Some aspects of tight-binding approach in chemisorption theory

The quantum-statistic problem of atoms chemisorption with hydrogen-like electronic structure on semiconductor and dielectric crystalline surfaces is considered. The superficial states are described in tight-binding approximation. The self-consistent system of equations for calculating the charge...

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Дата:	2001
Автори:	Rudavskii, Yu., Ponedilok, G., Petriv, Yu.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2001
Назва видання:	Condensed Matter Physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/119775
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Цитувати:	Some aspects of tight-binding approach in chemisorption theory / Yu. Rudavskii, G. Ponedilok, Yu. Petriv // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 133-140. — Бібліогр.: 7 назв. — англ.

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spelling	irk-123456789-1197752017-06-09T03:05:06Z Some aspects of tight-binding approach in chemisorption theory Rudavskii, Yu. Ponedilok, G. Petriv, Yu. The quantum-statistic problem of atoms chemisorption with hydrogen-like electronic structure on semiconductor and dielectric crystalline surfaces is considered. The superficial states are described in tight-binding approximation. The self-consistent system of equations for calculating the charge and spin-polarized states of the adsorbed atom in Hartree-Fock approximation is derived. Розглядається квантово-статистична задача хемосорбцiї атома з водневоподiбною електронною структурою на кристалiчних поверхнях напiвпровiдникiв та дiелектрикiв. Поверхневi стани описуються наближенням сильного зв’язку. В наближеннi Хартрi-Фока отримано самоузгоджену систему рiвнянь для розрахунку величини зарядового та спiн-поляризованого станiв адсорбованого атома. 2001 Article Some aspects of tight-binding approach in chemisorption theory / Yu. Rudavskii, G. Ponedilok, Yu. Petriv // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 133-140. — Бібліогр.: 7 назв. — англ. 1607-324X PACS: 73.20.H, 82.65.Y DOI:10.5488/CMP.4.1.133 http://dspace.nbuv.gov.ua/handle/123456789/119775 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The quantum-statistic problem of atoms chemisorption with hydrogen-like electronic structure on semiconductor and dielectric crystalline surfaces is considered. The superficial states are described in tight-binding approximation. The self-consistent system of equations for calculating the charge and spin-polarized states of the adsorbed atom in Hartree-Fock approximation is derived.
format	Article
author	Rudavskii, Yu. Ponedilok, G. Petriv, Yu.
spellingShingle	Rudavskii, Yu. Ponedilok, G. Petriv, Yu. Some aspects of tight-binding approach in chemisorption theory Condensed Matter Physics
author_facet	Rudavskii, Yu. Ponedilok, G. Petriv, Yu.
author_sort	Rudavskii, Yu.
title	Some aspects of tight-binding approach in chemisorption theory
title_short	Some aspects of tight-binding approach in chemisorption theory
title_full	Some aspects of tight-binding approach in chemisorption theory
title_fullStr	Some aspects of tight-binding approach in chemisorption theory
title_full_unstemmed	Some aspects of tight-binding approach in chemisorption theory
title_sort	some aspects of tight-binding approach in chemisorption theory
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2001
url	http://dspace.nbuv.gov.ua/handle/123456789/119775
citation_txt	Some aspects of tight-binding approach in chemisorption theory / Yu. Rudavskii, G. Ponedilok, Yu. Petriv // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 133-140. — Бібліогр.: 7 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT rudavskiiyu someaspectsoftightbindingapproachinchemisorptiontheory AT ponedilokg someaspectsoftightbindingapproachinchemisorptiontheory AT petrivyu someaspectsoftightbindingapproachinchemisorptiontheory
first_indexed	2025-07-08T16:34:43Z
last_indexed	2025-07-08T16:34:43Z
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fulltext	Condensed Matter Physics, 2001, Vol. 4, No. 1(25), pp. 133–140 Some aspects of tight-binding approach in chemisorption theory Yu.Rudavskii, G.Ponedilok, Yu.Petriv Lviv Polytechnic State University 12 S.Bandera Str., 79013 Lviv, Ukraine Received August 26, 2000 The quantum-statistic problem of atoms chemisorption with hydrogen-like electronic structure on semiconductor and dielectric crystalline surfaces is considered. The superficial states are described in tight-binding approxi- mation. The self-consistent system of equations for calculating the charge and spin-polarized states of the adsorbed atom in Hartree-Fock approxi- mation is derived. Key words: adatom, chemisorption, surface, chemisorption energy PACS: 73.20.H, 82.65.Y 1. Introduction The quantum atom chemisorption theory takes an important place in the physics of metal surface, semiconductors and detectors. This problem has been considered in numerous papers (see [1–4] and their references). Based on different theoretical models and methods, essential success in understanding the chemisorption mech- anisms and in qualitative explanation of the seen phenomena was achieved. The most frequently used are the method of density functional, molecular orbital meth- ods, clusters models. But still a lot of principal questions of quantum chemisorption theory remain open. In particular, it corresponds to construction both of adequate microscopic models and the development of effective approximate analysis methods of these models condition of the absence of small parameters, that does not permit to employ a standard perturbation theory. From this point of view, the microscopic models, which are built on the ideas of the well-known in quantum solid state theory Anderson model [5], are perspective. Firstly, in such models the physical meaning of parameters is quite obvious, that permits to choose and estimate them for spe- cific systems. Secondly, the Anderson model is well established in many aspects. It permits to transmit the approved results and methods on chemisorption theory prob- lems. The microscopic model of chemisorption, based on the main ideas of Anderson model was proposed in Newns paper [6]. The problem is solved using the Hartree- Fock approximation: the chemisorption energy and the charge state of chemisorbed c© Yu.Rudavskii, G.Ponedilok, Yu.Petriv 133 Yu.Rudavskii, G.Ponedilok, Yu.Petriv atom are calculated. The model, suggested initially in [6], was generalized later. In particular, the Coulomb electron-electron interaction was taken into consideration, the effects of non-orthogonality of adatom orbital to atom surface orbital and over- filling of one-site basis were studied. The chemisorption theory studied at papers [6,7], was built on supposition that the substrate is found in paramagnetic state. In present paper the extended version of microscopic Newns-Anderson [6] mod- el is taken as the base. The space non-homogeneousness of the model parameters, Coulomb and exchange interactions in the “the surface – adatom” system, the local potential-field effects of surface layer ions additionally are taken into consideration. The system of equations for calculating the charge and spin-polarized states of the adsorbed atom and binder energy of adatom with the surface in Hartree-Fock ap- proach are derived. Unlike Newns paper [6], where the chemisorption on the surface of metals was investigated, in our model we take into account the correlation effects, which permit to extend the model to atoms chemisorption problem on dielectric and semiconductor surfaces. 2. The microscopic model of chemisorption A laboratory system of coordinates is competent so that a plane xOy is situ- ated on half-limited crystal surface. In half-space Z > 0 a separate atom is found, on electronic structure of valence envelope of which has hydrogen-like structure. An electronic surface structure of limited environment is described in tight-binding approach. It means that a dominating role in the course of atom chemisorption is played by the electronic states of surface atoms. Let the Rj = Rxex+Ryey, j = 1, N are the coordinates of surface atoms, and R0 = Z0ez is the adatom position. A Hamilton operator of “surface–adatom” system in the secondary quantization image has the following form Ĥ = C + Ĥ0 + Ĥcorr. (1) The effective one-particle part of operator (1) has the following structure Ĥ0 = ∑ 16j6N ∑ σ=↑,↓ Ej,σa+jσajσ + ∑ σ=↑,↓ Eσd + 0σd0σ + ∑ 16i 6=j6N ∑ σ=↑,↓ Tija + iσajσ + ∑ 16i6N ∑ σ=↑,↓ (Vi,0a + iσd0σ + V0,id + 0σaiσ). (2) Here a+jσ (ajσ) and d+0σ (d0σ) are electrons Fermi creation (annihilation) operators on the Rj-th surface atom and on the adatom, accordingly, which is described by s-type atom wave functions Ψ(\|r−Ri\|) and ϕ(\|r−Ri\|). The respective energy levels are the ε0 and E0. Further in the paper we employed the notations n̂j,σ = a+jσajσ and n̂0,σ = d+0σd0σ for operators of the number of electrons that localized on atomic states with spin projection σ on the axis of quantization. Also, the n̂0 = n̂0,↑+ n̂0,↓ is the operator of the total number of electrons, localized on s-orbital of the adsorbed atom and the respective operator for j-th surface atom is n̂j = n̂j,↑ + n̂j,↓. 134 Tight-binding approach in chemisorption theory The renormalized energy of a localized electron state on the adatom has the following structure Eσ = E0 + U0〈n̂0,−σ〉+ N ∑ j=1 [ Φs(\|R0 −Rj\|) + U(\|R0 −Rj\|)〈n̂j〉 ] . (3) Here and further in the paper, the symbol 〈. . .〉 mean thermodynamically averaged by Gibbs distribution. The second term at the right part of (3) describes Hubbard level splitting due to the intensity of Coulomb interaction U 0. The latter term char- acterizes the adatom level shift taking into account the total action of surface atoms. The value Φs(\|Rj −R0\|) = ∫ ϕ2(r−R0)Vs(\|r−Rj \|) dr is the effective electron en- ergy in the potential field of Rj-th surface atom and Vs(\|r−Rj\|) = −Zse 2/\|r−Rj\|. The Coulomb integral U(\|R0 −Rj\|) = ∫ ∫ ϕ2(r−R0)W (\|r− r′\|)Ψ2(r′ −Rj) drdr ′ is proportional to the average energy of the electron at the field of effective surface atom charge and for s-orbital this energy depends only on the distance between the atoms. From expression (3) we can see that atomic level E0 of the adsorbed atom has got a shift under the action of total potential of surface atoms. The energy of localized electron state on Rj-th surface atom is as follows Ej,σ = E0 + Us〈n̂j,−σ〉+ Φa(\|Rj −R0\|) + U(\|Rj −R0\|)〈n̂0〉. (4) In this expression, the parameter Φa(\|Rj − R0\|) = ∫ Ψ2(r−Rj)Va(\|r−R0\|) dr characterizes level shift of j-th surface atom under adatom potential action, and Va(\|r−R0\|) = −Zae 2/\|r−R0\|. Here Vs(\|ri − Rj\|) and Va(\|ri − R0\|) are electron potential energy in Rj surface atom and adsorbed atom R0 field, accordingly. The two last terms in the right part of (4) describe the electrons correlation effects, like in expression (3). The parameter Us is the energy of Hubbard correlation of electrons that are localized on s-orbital of the surface atom. The operator of Coulomb electron-electron correlation in (1) is Ĥcorr = U0 2 ∑ σ=↑,↓ δn̂0σδn̂0,−σ + Us 2 ∑ 16j6N ∑ σ=↑,↓ δn̂jσδn̂j,−σ + ∑ 16j6N U(\|Rj −R0\|)δn̂jδn̂0. (5) In this expression we employed notes δn̂jσ = n̂jσ −〈n̂jσ〉, δn̂0σ = n̂0σ −〈n̂0σ〉 for the operators of fluctuations of number of electrons localized on the atoms. The values 〈n̂j〉 = 〈n̂j↑〉 + 〈n̂j↓〉, 〈n̂0〉 = 〈n̂0↑〉 + 〈n̂0↓〉 are the thermodynamically averaged number of electrons, localized on j-th surface atom and on the chemisorbed atom, accordingly. Non-operator part of Hamiltonian (1) C = −U0〈n̂0↑〉〈n̂0↓〉 − ∑ 16j6N Us〈n̂j↑〉〈n̂j↓〉 − ∑ 16j6N Φ(\|Rj −R0\|)〈n̂j〉〈n̂0〉 (6) is the effective electrostatic energy of correlation of electrons. 135 Yu.Rudavskii, G.Ponedilok, Yu.Petriv 3. Hartree-Fock approach. The case of crystalline surface Hartree-Fock approach permits a qualitative study of physics of chemisorption processes. However, in the approach, the microscopic models of chemisorption have been studied by now, if not to take into account directly the quantum-statistical calculation and computer simulation. In Hartree-Fock approach, in the operator (1), the energy of electron-electron correlation Ĥcorr is disregarded. The effective Hamilton operator for this approach is as follows: ĤHF = C + Ĥ0. (7) In the case of crystalline surface, it is convenient to take transformation directly in momentum space. Then Hamilton operator of the model in Hartree-Fock approach is Ĥ0 = ∑ k∈B ∑ σ=↑,↓ tka + kσakσ + ∑ σ=↑,↓ Eσd + 0σd0σ + ∑ k,q∈B ∑ σ=↑,↓ ∆q,σa + kσak−q,σ + ∑ k∈B ∑ σ=↑,↓ ( Vka + kσd0σ + V ∗ k d + 0σakσ ) . (8) The Fermi operators of annihilation and creation of electrons at {k, σ} states are as follows: akσ = 1√ N N ∑ j=1 ajσe −ikRj , a+kσ = 1√ N N ∑ j=1 a+jσe ikRj . (9) For a semi-limited media, as remarked in [6], the wave vector k consists of parallel to crystal surface components k‖, which acquires uninterrupted values within the first two-dimensional Brillouin zone, and perpendicular to crystal surface components k z, which acquires discrete values. In this section, under wave vector k we understand only parallel constituent of wave vector, skipping additional symbols in designations. Also, the perpendicular constituent of wave vector acquires only one discrete value is supposed. For physical sense, such a simplification is equivalent to supposition that in chemisorption processes only one zone is important that has lowermost energy of superficial states. Matrix elements of electron transfers between localized surface atom states are as follows: Tij = 1 N ∑ k tke ik(Ri−Rj), tk = N ∑ j=1 Tije −ik(Ri−Rj). (10) The value tk is the electrons spectrum of surface Hubbard subbands. The Fourier- coefficients of pseudopotential of electrons dispersion on non-homogeneous surface are as follows: ∆p,σ = 1 N ∑ 16j6N ∆jσe −ipRj . (11) The coefficient ∆j σ describes heterogeneous level shift of j-th surface atom under the action of correlation effects of Coulomb type ∆j,σ = Us 〈n̂j,−σ〉+ Φa (\|Rj −R0\|) + U (\|Rj −R0\|) 〈n̂0〉 . 136 Tight-binding approach in chemisorption theory Fourier-components of matrix potential elements, which characterize the hybridiza- tion processes of localized surface electrons and the electrons localized on the adatom are as follows: Vk = 1√ N N ∑ j=1 Vj,0e −ikRj . (12) For Hamilton operator Ĥ = C + Ĥ0 the following matrix has been calculated. G σ (ω) = ( 〈〈akσ ∣ ∣a+qσ〉〉ω 〈〈akσ ∣ ∣d+0σ〉〉ω 〈〈d0σ ∣ ∣a+qσ〉〉ω 〈〈d0σ ∣ ∣d+0σ〉〉ω ) , (13) elements of which are frequency images of two-time anticommutative one-electron Green functions 〈〈Â(t)\|B̂(t0)〉〉 = −iΘ(t− t0)〈[Â(t), B̂(t0)]+〉. Equation of motion for Green functions matrix elements (13) are as follows: (ω − tk) 〈〈akσ\|a+k′σ〉〉ω = 1 2π δk,k′ + Vk〈〈d0σ\|a+k′σ〉〉ω + ∑ q ∆qσ〈〈ak−q,σ\|a+k′σ〉〉ω, (ω − tk) 〈〈akσ\|d+0σ〉〉ω = Vk〈〈d0σ\|d+0σ〉〉ω + ∑ q ∆qσ〈〈ak−q,σ\|d+0σ〉〉ω, (ω − Eσ) 〈〈d0σ\|d+0σ〉〉ω = 1 2π + ∑ k V ∗ k 〈〈akσ\|d+0σ〉〉ω, (ω −Eσ) 〈〈d0σ\|a+k′σ〉〉ω = ∑ k V ∗ k 〈〈akσ\|a+k′σ〉〉ω. (14) From (14) the equations for Green functions 〈〈akσ\|a+k′σ〉〉ω are obtained (ω − tk) 〈〈akσ ∣ ∣a+k′σ〉〉ω = 1 2π δk,k′ + ∑ q Ωσ k,q〈〈aqσ ∣ ∣a+k′σ〉〉ω . (15) Effective pseudopotential of electrons dispersion is Ωσ k,q = VkV ∗ q ω − Eσ +∆k−q,σ. (16) The first term at (16) characterizes the intensity of resonant (non-elastic) dispersion, and the second term characterizes the elastic dispersion of band electrons on crystal heterogeneities. From equation (15) Green function of zero approximation is obtained Gσ 0 (k, ω) = 1 2π 1 ω − tk −∆σ − \|Vk\|2 ω −Eσ . (17) The parameter ∆σ = lim k→0 ∆k,σ = Us〈n−σ〉+∆Es(Z0), where term ∆Es(Z0) = 1 N N ∑ j=1 [Φa(\|Rj −R0\|) + U(\|Rj −R0\|)〈n0〉] (18) 137 Yu.Rudavskii, G.Ponedilok, Yu.Petriv Figure 1. Energetic scheme of chemisorption. describes homogeneous shift of surface band center E0 = N−1 ∑ k tk. Heterogeneous in surface plane wrap of conductivity band appears to more exact solution of integral equation (15). Electron dispersion law in this approximation is Eσ 1,2(ω) = 1 2 [ tk +∆σ + Eσ ± √ (tk +∆σ −Eσ)2 + 4\|Vk\|2 ] (19) For Green function Gσ 00(ω) = 〈〈d0σ\|d+0σ〉〉ω from the system of equation (14) by dint of iteration we find an expression Gσ 00(ω) = 1 2π 1 ω − Eσ − Σσ(ω) . (20) Self-energy part of Green functions is the series by power of pseudopotential disper- sion Σσ(ω) = ∑ k \|Vk\|2 ω − tk + ∑ k,q(k 6=q) V ∗ k∆k−q,σVq (ω − tk) (ω − tq) + ∑ k,q,p(k 6=q 6=p) V ∗ k∆k−q,σ∆q−p,σVq (ω − tk) (ω − tp) (ω − tq) + . . . . (21) In the easiest approach, let us suggest that ∆k,σ = ∆σδk,0, where δk,0 is a Kronecker symbol. Such an approximation is equivalent in the theory of structural disordered system to the approximation of homogeneous effective media. For this condition, the series (20) may be resumed and expression for Green function is derived Gσ 00(ω) = 1 2π [ ω − E0 − U0na,−σ −∆Ea(Z0)− ∑ k \|Vk\|2 ω − tk − Usns,−σ −∆Es(Z0) ]−1 . (22) 138 Tight-binding approach in chemisorption theory At this expression, the ∆Ea(Z0) = N ∑ j=1 [Φs(\|R0 −Rj \|) + U(\|R0 −Rj \|)〈nj〉] is the adatom local level energy shift under the surface atoms total potential. At (22) na,−σ = 〈n̂0,−σ〉 is the averaged thermodynamic value of electrons number with spin projection −σ that is localized on the adsorbed atom. Doing at (22) the analytic continuation ω → E + iε for Green function, we obtained Gσ 00(ω) = 1 2π [E −E0 − U0na,−σ −∆Ea(Z0)− Λ′(E) + iΛ′′(E)] −1 . (23) where Λ′ = 1 π P +∞ ∫ −∞ Λ′′(E ′)dE ′ E − E ′ − Usns,−σ −∆Es(Z0) Λ′′ = −Im ∑ k \|Vk\|2 E −Ek − Usns,−σ −∆Es(Z0) + iǫ = π ∑ k \|Vk\|2 δ(E − tk − Usns,−σ −∆Es(Z0)) and P mean main sentence value of integral in Cauchy sense. The adatom level position is received from transcendental equation E = E0 + U0na,−σ +∆Ea(Z0) + ∑ k \|Vk\|2 E − tk − Usns,−σ −∆Es(Z0) . (24) The width of the atomic energetic level is expressed by the imaginary part of Green function pole (23). Adatom state is defined by charge n = na,↑ + na,↓ and by the value of electron spin polarization m = (na,↑ − na,↓)/2. We can obtain these values from density of adatom states ρσa (E) = 1 π ImGσ 00(E) = 1 π Λ′′(E) (E − E0 − U0na,−σ −∆Ea(Z0)− Λ′(E))2 + (Λ′′(E))2 , (25) which have the form of Lorentz distribution. Then the averaged thermodynamic value of the number of electrons, localized at adatom is na,σ = µ ∫ −∞ ρσa (E) dE, where µ is the chemical potential of electron subsystem. 139 Yu.Rudavskii, G.Ponedilok, Yu.Petriv 4. Conclusions In the present paper, the generalized model of atoms chemisorption with hydro- gen-like electronic structure of valence envelope on crystalline surfaces based on microscopic Newns-Anderson model [6] is suggested. Our model differs from simi- lar studies of other authors in the following aspects. Firstly, the superficial states are taken into account in the approach of tight-binding, that makes it possible to extend the model for chemisorption phenomenon description to dielectric and semiconductor surfaces with narrow surface bands. Secondly, the superficial spatial- heterogeneousness of model parameters, the effects of local potential field of surface layer atoms, the Coulomb interaction at the “surface–adatom” system are taken in- to account. Finally, a self-consistent system of equations for calculating the charged and spin-polarized states of the adsorbed atom in Hartree-Fock approach is derived. References 1. Gomer R. Some Problems of the Chemisorption Theory. – In: Surface Science. Cleve- land, CRC Press, Inc., 1974. 2. Theory of Chemisorption. Edited by Smith J.R. Berlin–New York, Springer-Verlag, 1980. 3. Green M. Superficial Properties of Solid States. Moscow, Mir, 1980. 4. Zangwill E. Physics of Surface. Moscow, Mir, 1980. 5. Anderson P.W. // Phys. Rev., 1961, vol. 124. No. 1, p. 41–53. 6. Newns D.M. // Phys. Rev., 1969, vol. 178. No. 3, p. 1123–1135. 7. Schrieffer R., Gomer R. // Surf. Science, 1971, vol. 25, p. 315. Деякi аспекти наближення сильного зв’язку в теорії хемосорбцiї Ю.Рудавський, Г.Понеділок, Ю.Петрів Державний університет “Львівська політехніка” 79013 Львів, вул. С.Бандери, 12 Отримано 26 серпня 2000 р. Розглядається квантово-статистична задача хемосорбцiї атома з водневоподiбною електронною структурою на кристалiчних поверх- нях напiвпровiдникiв та дiелектрикiв. Поверхневi стани описуються наближенням сильного зв’язку. В наближеннi Хартрi-Фока отримано самоузгоджену систему рiвнянь для розрахунку величини зарядово- го та спiн-поляризованого станiв адсорбованого атома. Ключові слова: адатом, хемосорбція, поверхня, енергiя хемосорбцiї PACS: 73.20.H, 82.65.Y 140

Some aspects of tight-binding approach in chemisorption theory

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