Theory of atom-metal interactions

SURFACE

SCIENCE 6 (1967) 133-158 0 North-Holland

THEORY

OF ATOM-METAL

Publishing Co., Amsterdam

INTERACTIONS*

I. ALKALI ATOM ADSORPTION J. W. GADZUK Department of Mechanical Engineeringt and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.

Received 22 June 1966; revised manuscript

received 22 August 1966

The interaction of a metal with an alkali atom is considered from first principles. It is shown that treating the interaction of the metal with the alkali atom through perturbation theory is a meaningful approach. It is seen that the interaction causes a shift and broadening of the valence level of the alkali atom. Furthermore, it is seen that electron transitions between virtual atomic and metal states are formally equivalent to standard rearrangement processes. The first-order ns energy-level shift of the alkalis are calculated and the results given in standard form. The shift is found to be - + 0.3 eV. The natural broadening of the originally discrete ns level is calculated in closed form by using time-dependent perturbation theory. The theoretical bandwidth for an alkali atom adsorbed on a metal is found to be 5 1 eV. These results are discussed in relation to previously suggested values for the shifted and broadened level. It is found that the position of the shifted and broadened level relative to the conduction band of the metal is such that ionic bonds are formed between adsorbate and substrate. The possibility and implications of localized electrons in the interior of the metal near the surface and also around the alkali ion cores are discussed. It is found that electrons do tend to localize at the surface in the presence of alkali ion cores. The resulting surface dipoles are then discussed in relation to their effects on electron emission properties of the surface. Temperature dependences are included in a semi-quantitative manner and the origin of thermal depolarization effects is displayed.

1. Introduction The fact that alkali metals have the lowest ionization potential of all elements makes them very attractive for possible application in devices requiring low-energy plasmas and ion beams. This fact is being exploited in such areas as advanced energy conversion and ion propulsion. As a result of * This work was supported 039 AMG03200 (E)). TPresently with Department

by the Joint Services Electronics of Physics. 133

Februari

1967

Program (Contract DA36-

134

J. W. GADZUK

their low ionization potential, it is also known that if a monolayer of an alkali is adsorbed on a metal surface, the electron and ion emission properties of the surface are drastically altered. This fact is also being exploited by workers in such areas as thermionics and physical electronics. In all cases a better theoretical understanding of the basic processes occurring in the alkali atom-metal interaction is desirable. Such is the aim of this paper. Since Taylor and Langmuir’sl) classic study of cesium adsorption on tungsten there have been a few attempts to formulate theories on alkali adsorption. One grouping of theoretical work has been a natural offspring of the original Taylor and Langmuir work. Realizing that the ionization potential of unperturbed cesium is less than the work function of metals such as tungsten and that cesium is physically a big atom has led to the following simple picture. The alkali is adsorbed on a metal surface basically as an ion. The ion-metal interaction is described by the classical image force. The resulting system of metal with an adsorbed dipole layer that lowers the electron work function of the surface satisfactorily correlates the emission data with a tenable theoretical picture. This point of view was adopted by De Boer2) who further developed this model. More recently39 4~5, additional sophistications have been added to this model in order to more fully interpret data such as that of Taylor and Langmuir. This picture has also been useds) to interpret potassium on tungsten data. Since there are many assumptions and simplifications in applying this model, some of which have been critically examined by Rasors) and Gomers), there is a need for further study. Another approach has been taken by Gyftopoulos and Levine7). They view the adsorbate-substrate interaction from the chemist’s point of view and propose that cesium is chemisorbed by the formation of a partially ionicpartially covalent bond with the four substrate atoms upon which the cesium sits, thus forming essentially a CsW, molecule. Toyas) has presented one of the few treatments of atom-metal interactions starting from first principles. Gurneys) originally noted that the interaction of an atom with a metal causes the valence level of the atom to become broadened. He then noted that this broadening could result in the formation of polar bonds between the adsorbate and substrate which where not necessarily ionic in character. Dobretsovre) developed further the ideas presented by Gurney. The work of Gomer and Swansonil) has provided criteria, based on Gurney’s suggestion, for establishing the nature of the adsorbate-substrate bond, that is, whether it is ionic, polar metallic, or covalent. They point out that the position of the shifted and broadened ns level (n is the principal quantum number of the valence electron of the particular alkali in question) relative to the occupied portion of the conduction band largely determines

ATOM-METAL

INTERACTIONS.

I.

135

the nature of the bond. This point has been discussed further by the author 12) who based his work on calculations quite similar to the original calculations of Sternbergrs) who was concerned with resonance ionization and neutralization of metastable hydrogen atoms. A perturbation theoretic technique will be developed here for treating alkali atom-metal interactions. At times, absolute accuracy is sacrificed in favor of closed-form results. This point is discussed when the approximations are made and justification is given. This work is a first-order perturbation treatment using zero-order wave functions. More accurate numerical results would of course be obtained by using better wave functions but it is felt that the results obtained in first-order calculations are sufficiently accurate to permit the present conclusions. Furthermore, if exact solutions to the atommetal interaction problem were obtainable, there would be no need for perturbation theory. The structure of the paper is as follows. In section 2 the atom-metal interaction is discussed in a general manner. The model used for ensuing calculations is presented. The possibility and significance of virtual bound electron states at the surface deriving from the ns states of the alkali is discussed. In section 3, the shift of the ns level caused by the atom-metal interaction is computed. As originally pointed out by Gurneys), this level experiences a natural broadening as a result of the uncertainty principle and the finite lifetime of the atomic state when the atom is allowed to interact with the metal. This lifetime and bandwidth are calculated in section 4. The equivalence of the atom-metal interaction with a rearrangement collision is noted. This fact is used to simplify calculations. The consequences of these calculations are then discussed in section 5. The types of allowed bonding between alkali adsorbates and metals in the light of the calculations presented here are discussed and the effects on electron emission properties are pointed out. 2. Atom-metal

interaction

To consider, in its most general form, the problem of an atom interacting with a metal requires solution of the Schrodinger equation with the total Hamiltonian: IL

= HM + Ha + Hint 9

(1)

in which HM describes the unperturbed metal, Ha the unperturbed atom, and Hint the complete coupling of the atom with the metal. In the following analysis, zero temperature and the Born-Oppenheimer approximation are assumed. Thus nuclear kinetic energies do not appear and eq. (1) is taken to be the electron Hamiltonian. In addition, H,,, is taken to be spin-independent.

136

J. W.

First, consider

the unperturbed

GADZUK

metal electrons

&IW

that must satisfy

= E,IW.

(59

A free N-electron metal is assumed. Thus the total anti-symmetric state, a solution of eq. (2) is given by IW

= (W-“C(-

l>“g{nr(rr)r+(rz)...

r~,(r,,>> y

electron

(3)

P

where 9 is the usual permutation operator, the sum is over all permutations and uj( r) is the single free-electron eigenfunction with kinetic energy Ej such that O< Ej I ,u + 4, for a bound electron, with p= the Fermi energy and 4, = the electron work function, Because of spin degeneracy, at zero temperature the energy levels are filled only up to ,u. For the assumed free electron metal, the single electron wave functions are uj

=

(l/k,Lt)ei(ko1jx’ko2jv) ((k’,,3j + k,,3j)eik’osjt + (kfosj - k03j)e-ik’03j~} for 5 < 0,

nj = (l/k,L3)ei(kolj”+kozjy)2k’03jeiko’jr for<>O. In the UPS, the following

nomenclature

(4)

has been adapted:

V,, = p + 4, = A2kf/2m kij = ki,j

+ k,2,j + k~3j

k133j = k~3j + k,2

(5)

k’gj = kij + k,2 L is the length of the large but finite cubic metal. The indices 1, 2 denote the x, y directions on the plane of the surface and 3 denotes the z or 5 direction that is normal to the surface and positive outside the metal. Primes above the k indicate that the energy is measured from the bottom of the well. The subscript 0 stands for a bound metallic state. Note also that k,j and k,, j are both positive imaginary numbers. Low-lying levels below the conduction band are ignored. Hagstrumrd) has shown through experimental and theoretical work pertaining to Auger processes at surfaces, that the density of states function for the conduction band electrons is not drastically different in the surface region than in the interior of the metal. Bardeen ls) and more recently Loucks and Cutler 16) have calculated the form of the surface potential barrier. They find that the major portion of the surface barrier results from exchange and correlation effects and not from an electrostatic double layer, The dipole barrier is small which means that the electron wave functions are not greatly distorted in the surface region. In

ATOM-METAL

fact, Bardeen

and Loucks

INTERACTIONS.

and Cutler

137

I.

use perfect,

unperturbed

sine wave-

functions for 5 < 0 which have a nodal plane at the surface and which are taken to be zero for 5 > 0. They demonstrate that a metal with electrons in these states, the Sommerfeld metal, has a surface potential barrier of nearly the same form as is expected for a real metal. From this they are able to conclude that electron redistribution in the surface region is not nearly as significant as might be thought. Consequently the Sommerfeld model provides a reasonably adequate description for surface processes. Based on Hagstrum’s work and the calculations of Bardeen, Loucks, and Cutler, it will be assumed that for the present purposes, we can use density of states functions from the Sommerfeld model. This is an assumption consistent with the present state of surface theory. Since we are using decaying exponential functions in the surface region, the present calculations should be even more accurate than for a perfect Sommerfeld model. For the alkali atom it is assumed that the valence or JZSelectron moves in an average, self-consistent Coulomb potential of the nucleus and closed shell electrons, thereby reducing the alkali to a hydrogen-like single electron atom. The ns electron in its ground state must satisfy

where Vi is the ionization potential of the atom in question. In the ensuing calculations, hydrogen 2s wave functions are fitted to exactly) alkali wave functions by suitable adjustment of the constant a appearing below. Thus $,,, = (u/7r)%c(l - ur)eVLlr.

(6)

For cesium, a=0.99AP1. For potassium a= 1.16&l. This assumption is not as gross as it first seems. All calculations require wavefunctions only as a

1 ,

2

4

6

8

Fig. 1. Electron probability density versus radius for a cesium 6s electron. The curves compare the actual probability density with that which is obtained using eq. (6).

138

I.

W. GADZUK

measure of the charge density at a point and not as a energy. That eq. (6) gives an acceptable measure of this which 1$6,S(2r2 is plotted against radius for eq. (6) along pression in which the exact wavefunctions for the sample

measure of kinetic is seen in fig. 1 in with the same excase of cesium are

used. “z 0 ‘ ---7V. -AE

t

f-5

Fig. 2.

_

II s ~~~ *

z=-s

z=o

.f=o

(=S

Model for the atom-metal

interaction.

As the atom and metal are allowed to interact, fig. 2 must be considered. In this picture, s is the distance of the ion core from the surface, and E is the position of the ns level relative to the Fermi level of the metal. Note that this drawing has been drawn to imply a potential barrier between the atom and metal. This certainly is the case at large separations but requires some justification when considering separations of the order of angstroms as is the case for an adsorbed particle. As Gomers) points out, the potential seems to go below the Fermi level for small separation. But he was considering a one dimensional case whereas the considerations here are three dimensional. Thus even at small separations such as a few angstroms, much of the barrier between atom and metal remains above the Fermi level. This means that most electrons must tunnel through the region between the ion core and the metal. This point will be discussed more fully in another section. Now consider the potential felt by an electron in the region outside the metal. For <>O, the interaction of an electron with an unperturbed metal surface is represented by the classical image potential. The results of Cutler’s detailed quantum mechanical treatment of the electron-surface interaction suggest that the image potential is an adequate approximation to the true interaction even at distances of the order of an angstrom from the surface16,1*). This interaction takes the functional form, -q2/4d, where d, is the distance between the electron and the surface.

ATOM-METAL

INTERACTIONS.

139

I.

The presence of an alkali ion core in the surface region perturbs the metal and thus alters the electron-metal interaction. This is most easily understood in the classical picture as shown in fig. 3. The perturbation on the metal gives

Fig. 3.

Classical picture of the atom and the image charges it induces in the metal.

rise to a repulsive interaction +q2/R, between the image of the positive ion core and the electron. R is the distance between the electron and the image of the ion core. The electron also feels the Coulomb attraction of the alkali ion core - q2/r. With these considerations and the arguments presented in appendix A for reducing the total Hamiltonian to an effective one electron Hamiltonian it is seen that the single electron Hamiltonian takes the form: H=

- A2 -v2 2m

- v,

for 5 < s, (7)

H = $:V2

- q2/4d, + q2jR - q2/r

for 4 > s,

where S, is chosen such that - V, =(q2/4d, + q2/R)c=,,. This sort of interaction has been discussed by otherslg). The interactions in the single electron Hamiltonian may be split into a soluble unperturbed Hamiltonian plus a perturbation. This may be done in two different manners for 5 > S,

H’a--m= - q2/4d, + q2iR,

140

or

J. W. GADZUK

-

H, = __

A2

2m

V2 - q2i4d,

+ q2iR,

H’m--a= - q2/r. Clearly H= Ha + Hi_, = H,,, + HA_,. Eq. @a) describes an ns alkali electron, existing as a solution of Ha that is perturbed by the metal through the perturbation HL_,. On the other hand, eq. (8b) describes an electron moving in the metal which has been perturbed by the presence of the alkali ion core. This fact is reflected by the inclusion of the q2/R interaction in the unperturbed Hamiltonian of the test charge. This potential represents the effects of charge redistribution of all other metal electrons but the one in question, as a result of the alkali ion core. The metal electron described by eq. (8b) is made to feel the effect of the alkali ion core by turning on the potential H&_,. Thus, the potential which gives rise to binding in one case is viewed as a perturbation in the other and vice versa. This situation is familiar in the study of rearrangement collisions in which transitions occur between states which are eigenfunctions of different unperturbed Hamiltonians 20, si). In these so-called rearrangement collisions, as in the present case, the binding potential before the collision or transition is the perturbing potential after the transition. The formalism which exists for treating rearrangement transitions properly will be applied in section 4. It should be noted that since the eigenfunctions of Ha and H,,, are not eigenfunctions of the full single electron Hamiltonian, the states described by these functions are virtual states. The lifetime of these virtual states is determined by the strength of the coupling between the particle in the virtual state and the remainder of the atom-metal system. The usefulness of the division of the full Hamiltonian into a soluble part and a perturbation in the manner of eqs. (8a) or (8b) depends on the lifetime or natural width of the resulting virtual state. If the natural width of the virtual state localized in the region of the ion core, a solution of Ha, is much less than the width of the conduction band, then the subdivision as done in eqs. (8) can be useful in that it provides an easily visualised physical picture of an electron resonating between metal and atomic virtual states. On the other hand as the natural width of the virtual state approaches the width of the conduction band, the picture of a relatively localized virtual state becomes meaningless. In section 4 it will be seen that the width of the localized virtual state, an eigenfunction of Ha, but not of the full Hamiltonian, is small enough so that the idea of a perturbed atom is a useful concept. For this to occur, the amplitude of the atom-metal wavefunction, an eigenfunction of the full Hamiltonian, must be smaller in the region between the ion core and the metal than in either the metal or around the ion core. This is the physical reason why large chemisorbed atoms

ATOM-METAL

can be thought outwards

of in a different

another

INTERACTIONS.

141

I.

way than just as an extension

of the solid

layer. 3. Energy level shift

As an alkali atom is allowed to interact with a metal via the interaction given by eq. (8a), there is a shift in the atomic energy level relative to the Fermi level of the metal and the vacuum potential resulting in a perturbed eigenvalue. The first-order shift with the zero-order wavefunctions used, is given by AE

=

(6 SIf&L 14 s> (n,sln,s> *

(9)

The coordinate system used in the calculations of the energy shift is one with the origin, z=O, at the center of the ion core and with z increasing positively away from the metal. In this system, shown in fig. 3, the distance of the electron from the metal surface is given by d, = s + rcos 0 = s + z. The distance between the electron and the ion core image is given by R = 2s + rcos 8 =2s+z. This is a very good approximation for small r which is the only region in which the electron charge density is significant. Combining these ideas with eqs. (6) and (8a) allows eq. (9) to be written as a,

+CC

s( dz

dE = -_(s-SC)

_.q:-_??

2s+z o.

4(s+z) +m dz

s -(s-SC)

rss

>Iss

dx dy(1 - 2ar + u2r2)e-211r

-m

(10)

dx dy(1 - 2ar + u2r2)e-2ar

-co

where the integration has been restricted to the region outside the metal. Mathematical details of the integration appear in appendix B so only the end result is given here in an extremely cumbersome but exact, standard form. It is found that 4as

AE = 1.8

- 30a2s2 - 3 + 4e-4as(32a3s3

- 8a2s2 + 4as + 1)

d,;+ s

Za(s+s,)

2ns

+ e-20s(4a3s3

- 2a2s2 + 2as + 1)

dt :’ _ e- 2a(s-sc)(2a2s,2 _ as, + s

Zas, +

3

+ 6)

+

(11)

6a2s2 + 3as - 6a2s,s) + e-2as(56a2s2 + 4as + 2 -

e-2+scya3(s

-

s,)3

+

u2

(s

-

s,)’

+

$a (s - sJ + 1)

142

J. W.

GADZUK

where AE is given in electron volts, s in angstroms, and a, which is a characteristic of the particular alkali in question, in inverse angstroms. Numerical values for the exponential integral are tabulated as a function of the upper limit es). Eq. (11) is evaluated as a function of distance from the surface for the specific cases of potassium on platinum and cesium on tungsten and the graph of this function appears in fig. 4. Note that both curves exhibit the

o.8-0.6

‘3 J 0.4 W 4

0.2

Fig. 4.

:

‘:-;::::I

Energy shift determine from eq. (11) as a function of atom-metal for the cases of cesium-tungsten and potassi~-p~tinum.

separation

shape suggested by Gomerssrl) at distances greater than approximately the sum of one-half the lattice constant plus the alkali ionic radii-3-4A. This is the atom-metal separation when the atom becomes adsorbed. The results are not really meaningful at smaller separations as these separations would physically be prohibited by other repulsive forces not included in this formalism. Note that the energy level has been shifted upward in both cases by approximately 0.3eV for an adsorbed alkali. This is in accord with the value of 0.35eV suggested by Rasora) and Levine7). Since the calculated energy shift AE is small relative to the ionization potential, it might be expected that the perturbation theory treatment is not a bad approximation. 4. Level broadening The interaction of the metal with the alkali atom has another effect on the discrete ns level in addition to shifting its position. As was first pointed out by Gurneys), when the atom is brought closer to the metal, the originally sharp IZSlevel broadens into a band with finite width. Formally the explanation for this occurence is as follows. If there is some interaction that couples the space of atomic states with the space of metal states, then neither the atomic nor the metal state is a stationary state of the system. An approximation to the

ATOM-METAL

INTERACTIONS.

143

I.

actual state of the atom-metal system can be viewed as one in which an atom electron at the shifted energy level resonates between the atom and metal state. Because of the uncertainty principle, the finite lifetime results in a broadening of the originally discrete level into a band of finite width. In less formal terms, this process can be viewed as electron tunneling between the atom and metal. It is important to realize that if the shifted discrete level lies above the Fermi level at zero temperature, this state would not be occupied and thus it would theoretically be impossible to have an electron in this so-called resonance state at zero temperature. The finite bandwidth does not imply that there actually is an electron in this state. It only says that if there were one in this state, it would have a finite bandwidth. Since the amount that the shifted and broadened level overlaps the conduction band determines the form of the chemical bond between the adsorbed alkali and the metal11,12) it is very important to know the bandwidth. Furthermore, rough calculations performed by Rasors) indicate that there is little or no overlap, a situation that is favorable for simple ionic bonding whereas equally rough calculations of Gomersy 23) indicate a great deal of overlap which is favorable for polar metallic bonding. Thus there is a need to look deeper into this problem. Consider the atom and metal to be interacting as shown in fig. 2. The electron in the resonant atom-metal state is characterized by a reaction of the form: A++M-+A+M. (12) The transition frequency for an electron in this resonant given by a modified first order Golden Rule as: w=c deg

atom-metal

state is

(13)

where the sum is over all degeneracies of the metal state on the energy shell and the transition matrix element is discussed below. Calculations along the lines of those in this section have been made by others for such processes as Auger neutralization of ions by metals and resonance ionization of metastable atoms by metals.13v14y l9, 24,25 ). The present approach will follow closely that of Sternbergl3). Discussion of Golden Rule usage appears in those works and in another paper by the present author which shall be referred to as 1126). Formally, the process of resonance ionization and neutralization of an alkali atom by a metal, as described by eq. (12), is equivalent to a standard rearrangement collision in which the initial and final electron states are not eigenfunctions of the same Hamiltonian. Note that in the reaction going from

144

J.

W. GADZUK

right to left, corresponding to resonance ionization, the initial state of the electron is the ns atom state and the perturbation is Hi_,,, as is seen from eq. (8a). On the other hand, in the reaction going from left to right, corresponding to resonance neutralization, the initial state of the electron is a metal state and the perturbation HA _a = - q2/r draws the resonating electron out from the metal as is seen from eq. (8b). We treat the case where detailed balance requires that the transition rate from atom to metal be equal to the transition rate from metal to atom. One form of the transition matrix element for processes of this type has been shown to be given by T,, =

3

(14)

where xi is the initial state of the electron, xr is the final state, Hi is the perturbation on the electron in the initial state as determined from eqs. (8a) or (8b), and xi is an operator which projects onto the initial stateas). Since in these rearrangement events, the initial and final states are not orthogonal, xi allows removal of that portion of the matrix element which results from the nonorthogonality of the initial and final states which are not eigenfunctions of the same Hamiltonian. Since detailed balance requires that the transition rate for the process described by eq. (12) be the same in both directions, the usage of eqs. (8a) or (8b) to describe the final state is formally equivalent. As a result of the orthogonalization procedure described in II, it is seen that correct results are obtained most readily by dividing the single electron Hamiltonian in the manner of eq. (8b). Thus the initial state in the matrix element is chosen to be the metal state which is a solution of H,,, in eq. (8b). To the order in which we are working, the metal states given by eq. (4) are taken to be solutions of H,,,; thus Ixi)= 1~~). This approximation is discussed in II. It follows immediately from eq. (8b) that the initial state perturbation upon the metal state is HA_,. That this is the sensible, although perhaps not intuitively obvious choice of initial state and initial state interaction is shown in II. Furthermore, this is the form of the matrix element which has been used by others13*27). The final state is the unperturbed atomic state described by eq. (6). Consequently 1~~)= In,s). Upon following the procedure of II, it is seen that to a very good approximation the transition matrix element is given by:

Using the expression

for HA-a of eq. (8b) and writing

T,r as an integral

}

ti,*,,,(r)dxdy dz.

(15)

ATOM-METAL

The method

of integration

the value of the matrix

INTERACTIONS.

of the matrix

element

I.

element

is also detailed

is given in eq. (22) of II. Upon

145

in II where integration,

eq. (15) becomes: T- = ~~~q2k03ats2e~F” 1.f k YLtF2

{I + ;;

(f-“)},

(16)

where F2 = a2 + kg 1+ k,,2 and the wave vectors are defined by eq. (5). It is interesting to note that as s becomes large, the matrix element is approximately equal to the terms in front of the brackets. This large s expression is equivalent to the results obtained by Sternbergls). To obtain the transition rate assume that the metal is almost infinite, use the usual density of states allowing two electrons per level because of spin, and go from a sum to an integral. Then eq. (13) is written as 2n L3m

hl

3x

d+

w=2nzfi3s 0

sin CYI/~,~~’de,

(17)

s 0

where k’, is the wave number equivalent of the energy of the metal electron at the energy of transition. Note that the limits of the fI integration correspond only to final momentum states directed into the metal. Knowing the matrix element for the transition permits eq. (17) to be written

+ A2 The integration of magnitudes given below.

(k& + ki,)‘)

sin 9 d0.

is done over all possible directions of the final state. In terms and directions, the k vectors and associated quantities are kz, + ki2 = 12’; sin’ 0, k’ o3 = k’, cos0, F2 = a2 + k’i sin28, e -2Fs = e -2s(a*+k’o*

sin*@+

N

e-

20s e-ko2sa92/a

(19)

Because of the dominating exponential dependence on angle, the integral of eq. (18) is significant only for small 8. Thus the usual simplifying assumptions sin”0 N O”, case N 1, and an + b sin”8 N CZ~ are made. The value of the integral is quite insensitive to the magnitude of the upper limit if the upper limit is much greater than zero. Thus the upper limit can reasonably be taken at

146

infinity.

J. W.

With these assumptions,

GADZUK

use of (19) in (18) gives

(20) Straightforward integration of eq. (20) gives the final result for the transition probability as a function of alkali-metal separation: 2a4:‘!(l w=--7 ” 0

_ 4/a2s2 + g/a4s4)

(21)

In this expression, ao=Fz2/mq2 and k’ is the wavenumber equivalent of the electron energy, with the zero of energy at the bottom of the conduction band. Remember that k’ is determined by the position of the shifted ns level and thus is a function of s also. The lifetime z, of the atomic state before ionization occurs, is found by taking the reciprocal of the transition rate. In the limit of s approaching infinity, corresponding to no interaction, the transition rate goes to zero and the atomic state is a true stationary state of the ns electron, as it should be. On the other hand, as s goes to zero, the transition rate becomes infinite and thus the atom lifetime goes to zero as it should. Under these circumstances, the atom becomes part of the metal and a “surface alloy” is formed. Clearly it is in this case that the idea of a localized virtual state is of no use. Thus a check on the validity of a perturbation theory treatment is built into the theoretical model presented here. The finite lifetime, given by the reciprocal of w, implies an energy uncertainty or finite bandwidth given by r(S) N

tiw.

(22)

Using appropriate numerical values for the constants and the results of section 3 for the energy of the shifted ns level, enables evaluation of eq. (21) and its reciprocal and eq. (22). These quantities are shown in a hybrid but unambiguous fashion in fig. 5 for the special cases of cesium on tungsten and potassium on platinum. It is interesting to note that the results of these calculations imply a natural broadening of the order of 1 eV for an adsorbed alkali. This is smaller than suggested by Gomer. On the other hand this value is greater than the estimate of Rasor. The calculated value presented here should have a tendency to slightly overestimate the bandwidth for various reasons. First the calculation is done in first order perturbation theory which tends to overestimate relevant matrix elements. Second, the perturbation provided by the ion core

ATOM-METAL

INTERACTIONS.

147

I.

-10-3

-10-2

7 _, z?! --i IO ;; / iz

-15 IO IO0 _ ~

f

-16

IO

I

0

I

2

L

I

t

4

1

6

/

I

8

I,

I IO

s (i,

Fig. 5. Atom-to-metal transition probability and atom bandwidth determined from eqs. (21) and (22) as a function of atom-metal separation for cesium-tungsten and potassiumplatinum.

has been taken to be a bare Coulomb potential. In reality, some sort of shielded Coulomb potential results in the region between the ion and the metal. This reduces the interaction in the transition matrix element. These factors raise no fundamental objection but instead serve to remind us that the transition rate given by eq. (21) is an upperlimit. Implications of the level broadening in relation to adsorption theories and further discussion appear in the next section. 5. Discussion and conclusions At this point, the ideas and results of the previous sections are put together, thereby forming a self-consistent picture of the alkali atom-metal interaction. The implications of this interaction are then noted. Throughout this work, the implicit assumption has been made that the alkali can be treated as some sort of perturbed atom. This assumption is

148

J. W.

GADZUK

quite valid provided a potential barrier exists between the atom and metal of sufficient height to significantly reduce the amplitude of the wavefunction of an electron of the combined atom-metal state in the region between the metal and the ion core. If this condition obtains, then the true stationary atom-metal state can reasonably be divided into a virtual atomic state with a very small but non-vanishing amplitude extending into the metal and a virtual metal state with a small but non-vanishing amplitude extending into the ion core. It is the small overlap of these virtual states which gives rise to the possibility of tunneling or transition from one virtual state to the other. As mentioned earlier, the usefulness of this sort of splitting of the true eigenstates into two virtual states depends on the lifetime of the virtual states and thus the energy broadening of the virtual level. As the ratio of the virtual level width to the conduction band width becomes less than one, this way of viewing the surface impurity state becomes more sensible. In the present case, the virtual state width is of the order of one eV whereas the conduction band width is ten or more eV’s so the ratio I/2p < 0.1~ < 1, the inequality is well satisfied and the division of the true eigenstate into a fairly localized virtual atomic state and a virtual metal state with a slight amount of overlap between the states is reasonable. One can always consider any arbitrary quantum mechanical state which is not a stationary state of a system and follow the time development of such a state when allowed to evolve in time according to the full Hamiltonian of the system. If the change of state in time is slow enough so that the energy uncertainty is small compared to other relevant energies of the system such as the Fermi energy, then the original state can be thought of as a quasistationary state and the time development of a particle in such a state can be described in a self consistent manner in terms of discrete transitions between such quasi-stationary states. This of course is well discussed in the literature of many-body theoryss). Conceptually this is exactly the approach which has been taken in the present work where it was shown that the natural lifetime of an electron in a quasi-stationary state, say the virtual atomic state, is long enough to allow for specification of the complex energy within one eV. It is for this reason that the very physical approach of some3) in which they talk of different neutral and positively charged states for an adsorbed alkali on a metal surface has justification. It is not meant to be implied that the actual state of particles on a surface is a mixture of pure atoms and pure ions but instead that the true state of the system has the same observable properties as one in which electrons make resonance transitions between localized virtual atomic states and virtual conduction band states. Henceforth, reference to localized atomic states is to be regarded in this light. The actual form of the full system wavefunctions are shown in fig. 6 for

ATOM-METAL

INTERACTIONS.

I.

149

two different energies. The wavefunction labeled I), is that one for an electron with an energy not falling within the range of the broadened and shifted atomic state. This wavefunction is an oscillatory function in the metal and a decaying exponential in the region outside the metal with a slight increase in amplitude in the ion core region. The second wavefunction $‘,, is that which describes an electron with energy falling within the range of the broadened virtual level. There is a considerable amplitude in the region of the ion core where the wavefunction resembles an atomic wavefunction and reduced amplitude in the metal where uniform oscillations occur. The dashed lines schematically show how the true wavefunction is a combination of slightly overlapping unperturbed atom and metal wavefunctions.

Fig. 6. Wavefunctions for electrons at various energies. I+&,, is a wavefunction for an electron not within the energy of the broadened band. Ccr,’is a possible wavefunction for an electron within the band.

From fig. 6, it can be seen that the charge state of the alkali depends upon the occupation of the various allowed electron states. If many of the n metal electrons plus 1 alkali electron are in states similar to $‘,, then there would be a considerable electronic population in the region of the alkali and there would be no net charge on the adsorbate. At zero temperature this situation occurs if the shifted and broadened level falls below the Fermi level as shown in fig. 7a. In this case, the bond formed between the metal and atom is a metallic bond. This metallic bond might be polar, but it seems that the resulting dipole formed in such a polar metallic bond would not be sufficient to account for the drastic lowering of the electron work function occurring when a partial monolayer of an alkali is adsorbed on a metal surface. The next case of importance is when the broadened level partially overlaps the Fermi level as shown in fig. 7b. In this case a small number of the n+ 1 electrons will be in states such as II/‘,, these being in the occupied portion of the partially filled atomic band. Most of the n+ 1 electrons, the ones lying

150

I.

W.

GADZUK

Fig. 7. (a) Position of broadened level required for metallic bond. (b) Position of broadened level required for partially metallic-partially ionic bond. (c) Position of broadened level required for ionic bond. All cases are for zero temperature.

below the atomic band, will be in states $,. Thus only partial charge neutralization of the alkali ion occurs. Consequently the fractional ion core is bonded to the surface by a partially ionic-partially metallic bond, somewhat related to the partially ionic-partially covalent bond discussed by Gyftopoulos and Levine 7). The most important case occurs when the shifted and broadened level lies totally above the Fermi level as shown in fig. 7c. In this case all occupied electron states are of the form II/,,, and there is effectively no neutralization of the ion core. Consequently a purely ionic bond between the alkali and the metal is formed. As a result of the calculations in sections 3 and 4, it appears that this is what occurs for alkalis adsorbed on metals. Since there are n+ 1 electrons described by $, and n positive charges in the metal plus one positive charge at the alkali core, the net effect is as if one electron were localized at some position n+t

+m

k=O -L

ATOM-METAL

and one positive

INTERACTIONS.

charge were localized

I.

151

at

n

(Zp) =

&

c

Pi(z)zi + s,

i=l

where pi is the charge distribution of ion cores within the metal and zi is their position. The overall effect of this localization is equivalent to the formation of a dipole moment as a result of the alkali adsorption. This moment is given by Wl = 4KzJ

=qs+

- > n

(c

pi(z)Zi -,>

;

i=l

n+l fm

C 1$ZlkztiIllk dr)* k=O

(23)

-L

Eq. 23 will not be calculated explicitly here but instead will appear in a future work. However we can note that the quantity in brackets, if correctly evaluated would be given by an expression of the form q&(s) where E is some effective location for the polarization charge of the positive ion core and has a numerical value greater than s for small S. Thus we may write MO = q(s + E(S)). The change of electron work function of a metal resulting from a partial monolayer of discrete dipoles on the surface is given by d~$, = 27rcoA4,where cr is the dipole density and A4 the dipole strength. It has been shown 3, 4, that if the dipole strength of the ionic cesium-tungsten bond is given by an expression of the form M=ql, then 1 must be of the order of 4A to account for the drastic change in work function, This value is compatible with the theoretical picture provided in this paper. In order to obtain a dipole of sufficient strength to explain the experimental data, an ionic bond of the type predicted in this paper must be formed between adsorbate and substrate. This also indicates that the physical approach taken by some authorsi-5) is actually a justifiable approach. It is interesting to examine the consequences of non-zero temperatures on the perfect ionic bond. As the temperature is raised above zero, some electrons can be found at any energy above the Fermi level. Thus there will be a slight population of the atom-metal levels which have energies falling within the band of virtual atomic states. This is shown in fig. 8 where the finite temperature Fermi distribution determines the occupation probabilities. The effects of the finite temperature are easily understood in terms of the system wavefunctions of fig. 6. At zero temperature all electrons were described by wavefunctions of the form $,. As the temperature is raised, a temperature dependent fraction of the n + 1 electrons will be

152

I. W.

GADZUK

n(E) Fig. 8.

Relation between broadened atom-level and Fermi-Dirac for non-zero temperatures.

distributed

electrons

in states with energies within the broadened virtual atomic band. Those electrons, originally with $,,, type wavefunctions, are now allowed to have $,’ wavesfunction. By including thermal effects, eq. (23) must be modified to read : ” M,=(qs+f

CPiCzijzi)+ i=l

(24)

- UC/.(Ek)‘S~~~zl:,dr/~~o(Ek), ksl-

-L

ker

where fo(Ek) is the Fermi function, T$k means that the states within the virtual atomic band r are not included in the sum over k and kEI’ signifies summation over those values of k included in r only. For zero temperature, eq. (24) reduces to eq. (23). The thermal effects populate $,’ levels. But electrons in states y?=’tend to neutralize the alkali ion core by concentrating more electron charge in the alkali region. This is just the thermal depolarization effect which was mentioned by Rasor and Warner3) in a slightly different manner. Thus it is seen that a very significant temperature dependence is implicit in the present analysis. This fact may be useful in experimental investigations as a means for altering the population of electron states in a measureable way since the broadened band lies only of order of tenths of electron volts above the Fermi level which is of the same order as thermal energies.

ATOM-METAL

In summary,

INTERACTIONS.

it is seen that the interaction

I.

153

of an alkali atom with a metal

has been considered, starting from a very basic point. First-order perturbation techniques were used to obtain a standard-form expression for the ns level shift. Previously suggested shifts for adsorbed alkalis are consistent with those calculated herein. It has been shown that the atom-metal interaction causes an upward shift of about 0.3 eV in the perturbed 11seigenvalue of adsorbed alkalis provided it is meaningful to consider the interaction in perturbation theory. Phenomenological and qualitative justifications for this have been given. It has been shown that formally the interaction of the ns electron of the alkali atom with the metal is equivalent to a rearrangement collision. Well known standard techniques for dealing with rearrangement processes have been applied to the atom-metal interaction. By using these procedures and time dependent perturbation theory, the lifetime and thus the natural broadening of the perturbed IZSlevel have been calculated. Theoretically it is found that the perturbed level is less than 1 eV in width for adsorbed alkalis. This is considerably sharper than the width suggested by Gomerll*zs). The calculated value of the bandwidth does approach that value required for a straightforward application of the simple ionic binding model as suggested by Rasor and Warner. Due to the somewhat arbitrary assignment of the distance from the surface of an adsorbed particle, there have purposely not been more than approximate order of magnitude statements pertaining to numerical values of level shifts and widths. In all cases the atom-metal separation has been taken to be one half the lattice constant plus the alkali radius for it seems that the only well defined distance is that one between alkali atom and substrate atom nuclei. It has been shown that the ionic bond picture for alkalis adsorbed on metals seems to be justified. Polar metallic bonds do not produce a sufficient dipole moment to account for work function changes of metals with an adsorbed alkali monolayer. Net charge transfer from the atom into the metal is required to form a strong enough surface dipole. The charge that is transferred to the metal effectively localizes in the interior of the metal, but near the surface, forming a polar, ionic bond with the alkali ion core. If the temperature of the metal is raised above zero, there will be some electrons with energies overlapping the perturbed its band. It was then shown that population of these states tends to reduce the dipole moment of the metaladsorbed alkali configuration. This is simply a thermal depolarization effect. Finally it has been shown that thinking of the full atom-metal system as a system of interacting virtual atomic and metal states is useful in providing both physical insight and quantitative information pertaining to the true interacting atom-metal system.

154

J. W.

GADZUK

Acknowledgements The author had the pleasure of helpful discussions with Professors E. N. Carabateas, R. E. Stickney, and Dr. N. S. Rasor. Numerical computations were done at the MIT Computation Center. Appendix A The purpose of this appendix full many-body Hamiltonian particle Hamiltonian given by that the total Hamiltonian can H,,, = - ;

is to explicitly show the manner in which the of eq. (1) is reduced to an effective single eq. (7). Making reference to fig. 3, it is seen be written: n A2 V: T/(ri - RJ_) + Fyi + 2m* c c

Vr’ - &

i,L

i=l

+

Hphonon

_

5

_

-$+&-;I.

__4_

(Al)

re

The terms in the full Hamiltonian = kinetic

of eq. (Al) are: energy of electron

involved

in the bonding,

kinetic energy of alkali ion core,

A2 2m*

c

Vr’{

kinetic

energy of all IZ conduction

band electrons,

i=l

v(f+i

-

c i,L

44

periodic lattice electrons,

potential

H plWIl0n

lattice Hamiltonian,

- cl2

attraction

between

felt by conduction

atomic

electron

band

and alkali

ion

core, attraction image, repulsion image,

cl2 - 4s

= attraction

between

between

atomic

atomic

electron

electron

and

its own

and the alkali

of ion core with its own image,

ATOM-METAL

=

INTERACTIONS.

I.

155

repulsion between alkali ion and image of atomic electron,

q2

= attraction

r’

image.

between ion image and atomic electron

Since we are interested in processes involving the atomic electron, only those terms in the full Hamiltonian which are functions of the coordinate re are relevant. First consider the q’/IR’I term. Since the ion mass M is much greater than the electron mass m, the ion does not follow the detailed motion of the electron image but instead interacts with this image in its time average position which is at a point --s into the metal. This is just a Born-Oppenheimer approximation. Thus ~o = q2/2s.

W)

The interaction -q2/r’ is more involved. This effective interaction describes the conduction band electrons which have rearranged to screen the ion core, interacting with the conduction band electrons which have rearranged themselves to screen the atomic electron. Thus this interaction term is written as n Cl2 -= Cf(lri - fjI,S) (A3) r’

i
where the sum is over all occupied conduction band states only. The precise form off()r, - rjJ, 8) could be obtained only through a detailed analysis which we shall not pursue here. The point is that this interaction term in reality depends only on the band electron coordinates and the average position of the atom. Since the perturbation of the metal on the atom serves to shift the atomic level an amount AE, the perturbation of the atom on the metal should shift the conduction band states. This is the role played by this interaction term. Since each of the IZoccupied states is perturbed, the shift per state is lim n_aoO(AE/n)+O. Thus if we consider only a single band state, it is seen that the effects of the -q2/r’ interaction are small. Using eqs. (A2) and (A3) in a rearranged eq. (Al) yields:

-- h2 2m*

cvz*+c

V(ri - RJ +

i, L

c ”

f

i
(Iri - rjl,

s>+ Hphonon

156

I.

W. GADZUK

or H,,,= Wr,,s)+ Hion + fLet(ri&,s). W e will be concerned with single electron transitions between states $,,,(r,) and ui(re) or shifts of these states. The only non-vanishing matrix elements using the above Hamiltonian are CUiCre>

its hermitian

lHt0tl$~,s(~J) = C”iCre> Iff(re9s)l $,,s(~J)

3

conjugate, (rLS(re)

and

lHtotl 4ks(reD = ($,,,(rJ

Iff(r,,s)

W,,,(rJ>

when the wavefunctions are orthogonalized. With this separation of coordinates, the single electron Hamiltonian for an electron of unperturbed kinetic energy p + 4, - Vi, moving in the field provided by the perturbed metal plus the alkali ion core at position s is given by

which is the expression

cl2

42 Vz_ - $ 4llsl-r,.i,l e

H(r,, s) = $

+

12s

-

r,I’

used in eq. (7).

Appendix B The basic procedure for evaluating the integrals given. The following equalities are obtained

e

IT2 -JJJ

-

re

-2nr

zaeiwr

1

-2ar _

1 =n

sss

W)

(&Tq2dQ

(t2” - ‘“) (4a2 + X2)3

eiwr

rc2 sss

(42

dSX

W)

’

(4a3-ax2)

2 -2ar=y re

of eq. (10) will now be

iwrd3%

+ X2>”e

by taking the inverse of the Fourier transform Thus the numerator of eq. (10) can be written

’

of the particular

(B3) function.

24a2 (4a3 - ax2) + -- (42 Realizing

that je i XiXidxi= 27rnS(x,), consistent

+ 3c2)4

with the normalization

of the

ATOM-METAL

Fourier

transform,

INTERACTIONS.

and interchanging

+

157

I.

the orders of integration

96a2(4a3 - ax’) (4a2 + %2)4

yields

ix,2 6 k)

6 (XY)

d3%

.

e

>

Since Sf (xi) 6 (xi) dxi = f (0) 9

rl=

f

dzH’(z){~((4ai8+a~f)i-8~~~Ta:~~‘+

-_(s-SC) 96a2(4a3 +

- ax;) (B4)

(4u2 + Xz”)”

The x, integral can be done in the complex plane by choosing a contour along the real axis from - co to + cc and closing it with a semi-circle in the upper half plane for convergence. Poles of order 2, 3, and 4 at x,=2iu are enclosed in this contour. Performing the extremely tedious task of calculating the residues results in the following expression for eq. (B4):

q=

1

2a’=‘(4u3

dzH(z)Ee-

1213

-

2u2 1zj2 + 2u IZJ + 1) .

(B5)

-(s-so)

Note that by taking absolute values for z, use of a single contour for the x, integration which is valid for the full range of positive and negative values of z, is permissible. For the z integration, two separate integrals are done, one for each term in the perturbation. In the first case, a new variable t1 =2s+z is defined and in the second <, =s+z. Making the straightforward transformation and the extremely laborious but simple integrations gives the numerator appearing in eq. (11). The denominator is much simpler. Just remove H’(z) from eq. (B5) and integrate straightforwardly. Thus m

D=

4? e-2a’zJ(4u3 (z13 - 2u2 1~1’ + 2u lzJ + 1) U2

-(s-Q

Dividing

gives AE= x/D which is the result shown in eq. (11).

.

158

1. W. GADZUK

References 1) J. B. Taylor and I. Langmuir, Phys. Rev. 44 (1933) 423. 2) J. H. De Boer, in: Advances in Catalysis, Vol. VIII (Academic Press, Inc., New York, 1956). 3) N. S. Rasor and C. Warner, III, J. Appl. Phys. 35 (1964) 2589. 4) J. W. Gadzuk and E. N. Carabateas, J. Appl. Phys. 36 (1965) 357. 5) J. R. MacDonald and C. D. Barlow, Jr., J. Chem. Phys. 40 (1964) 1535; 43 (1965) 2575; 44 (1966) 202. 6) L. Schmidt and R. Gomer, J. Chem. Phys. 42 (1965) 3573. 7) E. P. Gyftopoulos and J. D. Levine, J. Appl. Phys. 33 (1962) 67; Surface Sci. 1 (1964) 171, 225, 349. 8) T. Toya, J. Res. Inst. Catalysis, Hokkaido Univ. VI (1958) 308; VIII (1961) 209. 9) R. W. Gurney, Phys. Rev. 47 (1935) 479. 10) L. V. Dobretsov, Electron and Ion Emission (NASA Technical Translation F-73, 1952). 11) R. Gomer and L. W. Swanson, J. Chem. Phys. 38 (1963) 1613. 12) J. W. Gadzuk, in: Proc. of the 25th Ann. Co& on Phys. Electronics, MIT March, 1965. 13) D. Sternberg, Ph. D. Thesis, Dept. of Physics, Columbia University 1957 (unpublished). 14) H. D. Hagstrum, Phys. Rev. 96 (1954) 336; 122 (1961) 83. 15) J. Bardeen, Phys. Rev. 49 (1936) 653. 16) T. L. Loucks and P. H. Cutler, J. Phys. Chem. Solids 25 (1964) 105. 17) F. Herman and S. Skillman, Atomic Structure Calculations (Prentice Hall, Inc , Englewood, N.J., 1963). 18) P. H. Cutler and J. C. Davis, Surface Sci. 1 (1964) 194. 19) F. M. Propst, Phys. Rev. 129 (1962) 7, and Report R-161 (Coordinated Science Laboratory University of Illinois, Feb. 1963). F. M. Propst and E. Luscher, Phys. Rev. 132 (1963) 1037. 20) M. H. Mittleman, Phys. Rev. 122 (1961) 1930. 21) T. Y. Wu and T. Ohmura, Quantum Theory of Scattering (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962) p. 211. 22) E. Jahnke and F. Emde, Tables oj’Functions (Dover Publishing Co., New York, 1945). 23) H. Utsugi and R. Gomer, J. Chem. Phys. 37 (1962) 1720. 24) H. S. W. Massey, Proc. Cambridge Phil. Sot. 26 (1930) 386; 27 (1931) 469. 25) W. I. Harrison, Phys. Rev. 123 (1961) 85. 26) J. W. Gadzuk, Surface Sci. 6 (1967) 159. 27) D. R. Bates, A. Fundaminsky and H. S. W. Massey, Phil. Trans. Roy. Sot. London 243 (1950) 93. 28) P. Nozieres, The Theory of Interacting Fermi Systems (W. A. Benjamin, Inc., New York, 1964).