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Effect of ordered overlayer on field emission from a substrate
Surface Science 70 (1978) 101-113 0 North-Holland Publishing Company
EFFECT OF ORDERED OVERLAYER SUBSTRATE *
ON FIELD EMISSION FROM A
Nikhiles KAR Department of Physics and Laboratory for Research of the Structure Pennsylvania, Philadelphia, Pennsylvania 19104, USA
of Matter, University of
Received 2 February 1977
For adsorbate covered metallic surface, Field Emission Energy Distributions (FEED) have been used primarily to study the properties of the adsorbate. In the case of intrinsic surface states/resonances for the substrate, it may be important to investigate the effect of chemisorption on the field emission from substrate. In this paper, we develop a formalism to treat the case of field emission from an ordered overlayer covered surface, from which the effects of surface states/resonances can be explicitly demonstrated. An application to a practical case of (1 X 1) gold covered tungsten (100) face is shown to give reasonable results.
1. Introduction Field emission from adsorbate covered metallic surfaces has been the object of experimental and theoretical study for many years. Field emission Energy Distributions (FEED) have been used to probe the chemisorption process, primarily by studying the spectroscopy of the adsorbate. An excellent account of these studies may be found in the review article by Gadzuk and Plummer [I]. The emphasis in theoretical work has been on the adsorbate spectrum rather than the change in the substrate emission due to chemisorption. Duke and Alferieff [2] were the first to point out the role of elastic resonance tunneling through virtual energy levels of atoms and molecules adsorbed on a metal surface. The theoretical picture of resonance tunneling was further developed by Gadzuk, Plummer and others. [3]. Penn, Comer and Cohen [4] reformulated the problem in terms of the Anderson model [5] and derived expressions for the energy resolved field emission current. On the other hand, there are some cases where it is important to look for the effect of the adsorbate on the field emission current from the substrate. One wellknown case is the so-called Swanson peak, seen in both photoemission [6] and * This work was supported in part by the National Science Foundation through the University of Pennsylvania Materials Research Laboratory; DMR 76-00678 and by the grant DMR 73-07682-A03. 101
102
N. Kar / Ordered overlayer and field emission from substrate
field emission [7] from the (100) face of tungsten. Various experiments [8] showed that the structure disappeared upon adsorption of CO, 02 as well as inert gases Kr and Ar. However, recent field emission experiments by Richter and Comer [9] on tungsten with adsorbed gold and photoemission experiments by Egelhoff et al. [lo] on tungsten with adsorbed mercury have shown that the structure is not destroyed. The Swanson peak is now believed to be due to a band of surface states/resonances [l l-l 31 on tungsten (100) face and is thus essentially connected with the properties of the substrate. Consequently, in order to study theoretically the experimental results mentioned above, it is necessary to describe the substrate as well as the overlayer reasonably accurately. In an earlier paper [ 121, we showed that in the context of field emission from (100) face of tungsten, geometrical structure of the overlayer may play an important part. In this paper, we shall develop a formalism, utilizing the scattering properties of the overlayer to obtain an expression for the field-emission current, from which we can explicitly demonstrate the effect of the overlayer on the intrinsic surface states/resonances of the substrate.
2. Formalism The model considered is an extension of the one used for clean surfaces by Nicolaou and Modinos [ll] and by Kar and Soven [12,13]. In that model it is assumed that to the left of a plane defined as z = a, the potential is the periodic potential of the bulk solid. To the right of the plane, the potential is assumed to depend only on z. When dealing with an ordered overlayer, we consider the last layer to be a layer of the adsorbed atoms with the appropriate geometry. The model is shown schematically in fig. 1. The vacuum-overlayer interface is assumed to be the plane tangential to the overlayer atomic muffin-tins and the external electric field is assumed to start from this plane. Using a transfer-hamiltonian technique, one may derive an expression for the energy resolved field-emission current [ 11-l 3,151
i’Q = j- d2k,, I &=o I2F(E, kll) NE, k,$ ,
(1)
SRZ
where F is a slowly varying function of E and kti, D(E, kll) is an exponential tunneling factor and Gg=e is the first transverse Fourier component of the wave function at the surface. In this paper, we shall describe the technique of computing the wave function at the overlayer-vacuum interface and specifically consider how the effects of surface states and/or resonances can be incorporated in the evaluation of the energy-resolved current. In general the overlayer will be considered to have a lower transverse symmetry than the substrate. The surface Brillouin zone (SBZ) for the overlayer would conse-
IV. Kar / Ordered overlayer and field emission from substrate
SUBSTRATE
103
OVERLAYER
VACUUM
(bx
ii:
T
(p’ x’
z’
YT
Z= -a
Z=0
Fig. 1. Schematic representation of the muffin-tin model. The last row (dotted circles) represent the overlayer muffin-tins. The matching plane is tengential to the last row of muffin-tins. Fig. 2. Schematic representation
of the matching planes for an overlayer covered surface.
quently be smaller. Let us denote by k} the (two-dimensional) reciprocal lattice vectors for the overlayer and by(G) those for the substrate SBZ. In fig. 2, we schematically show the problem of the wave function matching. With the origin at the centre of the overlayer, the two matching planes are at z = %z, where a is the muffin-tin radius of the overlayer atoms. The scattering properties of the overlayer will be used to do the matching. To describe the substrate, we shall use complex band wave functions, The technique of determining these wave functions has been discussed by Pendry [ 141 in some detail. Nicolaou and Modinos [ 1 l] were the first to use these complex band wave functions in the calculation of field emission current. We may write the wave function for a complex band, denoted by the index CY, as
and the derivative with respect to z as d’&l’z) dz
= C&&)
exp[i(kll + G) * ~111.
G
It may be shown that state, then
[ 131 if either at or CY’refer to a non-propagating
(3) Bloch
(4)
104
N. Kar / Ordered overlayer and field emission from substrate
with the proviso that when both functions are non-propagating, they must be decaying in the same sense. For 01and (Y’both propagating, it may be shown [13] that the proper ortho-normalization condition is
For the overlayer, using the transverse symmetry, hand side matching plane may be written as:
the wave function
on the right
~=Cx,exp[i(kll+g).rlll, g
(6)
and on the left-hand side
i=~x,exp[i(k,,+g).r~~l.
(7)
The functions j&, G have dependence on the normal co-ordinate. For convenience, the dependence has been left implicit; further, G is the value at z = a, and xR the value at z = -a, with the origin now considered to be at an atomic site for the overlayer. We now introduce amplitudes [14] ug, ug on the left-hand side and ii,, i$ on the right-hand side which are related to Q and & respectively. The relations are, for the left-hand side: xg = ug exp(-iKgla)
+ ug exp(iKgla) ,
XL = iKglUg exp(-iK,la)
(84
- iKglu g exp(iKcqu) ,
@b)
and on the right-hand side 5$ = Ligexp(irc,la)
t Cq exp(-iK,la)
2; = iKglUg exp(irc,la)
,
- iKgLiJgexp(--irc,la)
Pa) ,
@b)
where xi, x; are the derivatives of the functions with respect to the normal coordinate z and evaluated at z = -a and z = +a respectively. The quantity Kg1 is defined as (E - l&l + gl 2) l/2
for
E > lkll +g12 ,
1 i(lkll +g12 -- E)1/2
for
E < lkll +gl* .
(10)
Kg1 =
The scattering properties of the layer may be expressed in the form [ 141: (1 la)
I& =
5
(Rg&ig,
+ Tgg,ug,)
(1lb)
N. Kar / Ordered overlayer and field emission from substrate
105
where the matrices T and R incorporate the scattering properties of the atoms in the layer. The expressions for T and R are well-known and a derivation may be found in the book by Pendry [14]. Considering x, x, u, u, etc. to be vectors with components G, j&, ug, uR etc. T and R to be matrices with components Tgg,, R,, and defining matrices Egg’ =
exp(irc,la)
6,,gg ,
Kg,gg = iKgl 6,,t
(12)
,
we can write eqs. (8), (9), and (11) in matrix form. Eq. (11) becomes
which may be manipulated
to yield (13)
with
Q, =7-l,
Qrr = -T-‘R
QIII=R~‘,
Qrv=
,
T-RT-‘R.
Utilizing these expressions we can relate x and x’ to zand On the vacuum side the wave function may be written
(14)
x’.
so that at the plane z = a, ti = ?Bg
exp]i&
+g)
~111 ,
(15)
and the derivative in the normal direction IIl’ =F
Y&& exp [@It + g)
ql ,
where 7, is the logarithmic derivative at z = a for LQ. We now consider the wave function matching problem; layer-vacuum interface plane, one gets Zg=Bg,
(16) first, across the over-
< = ygBg .
If we consider y to be a diagonal matrix with components write the above equations in a matrix form,
Y~,~’= Y$~,~,, we can
(17)
N. Kar / Ordered overlayer and field emission from substrate
106
Now using eq. (17) in conjunction with (8), (9) and (13), we can write the left-hand side wave function and derivative x and x’ in terms of .the amplitude B. One may write x’=
x=WB,
W’B ,
(18)
with W=;[(E-lQI+EQur)(l
tyk-‘)E-’
W’=&Y[(E-lQI-EQuI)(l
t(E-‘QutEQIv)(l
+yk-‘)E-l+(E-lQII-EQ,v)(l
-^/K-r)171, -yk-‘)I!?] .
(19a) (19b)
Let us define a matrix r such that x’=rx.
(20)
It may be easily proved that the matrix r is hermitian. (18) and (20) that r = w’w-’
.
It is clear from comparing
(21)
We now have to satisfy the matching conditions across the substrate overlayer interface. Let us assume that the SBZ for the overlayer is smaller than the SBZ for the substrate. For simplicity, let us assume that for each point in the overlayer SBZ, there are two points in the substrate SBZ, which would be connected by a reciprocal lattice vector of the overlayer. This would be true for a (2 X 1) or c(2 X 2) structure. Thus, some point ktt within the overlayer SBZ is equivalent to two points, /cl,, kll +gl, both inside the substrate SBZ. The generalization of the following to a structure of lower symmetry is obvious. The matching conditions at the substrate-overlayer matching plane are
t c ~&,,) GYP
=
F x,
{Rp,
+ R&t)
exp [Gil
exp[i(klj + g) . 711I .
+ C)
r
III
(24
In the above equation $G~‘s are the components of the incident wave, i.e. the wave function corresponding to a real Bloch wave travelling from left to right; cpcp’s are components of the wave functions for complex band states, i.e. they propagate (in the case of real Bloch states) or decay (in the case of complex Bloch states) from right to left. There is an equation similar to (22) with the primed quantities for the derivatives. We shall consider the set (g} such that &} includes {G) and {G’ + gr}. We note
N. Kar / Ordered overlayer andfield emissionfrom substrate
107
that the number indicated by the index @is the number of G’s in {G} and the nymber indicated by p’ is the number of reciporcal lattice vectors in the set {G +g$). Let us make the following definitions for convenience. for
g =G ,
SE @I >
I0
for
g=g,+G’,
sE{a),
I
for
g=G,
6 E $3’) f
GG6 Ug&=
0
Considering V, and ~$6 to be now ~~dimension~ column vectors (since the sum of the numbers in the set {G) and {G’ + gl} equal the number is the set j’g}which is NJ, we may write down eq. (22) and the similar one for the derivatives as: % =--P&+x&,
v; = --P-h + T‘xs
9
where the matrices P and P’ are defined by (26a)
(26b) From (25) one may easily show that (r - p’P-l>xs
= @;i - P’P-fu&) )
i.e. XS=F-l&
(27)
3
with F=l-‘-P’P-‘,
ug =I& - P’F-‘us
,
(28)
108
N. Kar / Ordered overlayer and field emission from substrate
Since xs = W& 9 we may write Bs = W-‘F1u6
.
(29)
Let us define a matrix M such that M
c g’g” = 6
Ug’6$“6 ,
(30)
where the prime over the sum indicates that the sum would be carried out only for indices 6 corresponding to real Bloch waves. It is then easy to show that F’
&,a(* = (W-lF-‘M(F+-‘(W+-l)g,g.
(31)
NOW it may be shown that [13] for most important cases, it is possible to express M in terms of the matrix F, by utilizing the ortho-normalization conditions:
M= &p-F).
(32)
0
where So is the unit cell area. From (30) and (32) one gets I$ l&l2
[ w-‘(F+)-‘(w+)-’
=&
- W-‘FyFv+-‘],,,
.
0
(33)
If we define a matrix A such that A=W+FW,
(34)
then in terms of A, we have
F’ IBg612=-&
[(A+)-’ - A-‘lglg.
(35)
0
We may also note that for the amplitudes we may write
F’
lxgsl* = -
&
((F+)-’ - F-l).
x& at the substrate-overlayer
interface,
(36)
3. Discussion Let us now go back to eq. (1) for the expression of energy resolved field emission current. The important term there is I tiFo I* and that is equal to ZZb]8oa I*. Eq.
N. Kar / Ordered overlayer and field emission from substrate
109
(35) gives us a simple expression for zl;i l&s I2 in terms of the matrix A = I@f%’ and informations about surface states and/or resonances may be derived from considering the properties of the matrix A, or more correctly, F. An important case is for an energy E and transverse wave-vector kit for which there are no propagating Bloch states, i.e. E is in a band gap. It may be shown in that case F+ = F and consequently A+ = A, and so from (35) and (36) in general, both 1~1 and : lBg12are zero. However, if F is a singular matrix, i.e. det F = 0 for some E in a band gap, then F is no longer invertible and eqs. (35) and (36) would no longer apply. Both 1~~1’ and l&l2 could in that case be non-zero, and the condition det F= 0 corresponds to the existence of a surface state at the substrate-overlayer interface. Using the proper normalization condition for a bound state, one can then compute the current from the surface state. We may note two points in this connection: first, at some symmetry point on the surface Brillouin zone, real bands of a certain symmetry character may be absent, but complex bands of that symmetry may give rise to a surface state. In that case, even if there are real bands of other symmetry character, there would not be any mixing and pure surface states may exist even in presence of real bands. Secondly, it may be possible that although a surface state exists, from symmetry properties its first transverse fourier component may vanish. In that case, the surface state would not make any significant contribution to field emission current. It seems clear that the matrix F’ is the scattering matrix whose poles give the bound states. In presence of bands, and when not forbidden by symmetry, these bound states would broaden into resonances. That would mean a pole in the complex energy plane with a small imaginary part. For a true surface state, the condition is det F = 0, or det A = 0. If the matrix A is diagonalized and the eigenvalues are determined as a function of E for a fixed ktr, the surface state energy would be that value of E for which an eigenvalue vanishes. For the case of the resonance, at some energy, one of the eigenvalues will become very small in absolute value, and we may identify the energy of resonance by identifying the energy at which the real part of the eigenvalue becomes zero and the imaginary part is small. To incorporate the effect of surface resonances in the field emission current, we consider the matrix A to be diagonalized by the similarity operation with a matrix S, i.e. S-‘AS
= A,
(37)
where A = {h,} is the diagonal form. Let us denote the elements those of S-’ by Ji,. Then writing A=SAS-‘, and therefore A-’
= SA-‘S-i
(A+)-’
= (S-‘)+(A+)++
,
of S by S, and
110
N. Kar / Ordered overlayer and field emission from substrate
we can show that
After doing some simplification,
the above relation becomes:
where A, = AL + ihk. At resonance, one of the eigenvalues, say X, becomes very small in absolute value and the real part of X, is zero. It is then sufficient to keep in the sum only the term in X,. Further, assuming Xl: to be so small that the Gaussian form can be approximated by a delta function, it is possible to rewrite (38) as (39) where ER is the resonance energy, i.e. the energy at which Xh = 0. We may in the case of Xi = 0, i.e. in the case of a true surface state the expression over exactly into that for a true surface state with proper normalization. therefore use this method to identify surface states and/or resonances and the filed emission current from those states explicitly.
note that (39) goes We may compute
4. Application As an application of the above formalism, we shall consider the case of (1 X 1) overlayer of gold on (100) face of tungsten. Before that, briefly reviewing the case of clean tungsten, we note that the Swanson peak in the field emission energy distribution (FEED) curve has been attributed to a band of surface states/resonances according to recent calculations of Nicolaou and Modinos [ 1 l] and by Kar and Soven [12,13]. Symmetry considerations show that the dominant contribution to the structure comes from the interor of the SBZ and therefore the non-relativistic calculations mentioned above are sufficient for the purpose, since in the interior of the zone, spin-orbit coupling only makes quantitative changes in the band-structure. In fig. 3, we show the SBZ appropriate to the (100) face of tungsten. The crosshatched region indicates the region of the SBZ important in field emission. It should be noted that there are continuum states at all energies in this region of the SBZ. Nonetheless, true bound states or extremely narrow resonances also occur. On the symmetry lines, the bands (for energies in the region important for FEED) are antisymmetric (I%) or symmetric (I?$) with respect to reflection in the relevant mirror plane, while the surface states have opposite symmetry. In the interior of the zone, these states become resonances, but they are extremely sharp (the typical
N. Kar / Ordered overlayer and field emission from substrate
-0.05
-Ox)4 -0.03
-0.02
ENERGY
(Ry)
111
-0.01
E,
Fig. 3. Surface Brillouin zone for (100) face of tungsten; cross-hatched region is the one important for filed emission. Fig. 4. Calculated enhancement
curves for (a) clean and (b) (1 X 1) gold covered tungsten.
width being 10P4 Ry) since the mixing is nearly forbidden by symmetry. It may be noted that as long as the transverse periodicity is not disturbed and the external conditions are not drastically altered, the qualitative features of the surface states/ resonances are preserved. For example, although the position in energy of the surface states/resonance depend on the model, the symmetry character etc. are not changed whether we position the matching plane tangential to the last row of spheres or interesting those, and whether we consider the external potential to be smoothly varying or just a step barrier. On the other hand, as we showed in an earlier paper, an overlayer of lower transverse periodicity than the clean tungsten (100) face opens up new channels of decay for the bound states, and, in general, the states become resonances with such a large width that they are no longer observable in field emission. However, it is an interesting question whether the surface states/resonances would be preserved with a (1 X 1) overlayer, where the transverse periodicity is retrained. In view of the experimental data with gold overlayer, we investigated the case of a (1 X 1) overlayer of gold on tungsten (100) face. A brief report of this has already been published. The potential was constructed by superposing the atomic potentials of one layer of gold atoms and a few layers of tungsten. While the potential is not self-consistent, the gold phase shifts we calculate using it are not unreasonable. In particular, the gold d-wave resonance is well below the Fermi energy
112
N. Kar / Ordered overlayer and field emission from substrate
(determined by the tungsten substrate), suggesting that in the model a gold atom will have essentially the full complement of electrons. We treat the overlayer system by the formalism developed above. We identify the surface states/resonances and separately compute the current from them. We find that the situation is qualitatively identical to the clean tungsten case. There are pure surface states along i? and i% and resonances at general points in the zone. These states are quite close in energy to the clean tungsten bound states, although roughly shifted by O.OlRy. The strengths of these resonances as measured by their contribution to the field emission current is slightly less than in clean tungsten, and they tend to be somewhat broader as well. The enhancement curve done with a full overlayer is shown in fig. 1. The dotted curve is the one for clean tungsten and we see that the curves are similar in nature, although they are separated in energy by about 0.01 Ry. Experimental results suggest that the separation is smaller than our theoretical result; however, from our fairly simple model, we do not expect accurate quantitative agreement, but a qualitative explanation of the experimental data. These theoretical results seem quite plausible. When the case of clean tungsten is considered, it is observed that the external potential does not have a very strong influence on the surface states/resonances; in particular, the physical origin of these states are more strongly tied to the band structure of tungsten than to external conditions. We might therefore expect that within limits, the surface states will exist for any reasonable potential with the same tranverse symmetry as the substrate. However, although our formalism would predict the existence of the structure with gold adsorbed on tungsten with (1 X 1) structure, it would also predict a shift. The fact that experimental results show little, if any, shift may indicate that the model for calculation is not accurate enough. In conclusion, we have developed a formalism for treating an ordered overlayer on a semi-infinite solid. Although the phenomenon of field emission is the one that has been considered explicitly and an application to an actual case is shown to give reasonable results, this formalism can be used to treat other phenomena like photoemission, for example, especially where surface states and/or resonances play an important role. Acknowledgements The author is grateful to Professor Paul Soven for suggesting this problem and for his helpful suggestions. The author also wishes to thank Professor E.W. Plummer for many illuminating discussions.
References [l] J.W. Gadzuk and E.W. Plummer, Rev. Mod. Phys. 45 (1973) 487. [2] C.B. Duke and M.E. Alferieff, J. Chem. Phys. 46 (1967) 923.
N. Kar /Ordered
overlayer and field emission from substrate
113
[ 31 E.W. Plummer, J.W. Gadzuk and R.D. Young, Solid Staie Commun. 7 (1969) 487; J.W. Gadzuk, Phys. Rev. Bl (1970) 2110;Phys. Rev. B3 (1971) 1722. [4] D.R. Penn, R. Comer and M.H. Cohen, Phys. Rev. Letters 27 (1971) 26; Phys. Rev. BS (1972) 768. [5] P.W. Anderson, Phys. Rev. 124 (1961)41. [6] L.W. Swanson and L.C. Crouser, Phys. Rev. Letters 16 (1965) 389; Phys. Rev. 163 (1967) 622. [7] B.J. Waclawski and E.W. Plummer, Phys. Rev. Letters 29 (1972) 783. [SJ E.W. Plummer and A.E. Bell, J. Vacuum Sci. Technol. 9 (1972) 583; C. Lea and R. Comer, J. Chem. Phys. 54 (1971) 3394. [9] L. Richter and R. Gomer, Phys. Rev. Letters 37 (1976) 763. [lo] W.E. Egelhoff, D.L. Perry and J.W. Linnett, Surface Sci. 54 (1976) 670.
[ 111 N. Nicolaou [ 121 [ 131 [ 141 [15]
and A. Modinos, Phys. Rev. Bll (1975) 3587; A. Modinos and N. Nicolaou, Phys. Rev. B13 (1976) 1536. N. Kar and P. Soven, Solid State Commun. 20 (1976) 977. N. Kar, Ph.D. Thesis, University of Pennsylvania. J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, D.R. Penn, Phys. Rev. B14 (1976) 849.
1974).
EFFECT OF ORDERED OVERLAYER SUBSTRATE *
ON FIELD EMISSION FROM A
Nikhiles KAR Department of Physics and Laboratory for Research of the Structure Pennsylvania, Philadelphia, Pennsylvania 19104, USA
of Matter, University of
Received 2 February 1977
For adsorbate covered metallic surface, Field Emission Energy Distributions (FEED) have been used primarily to study the properties of the adsorbate. In the case of intrinsic surface states/resonances for the substrate, it may be important to investigate the effect of chemisorption on the field emission from substrate. In this paper, we develop a formalism to treat the case of field emission from an ordered overlayer covered surface, from which the effects of surface states/resonances can be explicitly demonstrated. An application to a practical case of (1 X 1) gold covered tungsten (100) face is shown to give reasonable results.
1. Introduction Field emission from adsorbate covered metallic surfaces has been the object of experimental and theoretical study for many years. Field emission Energy Distributions (FEED) have been used to probe the chemisorption process, primarily by studying the spectroscopy of the adsorbate. An excellent account of these studies may be found in the review article by Gadzuk and Plummer [I]. The emphasis in theoretical work has been on the adsorbate spectrum rather than the change in the substrate emission due to chemisorption. Duke and Alferieff [2] were the first to point out the role of elastic resonance tunneling through virtual energy levels of atoms and molecules adsorbed on a metal surface. The theoretical picture of resonance tunneling was further developed by Gadzuk, Plummer and others. [3]. Penn, Comer and Cohen [4] reformulated the problem in terms of the Anderson model [5] and derived expressions for the energy resolved field emission current. On the other hand, there are some cases where it is important to look for the effect of the adsorbate on the field emission current from the substrate. One wellknown case is the so-called Swanson peak, seen in both photoemission [6] and * This work was supported in part by the National Science Foundation through the University of Pennsylvania Materials Research Laboratory; DMR 76-00678 and by the grant DMR 73-07682-A03. 101
102
N. Kar / Ordered overlayer and field emission from substrate
field emission [7] from the (100) face of tungsten. Various experiments [8] showed that the structure disappeared upon adsorption of CO, 02 as well as inert gases Kr and Ar. However, recent field emission experiments by Richter and Comer [9] on tungsten with adsorbed gold and photoemission experiments by Egelhoff et al. [lo] on tungsten with adsorbed mercury have shown that the structure is not destroyed. The Swanson peak is now believed to be due to a band of surface states/resonances [l l-l 31 on tungsten (100) face and is thus essentially connected with the properties of the substrate. Consequently, in order to study theoretically the experimental results mentioned above, it is necessary to describe the substrate as well as the overlayer reasonably accurately. In an earlier paper [ 121, we showed that in the context of field emission from (100) face of tungsten, geometrical structure of the overlayer may play an important part. In this paper, we shall develop a formalism, utilizing the scattering properties of the overlayer to obtain an expression for the field-emission current, from which we can explicitly demonstrate the effect of the overlayer on the intrinsic surface states/resonances of the substrate.
2. Formalism The model considered is an extension of the one used for clean surfaces by Nicolaou and Modinos [ll] and by Kar and Soven [12,13]. In that model it is assumed that to the left of a plane defined as z = a, the potential is the periodic potential of the bulk solid. To the right of the plane, the potential is assumed to depend only on z. When dealing with an ordered overlayer, we consider the last layer to be a layer of the adsorbed atoms with the appropriate geometry. The model is shown schematically in fig. 1. The vacuum-overlayer interface is assumed to be the plane tangential to the overlayer atomic muffin-tins and the external electric field is assumed to start from this plane. Using a transfer-hamiltonian technique, one may derive an expression for the energy resolved field-emission current [ 11-l 3,151
i’Q = j- d2k,, I &=o I2F(E, kll) NE, k,$ ,
(1)
SRZ
where F is a slowly varying function of E and kti, D(E, kll) is an exponential tunneling factor and Gg=e is the first transverse Fourier component of the wave function at the surface. In this paper, we shall describe the technique of computing the wave function at the overlayer-vacuum interface and specifically consider how the effects of surface states and/or resonances can be incorporated in the evaluation of the energy-resolved current. In general the overlayer will be considered to have a lower transverse symmetry than the substrate. The surface Brillouin zone (SBZ) for the overlayer would conse-
IV. Kar / Ordered overlayer and field emission from substrate
SUBSTRATE
103
OVERLAYER
VACUUM
(bx
ii:
T
(p’ x’
z’
YT
Z= -a
Z=0
Fig. 1. Schematic representation of the muffin-tin model. The last row (dotted circles) represent the overlayer muffin-tins. The matching plane is tengential to the last row of muffin-tins. Fig. 2. Schematic representation
of the matching planes for an overlayer covered surface.
quently be smaller. Let us denote by k} the (two-dimensional) reciprocal lattice vectors for the overlayer and by(G) those for the substrate SBZ. In fig. 2, we schematically show the problem of the wave function matching. With the origin at the centre of the overlayer, the two matching planes are at z = %z, where a is the muffin-tin radius of the overlayer atoms. The scattering properties of the overlayer will be used to do the matching. To describe the substrate, we shall use complex band wave functions, The technique of determining these wave functions has been discussed by Pendry [ 141 in some detail. Nicolaou and Modinos [ 1 l] were the first to use these complex band wave functions in the calculation of field emission current. We may write the wave function for a complex band, denoted by the index CY, as
and the derivative with respect to z as d’&l’z) dz
= C&&)
exp[i(kll + G) * ~111.
G
It may be shown that state, then
[ 131 if either at or CY’refer to a non-propagating
(3) Bloch
(4)
104
N. Kar / Ordered overlayer and field emission from substrate
with the proviso that when both functions are non-propagating, they must be decaying in the same sense. For 01and (Y’both propagating, it may be shown [13] that the proper ortho-normalization condition is
For the overlayer, using the transverse symmetry, hand side matching plane may be written as:
the wave function
on the right
~=Cx,exp[i(kll+g).rlll, g
(6)
and on the left-hand side
i=~x,exp[i(k,,+g).r~~l.
(7)
The functions j&, G have dependence on the normal co-ordinate. For convenience, the dependence has been left implicit; further, G is the value at z = a, and xR the value at z = -a, with the origin now considered to be at an atomic site for the overlayer. We now introduce amplitudes [14] ug, ug on the left-hand side and ii,, i$ on the right-hand side which are related to Q and & respectively. The relations are, for the left-hand side: xg = ug exp(-iKgla)
+ ug exp(iKgla) ,
XL = iKglUg exp(-iK,la)
(84
- iKglu g exp(iKcqu) ,
@b)
and on the right-hand side 5$ = Ligexp(irc,la)
t Cq exp(-iK,la)
2; = iKglUg exp(irc,la)
,
- iKgLiJgexp(--irc,la)
Pa) ,
@b)
where xi, x; are the derivatives of the functions with respect to the normal coordinate z and evaluated at z = -a and z = +a respectively. The quantity Kg1 is defined as (E - l&l + gl 2) l/2
for
E > lkll +g12 ,
1 i(lkll +g12 -- E)1/2
for
E < lkll +gl* .
(10)
Kg1 =
The scattering properties of the layer may be expressed in the form [ 141: (1 la)
I& =
5
(Rg&ig,
+ Tgg,ug,)
(1lb)
N. Kar / Ordered overlayer and field emission from substrate
105
where the matrices T and R incorporate the scattering properties of the atoms in the layer. The expressions for T and R are well-known and a derivation may be found in the book by Pendry [14]. Considering x, x, u, u, etc. to be vectors with components G, j&, ug, uR etc. T and R to be matrices with components Tgg,, R,, and defining matrices Egg’ =
exp(irc,la)
6,,gg ,
Kg,gg = iKgl 6,,t
(12)
,
we can write eqs. (8), (9), and (11) in matrix form. Eq. (11) becomes
which may be manipulated
to yield (13)
with
Q, =7-l,
Qrr = -T-‘R
QIII=R~‘,
Qrv=
,
T-RT-‘R.
Utilizing these expressions we can relate x and x’ to zand On the vacuum side the wave function may be written
(14)
x’.
so that at the plane z = a, ti = ?Bg
exp]i&
+g)
~111 ,
(15)
and the derivative in the normal direction IIl’ =F
Y&& exp [@It + g)
ql ,
where 7, is the logarithmic derivative at z = a for LQ. We now consider the wave function matching problem; layer-vacuum interface plane, one gets Zg=Bg,
(16) first, across the over-
< = ygBg .
If we consider y to be a diagonal matrix with components write the above equations in a matrix form,
Y~,~’= Y$~,~,, we can
(17)
N. Kar / Ordered overlayer and field emission from substrate
106
Now using eq. (17) in conjunction with (8), (9) and (13), we can write the left-hand side wave function and derivative x and x’ in terms of .the amplitude B. One may write x’=
x=WB,
W’B ,
(18)
with W=;[(E-lQI+EQur)(l
tyk-‘)E-’
W’=&Y[(E-lQI-EQuI)(l
t(E-‘QutEQIv)(l
+yk-‘)E-l+(E-lQII-EQ,v)(l
-^/K-r)171, -yk-‘)I!?] .
(19a) (19b)
Let us define a matrix r such that x’=rx.
(20)
It may be easily proved that the matrix r is hermitian. (18) and (20) that r = w’w-’
.
It is clear from comparing
(21)
We now have to satisfy the matching conditions across the substrate overlayer interface. Let us assume that the SBZ for the overlayer is smaller than the SBZ for the substrate. For simplicity, let us assume that for each point in the overlayer SBZ, there are two points in the substrate SBZ, which would be connected by a reciprocal lattice vector of the overlayer. This would be true for a (2 X 1) or c(2 X 2) structure. Thus, some point ktt within the overlayer SBZ is equivalent to two points, /cl,, kll +gl, both inside the substrate SBZ. The generalization of the following to a structure of lower symmetry is obvious. The matching conditions at the substrate-overlayer matching plane are
t c ~&,,) GYP
=
F x,
{Rp,
+ R&t)
exp [Gil
exp[i(klj + g) . 711I .
+ C)
r
III
(24
In the above equation $G~‘s are the components of the incident wave, i.e. the wave function corresponding to a real Bloch wave travelling from left to right; cpcp’s are components of the wave functions for complex band states, i.e. they propagate (in the case of real Bloch states) or decay (in the case of complex Bloch states) from right to left. There is an equation similar to (22) with the primed quantities for the derivatives. We shall consider the set (g} such that &} includes {G) and {G’ + gr}. We note
N. Kar / Ordered overlayer andfield emissionfrom substrate
107
that the number indicated by the index @is the number of G’s in {G} and the nymber indicated by p’ is the number of reciporcal lattice vectors in the set {G +g$). Let us make the following definitions for convenience. for
g =G ,
SE @I >
I0
for
g=g,+G’,
sE{a),
I
for
g=G,
6 E $3’) f
GG6 Ug&=
0
Considering V, and ~$6 to be now ~~dimension~ column vectors (since the sum of the numbers in the set {G) and {G’ + gl} equal the number is the set j’g}which is NJ, we may write down eq. (22) and the similar one for the derivatives as: % =--P&+x&,
v; = --P-h + T‘xs
9
where the matrices P and P’ are defined by (26a)
(26b) From (25) one may easily show that (r - p’P-l>xs
= @;i - P’P-fu&) )
i.e. XS=F-l&
(27)
3
with F=l-‘-P’P-‘,
ug =I& - P’F-‘us
,
(28)
108
N. Kar / Ordered overlayer and field emission from substrate
Since xs = W& 9 we may write Bs = W-‘F1u6
.
(29)
Let us define a matrix M such that M
c g’g” = 6
Ug’6$“6 ,
(30)
where the prime over the sum indicates that the sum would be carried out only for indices 6 corresponding to real Bloch waves. It is then easy to show that F’
&,a(* = (W-lF-‘M(F+-‘(W+-l)g,g.
(31)
NOW it may be shown that [13] for most important cases, it is possible to express M in terms of the matrix F, by utilizing the ortho-normalization conditions:
M= &p-F).
(32)
0
where So is the unit cell area. From (30) and (32) one gets I$ l&l2
[ w-‘(F+)-‘(w+)-’
=&
- W-‘FyFv+-‘],,,
.
0
(33)
If we define a matrix A such that A=W+FW,
(34)
then in terms of A, we have
F’ IBg612=-&
[(A+)-’ - A-‘lglg.
(35)
0
We may also note that for the amplitudes we may write
F’
lxgsl* = -
&
((F+)-’ - F-l).
x& at the substrate-overlayer
interface,
(36)
3. Discussion Let us now go back to eq. (1) for the expression of energy resolved field emission current. The important term there is I tiFo I* and that is equal to ZZb]8oa I*. Eq.
N. Kar / Ordered overlayer and field emission from substrate
109
(35) gives us a simple expression for zl;i l&s I2 in terms of the matrix A = I@f%’ and informations about surface states and/or resonances may be derived from considering the properties of the matrix A, or more correctly, F. An important case is for an energy E and transverse wave-vector kit for which there are no propagating Bloch states, i.e. E is in a band gap. It may be shown in that case F+ = F and consequently A+ = A, and so from (35) and (36) in general, both 1~1 and : lBg12are zero. However, if F is a singular matrix, i.e. det F = 0 for some E in a band gap, then F is no longer invertible and eqs. (35) and (36) would no longer apply. Both 1~~1’ and l&l2 could in that case be non-zero, and the condition det F= 0 corresponds to the existence of a surface state at the substrate-overlayer interface. Using the proper normalization condition for a bound state, one can then compute the current from the surface state. We may note two points in this connection: first, at some symmetry point on the surface Brillouin zone, real bands of a certain symmetry character may be absent, but complex bands of that symmetry may give rise to a surface state. In that case, even if there are real bands of other symmetry character, there would not be any mixing and pure surface states may exist even in presence of real bands. Secondly, it may be possible that although a surface state exists, from symmetry properties its first transverse fourier component may vanish. In that case, the surface state would not make any significant contribution to field emission current. It seems clear that the matrix F’ is the scattering matrix whose poles give the bound states. In presence of bands, and when not forbidden by symmetry, these bound states would broaden into resonances. That would mean a pole in the complex energy plane with a small imaginary part. For a true surface state, the condition is det F = 0, or det A = 0. If the matrix A is diagonalized and the eigenvalues are determined as a function of E for a fixed ktr, the surface state energy would be that value of E for which an eigenvalue vanishes. For the case of the resonance, at some energy, one of the eigenvalues will become very small in absolute value, and we may identify the energy of resonance by identifying the energy at which the real part of the eigenvalue becomes zero and the imaginary part is small. To incorporate the effect of surface resonances in the field emission current, we consider the matrix A to be diagonalized by the similarity operation with a matrix S, i.e. S-‘AS
= A,
(37)
where A = {h,} is the diagonal form. Let us denote the elements those of S-’ by Ji,. Then writing A=SAS-‘, and therefore A-’
= SA-‘S-i
(A+)-’
= (S-‘)+(A+)++
,
of S by S, and
110
N. Kar / Ordered overlayer and field emission from substrate
we can show that
After doing some simplification,
the above relation becomes:
where A, = AL + ihk. At resonance, one of the eigenvalues, say X, becomes very small in absolute value and the real part of X, is zero. It is then sufficient to keep in the sum only the term in X,. Further, assuming Xl: to be so small that the Gaussian form can be approximated by a delta function, it is possible to rewrite (38) as (39) where ER is the resonance energy, i.e. the energy at which Xh = 0. We may in the case of Xi = 0, i.e. in the case of a true surface state the expression over exactly into that for a true surface state with proper normalization. therefore use this method to identify surface states and/or resonances and the filed emission current from those states explicitly.
note that (39) goes We may compute
4. Application As an application of the above formalism, we shall consider the case of (1 X 1) overlayer of gold on (100) face of tungsten. Before that, briefly reviewing the case of clean tungsten, we note that the Swanson peak in the field emission energy distribution (FEED) curve has been attributed to a band of surface states/resonances according to recent calculations of Nicolaou and Modinos [ 1 l] and by Kar and Soven [12,13]. Symmetry considerations show that the dominant contribution to the structure comes from the interor of the SBZ and therefore the non-relativistic calculations mentioned above are sufficient for the purpose, since in the interior of the zone, spin-orbit coupling only makes quantitative changes in the band-structure. In fig. 3, we show the SBZ appropriate to the (100) face of tungsten. The crosshatched region indicates the region of the SBZ important in field emission. It should be noted that there are continuum states at all energies in this region of the SBZ. Nonetheless, true bound states or extremely narrow resonances also occur. On the symmetry lines, the bands (for energies in the region important for FEED) are antisymmetric (I%) or symmetric (I?$) with respect to reflection in the relevant mirror plane, while the surface states have opposite symmetry. In the interior of the zone, these states become resonances, but they are extremely sharp (the typical
N. Kar / Ordered overlayer and field emission from substrate
-0.05
-Ox)4 -0.03
-0.02
ENERGY
(Ry)
111
-0.01
E,
Fig. 3. Surface Brillouin zone for (100) face of tungsten; cross-hatched region is the one important for filed emission. Fig. 4. Calculated enhancement
curves for (a) clean and (b) (1 X 1) gold covered tungsten.
width being 10P4 Ry) since the mixing is nearly forbidden by symmetry. It may be noted that as long as the transverse periodicity is not disturbed and the external conditions are not drastically altered, the qualitative features of the surface states/ resonances are preserved. For example, although the position in energy of the surface states/resonance depend on the model, the symmetry character etc. are not changed whether we position the matching plane tangential to the last row of spheres or interesting those, and whether we consider the external potential to be smoothly varying or just a step barrier. On the other hand, as we showed in an earlier paper, an overlayer of lower transverse periodicity than the clean tungsten (100) face opens up new channels of decay for the bound states, and, in general, the states become resonances with such a large width that they are no longer observable in field emission. However, it is an interesting question whether the surface states/resonances would be preserved with a (1 X 1) overlayer, where the transverse periodicity is retrained. In view of the experimental data with gold overlayer, we investigated the case of a (1 X 1) overlayer of gold on tungsten (100) face. A brief report of this has already been published. The potential was constructed by superposing the atomic potentials of one layer of gold atoms and a few layers of tungsten. While the potential is not self-consistent, the gold phase shifts we calculate using it are not unreasonable. In particular, the gold d-wave resonance is well below the Fermi energy
112
N. Kar / Ordered overlayer and field emission from substrate
(determined by the tungsten substrate), suggesting that in the model a gold atom will have essentially the full complement of electrons. We treat the overlayer system by the formalism developed above. We identify the surface states/resonances and separately compute the current from them. We find that the situation is qualitatively identical to the clean tungsten case. There are pure surface states along i? and i% and resonances at general points in the zone. These states are quite close in energy to the clean tungsten bound states, although roughly shifted by O.OlRy. The strengths of these resonances as measured by their contribution to the field emission current is slightly less than in clean tungsten, and they tend to be somewhat broader as well. The enhancement curve done with a full overlayer is shown in fig. 1. The dotted curve is the one for clean tungsten and we see that the curves are similar in nature, although they are separated in energy by about 0.01 Ry. Experimental results suggest that the separation is smaller than our theoretical result; however, from our fairly simple model, we do not expect accurate quantitative agreement, but a qualitative explanation of the experimental data. These theoretical results seem quite plausible. When the case of clean tungsten is considered, it is observed that the external potential does not have a very strong influence on the surface states/resonances; in particular, the physical origin of these states are more strongly tied to the band structure of tungsten than to external conditions. We might therefore expect that within limits, the surface states will exist for any reasonable potential with the same tranverse symmetry as the substrate. However, although our formalism would predict the existence of the structure with gold adsorbed on tungsten with (1 X 1) structure, it would also predict a shift. The fact that experimental results show little, if any, shift may indicate that the model for calculation is not accurate enough. In conclusion, we have developed a formalism for treating an ordered overlayer on a semi-infinite solid. Although the phenomenon of field emission is the one that has been considered explicitly and an application to an actual case is shown to give reasonable results, this formalism can be used to treat other phenomena like photoemission, for example, especially where surface states and/or resonances play an important role. Acknowledgements The author is grateful to Professor Paul Soven for suggesting this problem and for his helpful suggestions. The author also wishes to thank Professor E.W. Plummer for many illuminating discussions.
References [l] J.W. Gadzuk and E.W. Plummer, Rev. Mod. Phys. 45 (1973) 487. [2] C.B. Duke and M.E. Alferieff, J. Chem. Phys. 46 (1967) 923.
N. Kar /Ordered
overlayer and field emission from substrate
113
[ 31 E.W. Plummer, J.W. Gadzuk and R.D. Young, Solid Staie Commun. 7 (1969) 487; J.W. Gadzuk, Phys. Rev. Bl (1970) 2110;Phys. Rev. B3 (1971) 1722. [4] D.R. Penn, R. Comer and M.H. Cohen, Phys. Rev. Letters 27 (1971) 26; Phys. Rev. BS (1972) 768. [5] P.W. Anderson, Phys. Rev. 124 (1961)41. [6] L.W. Swanson and L.C. Crouser, Phys. Rev. Letters 16 (1965) 389; Phys. Rev. 163 (1967) 622. [7] B.J. Waclawski and E.W. Plummer, Phys. Rev. Letters 29 (1972) 783. [SJ E.W. Plummer and A.E. Bell, J. Vacuum Sci. Technol. 9 (1972) 583; C. Lea and R. Comer, J. Chem. Phys. 54 (1971) 3394. [9] L. Richter and R. Gomer, Phys. Rev. Letters 37 (1976) 763. [lo] W.E. Egelhoff, D.L. Perry and J.W. Linnett, Surface Sci. 54 (1976) 670.
[ 111 N. Nicolaou [ 121 [ 131 [ 141 [15]
and A. Modinos, Phys. Rev. Bll (1975) 3587; A. Modinos and N. Nicolaou, Phys. Rev. B13 (1976) 1536. N. Kar and P. Soven, Solid State Commun. 20 (1976) 977. N. Kar, Ph.D. Thesis, University of Pennsylvania. J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, D.R. Penn, Phys. Rev. B14 (1976) 849.
1974).