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The local electronic densities of states at the surface of disordered alloys
Surface Science 57 (1976) 540-558 0 North-Holland Publishing Company
THE LOCAL ELECTRONIC
DENSITIES OF STATES AT
THE SURFACE OF DISORDERED
J. L. MORAN-LbPEZ
ALLOYS *
* * . G. KERKER and K.H. BENNEMANN
Institute for Theoretical Physics, Freie Universitdt Berlin, I Berlin 33, Arnimallee 3, Germany
Received 12 January 1976; manuscript received in fiial form 27 February 1976
The local densities of states at a clean as well as at an adsorbate-covered surface of a disordered alloy is studied within the tight-binding approximation using the first terms in a continued fraction series for the electronic Green’s function. The dependence of the adatom local densities of states on the alloy composition, adatom position, special local atomic environment around the adatom and the hopping integral between the adatom and its nearest surface atoms is given. Also the binding energy of an adatom is calculated as a function of the alloy composition. We discuss the significance of electron charge transfer for chemisorption and in general for the determination of the electronic density of states at the surface.
1. Introduction
It has been widely suggested portant
role in determining
have shown
that the d-electrons
catalytic
[5,6] that transition
properties
of transition
of metal surfaces
metals
play an im-
[l-4].
Experiments
metal alloys like CuNi alloys exhibit
haviour with interesting technical applications. Recently, the behaviour of d-electrons near transition
metal surfaces
catalytic
be-
has been
[7-l 1] as well as the problem of chemisorption metals [ 12-171. None of these calculations have been extended to alloy surfaces. The continued fraction technique for determining the electronic Green function is the most appropriate method to treat alloy surfaces. In the following we present simple calculations [ 181 of the local densities of states at clean as well as at adsorbate covered surfaces of disordered substitutional alloys. The local density of states (LDS) is given as a function of alloy composition, special local atomic environment near the adatom and for different positions of the adatom. The results obtained at a pure metal surface agree with those given by Cyrot-Lackmann et al. [lo] using studied
* Supported by DFG BE 58514. ** Supported in part by Consejo National de Ciencia y Tecnologia (MBxico).
a more elaborate moment calculation. We show how electronic charge-transfer affects the d-electronic energies cd and the LDS of the first three layers at the surface /19,20]. Finally, we also c~c~~la~ethe binding energy of an adatom as a function of alloy compositja~. In section 2 we briefly describe the theory for the electronic structure at the SWface of transition metal alloys which was presented in an earlier paper [t 81. In section 3 we present the results obtained by our method for the LDS at a clean (100) alloy surface of a cleaved face centered cubic lattice. Results for the LDS on adsorbate atoms at an alloy surface are given in section 4. In section 5 we present results for the dependence of the binding energy on alloy composition. In section 6 we discuss QUr results.
2. Theory far d-ckctrons near an alioy surface In an earlier paper [18f we have developed a tl~t”b~nd~ng type theory to determine the electronic structure at alloy surfaces. In the following we s~rnmar~~c briefly this theory, extend it and discuss the appro~jmat~ons used. The electrons in a binary alloy A,B,_, with substitutional disorder are described by the usuat tightbinding H~~niltonian (2.1) where the electronic energy levels pi = c: + U&?ti+ Udsn: ,
(2.2)
sari take the values cA or fB defending on whether the site i is occupied by an Aatom or a B-atum. Here, Ui denotes the intra-atomic Coulomb interaction on site i with ni d-electrons and Ud, denotes the Coulomb interaction between the d- and s-electrons on site i. The energies E: are supposed to take into account s-d hybridization The hopping integaals tij which describe electronic transitians between states centered on atomic sites i and i are assumed to be concentration ~dependent. cl and ci are the usual creation and ~~h~lat~on operators for d-electrons on site i. The focal electronic Green function Gij describing d-electrons on the atom i is given by G&‘) = (E - ei - A,)- 1 ,
(2.3)
where the electron self-energy Aj resulting from the hopping of the electrons between the atoms is given by (2.4a)
which results from aIf electron ho~~~n~ processes starting and ending rat the site j and avoiding the sites i and j in-between. The self-energies A! and AZ result from delectron hopping processes avoiding sites i, j, I, m and are given by expressions which are similar to the one for A; 12 i.-23 1. Note, that the coherent potential approximation (CPA) for determining G&Z) in bulk alJoys is obtained by replacing the electronic energies er by an average energy Z [23j in eqs. (2.4). furthermore, it is ihteresting to note that vve find (see section 3) essentially the same resufts for the bulk electron density of states as obtained by Velicky,et a1. f25] using CPA if we keep only the t2=terms in the expansions for Ai and A; and use the approximation Ag=A{. The above theory is now applied to d-electrons at the surface of transition metal alloy. Near the surface the Green function Gli(~) depends not only in the type i of atom but also on the perpendicular distance from the surface. Therefore, we use for the Green functions the notation GL,i which describes d-electrons at atams of sort i in the f;-th atomic layer parallel to the surface. The average electron Green’s function in the first atomic layer at the surface is G, = “Gl,A f (1 -XI G,,,
GI i(E) = (E -
fI,f
-
f2.5)
7
AI)-”
I
f2.6)
Here, eI,;denotes
ei for atomic sites in the first layer. Sim~arly~ the Green’s fun~ti~l~s GIi? GH$ GIIK and GIIId refer to the d-electrons in the second and third atomic layer away from the surface. For s~rn~l~cit~ we assume now that the effects due to the surface are significant only within the first three surface layers. Therefore, we use for the atomic sites beyond the first three surface layers the bulk Green’s function (2.7)
G=xGA+(l-x)Gn, where G,-(E) = (I!?- f; - A&l
,
and Ar, is given by eq. (2-4:) for the case of a bulk atom. The expression for the electron selfenergy A, referring to a site in the first layer is easily obtained from “4% (2.4a). Neglecting terms of higher order than f-Xin eq. (2.4a) and performing the summation over the nearest neighbour atomsj of sort A and f3 En the first and second layers one finds the expression A, = rot2
J.L. Morh-LdpezlLocal
electronic density of states at surface of disordered
alloys
543
Here zO denotes the number of nearest neighbour atoms j lying also in the first layer and z1 the number of nearest neighbour atoms lying in the second surface layer. Note, A; s Ai if i and j refer to atomic sites in the first surface layer. Similarly, A; c AfI if i refers to the first surface layer and j to the second surface layer. We put tii s t for nearest neighbour atoms. The electronic energies eI,*, E~,~, eII,*, etc., denote the energy Ei for atoms A and B in the first surface layer I, second surface layer II, etc. Note, in the derivation of eq. (2.8) we have taken into account that we get different contributions to A, and Ai if the nearest neighbour sites j are in the first or in the second layer. Expressions similar to AI are obtained for AI, and AIII again by using eq. (2.4a). For determining A; we use the approximation A?= Lu; which is exact for a Bethe lattice [24]. To get a feeling for this approximation one should remember that A7 is obtained from A{ if one avoids in determining A/ hopping processes which involve the atomic site i. Thus, the approximation on Af should become better at the surface, since the number of neglected paths is smaller. Then we get straightforwardly from eq. (2.4b) [18] a set of six coupled equations with Ai, A:I, Aif, AI’, AkI and AFI as unknowns in terms oft, z,,, zl, eI i, EII,i. EIII,i, the bulk energy levels ei and A:. For the self-energy Al, referring to the bulk one finds from eq. (2.4a) the expression Ab = (2zltz0)t2
x E-eA-A;
E-en-At
(2.9)
Thus, the electronic Green functions GI,i, GI, i and GI,I i are determined by solving the six coupled equations for the unknowns Ai, AiIi etc. Then the electronic density of statesfor an atom of type i within the j-th layer is determined from the electronic Green function by the formula Nj,i(E) = - i Im Gi,i(E) . Note, in determining the d-electron Green functions Gj,i we have assumed that surface effects occur only in the first three surface layers. This assumption seems reasonable in view of previous tight-binding type calculations at pure metal surfaces and it simplifies considerably the numerical calculations. However, note that the theory for alloy surfaces can easily be extended to the case where we allow also the 4th and Sth, etc., surface layer to be affected by the surface. Furthermore note, that in determining Gi i we have neglected the contributions to Ai and A; which are of higher order than t2. It is known from calculations for the bulk, that these approximations affect essentially only the fine structure in Ni,i(E), but should in general not be of essential significance for an exploratory model calculation and for determining the concentration dependence Of Ni,i(E), the binding energy of chemisorbed atoms, electron charge transfer, etc. Indeed, we obtain with these approximations for Nj i(E) essentially the same results as found for bulk calculations for CuNi alloys using CPA which sums up all terms in eq. (2.4a). However, depending on the crystal structure terms of higher order than t2 might be of importance for
544
J.L. MothL6pezlLocul
electronic
density of states at surface of disordered alloys
determining the variation of surface properties obtained for different surface crystal planes. The approximation tii F t for nearest neighbour atoms i,j should be valid if the d-band width of the pure metals A and B is not too different. Finally, the approximation A? = A/ leads to similar bulk densities of states as those obtained by CPA calculations with a Hubbard density of states. At the surface this approximation should become better than in bulk, since the error which is introduced, is smaller because the number of possible electron hopping paths is reduced.
3. Numerical results for the LDS near the surface of a transition metal alloys In this section we present the average density of states and LDS in the first three laye layers of a (100) cleaved fee lattice. From the agreement between experiment and theory (CPA, etc.) we conclude that the approximations used in our alloy surface theory are valid for CuNi alloys. Therefore, in our calculations we use parameters corresponding to CuNi alloys [26]. Note that for CuNi alloys one can neglect electronic charge-transfer between the Cu and Ni atoms. This is plausible since the Fermi energies of Cu and Ni are almost the same. Furthermore, by neglecting electronic charge-transfer in CuNi alloys one obtains good agreement between experimental and numerical results for the bulk electronic density of states [26].
AE_AE
-20
(4
Ao2Boa
tb)
A0.4B0.6
Fig. 1. Average density of states N,(E) of the first three surface layers (i = I, II, III) and a bulk layer b of an alloy A,B1, corresponding to Cu,Nil,: (a) x = 0.2, (b) x = 0.4. We express the energy in all figures in units of 0.1227 Ry, which is half of the Cu d-band width. We choose the zero of energy such that for the bulk ENi’ 0.51 = 612 and eCU = -612.
J.L. Mor&Ldpez/Local electronic density of states at surface of disordered alloys
545
NE)
i* )
Cb)
Ao.2Bo.a
Ao4Bo.6
Fig. 2. Local density of states per atom Ni(E) for different concentrations in an alloy corresponding to Cu,Ni,,. The labels Cu, Ni and Ave refer to a Cu site, a Ni site and an average site, respectively. The curves labeled I and b refer to the surface layer and the bulk, respectively.
In the following we express the energy in units of 0.1227 Ry which is half of the CU d-band width. We take for 6 = ENi- eCu the value 1.02 and use f = 0.206 in accordance with prior bulk calculations. For the alloy averaged electron self-energy Ab, at a bulk site one obtains from eq. (2.4b) the expression l-x
x
A; =(2z1+zo-l)t2
E-eA-A~‘E-eB-A~ which can be rewritten
as a third order equation
(At)3 - 2E(Ak)2 + [E2 + EAEB + (2zl+zo
1 ’
(3.1)
for Ab, [24,27] -1) t2]Ai (3.2)
Note, in deriving eq. (3.1) we used AT =:A{ and neglected terms of higher order than t2. The expression for Ai given by eq. (3.2) is equivalent to the one used in the coherent potential approximation (Velicky et al. [25], eq. (4.32)). Then with these parameters, we solve by iteration the six coupled equations for At, A,;, etc., which were discussed in section 2. The Fermi energy E, is calculated selfconsistently by using [28] EF s
..-m
iVav(E)dE=XNce+
(l-x)NNi
2
(3.3)
where N,,(E) is the average bulk density of states, Arti = 10 and NNi = 9.45 electrons. The approximations used are essentially the same used in CPA calculations on bulk CuNi alloys and should have the same validity. These approximations should
J.L. Morh-LdpezlLocal
546
(‘1
electronic density of states at surface of disordered alloys
cb)
Ao.2Bo.a
A~+BO6
Fig. 3. Average density of states N&5), for two different concentrations, of the first three surface layers (i = I, II, III) and a bulk layer b of an alloy Cu,Nil,, assuming no electron charge transfer between the surface layers.
of other transition metal alloys like for example RhNi, AgPd, etc. In fig. 1 we show results for the average local d-electron densities of states for concentrations x = 0.2 and x = 0.4. In fig. 2 we show the LDS on Cu, Ni and average sites for the same concentrations. All these results are obtained by using hj = fi,A - ei,B = &bulk, i = 1,2,3. Note, this assumption implies electron charge-transfer between the atomic layers at the surface. The average number of electrons per site in the i-th surface layer is namely then different from the corresponding number at a bulk site. This is found from ni = IdEN,( Consequently, allowing no d-electron charge-transfer between the surface layers +d,i = nd,bulk) [ 19,201 implies layer dependent shifts in the difference hj = Ei,A - ej,B of the energy levels. Here, 6i refers to the ith atomic surface layer. These shifts in 6j are then calculated by using apply as well to a number
EF s
dE [&@&
h,,, &,I,) -Nbdk(E)j
= 0,
i= I,II,III,
(3.4)
-cc
for the first three layers. In figs. 3 and 4 we show how the LDS shown in figs. 1 and 2 are modified for the case where no d-electron charge transfer is allowed between the layers. In fig. 5 we show how 6i = ei,A - fi,n , i = 1,2, changes with concentration.
J. L. Mordn-LdpezjLocal electronic demiry of states at surface of disordered alloys
547
NEJ
E
.
la1
AO ?O
E
(b)
8
Ao4Bos
Fig. 4. Local density of states of the surface layer, NJ,~(E), and for the bulk, Nb(E) m ;i Cu site, on an average site (Ave), respectively. No electron charge transfer between the surface layers is allowed.
_.-._
.- -.___
= -.--I-*
-. %
1.2(
1.10
1.M
I-
--I
_---
%I
-0-
0.0
0.2
0.6
0.4
0.8
1.0
X
Fig. 5. Concentration dependence of61 = EI,Ni - q,cu and Sff= E[I,Nj- EII,~~ in energy units of half of the Cu d-band width. We take for the bulk &b = 1.02.
548
J.L. MorhLdpezfLocal
electronic density of states at surface of disordered alloys
Note, in order to calculate the change Ani in ni due to electronic charge-transfer we have to determine Ei self-consistently using Ei = ep+ Uni. This does not change the theory presented but only complicates the numerical analysis. Note, d-electron conservation implies ‘ci Ani = 0. If s- and d-electrons are treated together than one obtains ci(A,P + An2 = 0, where An; is the change in the number of s-electrons within the Wigner-Seitz cell of the i-th atom due to s+d and d-+d charge-transfer.
4. Local electron density of states on adsorbed atoms To avoid the numerical complications due to electron charge-transfer and Coulomb interactions we restrict ourselves to adatoms of the same kind as those of the substrate. We study how the LDS on an adatom changes with (a) the alloy composition of the substrate, (b) the local atomic environment of the substrate around the adatom and (c) for different adatom positions. We calculate the LDS on the adatom as follows. The Green’s function for the adatom is given by G,(E) = (E - e, - A,)-’
,
(4.1)
where A, is calculated by using the expansion
Cl
Ave
cu
c2
N,
(2.3a)
c3
c4
Fig. 6. Different adatom (a) positions A, B and C on a (lOO)-fee lattice surface are shown. Also the different adatom-substrate configurations Cl, C2, C3 and C4 are shown. The black circles refer to Cu atoms, the open to Ni and the dashed to average atoms, respectively.
J. L. Morbn-Ldpez/i,ocaI electronic density of stutes at surface of disordered alloys
549
*02B08
‘Ni
Fig. 7. Locat density of statesNa(E)on the adatom for its positionsA, B, C and for the alloy concentrationx = 0.2. We put ca = ENi,b, t’= t, 6i = 6b and the units are those given in tlg. 1.
A,=x
t’2 + r:
i+aE-Ei-Ai
d2t
jziaa (E-fi-Ai)(~-Ei-A~)‘ ,
(4.2)
Here ea is the adatom electronic energy and t’ is the hopping integral between the a datom and its nearest neighbour atoms. Ai is the first layer renormalization energy given by eq. (2.8). To get more detailed information on the shape of the local density of states at the adatom we have included t3 terms in the expansion (4.2) for the local self-energy. In fig. 6 we show the various adatom positions (A), (B), (C) and the local atomic substrate configurations (Cl), (C2), (C3) and (C4) which we have studied. In figs. 7 and g results are presented for the LDS of the adatom when it is in the positions (A), (B) and (C). We use ea = eNi , t’= t and put x = 0.2 and x = 0.4. In figs. 9 and IO we show results for the adatom LDS for the substrate configurations (Cl), (C2), (C3) and (C4). It is interesting to note, see fig. 11, that our results for x = 0 reduce to those obtained by Cyrot-Lackmann et al. [IO] using the moment expansion method. We conclude from this fact that the local density of states on the adatom does not de-
550
J.L. Mordn-Ldpez/Locai
A
0.4
electronic density of states at surface of disordered alloys
B 0.6
1.
-2.0
‘*E
0.0
r
2.0
‘Nt
Fig. 8. Local density of states Na(E) on the adatom for its positions A, B, and C. We use Ea = eNi,b, t’= t, 6i= 6b and the alloy concentration x = 0.4.
pend very sensitively on the fine details of the substrate density of states but more on (a) the position (A),(B) or (C) of the adatom, (b) the hopping integral t’ between adatom and substrate and (c) the position of E, relative to eI i of the substrate. By alloying we are changing cont~uous~y the relative position of ea and eI,i of the substrate. As a result the LDS on the adatom becomes more localized as we go from x = 0 to x = 1 .O. This effect can be best seen for the configuration (C4) when the adatom has four Cu nearest neighbours.
5. The dependence of the binding energy of an adatom on alloy composition The binding energy EB for a chemisorbed atom is given by the energy of the semiinfinite crystal alloy plus the isolated atom minus the energy of the semi-infinite crystal alloy plus the adsorbed atom. This gives for E, the expression
J.L. Mom%-LdpezlLocalelectronic density of states at surface of disordered alloys
551
1’=1
A0.2B0 8
Ea=Eb,N,
EF
I
-2.0
0.0
r
I
2.0
E
‘Ni
Fig. 9. Local density of states Na(E) on the adatom for the different substrate confgurations Cl, C2, C3 and C4. We use ea= ENi,b, t’= t, 6i= sb and the alloy concentration x = 0.2.
where rrz = JdE N:(E) and na= IdEN, denote the number of electrons on the adatom. ni refers to the case where no electron charge-transfer occurs. Ua denotes the effective intra-atomic Coulomb interaction between the electrons on the adatom. The LDSA$E) refers to the case where no adatom is present. The LDS N,(E) and Ni(E) take into account the interaction between the adatom and the substrate. EL is the contribution to EB which results from inter-atomic Coulomb interactions and includes the Madelung energy. If we neglect inter-atomic Coulomb interaction and electron charge-transfer between the substrate atoms and the adatom, then the binding energy is given by
552
J. L. Morh-Ldpez/Local
electronic density
of statesat surface
of disordered alIoys
1” t
Ao.4Bo.6
Ea=Eb,NI
0.0
-2.0
2.0
E
t ENi
Fig. 10. Local density of states @a(E) on the adatom for the different substrate conf~urations Cl, C2, C3 and C4. We use ea = ENi,b, t’= t, 6i = Sb and alloy concentration x = 0.4.
E, =
n,e,+
sEFC~~(E)EdE-~ [Eli +N,(E)] .-co
i
EdE .
(5.2)
-m
In fig. 13 we show how the adatom binding energy EB(x) changes as a function of concentration x for different adatom-substrate configurations. The results are obtained by assuming that no electron charge-transfer occurs. While in general electron charge-transfer may be important for determining E, in the case of CuNi alloy surface electron charge-transfer are presumably small and should not affect much the concentration dependence of E&). 6. Discussion We have presented a microscopic theory for the electronic structure at transitionmetal alloy surfaces, which play an important role in catalysis. Our theory is relatively
J.L. MorhLbpezfLocal
electronic density of states at surface of disordered alloys
553
N,(E)
t
t’=t Ea=Eb.Ni
1
-20
2.0
0.0 t
Fig. 11. Local density of states Na(E) of the adatom on pure Ni for the positions A, B and C. We USe Ea= CZNi,b, t’= I and 6i’ 6b.
simple and needs to be improved, for example, in order to study in detail the electronic density of states at the surface of alloys and important effects due to the interplay of the s-electron and d-electron charge redistribution at the surface. Nevertheless our results are instructive for what one may expect in a more realistic theory and demonstrate the significance of various electronic parameters for chemisorption at alloy surfaces. Due to the various approximations used in the theory presented here our results should apply best to CuNi and similar alloys. Due to the approximations Ay = A{and the neglection of terms of order higher than t2 in Ai and A{, etc., we obtain simplified LDSNi(E) without fine structure. Consequently, we have to extend our theory in order to determine surface properties which depend on the detailed structure present in N,(E). However, the good agreement between the results obtained for the bulk Ni(E) with our theory and CPA suggest that our theory should give correctly the overall shape of the LDS. Furthermore, the good agreement of our results obtained for pure Ni with the above approximations and those obtained by Cyrot-Lackmann et al. [lo] using a moment expansion technique indicates that the approximations used in this paper do not restrict too much the validity of the
554
J.L. Morrin-Ldpez/Local
electronic density of states at surface of disordered allo.vs
results obtained for the LDS and E, and in particular their dependence on the alloy composition. However, if electron charge-transfer between adatom and substrate occurs, then EB(x) needs to be determined by calculating EL and determining E, = e,” + (/ana and Ei = ep + Uini self-consistently. This is in general also important for calculating EB(x), since the electron charge-transfer will vary with the concentration x. However, for Ni and Cu adatoms on CuNi alloys and similar alloys for which electron charge-transfer are presumably small the results presented for EB(x) should be valid approximately. First in figs. l-4 we show results for the electronic density of states N(E) at the surface of an alloy corresponding to Cu,Nil_, . One sees that mainly only Ni(E) of the first two surface layers is changed with respect to the bulk. As expected, one observes that due to the surface N(E) tends to “narrow” and to split into two peaks. The surface has a more dramatic effect on N,i(EF) than on Ncu(EF). Also as shown by fig. 1 (which shows results referring to 6, = Gbulk, i = 1,2,3, which implies d-electron charge-transfer between the atomic layers at the surface) and fig. 3 (which shows results for 6,f6,,k such that no d-electron charge-transfer occurs) electron cnarge-transfer affects mainly only NNi(E). This will also be the case if the electron charge-transfer result from adatoms or s-d transitions. Note, if s-d and d-d electron charge-transfer are small at the surface, then one expects the results shown in figs. 3 and 4. The results shown in figs. 1 and 2 should correspond to the situation where s-d and d-d electron charge-transfer occur. Fig. 5 shows the layer dependent shifts in 6i = Ei,A -ei,B which should result if no d-electron charge-transfer between the layers occurs. Note, the magnitude of the shifts and their concentration dependence. In figs. 7-10 results are shown for the electronic density of statesAra of an adatom. The various results show how the shape ofNa(E) and E, depend on the adatom position, the substrate composition and local atomic environment effects (figs. 9, 10). The results obtained for the various adatom positions A, B, C may shed some light on the significance of atomic disorder, crystal irregularities, etc., at the surface for chemisorption. The results show that the adatom couples the stronger (see change of the peak and width in N,(E)) to the crystal surface the more nearest neighbours it has. Similar results are found if we increase the difference (t’ - t). In figs. 9,10 one observes that for E, 2 ENi the adatom couples more strongly to Ni surface atoms. See, for example, that for configurations C4 N,(E) is most similar to 6(E- ~a). We find that only the local atomic environment effects due to the nearest neighbours of the adatom are important. The results indicate that cluster effects and segragation at alloy surface are important for chemisorption. To understand the shifts in E,, which are shown in figs. 7-l 1, note that they result primarily from the second term in eq. (4.2). Consequently, no shift is observed for the adatom position A for which the second term in eq. (4.2) does not appear within the approximation used. In contrast, for the adatom positions B and C the second term in eq. (4.2) is important and leads to shifts in the center of gravity of the adatom electronic states. Since more terms of order t3 occur for the adatom position C than for B, one observes the largest shift for E, for the adatom position C.
J.L. Monk-Ldpez/Local electronic density of states at surface of disordered alloys
555
..40
d’ ..20
EF
t -2.0
0.0 t t E(l) E(2)
Fig. 12. Local density of states N,(E) on the adatom on pure Ni with different Ef,Ni. The energylevel ~on~g~ations (1) and (2) refer to the cases where electron charge transfer between the SUrfaCelayer iS allowed (EIBi = Eb,Nj) and iS not allowed (eI,Ni= eb~jj) and is not allowed (q,~j +; eb Ni), respectively. We Use ea = 0.5 1 and t’= t. Note the energy scale in the insett is different thao’in the main figure.
The changes in the shape ofilra@) observed for the various adatom-substrate con~gurations correspond to the occurrance of bonding and anti-bonding states when the adatom couples to the metal surface. In fig. 11 we show results for a pure Ni surface. Note, these results agree excellently with those obtained by Cyrot-Lackmann et al. [lo] taking into account a more detailed realistic substrate density of states N(E). This could indicate that our results for N,(E) may not change much if we would use a more realistic substrate density of states at the alloy surface. Fig. 12 demonstrates how the results on N,(E) depend on (ea - fI,Ni). Note, configurations (1) and (2) with eI,Ni > e, may correspond to the cases when d-d electron charge-transfer and no d-electron charge-transfer occur between the surface layers, respectively. Since for Er,Ni(2) > f, one expects a partial rising of the adatom electronic States towards Er,Ni, the shift of the peak in N,(E) should be larger for the
556
J.L. Morh-L6pezfLocal
electronic density of states at surface of disordered alloys
I
“J(f :F)
I
Ni
“I”
:.-Cl
\
(EF=EF,N,)
0.4-
0.2 -
0.0 Ni
+..,
t’=t %= Lb,N,
0.2
..
*. . ‘...
., . .
. .. .
0.4
0.6 X
0.8
1.0 CU
Fig. 13. The biding energy of an adatom as a function of alloy composition. We put the adatom in the central position C and assume substrate configurations Cl, C3 and C4. We use ea = ENi, t’ = t and the same energy units as in all other figures. All curves are calculated by taking into account the dependence of EF(x) on x, except Cl (EF= EF,Ni) which is calculated for COnStaM EF. The insert shows the average and the Ni surface electron density of states at EF.
configuration (2) as is observed in fig. 12. Also, iince for EI,Ni > ea adatom electronic states are pulled away from E, by the metal background density of states (for example, towards Er,Ni(2)) more strongly than for the configuration (l), the height of the peak in N,(E) is more strongly reduced for configuration (2) than for (1). Fig. 13 shows results on the binding energy Es. For increasing the difference t’- 1 one finds that EB increases, but the characteristic dependence of E&x) on x remains nearly unchanged. Comparing the results obtained for EF(x) (EF (x) < EF(Ni)) calculated from eq. (3.3) and for EF=EF(Ni) indicate that&(x) depends sensitively on (&-@and hence on electron charge-transfer. Note, EF(x) & E&Ni) is expected if one includes s-electrons in calculating E&) and also from work function data (piecu = 0.04 eV). The results for E, shown in fig. 13 have been obtained by assuming E, = ENi,bulk. The
J.L. Morh-Ldpez/Local
electronic density of states at surface of disordered
alloys
557
curves C4, Cl, and C3 with EP(x) determined selfconsistently result if no electron charge-transfer occurs. The curve Cl with E, = E,(Ni) implies electron charge-transfer and a filling of the Ni d-band at x y 0.6. In reality such a charge-transfer may result from the electrons of the adatom as well as from the redistribution of the s-electrons which tend to leak out of the metal. Of course, for a quantitative analysis, electron charge-transfer effects should be calculated selfconsistently by taking into account Madelung energies and that the energies ed depend on nd. The dependence of E, on the concentration x arises mainly from the following facts. We note that E, is mainly determined by the contribution JdE(E - e,)N,(E). Consequently, since EP(x) 2 EF(0) +x X const. , and N,(E)
= n.&E - c& + x AA’?(E)
+ (1 -x) A?(E)
,
see eqs. (4.1) and (4.2), one expects EB(x) to vary approximately linearly with (1 -x> for an average alloy surface. Here, we denote the change in N,(E) due to Cu atoms and Ni atoms by ANCU and ANF, respectively. The results obtained for C4 indicate that ANCu < AiN’. The risults obiained for the local atomic environment C3 and C4 indicate that the binding of the adatom is strongly determined by the alloy atoms which are nearest neighbours of the adatom. Assuming that the activation energy Ea for a chemical reaction at the surface is proportional to E,, then the results for E, which were obtained for the average substrate configuration Cl with EF(x) yield lnK=a+bx, which agrees with experiment [29,30]. Here K is the catalytic activity and a and b are constants. The results shown in fig. 13 were obtained without using the rigidband approximation. Similar results for EB(E) are expected for other transitionmetal alloys like Ag,Pd,_,, Ag,Rh,_,, etc. For alloys Ni,Pd,_,, Rh,Pdl_x, etc., we expect E(x) not to vary much with x. Our results indicate that E, is strongly determined by the perturbation of N,(E) by the surface. This perturbation is large if the local density of states of the surface atoms in the vicinity of the adatom is large for E = E,. Therefore, by affecting NI,~(E) for E 2 E, magnetism may have a strong effect on E,. If electron charge-transfer is important then the Madelung energy could contribute significantly to E,. Such effects could be calculated directly by extending straightforwardly the theory presented here, for example. It would be more difficult to treat such effects within the Debye-Htickel theory.
558
J.L. Morcin-Ldpez/Local
electronic density of states at surface of disordered
alloys
References fl] [2] [ 31 [4] [5] [6] [7] [8] [9] [JO]
D.A. Dowden, J. Chem. Sot. (1950) 242. G. Rienacker, Abh. Deut. Akad. Wiss. Berlin, Kl. Chem. Geol. Biol. (1955) 3. D.D. Eley, 2. Elektrochem. 60 (1956) 797. F.F. Wolkenstein, The Electronic Theory of Catalysis (Pergamon, Oxford, 1963). J.H. Sinfelt, J. Catalysis 24 (1972) 283. J.H. Sinfelt, J. Catalysis 29 (1973) 308. D. Kalkstein and P. Soven, Surface Sci. 26 (1971) 85. R. Haydock and M.J. Kelly, Surface Sci. 38 (1973) 139. M.J. Kelly, Surface Sci. 43 (1974) 587. F. Cyrot-Lackmann, MC. Desjonqueres and J.P. Gaspard, J. Phys. C (Solid State Phys.) 7 (1974) 925. [ll] D.M. Newns, Phys. Rev. 178 (1969) 1123. [12] T.B. Grimley and B.J. Thorpe, J. Phys. F (Metal Phys.) l(l971) L4. 1131 J.R. Schrieffer, J. Vacuum Sci. Technol. 9 (1972) 561. [ 141 G. Allan and P. Lenglart, J. Phys. (Paris) Suppl. Cl (1970) 93. [ 151 F. Cyrot-Lackmann and F. DucastelJe, Phys. Rev. B4 (197 1) 2406. [16] T.L. Einstein and J.R. Schrieffer, Phys. Rev. B7 (1973) 3625. [ 171 N.F. Berk,, Surface Sci. 48 (1975) 289. [ 181 J.L. Moran-Ldpez, G. Kerker and K.H. Bennemann, J. Phys. F (Metal Phys.) S (1975) 1277. [19] G. Allan and P. Lenglart, Surface Sci. 30 (1973) 641. [20] MC. Desjonqueres and F. Cyrot-Lackmann, J. Phys. F (Metal Phys.) 5 (1975) 1368. 1211 K.M. Watson, Phys. Rev. 105 (1957) 1388. [22] P.W. Anderson, Phys. Rev. 109 (1958) 1492. [23] F. Brouers, M. Cyrot and F. Cyrot-Lackmann, Phys. Rev. B7 (1973) 4370. 1241 R. Jacobs, J. Phys. F (Metal Phys.) 3 (1973) 933. [25] B. Velicky, S. Kirkpatrick and H. Ehrenreich, Phys. Rev. 175 (1968) 747. [26] G.M. Stocks, R.W. Williams and J.S. Faulkner, Phys. Rev. B4 (1971) 4390. [27] F. Brouers, P. Ledered and M. Heritier, in: Proc. Intern. Symposium on Amorphous Magnetism, Detroit, Mich., 1972, p. 51. [ZS] C.D. Gelatt and H. Ehrenreich, Phys. Rev. BlO (1974) 398. [29] T. Yamashina, Japan. J. Appl. Phys. Suppl. 2, Part 2 (1974) 493. [ 301 Y. Takasu et al., Conf. on Solid Surfaces.
THE LOCAL ELECTRONIC
DENSITIES OF STATES AT
THE SURFACE OF DISORDERED
J. L. MORAN-LbPEZ
ALLOYS *
* * . G. KERKER and K.H. BENNEMANN
Institute for Theoretical Physics, Freie Universitdt Berlin, I Berlin 33, Arnimallee 3, Germany
Received 12 January 1976; manuscript received in fiial form 27 February 1976
The local densities of states at a clean as well as at an adsorbate-covered surface of a disordered alloy is studied within the tight-binding approximation using the first terms in a continued fraction series for the electronic Green’s function. The dependence of the adatom local densities of states on the alloy composition, adatom position, special local atomic environment around the adatom and the hopping integral between the adatom and its nearest surface atoms is given. Also the binding energy of an adatom is calculated as a function of the alloy composition. We discuss the significance of electron charge transfer for chemisorption and in general for the determination of the electronic density of states at the surface.
1. Introduction
It has been widely suggested portant
role in determining
have shown
that the d-electrons
catalytic
[5,6] that transition
properties
of transition
of metal surfaces
metals
play an im-
[l-4].
Experiments
metal alloys like CuNi alloys exhibit
haviour with interesting technical applications. Recently, the behaviour of d-electrons near transition
metal surfaces
catalytic
be-
has been
[7-l 1] as well as the problem of chemisorption metals [ 12-171. None of these calculations have been extended to alloy surfaces. The continued fraction technique for determining the electronic Green function is the most appropriate method to treat alloy surfaces. In the following we present simple calculations [ 181 of the local densities of states at clean as well as at adsorbate covered surfaces of disordered substitutional alloys. The local density of states (LDS) is given as a function of alloy composition, special local atomic environment near the adatom and for different positions of the adatom. The results obtained at a pure metal surface agree with those given by Cyrot-Lackmann et al. [lo] using studied
* Supported by DFG BE 58514. ** Supported in part by Consejo National de Ciencia y Tecnologia (MBxico).
a more elaborate moment calculation. We show how electronic charge-transfer affects the d-electronic energies cd and the LDS of the first three layers at the surface /19,20]. Finally, we also c~c~~la~ethe binding energy of an adatom as a function of alloy compositja~. In section 2 we briefly describe the theory for the electronic structure at the SWface of transition metal alloys which was presented in an earlier paper [t 81. In section 3 we present the results obtained by our method for the LDS at a clean (100) alloy surface of a cleaved face centered cubic lattice. Results for the LDS on adsorbate atoms at an alloy surface are given in section 4. In section 5 we present results for the dependence of the binding energy on alloy composition. In section 6 we discuss QUr results.
2. Theory far d-ckctrons near an alioy surface In an earlier paper [18f we have developed a tl~t”b~nd~ng type theory to determine the electronic structure at alloy surfaces. In the following we s~rnmar~~c briefly this theory, extend it and discuss the appro~jmat~ons used. The electrons in a binary alloy A,B,_, with substitutional disorder are described by the usuat tightbinding H~~niltonian (2.1) where the electronic energy levels pi = c: + U&?ti+ Udsn: ,
(2.2)
sari take the values cA or fB defending on whether the site i is occupied by an Aatom or a B-atum. Here, Ui denotes the intra-atomic Coulomb interaction on site i with ni d-electrons and Ud, denotes the Coulomb interaction between the d- and s-electrons on site i. The energies E: are supposed to take into account s-d hybridization The hopping integaals tij which describe electronic transitians between states centered on atomic sites i and i are assumed to be concentration ~dependent. cl and ci are the usual creation and ~~h~lat~on operators for d-electrons on site i. The focal electronic Green function Gij describing d-electrons on the atom i is given by G&‘) = (E - ei - A,)- 1 ,
(2.3)
where the electron self-energy Aj resulting from the hopping of the electrons between the atoms is given by (2.4a)
which results from aIf electron ho~~~n~ processes starting and ending rat the site j and avoiding the sites i and j in-between. The self-energies A! and AZ result from delectron hopping processes avoiding sites i, j, I, m and are given by expressions which are similar to the one for A; 12 i.-23 1. Note, that the coherent potential approximation (CPA) for determining G&Z) in bulk alJoys is obtained by replacing the electronic energies er by an average energy Z [23j in eqs. (2.4). furthermore, it is ihteresting to note that vve find (see section 3) essentially the same resufts for the bulk electron density of states as obtained by Velicky,et a1. f25] using CPA if we keep only the t2=terms in the expansions for Ai and A; and use the approximation Ag=A{. The above theory is now applied to d-electrons at the surface of transition metal alloy. Near the surface the Green function Gli(~) depends not only in the type i of atom but also on the perpendicular distance from the surface. Therefore, we use for the Green functions the notation GL,i which describes d-electrons at atams of sort i in the f;-th atomic layer parallel to the surface. The average electron Green’s function in the first atomic layer at the surface is G, = “Gl,A f (1 -XI G,,,
GI i(E) = (E -
fI,f
-
f2.5)
7
AI)-”
I
f2.6)
Here, eI,;denotes
ei for atomic sites in the first layer. Sim~arly~ the Green’s fun~ti~l~s GIi? GH$ GIIK and GIIId refer to the d-electrons in the second and third atomic layer away from the surface. For s~rn~l~cit~ we assume now that the effects due to the surface are significant only within the first three surface layers. Therefore, we use for the atomic sites beyond the first three surface layers the bulk Green’s function (2.7)
G=xGA+(l-x)Gn, where G,-(E) = (I!?- f; - A&l
,
and Ar, is given by eq. (2-4:) for the case of a bulk atom. The expression for the electron selfenergy A, referring to a site in the first layer is easily obtained from “4% (2.4a). Neglecting terms of higher order than f-Xin eq. (2.4a) and performing the summation over the nearest neighbour atomsj of sort A and f3 En the first and second layers one finds the expression A, = rot2
J.L. Morh-LdpezlLocal
electronic density of states at surface of disordered
alloys
543
Here zO denotes the number of nearest neighbour atoms j lying also in the first layer and z1 the number of nearest neighbour atoms lying in the second surface layer. Note, A; s Ai if i and j refer to atomic sites in the first surface layer. Similarly, A; c AfI if i refers to the first surface layer and j to the second surface layer. We put tii s t for nearest neighbour atoms. The electronic energies eI,*, E~,~, eII,*, etc., denote the energy Ei for atoms A and B in the first surface layer I, second surface layer II, etc. Note, in the derivation of eq. (2.8) we have taken into account that we get different contributions to A, and Ai if the nearest neighbour sites j are in the first or in the second layer. Expressions similar to AI are obtained for AI, and AIII again by using eq. (2.4a). For determining A; we use the approximation A?= Lu; which is exact for a Bethe lattice [24]. To get a feeling for this approximation one should remember that A7 is obtained from A{ if one avoids in determining A/ hopping processes which involve the atomic site i. Thus, the approximation on Af should become better at the surface, since the number of neglected paths is smaller. Then we get straightforwardly from eq. (2.4b) [18] a set of six coupled equations with Ai, A:I, Aif, AI’, AkI and AFI as unknowns in terms oft, z,,, zl, eI i, EII,i. EIII,i, the bulk energy levels ei and A:. For the self-energy Al, referring to the bulk one finds from eq. (2.4a) the expression Ab = (2zltz0)t2
x E-eA-A;
E-en-At
(2.9)
Thus, the electronic Green functions GI,i, GI, i and GI,I i are determined by solving the six coupled equations for the unknowns Ai, AiIi etc. Then the electronic density of statesfor an atom of type i within the j-th layer is determined from the electronic Green function by the formula Nj,i(E) = - i Im Gi,i(E) . Note, in determining the d-electron Green functions Gj,i we have assumed that surface effects occur only in the first three surface layers. This assumption seems reasonable in view of previous tight-binding type calculations at pure metal surfaces and it simplifies considerably the numerical calculations. However, note that the theory for alloy surfaces can easily be extended to the case where we allow also the 4th and Sth, etc., surface layer to be affected by the surface. Furthermore note, that in determining Gi i we have neglected the contributions to Ai and A; which are of higher order than t2. It is known from calculations for the bulk, that these approximations affect essentially only the fine structure in Ni,i(E), but should in general not be of essential significance for an exploratory model calculation and for determining the concentration dependence Of Ni,i(E), the binding energy of chemisorbed atoms, electron charge transfer, etc. Indeed, we obtain with these approximations for Nj i(E) essentially the same results as found for bulk calculations for CuNi alloys using CPA which sums up all terms in eq. (2.4a). However, depending on the crystal structure terms of higher order than t2 might be of importance for
544
J.L. MothL6pezlLocul
electronic
density of states at surface of disordered alloys
determining the variation of surface properties obtained for different surface crystal planes. The approximation tii F t for nearest neighbour atoms i,j should be valid if the d-band width of the pure metals A and B is not too different. Finally, the approximation A? = A/ leads to similar bulk densities of states as those obtained by CPA calculations with a Hubbard density of states. At the surface this approximation should become better than in bulk, since the error which is introduced, is smaller because the number of possible electron hopping paths is reduced.
3. Numerical results for the LDS near the surface of a transition metal alloys In this section we present the average density of states and LDS in the first three laye layers of a (100) cleaved fee lattice. From the agreement between experiment and theory (CPA, etc.) we conclude that the approximations used in our alloy surface theory are valid for CuNi alloys. Therefore, in our calculations we use parameters corresponding to CuNi alloys [26]. Note that for CuNi alloys one can neglect electronic charge-transfer between the Cu and Ni atoms. This is plausible since the Fermi energies of Cu and Ni are almost the same. Furthermore, by neglecting electronic charge-transfer in CuNi alloys one obtains good agreement between experimental and numerical results for the bulk electronic density of states [26].
AE_AE
-20
(4
Ao2Boa
tb)
A0.4B0.6
Fig. 1. Average density of states N,(E) of the first three surface layers (i = I, II, III) and a bulk layer b of an alloy A,B1, corresponding to Cu,Nil,: (a) x = 0.2, (b) x = 0.4. We express the energy in all figures in units of 0.1227 Ry, which is half of the Cu d-band width. We choose the zero of energy such that for the bulk ENi’ 0.51 = 612 and eCU = -612.
J.L. Mor&Ldpez/Local electronic density of states at surface of disordered alloys
545
NE)
i* )
Cb)
Ao.2Bo.a
Ao4Bo.6
Fig. 2. Local density of states per atom Ni(E) for different concentrations in an alloy corresponding to Cu,Ni,,. The labels Cu, Ni and Ave refer to a Cu site, a Ni site and an average site, respectively. The curves labeled I and b refer to the surface layer and the bulk, respectively.
In the following we express the energy in units of 0.1227 Ry which is half of the CU d-band width. We take for 6 = ENi- eCu the value 1.02 and use f = 0.206 in accordance with prior bulk calculations. For the alloy averaged electron self-energy Ab, at a bulk site one obtains from eq. (2.4b) the expression l-x
x
A; =(2z1+zo-l)t2
E-eA-A~‘E-eB-A~ which can be rewritten
as a third order equation
(At)3 - 2E(Ak)2 + [E2 + EAEB + (2zl+zo
1 ’
(3.1)
for Ab, [24,27] -1) t2]Ai (3.2)
Note, in deriving eq. (3.1) we used AT =:A{ and neglected terms of higher order than t2. The expression for Ai given by eq. (3.2) is equivalent to the one used in the coherent potential approximation (Velicky et al. [25], eq. (4.32)). Then with these parameters, we solve by iteration the six coupled equations for At, A,;, etc., which were discussed in section 2. The Fermi energy E, is calculated selfconsistently by using [28] EF s
..-m
iVav(E)dE=XNce+
(l-x)NNi
2
(3.3)
where N,,(E) is the average bulk density of states, Arti = 10 and NNi = 9.45 electrons. The approximations used are essentially the same used in CPA calculations on bulk CuNi alloys and should have the same validity. These approximations should
J.L. Morh-LdpezlLocal
546
(‘1
electronic density of states at surface of disordered alloys
cb)
Ao.2Bo.a
A~+BO6
Fig. 3. Average density of states N&5), for two different concentrations, of the first three surface layers (i = I, II, III) and a bulk layer b of an alloy Cu,Nil,, assuming no electron charge transfer between the surface layers.
of other transition metal alloys like for example RhNi, AgPd, etc. In fig. 1 we show results for the average local d-electron densities of states for concentrations x = 0.2 and x = 0.4. In fig. 2 we show the LDS on Cu, Ni and average sites for the same concentrations. All these results are obtained by using hj = fi,A - ei,B = &bulk, i = 1,2,3. Note, this assumption implies electron charge-transfer between the atomic layers at the surface. The average number of electrons per site in the i-th surface layer is namely then different from the corresponding number at a bulk site. This is found from ni = IdEN,( Consequently, allowing no d-electron charge-transfer between the surface layers +d,i = nd,bulk) [ 19,201 implies layer dependent shifts in the difference hj = Ei,A - ej,B of the energy levels. Here, 6i refers to the ith atomic surface layer. These shifts in 6j are then calculated by using apply as well to a number
EF s
dE [&@&
h,,, &,I,) -Nbdk(E)j
= 0,
i= I,II,III,
(3.4)
-cc
for the first three layers. In figs. 3 and 4 we show how the LDS shown in figs. 1 and 2 are modified for the case where no d-electron charge transfer is allowed between the layers. In fig. 5 we show how 6i = ei,A - fi,n , i = 1,2, changes with concentration.
J. L. Mordn-LdpezjLocal electronic demiry of states at surface of disordered alloys
547
NEJ
E
.
la1
AO ?O
E
(b)
8
Ao4Bos
Fig. 4. Local density of states of the surface layer, NJ,~(E), and for the bulk, Nb(E) m ;i Cu site, on an average site (Ave), respectively. No electron charge transfer between the surface layers is allowed.
_.-._
.- -.___
= -.--I-*
-. %
1.2(
1.10
1.M
I-
--I
_---
%I
-0-
0.0
0.2
0.6
0.4
0.8
1.0
X
Fig. 5. Concentration dependence of61 = EI,Ni - q,cu and Sff= E[I,Nj- EII,~~ in energy units of half of the Cu d-band width. We take for the bulk &b = 1.02.
548
J.L. MorhLdpezfLocal
electronic density of states at surface of disordered alloys
Note, in order to calculate the change Ani in ni due to electronic charge-transfer we have to determine Ei self-consistently using Ei = ep+ Uni. This does not change the theory presented but only complicates the numerical analysis. Note, d-electron conservation implies ‘ci Ani = 0. If s- and d-electrons are treated together than one obtains ci(A,P + An2 = 0, where An; is the change in the number of s-electrons within the Wigner-Seitz cell of the i-th atom due to s+d and d-+d charge-transfer.
4. Local electron density of states on adsorbed atoms To avoid the numerical complications due to electron charge-transfer and Coulomb interactions we restrict ourselves to adatoms of the same kind as those of the substrate. We study how the LDS on an adatom changes with (a) the alloy composition of the substrate, (b) the local atomic environment of the substrate around the adatom and (c) for different adatom positions. We calculate the LDS on the adatom as follows. The Green’s function for the adatom is given by G,(E) = (E - e, - A,)-’
,
(4.1)
where A, is calculated by using the expansion
Cl
Ave
cu
c2
N,
(2.3a)
c3
c4
Fig. 6. Different adatom (a) positions A, B and C on a (lOO)-fee lattice surface are shown. Also the different adatom-substrate configurations Cl, C2, C3 and C4 are shown. The black circles refer to Cu atoms, the open to Ni and the dashed to average atoms, respectively.
J. L. Morbn-Ldpez/i,ocaI electronic density of stutes at surface of disordered alloys
549
*02B08
‘Ni
Fig. 7. Locat density of statesNa(E)on the adatom for its positionsA, B, C and for the alloy concentrationx = 0.2. We put ca = ENi,b, t’= t, 6i = 6b and the units are those given in tlg. 1.
A,=x
t’2 + r:
i+aE-Ei-Ai
d2t
jziaa (E-fi-Ai)(~-Ei-A~)‘ ,
(4.2)
Here ea is the adatom electronic energy and t’ is the hopping integral between the a datom and its nearest neighbour atoms. Ai is the first layer renormalization energy given by eq. (2.8). To get more detailed information on the shape of the local density of states at the adatom we have included t3 terms in the expansion (4.2) for the local self-energy. In fig. 6 we show the various adatom positions (A), (B), (C) and the local atomic substrate configurations (Cl), (C2), (C3) and (C4) which we have studied. In figs. 7 and g results are presented for the LDS of the adatom when it is in the positions (A), (B) and (C). We use ea = eNi , t’= t and put x = 0.2 and x = 0.4. In figs. 9 and IO we show results for the adatom LDS for the substrate configurations (Cl), (C2), (C3) and (C4). It is interesting to note, see fig. 11, that our results for x = 0 reduce to those obtained by Cyrot-Lackmann et al. [IO] using the moment expansion method. We conclude from this fact that the local density of states on the adatom does not de-
550
J.L. Mordn-Ldpez/Locai
A
0.4
electronic density of states at surface of disordered alloys
B 0.6
1.
-2.0
‘*E
0.0
r
2.0
‘Nt
Fig. 8. Local density of states Na(E) on the adatom for its positions A, B, and C. We use Ea = eNi,b, t’= t, 6i= 6b and the alloy concentration x = 0.4.
pend very sensitively on the fine details of the substrate density of states but more on (a) the position (A),(B) or (C) of the adatom, (b) the hopping integral t’ between adatom and substrate and (c) the position of E, relative to eI i of the substrate. By alloying we are changing cont~uous~y the relative position of ea and eI,i of the substrate. As a result the LDS on the adatom becomes more localized as we go from x = 0 to x = 1 .O. This effect can be best seen for the configuration (C4) when the adatom has four Cu nearest neighbours.
5. The dependence of the binding energy of an adatom on alloy composition The binding energy EB for a chemisorbed atom is given by the energy of the semiinfinite crystal alloy plus the isolated atom minus the energy of the semi-infinite crystal alloy plus the adsorbed atom. This gives for E, the expression
J.L. Mom%-LdpezlLocalelectronic density of states at surface of disordered alloys
551
1’=1
A0.2B0 8
Ea=Eb,N,
EF
I
-2.0
0.0
r
I
2.0
E
‘Ni
Fig. 9. Local density of states Na(E) on the adatom for the different substrate confgurations Cl, C2, C3 and C4. We use ea= ENi,b, t’= t, 6i= sb and the alloy concentration x = 0.2.
where rrz = JdE N:(E) and na= IdEN, denote the number of electrons on the adatom. ni refers to the case where no electron charge-transfer occurs. Ua denotes the effective intra-atomic Coulomb interaction between the electrons on the adatom. The LDSA$E) refers to the case where no adatom is present. The LDS N,(E) and Ni(E) take into account the interaction between the adatom and the substrate. EL is the contribution to EB which results from inter-atomic Coulomb interactions and includes the Madelung energy. If we neglect inter-atomic Coulomb interaction and electron charge-transfer between the substrate atoms and the adatom, then the binding energy is given by
552
J. L. Morh-Ldpez/Local
electronic density
of statesat surface
of disordered alIoys
1” t
Ao.4Bo.6
Ea=Eb,NI
0.0
-2.0
2.0
E
t ENi
Fig. 10. Local density of states @a(E) on the adatom for the different substrate conf~urations Cl, C2, C3 and C4. We use ea = ENi,b, t’= t, 6i = Sb and alloy concentration x = 0.4.
E, =
n,e,+
sEFC~~(E)EdE-~ [Eli +N,(E)] .-co
i
EdE .
(5.2)
-m
In fig. 13 we show how the adatom binding energy EB(x) changes as a function of concentration x for different adatom-substrate configurations. The results are obtained by assuming that no electron charge-transfer occurs. While in general electron charge-transfer may be important for determining E, in the case of CuNi alloy surface electron charge-transfer are presumably small and should not affect much the concentration dependence of E&). 6. Discussion We have presented a microscopic theory for the electronic structure at transitionmetal alloy surfaces, which play an important role in catalysis. Our theory is relatively
J.L. MorhLbpezfLocal
electronic density of states at surface of disordered alloys
553
N,(E)
t
t’=t Ea=Eb.Ni
1
-20
2.0
0.0 t
Fig. 11. Local density of states Na(E) of the adatom on pure Ni for the positions A, B and C. We USe Ea= CZNi,b, t’= I and 6i’ 6b.
simple and needs to be improved, for example, in order to study in detail the electronic density of states at the surface of alloys and important effects due to the interplay of the s-electron and d-electron charge redistribution at the surface. Nevertheless our results are instructive for what one may expect in a more realistic theory and demonstrate the significance of various electronic parameters for chemisorption at alloy surfaces. Due to the various approximations used in the theory presented here our results should apply best to CuNi and similar alloys. Due to the approximations Ay = A{and the neglection of terms of order higher than t2 in Ai and A{, etc., we obtain simplified LDSNi(E) without fine structure. Consequently, we have to extend our theory in order to determine surface properties which depend on the detailed structure present in N,(E). However, the good agreement between the results obtained for the bulk Ni(E) with our theory and CPA suggest that our theory should give correctly the overall shape of the LDS. Furthermore, the good agreement of our results obtained for pure Ni with the above approximations and those obtained by Cyrot-Lackmann et al. [lo] using a moment expansion technique indicates that the approximations used in this paper do not restrict too much the validity of the
554
J.L. Morrin-Ldpez/Local
electronic density of states at surface of disordered allo.vs
results obtained for the LDS and E, and in particular their dependence on the alloy composition. However, if electron charge-transfer between adatom and substrate occurs, then EB(x) needs to be determined by calculating EL and determining E, = e,” + (/ana and Ei = ep + Uini self-consistently. This is in general also important for calculating EB(x), since the electron charge-transfer will vary with the concentration x. However, for Ni and Cu adatoms on CuNi alloys and similar alloys for which electron charge-transfer are presumably small the results presented for EB(x) should be valid approximately. First in figs. l-4 we show results for the electronic density of states N(E) at the surface of an alloy corresponding to Cu,Nil_, . One sees that mainly only Ni(E) of the first two surface layers is changed with respect to the bulk. As expected, one observes that due to the surface N(E) tends to “narrow” and to split into two peaks. The surface has a more dramatic effect on N,i(EF) than on Ncu(EF). Also as shown by fig. 1 (which shows results referring to 6, = Gbulk, i = 1,2,3, which implies d-electron charge-transfer between the atomic layers at the surface) and fig. 3 (which shows results for 6,f6,,k such that no d-electron charge-transfer occurs) electron cnarge-transfer affects mainly only NNi(E). This will also be the case if the electron charge-transfer result from adatoms or s-d transitions. Note, if s-d and d-d electron charge-transfer are small at the surface, then one expects the results shown in figs. 3 and 4. The results shown in figs. 1 and 2 should correspond to the situation where s-d and d-d electron charge-transfer occur. Fig. 5 shows the layer dependent shifts in 6i = Ei,A -ei,B which should result if no d-electron charge-transfer between the layers occurs. Note, the magnitude of the shifts and their concentration dependence. In figs. 7-10 results are shown for the electronic density of statesAra of an adatom. The various results show how the shape ofNa(E) and E, depend on the adatom position, the substrate composition and local atomic environment effects (figs. 9, 10). The results obtained for the various adatom positions A, B, C may shed some light on the significance of atomic disorder, crystal irregularities, etc., at the surface for chemisorption. The results show that the adatom couples the stronger (see change of the peak and width in N,(E)) to the crystal surface the more nearest neighbours it has. Similar results are found if we increase the difference (t’ - t). In figs. 9,10 one observes that for E, 2 ENi the adatom couples more strongly to Ni surface atoms. See, for example, that for configurations C4 N,(E) is most similar to 6(E- ~a). We find that only the local atomic environment effects due to the nearest neighbours of the adatom are important. The results indicate that cluster effects and segragation at alloy surface are important for chemisorption. To understand the shifts in E,, which are shown in figs. 7-l 1, note that they result primarily from the second term in eq. (4.2). Consequently, no shift is observed for the adatom position A for which the second term in eq. (4.2) does not appear within the approximation used. In contrast, for the adatom positions B and C the second term in eq. (4.2) is important and leads to shifts in the center of gravity of the adatom electronic states. Since more terms of order t3 occur for the adatom position C than for B, one observes the largest shift for E, for the adatom position C.
J.L. Monk-Ldpez/Local electronic density of states at surface of disordered alloys
555
..40
d’ ..20
EF
t -2.0
0.0 t t E(l) E(2)
Fig. 12. Local density of states N,(E) on the adatom on pure Ni with different Ef,Ni. The energylevel ~on~g~ations (1) and (2) refer to the cases where electron charge transfer between the SUrfaCelayer iS allowed (EIBi = Eb,Nj) and iS not allowed (eI,Ni= eb~jj) and is not allowed (q,~j +; eb Ni), respectively. We Use ea = 0.5 1 and t’= t. Note the energy scale in the insett is different thao’in the main figure.
The changes in the shape ofilra@) observed for the various adatom-substrate con~gurations correspond to the occurrance of bonding and anti-bonding states when the adatom couples to the metal surface. In fig. 11 we show results for a pure Ni surface. Note, these results agree excellently with those obtained by Cyrot-Lackmann et al. [lo] taking into account a more detailed realistic substrate density of states N(E). This could indicate that our results for N,(E) may not change much if we would use a more realistic substrate density of states at the alloy surface. Fig. 12 demonstrates how the results on N,(E) depend on (ea - fI,Ni). Note, configurations (1) and (2) with eI,Ni > e, may correspond to the cases when d-d electron charge-transfer and no d-electron charge-transfer occur between the surface layers, respectively. Since for Er,Ni(2) > f, one expects a partial rising of the adatom electronic States towards Er,Ni, the shift of the peak in N,(E) should be larger for the
556
J.L. Morh-L6pezfLocal
electronic density of states at surface of disordered alloys
I
“J(f :F)
I
Ni
“I”
:.-Cl
\
(EF=EF,N,)
0.4-
0.2 -
0.0 Ni
+..,
t’=t %= Lb,N,
0.2
..
*. . ‘...
., . .
. .. .
0.4
0.6 X
0.8
1.0 CU
Fig. 13. The biding energy of an adatom as a function of alloy composition. We put the adatom in the central position C and assume substrate configurations Cl, C3 and C4. We use ea = ENi, t’ = t and the same energy units as in all other figures. All curves are calculated by taking into account the dependence of EF(x) on x, except Cl (EF= EF,Ni) which is calculated for COnStaM EF. The insert shows the average and the Ni surface electron density of states at EF.
configuration (2) as is observed in fig. 12. Also, iince for EI,Ni > ea adatom electronic states are pulled away from E, by the metal background density of states (for example, towards Er,Ni(2)) more strongly than for the configuration (l), the height of the peak in N,(E) is more strongly reduced for configuration (2) than for (1). Fig. 13 shows results on the binding energy Es. For increasing the difference t’- 1 one finds that EB increases, but the characteristic dependence of E&x) on x remains nearly unchanged. Comparing the results obtained for EF(x) (EF (x) < EF(Ni)) calculated from eq. (3.3) and for EF=EF(Ni) indicate that&(x) depends sensitively on (&-@and hence on electron charge-transfer. Note, EF(x) & E&Ni) is expected if one includes s-electrons in calculating E&) and also from work function data (piecu = 0.04 eV). The results for E, shown in fig. 13 have been obtained by assuming E, = ENi,bulk. The
J.L. Morh-Ldpez/Local
electronic density of states at surface of disordered
alloys
557
curves C4, Cl, and C3 with EP(x) determined selfconsistently result if no electron charge-transfer occurs. The curve Cl with E, = E,(Ni) implies electron charge-transfer and a filling of the Ni d-band at x y 0.6. In reality such a charge-transfer may result from the electrons of the adatom as well as from the redistribution of the s-electrons which tend to leak out of the metal. Of course, for a quantitative analysis, electron charge-transfer effects should be calculated selfconsistently by taking into account Madelung energies and that the energies ed depend on nd. The dependence of E, on the concentration x arises mainly from the following facts. We note that E, is mainly determined by the contribution JdE(E - e,)N,(E). Consequently, since EP(x) 2 EF(0) +x X const. , and N,(E)
= n.&E - c& + x AA’?(E)
+ (1 -x) A?(E)
,
see eqs. (4.1) and (4.2), one expects EB(x) to vary approximately linearly with (1 -x> for an average alloy surface. Here, we denote the change in N,(E) due to Cu atoms and Ni atoms by ANCU and ANF, respectively. The results obtained for C4 indicate that ANCu < AiN’. The risults obiained for the local atomic environment C3 and C4 indicate that the binding of the adatom is strongly determined by the alloy atoms which are nearest neighbours of the adatom. Assuming that the activation energy Ea for a chemical reaction at the surface is proportional to E,, then the results for E, which were obtained for the average substrate configuration Cl with EF(x) yield lnK=a+bx, which agrees with experiment [29,30]. Here K is the catalytic activity and a and b are constants. The results shown in fig. 13 were obtained without using the rigidband approximation. Similar results for EB(E) are expected for other transitionmetal alloys like Ag,Pd,_,, Ag,Rh,_,, etc. For alloys Ni,Pd,_,, Rh,Pdl_x, etc., we expect E(x) not to vary much with x. Our results indicate that E, is strongly determined by the perturbation of N,(E) by the surface. This perturbation is large if the local density of states of the surface atoms in the vicinity of the adatom is large for E = E,. Therefore, by affecting NI,~(E) for E 2 E, magnetism may have a strong effect on E,. If electron charge-transfer is important then the Madelung energy could contribute significantly to E,. Such effects could be calculated directly by extending straightforwardly the theory presented here, for example. It would be more difficult to treat such effects within the Debye-Htickel theory.
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electronic density of states at surface of disordered
alloys
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