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Two-electron bond approach to chemisorption
Solid State Communications,Vol. Printed in Great Britain.
51, No. 10, pp. 787-791,
0038-l
1984.
ELASTIC PLATES MODEL OF DOMAIN WALLS IN INTERCALATED P. Hawrylak, K.R. Subbaswamy Department
of Physics and Astronomy,
University
098/84 $3 .OO+ .OO Pergamon Press Ltd.
GRAPHITE
and G.W. Lehman
of Kentucky,
Lexington,
KY 40506, U.S.A.
(Received 8 May 1984 by A.A. Maradudin) The recently proposed model of thin elastic plates coupled via harmonic forces for the calculation of domain wall energies in intercalated graphite is extended to higher stages. It is found that whether coherent domains bind to each other depends on the degree of stagger. The effect of the loss of coherence and the interface between different stages are discussed.
1. INTRODUCTION THE IMPORTANCE of domains in understanding the physics of graphite intercalation has been underscored in recent experimental [l-6] as well as theoretical work [7-lo]. The domain model for staging proposed by Daumas and H&old [DH] [2] has received support from the transmission electron microscopy work of Thomas et al. [ 111, as well as the experiments on staging transformation of Clark and coworkers [3,4]. Furthermore, Flanderois et aZ. [6] deduced from their X-ray work the existence of intercalant islands with an average in-plane separation of lo-15 A, with X-ray line broadening indicating island sizes in the range of 100 to 200 A. Similar observations for the intercalation compound Ag-TiSz have also been reported [ 121. Elastic interactions leading to the formation of islands in graphite intercalation compounds have been studied extensively [7]. It has been shown that for low density of intercalant coherency strains induced in the infinite graphite matrix give rise to attractive interactions among intercalants lying in the basal plane and repulsive interactions for atoms displaced along the c-axis. This leads to the formation of domain structures such as the DH domains. During intercalation when the coherency strains are relieved the elastic interactions among intercalant islands lead to different behavior as demonstrated recently for stage 2 domains by Kirczenow [lo]. Kirczenow [lo] treats individual graphite layers as thin elastic plates coupled to each other via harmonic, nearest neighbor forces, and the intercalants as forming compact, continuous layers. An intercalant layer increases the separation between bounding carbon layers at the cost of elastic energy. When a dislocation occurs in an intercalant layer the distance between bounding layers is reduced in the vicinity of the dislocation, and the graphite layers undergo a bending distortion as well. The decrease in separation decreases the elastic energy, while bending increases the energy. The competition 787
between these two effects results in unusual oscillatory behavior of dislocation energy as a function of dislocation width. Calculations for stage 2 show that DH domains bind to each other and inplane domains have to overcome an elastic energy barrier before they can coalesce [lo]. Note that inplane domains are present in the sample for kinetic reasons during either intercalation or staging transformation. In this paper we examine the stage dependence of domain wall energetics of graphite intercalation compounds based on the elastic plates model. In Section 2 the model is described and the energies associated with coherent DH domains for stages 2-5 are calculated for staggered as well as matching domains. Also, the effect of loss of coherence is considered. In Section 3 the elastic energy of the interface between laterally coexisting stage 1 and stage 2 domains is considered in the continuum approximation. Implications for the stage 2 to stage 1 transformation are discussed. 2. ELASTIC ENERGIES OF DOMAIN WALLS We first consider DH walls. For every stage n there are several inequivalent configurations, which we denote by n/m, where m is the number of carbon layers by which neighboring stacks are staggered. A particular example, n/m = 4/l, is shown schematically in Fig. l(a). Exploiting the periodicity in the c-direction (taken to be along the z-axis) we need only consider n carbon layers. Let 5‘i(X),(i = 1,2, . . . n) describe the profiles of the YC layers, assumed to be independent of the third direction. The bending is assumed to occur over the region -R
798
TWO-ELECTRON BOND APPROACH TO CHEMISORPTION
where E~ stands for the adsorbate energy, E,n for that of the metal, Vam represents the hopping interaction between metal and adsorbate electrons and U is the Coulomb integral of the absorbate. The charges of the absorbate and metal atoms are given by qA
2 -b = OLam
2 +
q M = Olam
are Green's function which describe the propagation of two electrons. The image potential considered is a quantumm:d,anical generalization of the classic expression [12] 1
V/m - 4(R +/@~)
2a~ 2
(3)
20Lmm,
and the total charge will be =
N(E) = - - - l m ~ K i i ( E + i p )
2,
77
since we are considering only two electrons. This implies the following relation for the coefficients 2 +c~+
Olam
2
O~rn m
= 1.
Once the bond between the adsorbate and the metal atoms is described, the metal band-structure must be introduced. To that end a group orbital kbm) is defined (11) 1
kbm) = f f ~
Vaklk),
(4)
k
so that the metal could be considered reduced to an effective atom. The wave function for the system so defined will be of the form defined in [1 ], but with 1 OLam :
~
~,
VakOLak
(5)
l
~m,. : ~
Z V., Vo~.~*k' kk'
where, from normalization requirements IVak12 = I dEIV(E)I2P{E),
p2 = Z
(6)
k
and o(E) stands for the density of state of the clean metal. At this point we introduce a bond Green's function K(co), which describes the correlated motion of the two electrons and a joint density of states for both electrons. The bond Green's function is given by
v]
[a~°(co) t--;
v co-(2E. + u) + v,m 0 K(co) = o G2~(co)/
where both
1 i dE IV(E)I2p(E) area(co)
=
~.
co - E o (8)
Gram(co) = ~1 f f dE dE' V(E)V(E')p(E)p(E') co -- E-- E' + Vim
(9)
R being the adsorbate-metal atom separation. Hence, the bond density of states (BDOS) will be 1
qA + qM
Vol. 5 I, No. l0
= ~ Ni(E), i
(10)
where Ni(E) represents the contribution of the ith configuration or the configurational DOS. Therefore, the charge population will be the residue Io~ml2 =
ResKii(co)co=i,:,n,
if Kii(co ) has a pole at co =Em -- iTm. Of course both the Vam and Vim depend on the distance between the adsorbate and metal surface atoms, and the interaction Vain as a function of energy gives a measure of the localization of the absorption bond. If Vain is a smooth function of energy then the bond will tend to be localized. For the sake of illustration we shall take the strictly localized case i.e. the interaction is independent of the energy, V(E) = X. This will be the only parameter entering in this theory, just the hopping integral of the Anderson-Newns Hamiltonian. As an application of the model here presented the system H/(100) will be considered, hydrogen being chosen since, conceptually, it is the simplest adsorbate to study. An alternative, somewhat related approach to desorption (of protons) from transition metal surfaces using configuration-interaction techniques on finite clusters was reported by Melius et al. [14] where the importance of double excitation processes in the formation of H + ions was stressed. They give an alternative to direct excitation, suggesting a novel surface predissociation excitation path which involves direct dipole-allowed
11,
(7) l
transitions to states which then undergo curve crossing with double excited dissociative states. Our approach is rather different, the emphasis being placed here on H- ions. In Fig. 1 a schematic picture is given of the Bethe-lattice taken for both the bulk and the surface. The cluster considered in the bond with the hydrogen atom is shown as well.
Vol. 5 l, No. l0
TWO-ELECTRON BOND APPROACH TO CHEMISORPTION BULK
799
E (ev;
>
H%M
\
SURFACE
-
H-_M +
CLUSTER - BETHE LATTICE * HYDROGEN
2
Fig. 1. Schematic picture of the cluster-Bethe lattice considered to describe the metal partner in the chemisorption bond.
To calculate the BDOS, the hopping parameter was adjusted to produce a chemisorption energy of 3.3 eV, just the chemisorption energy for hydrogen on W(100) in bridge position. The BDOS so obtained, i.e., at the equilibrium distance, as can be seen in Fig. 2, presents three adsorbate bond states corresponding to the three basic singlet configurations (H-W, H - - W ÷ and H+-W -. The main component of the ground state (GS) is the H - W configuration, the H - - W ÷ being also important whereas the amount of H÷-W - in the ground state composition is irrelevant. The potential curves as a function of the distance between the adsorbate and the surface metal atom (BO) have been calculated for the three states above considered. They are presented in Fig. 3. Both the H - - M ÷ and H÷-M - curves are physically unattainable on the asymptotic limit, i.e. they exist near the surface only. These curves do correspond to ionization processes GROUND
N(E)
H--M*
H+_M -
I -5
~"'o'
5
i
10
15
E(eV)
Fig. 2. BDOS of the system H/W obtained at the equilibrium distance. Peaks are labelled with the configuration of major weight.
3
Fig. 3. Potential curves for the three basic configurations. on or in the proximity of the surface which should produce features in EELS measurements. Our calculation gives an excitation energy from the GS to the H - - W * ionic state of about 8 eV as can be seen in Figs. 2 and 3. Experimental data of EELS from clean and hydrogen covered W(001) surfaces point out that hydrogen absorption produces a loss just at 8.2 eV which develops continuously with increasing exposure [15]. Although this loss was attributed to excitation from the deep lying H level ls to empty levels at or near the Fermi level, we rather interpret it as due to the excitation from the GS to the H--W+ state. Similarly, the excitation from the GS to the H+-W - ionic state, whose calculated value is about 13.5 eV should give rise to ~ loss at about that energy. However, as the ionized adsorbate leaves the surface, the metal cannot stay ionized; sooner or later the excess charge would be screened out by the highly mobile electrons of the bulk, thereby the H--W+ curve going into the H - - W curve, not considered here. This curve would lie around 5 eV (W work function) below the H - - W * curve of Fig. 3, its asymptotic limit being -- 0.7 eV (H affinity). H- desorption by this curve would correspond to the process e + H - M -+ H - - M reported by Xiang and Lichtman [8] and would have a threshold energy o f about 4 eV, close to the experimental value o f ~ 5 eV [81. Likewise, H ÷ desorption by the curve H÷-M, which would lie at about 5 eV above the H+-M - curve of Fig. 3, would give a threshold energy of around 16 eV, in agreement with experimental data [8]. REFERENCES 1.
G.E. Moore,./.
Appl. Phys. 30, 1086 (1959).
800 2. 3. 4. 5. 6.
7.
TWO-ELECTRON BOND APPROACH TO CHEMISORPTION D. Menzel & R. Gomer, J. Chem. Phys. 41,3311 (1964); P.A. Redhead, Can. J. Phys. 42,886 (1964). M.L. Knotek & P.J. Feibelman,Phys. Rev. Lett. 40, 964 (1978). A.Kh. Ayukhanov & E. Yurmashev, Soviet Phys. Tech. Phys. 22, 1289 (1977). M.L. Yu,Phys. Rev. B19, 5995 (1979). J.L. Hock & D. Lichtman, Surf Sci. 77, L184 (1978); J.L. Hock, J.H. Craig, Jr. & D. Lichtman, Surf Sei. 85, 101 (1979);J.L. Hock, J.H. Craig, Jr. & D. Lichtman,Surf Sci. 85, L218 (1979); J.L. Hock, J.H. Craig, Jr. & D. Lichtman, Surf Sci. 85,101 (1979). Liu Zhen Xiang & D. Lichtman, Surf Sci. 114, 287 (1982).
8. 9.
10. ! 1. 12. 13. 14. 15.
Vol. 51, No. 10
kiu Zhen Xian & D. kichtman,Surf Sci. 125,490 (1983). J.D. Joannopoulos & F. Yndurain,Phys. Rev. B10, 5164 (1974); F. Yndurain, J.D. Joannopoulos, M.L. Cohen & L. Falicov, Solid State Commun. 15,617(1974). Roy McWeeny, Coulson's Valence, Oxford University Press (1979). D.M. Newns, Phys. Rev. 178, 1123 (1969). B. Bell & A. Madhukar,Phys. Rev. 14, 4281 (1976). J.W. Gadzuk, Surf Sci. 67,77 (1977). C.F. Melius, R.H. Stulen & J.D. Noell, Phys. Rev. Let(. 48, 1429 (1982). N.R. Avery,Y. Elec. Spectrosc. 15,207 (1979); N.R. Avery, Surjl SoL 111,358 (1981).
51, No. 10, pp. 787-791,
0038-l
1984.
ELASTIC PLATES MODEL OF DOMAIN WALLS IN INTERCALATED P. Hawrylak, K.R. Subbaswamy Department
of Physics and Astronomy,
University
098/84 $3 .OO+ .OO Pergamon Press Ltd.
GRAPHITE
and G.W. Lehman
of Kentucky,
Lexington,
KY 40506, U.S.A.
(Received 8 May 1984 by A.A. Maradudin) The recently proposed model of thin elastic plates coupled via harmonic forces for the calculation of domain wall energies in intercalated graphite is extended to higher stages. It is found that whether coherent domains bind to each other depends on the degree of stagger. The effect of the loss of coherence and the interface between different stages are discussed.
1. INTRODUCTION THE IMPORTANCE of domains in understanding the physics of graphite intercalation has been underscored in recent experimental [l-6] as well as theoretical work [7-lo]. The domain model for staging proposed by Daumas and H&old [DH] [2] has received support from the transmission electron microscopy work of Thomas et al. [ 111, as well as the experiments on staging transformation of Clark and coworkers [3,4]. Furthermore, Flanderois et aZ. [6] deduced from their X-ray work the existence of intercalant islands with an average in-plane separation of lo-15 A, with X-ray line broadening indicating island sizes in the range of 100 to 200 A. Similar observations for the intercalation compound Ag-TiSz have also been reported [ 121. Elastic interactions leading to the formation of islands in graphite intercalation compounds have been studied extensively [7]. It has been shown that for low density of intercalant coherency strains induced in the infinite graphite matrix give rise to attractive interactions among intercalants lying in the basal plane and repulsive interactions for atoms displaced along the c-axis. This leads to the formation of domain structures such as the DH domains. During intercalation when the coherency strains are relieved the elastic interactions among intercalant islands lead to different behavior as demonstrated recently for stage 2 domains by Kirczenow [lo]. Kirczenow [lo] treats individual graphite layers as thin elastic plates coupled to each other via harmonic, nearest neighbor forces, and the intercalants as forming compact, continuous layers. An intercalant layer increases the separation between bounding carbon layers at the cost of elastic energy. When a dislocation occurs in an intercalant layer the distance between bounding layers is reduced in the vicinity of the dislocation, and the graphite layers undergo a bending distortion as well. The decrease in separation decreases the elastic energy, while bending increases the energy. The competition 787
between these two effects results in unusual oscillatory behavior of dislocation energy as a function of dislocation width. Calculations for stage 2 show that DH domains bind to each other and inplane domains have to overcome an elastic energy barrier before they can coalesce [lo]. Note that inplane domains are present in the sample for kinetic reasons during either intercalation or staging transformation. In this paper we examine the stage dependence of domain wall energetics of graphite intercalation compounds based on the elastic plates model. In Section 2 the model is described and the energies associated with coherent DH domains for stages 2-5 are calculated for staggered as well as matching domains. Also, the effect of loss of coherence is considered. In Section 3 the elastic energy of the interface between laterally coexisting stage 1 and stage 2 domains is considered in the continuum approximation. Implications for the stage 2 to stage 1 transformation are discussed. 2. ELASTIC ENERGIES OF DOMAIN WALLS We first consider DH walls. For every stage n there are several inequivalent configurations, which we denote by n/m, where m is the number of carbon layers by which neighboring stacks are staggered. A particular example, n/m = 4/l, is shown schematically in Fig. l(a). Exploiting the periodicity in the c-direction (taken to be along the z-axis) we need only consider n carbon layers. Let 5‘i(X),(i = 1,2, . . . n) describe the profiles of the YC layers, assumed to be independent of the third direction. The bending is assumed to occur over the region -R
798
TWO-ELECTRON BOND APPROACH TO CHEMISORPTION
where E~ stands for the adsorbate energy, E,n for that of the metal, Vam represents the hopping interaction between metal and adsorbate electrons and U is the Coulomb integral of the absorbate. The charges of the absorbate and metal atoms are given by qA
2 -b = OLam
2 +
q M = Olam
are Green's function which describe the propagation of two electrons. The image potential considered is a quantumm:d,anical generalization of the classic expression [12] 1
V/m - 4(R +/@~)
2a~ 2
(3)
20Lmm,
and the total charge will be =
N(E) = - - - l m ~ K i i ( E + i p )
2,
77
since we are considering only two electrons. This implies the following relation for the coefficients 2 +c~+
Olam
2
O~rn m
= 1.
Once the bond between the adsorbate and the metal atoms is described, the metal band-structure must be introduced. To that end a group orbital kbm) is defined (11) 1
kbm) = f f ~
Vaklk),
(4)
k
so that the metal could be considered reduced to an effective atom. The wave function for the system so defined will be of the form defined in [1 ], but with 1 OLam :
~
~,
VakOLak
(5)
l
~m,. : ~
Z V., Vo~.~*k' kk'
where, from normalization requirements IVak12 = I dEIV(E)I2P{E),
p2 = Z
(6)
k
and o(E) stands for the density of state of the clean metal. At this point we introduce a bond Green's function K(co), which describes the correlated motion of the two electrons and a joint density of states for both electrons. The bond Green's function is given by
v]
[a~°(co) t--;
v co-(2E. + u) + v,m 0 K(co) = o G2~(co)/
where both
1 i dE IV(E)I2p(E) area(co)
=
~.
co - E o (8)
Gram(co) = ~1 f f dE dE' V(E)V(E')p(E)p(E') co -- E-- E' + Vim
(9)
R being the adsorbate-metal atom separation. Hence, the bond density of states (BDOS) will be 1
qA + qM
Vol. 5 I, No. l0
= ~ Ni(E), i
(10)
where Ni(E) represents the contribution of the ith configuration or the configurational DOS. Therefore, the charge population will be the residue Io~ml2 =
ResKii(co)co=i,:,n,
if Kii(co ) has a pole at co =Em -- iTm. Of course both the Vam and Vim depend on the distance between the adsorbate and metal surface atoms, and the interaction Vain as a function of energy gives a measure of the localization of the absorption bond. If Vain is a smooth function of energy then the bond will tend to be localized. For the sake of illustration we shall take the strictly localized case i.e. the interaction is independent of the energy, V(E) = X. This will be the only parameter entering in this theory, just the hopping integral of the Anderson-Newns Hamiltonian. As an application of the model here presented the system H/(100) will be considered, hydrogen being chosen since, conceptually, it is the simplest adsorbate to study. An alternative, somewhat related approach to desorption (of protons) from transition metal surfaces using configuration-interaction techniques on finite clusters was reported by Melius et al. [14] where the importance of double excitation processes in the formation of H + ions was stressed. They give an alternative to direct excitation, suggesting a novel surface predissociation excitation path which involves direct dipole-allowed
11,
(7) l
transitions to states which then undergo curve crossing with double excited dissociative states. Our approach is rather different, the emphasis being placed here on H- ions. In Fig. 1 a schematic picture is given of the Bethe-lattice taken for both the bulk and the surface. The cluster considered in the bond with the hydrogen atom is shown as well.
Vol. 5 l, No. l0
TWO-ELECTRON BOND APPROACH TO CHEMISORPTION BULK
799
E (ev;
>
H%M
\
SURFACE
-
H-_M +
CLUSTER - BETHE LATTICE * HYDROGEN
2
Fig. 1. Schematic picture of the cluster-Bethe lattice considered to describe the metal partner in the chemisorption bond.
To calculate the BDOS, the hopping parameter was adjusted to produce a chemisorption energy of 3.3 eV, just the chemisorption energy for hydrogen on W(100) in bridge position. The BDOS so obtained, i.e., at the equilibrium distance, as can be seen in Fig. 2, presents three adsorbate bond states corresponding to the three basic singlet configurations (H-W, H - - W ÷ and H+-W -. The main component of the ground state (GS) is the H - W configuration, the H - - W ÷ being also important whereas the amount of H÷-W - in the ground state composition is irrelevant. The potential curves as a function of the distance between the adsorbate and the surface metal atom (BO) have been calculated for the three states above considered. They are presented in Fig. 3. Both the H - - M ÷ and H÷-M - curves are physically unattainable on the asymptotic limit, i.e. they exist near the surface only. These curves do correspond to ionization processes GROUND
N(E)
H--M*
H+_M -
I -5
~"'o'
5
i
10
15
E(eV)
Fig. 2. BDOS of the system H/W obtained at the equilibrium distance. Peaks are labelled with the configuration of major weight.
3
Fig. 3. Potential curves for the three basic configurations. on or in the proximity of the surface which should produce features in EELS measurements. Our calculation gives an excitation energy from the GS to the H - - W * ionic state of about 8 eV as can be seen in Figs. 2 and 3. Experimental data of EELS from clean and hydrogen covered W(001) surfaces point out that hydrogen absorption produces a loss just at 8.2 eV which develops continuously with increasing exposure [15]. Although this loss was attributed to excitation from the deep lying H level ls to empty levels at or near the Fermi level, we rather interpret it as due to the excitation from the GS to the H--W+ state. Similarly, the excitation from the GS to the H+-W - ionic state, whose calculated value is about 13.5 eV should give rise to ~ loss at about that energy. However, as the ionized adsorbate leaves the surface, the metal cannot stay ionized; sooner or later the excess charge would be screened out by the highly mobile electrons of the bulk, thereby the H--W+ curve going into the H - - W curve, not considered here. This curve would lie around 5 eV (W work function) below the H - - W * curve of Fig. 3, its asymptotic limit being -- 0.7 eV (H affinity). H- desorption by this curve would correspond to the process e + H - M -+ H - - M reported by Xiang and Lichtman [8] and would have a threshold energy o f about 4 eV, close to the experimental value o f ~ 5 eV [81. Likewise, H ÷ desorption by the curve H÷-M, which would lie at about 5 eV above the H+-M - curve of Fig. 3, would give a threshold energy of around 16 eV, in agreement with experimental data [8]. REFERENCES 1.
G.E. Moore,./.
Appl. Phys. 30, 1086 (1959).
800 2. 3. 4. 5. 6.
7.
TWO-ELECTRON BOND APPROACH TO CHEMISORPTION D. Menzel & R. Gomer, J. Chem. Phys. 41,3311 (1964); P.A. Redhead, Can. J. Phys. 42,886 (1964). M.L. Knotek & P.J. Feibelman,Phys. Rev. Lett. 40, 964 (1978). A.Kh. Ayukhanov & E. Yurmashev, Soviet Phys. Tech. Phys. 22, 1289 (1977). M.L. Yu,Phys. Rev. B19, 5995 (1979). J.L. Hock & D. Lichtman, Surf Sci. 77, L184 (1978); J.L. Hock, J.H. Craig, Jr. & D. Lichtman, Surf Sei. 85, 101 (1979);J.L. Hock, J.H. Craig, Jr. & D. Lichtman,Surf Sci. 85, L218 (1979); J.L. Hock, J.H. Craig, Jr. & D. Lichtman, Surf Sci. 85,101 (1979). Liu Zhen Xiang & D. Lichtman, Surf Sci. 114, 287 (1982).
8. 9.
10. ! 1. 12. 13. 14. 15.
Vol. 51, No. 10
kiu Zhen Xian & D. kichtman,Surf Sci. 125,490 (1983). J.D. Joannopoulos & F. Yndurain,Phys. Rev. B10, 5164 (1974); F. Yndurain, J.D. Joannopoulos, M.L. Cohen & L. Falicov, Solid State Commun. 15,617(1974). Roy McWeeny, Coulson's Valence, Oxford University Press (1979). D.M. Newns, Phys. Rev. 178, 1123 (1969). B. Bell & A. Madhukar,Phys. Rev. 14, 4281 (1976). J.W. Gadzuk, Surf Sci. 67,77 (1977). C.F. Melius, R.H. Stulen & J.D. Noell, Phys. Rev. Let(. 48, 1429 (1982). N.R. Avery,Y. Elec. Spectrosc. 15,207 (1979); N.R. Avery, Surjl SoL 111,358 (1981).