approximation and in the correlation calculated in the open-shell approximation. (ii) The range of application of our method is dictated by the feasibility of constructing a quantumchemical model rather than by restrictions on the effective Hamiltonian (because an accurate description can be made for closed shells, for atomic spectra of adsorbates or impurities weakly affected by band electrons). The difference in the calculated thresholds for the Auger electron spectra between 2D and 3D graphite [see Eqs. (9), (10)], as well as their relation to the Fermi energy, shows that the experimental technique based on measuring the threshold energies is promising in the spectroscopy of materials based on 2D graphite, where the substrate and/or adsorbate can change the Fermi energy. ACKNOWLEDGMENTS This study was supported by the federal program “Controlled Synthesis of Fullerenes and Other Atomic Clusters.” REFERENCES 1. L. Calliari, G. Speranza, J. C. Lascovich, and A. Santoni, Surf. Sci. 501 (3), 253 (2002). 2. F. R. McFeely, S. P. Kowalczyk, L. Ley, R. G. Cavell, R. A. Pollak, and D. A. Shirley, Phys. Rev. B 9 (12), 5268 (1974). 3. P. Skytt, P. Glans, D. C. Mancini, J.-H. Guo, N. Wassdahl, J. Nordgren, and Y. Ma, Phys. Rev. B 50 (15), 10457 (1994). 4. J. E. Houston, J. W. Rogers, Jr., R. R. Rye, F. L. Hutson, and D. E. Ramaker, Phys. Rev. B 34 (2), 1215 (1986). 5. J. A. Pople, Nobel Prize Lecture (Stockholm, 1998); Usp. Fiz. Nauk 172 (3), 349 (2002). 6. S. S. Moliver, Fiz. Tverd. Tela (St. Petersburg) 42 (8), 1518 (2000) [Phys. Solid State 42, 1561 (2000)].

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7. S. S. Moliver, Fiz. Tverd. Tela (St. Petersburg) 41 (3), 404 (1999) [Phys. Solid State 41, 362 (1999)]. 8. S. S. Moliver and Yu. F. Biryulin, Fiz. Tverd. Tela (St. Petersburg) 43 (5), 944 (2001) [Phys. Solid State 43, 982 (2001)]. 9. S. S. Moliver, Doctoral Dissertation (Ul’yanovsk State Univ., Ul’yanovsk, 2001). 10. M. S. Dresselhaus, G. Dresselhaus, K. Sugihara, I. L. Spain, and H. A. Goldberg, Graphite Fibers and Filaments (Springer, Berlin, 1988). 11. A. Shluger and E. Stefanovich, Phys. Rev. B 42 (15), 9664 (1990).

12. M. Cini and C. Verdozzi, J. Phys.: Condens. Matter 1 (40), 7457 (1989). 13. C. Verdozzi, M. Cini, J. A. Evans, R. J. Cole, A. D. Laine, P. S. Fowles, L. Duo, and P. Weightman, Europhys. Lett. 16 (8), 743 (1991). 14. G. A. Zawatzky and A. Lanselink, Phys. Rev. B 21 (5), 1790 (1980). 15. W. Nolting, G. Geipel, and K. Ertl, Phys. Rev. B 45 (11), 5790 (1992).

Translated by A. Kazantsev

PHYSICS OF THE SOLID STATE

Vol. 46

No. 9

2004