Surface Science 76 (1978) 443-463 0 North-Holland Publishing Company
A MODEL OF THE ION SPUTTERING PROCESS
H.G. PRIVAL * The %QW company, Link Division, Silver Spring, ~a~~nd
Received 28 June 1977; manuscript received in final form 21 March 1978
This paper proposes a new model of the ion sputtering process. According to this model: (1) Sputtering atoms or ions originate almost entirely from the surface layer. (2) Atoms initially leave the surface in the same charge state in which they exist in the crystal, and are largely neutralized in traversing a region near the surface. (3) Neutralization can be expiained in terms of quantum mechanical electron transmission and reflection phenomena that occur when the atom is within about three lattice spacings from the surface. The model upon which the theory is based pictures a potential well at the metal surface caused by the electric field of the departing ion. This potential well completely cancels the effect of the potential barrier normally present at the surface of the undisturbed metal, and for certain distances of the ion from the surface, results in a very high probability of emission of an electron from the metal surface, which then immediately neutralizes the emitted ion. This high neutralization probability explains the very low (low2 to 10M6) sputtered ion yields observed experimentally. The ion sputtering yields for a number of elements were computed and compared with values obtained experimentally. It is found that the theory gives results compatible with experiment, and also provides an, at least qualitative, explanation for some of the heretofore rather puzzling experimentally observed features of ion sputtering.
The process of ion sputtering, or secondary ion emission has been the subject of considerable investigation over the past two decades [l-9], and especially since the invention of the ion microscope by Castaing and Slodzian [lo]. The ion sputtering process, i.e., the ejection of charged particles which survive to macroscopic distances from the ejection site, is governed by processes in addition to those which affect neutral sputtering, and which are considerably less well understood. Theories of neutral sputtering are available which appear to agree with experiment quite well [ 1l-141. The same cannot be said for ion sputtering. This paper describes a model of the ion sputtering process. It is assumed that those factors which determine the neutral yield of sputtered particles are ade* The work described in this paper was performed while the author was a guest worker with the Surface Microanalysis Section of the Analytical Chemistry Division of the National Bureau of Standards, Washington, DC, USA. 443
H.G. Prival /Model
of ion sputteting process
quately explained by existing theory. An excellent review of neutral sputtering is to be found in a paper by Oechsner [ 131. The model to be presented is an attempt to give an interpretation of the observed features of ion sputtering in terms of the physical properties of the materials involved. Only metals and semimetals were studied but it is believed the theory should be applicable to non-metals as well. The model developed is used to compute the ion sputtering yields for a number of metals and semimetals and the computed values are compared with values measured by the author and by another independent worker [ 151. Before describing the proposed model, we describe some of the salient features of ion sputtering, as observed by many workers in the field. First, and most striking is the extremely large range of sputtered ion yields from different materials [l-4]. The sputtered ion yields measured by the author for 20 keV primary argon ions bombarding 32 different pure elements, varied over a range of almost 106, with the lowest yields being obtained from gold and cadmium and the highest being obtained from aluminum, silicon, and indium. Second is the fact that the absolute value of the ion sputtering yields is very small, IO-* to 10d6 ions per incident ion. It is very much smaller than the total sputtering yield, that is, total ions plus neutrals per incident primary ion, which is of the order to unity. Further, the ion yield does not appear to have any obvious correlation with the total yield. Table 1 shows the normalized total and ion yields of a few metals normalized to that of
Table 1 A comparison of total bared by argon ions
yields of elements
Principal mass number
Absolute total sputtering yield (45 keV) a
Measured relative ion sputtering yield (20 keV)
51 56 58 63 98 106 109 120 181 184 195 197 208
V Fe Ni cu MO Pd
1.0 2.3 3.5 6.8 1.5 5.3 10.8 4.3 1.6 2.3 5.3 10.2 10.5
1.0 0.17 0.041 0.027 0.018 0.013 0.022 0.0089 0.028 0.024 0.0077 0.0024 0.038
Ag Sn Ta w Pt Au Pb
Range of values a From
Almen and Bruce
H.G. Prival /Model
of ion sputtering process
vanadium. For example, gold, which has one of the poorest ion sputtering yields, has a very high total yield. In other words, in the sputtering process, many more neutrals are sputtered than ions. Third, it has been found that the presence of surface compounds, notably oxides, profoundly influences sputtering yields. For example, with strong oxide formers such as aluminum or silicon, the ion yield from an oxide film can be 10 to 100 times larger than that from the clean metal . Fourth, variation of sputtering yield with target temperature is also noted, but is not very rapid, certainly less than the exponential variation one would except if a thermionic process were involved in ion sputtering’. Finally, it is noted that the ion sputtering yield varies over a much wider range from metal to metal than any of the commonly measured properties of a metal. Table 2 shows 14 such properties for the 32 elements considered. The property showing the largest variation is the mass number.
2. Proposed model In constructing a model of the ion sputtering process, we start with the Sommerfeld model  of a metal. This is the familiar picture of an array of ion cores situated in a regular three dimensional lattice immersed in a “gas” of free electrons contributed by the valence electrons of the individual atoms. The mean free path of these “free” electrons, between collisions with the ion cores, is long compared to the lattice spacing, so that even though an individual electron moves in a potential field which has a strong periodic component due to the regular array of ion cores, as well as a random component due to the fields of the other electrons, the assumption is made that within the metal, an electron sees a uniform potential. The process of ion sputtering is hypothesized to be the following: An incident primary ion collides with a surface atom (ion core). The primary ion and the struck atom, now both having relatively high energy, then suffer collisions with other lattice atoms. At each collision, the incident particle loses energy and, in general, changes its direction of motion until finally, as a result of this series of collisions, one or more surface atoms are ejected. The total number of collisions between the arrival of the primary ion and the collision which results in the ejection of a surface atom is small This can be seen by the fact that the range of ions in the lo-20 keV energy range in metals is relatively small (of the order of 100 A) and the rate of energy loss by elastic collision is quite high. The surface atom which is ejected is hypothesized to be one of the ion cores of the metal lattice and is an ion not by virture of a collisional charge transfer process, but simply because according to the Sommerfeld model, almost all atoms in a metal exist in an ionized state . The reason that sputtered atoms are initially emitted from the metal surface in the same charge state as they exist in the body of the metal is that the probability of charge exchange in the collisions between primary ions or
4.073 4.518 4.393
1.814 1.837 1.880
4.311 4.775 4.932 4.695 5.386 5.134
1.859 1.901 1.921 1.979 2,099 2,115
:harge Exchange Jistance
Ullctio~ : (eV)
Table 2 Properties of the elements
F elml EnergY
2.610 2.619 2.702
F.WlIi Energy EfW
Binding Energy EbW)
Table 2 (continued)
Yin :harge Exchange 3istance
5.79 ,371o .3306
Lrns, 3 x1029) ('2)
3.02 2.90 2.07 2.85 2.73
'0.14 '0.89 11.39 8.33 9.11
FC TETRAG TETRAG WOMB HEX KC BCC
3.49 9.70 WOMB
2.74 2.88 5.46
Ct F El
high energy knock-ons is quite low. This can be seen by considering the large disparity between the velocity of incident ions and bound metal electrons. Landau  and Zener  have shown that the charge transfer probability is a very rapid function of the velocity ratio. Of the atoms ejected from the metal surface, almost all are initially ions. However, as an ion moves away from the metal surface, it is, with very high probability, neutralized by electron exchange with the surface. It is only those ions which survive neutralization at the surface, or that are neutralized to a metastable excited state, and which then de-excite by auto-ionization after emission from the surface, that contributes to the observed sputtered ion yield. However, auto-ionization makes only a minuie contribution to the sputtered ion yield, as can be seen by examining the energy spectrum of emitted ions. For if this process were to make an appreciable contribution, a corresponding number of sputtered ions would be found to have energies lower than that corresponding to the accelerating potential used to collect them, i.e., would have an apparent “negative” energy. Since almost all of the emitted ions wind up being neutralized, it is important to consider theories of neutral sputtering, since a successful theory of neutral spufitering will explain the number of ions initially emitted from the metal surface. Rol and coworkers [ 11,121 and other have developed theories of neutral sputtering which agree with experimental data extremely well. Rol gives an expression for the total sputtering yield, S,, atoms per incident primary ion, as
where E is the bombarding energy, Mr and M2 are the masses of the collision partners, h(E;? is the mean free path for elastic collisions and K is a constant. The constant K is a function of the binding energy of an atom to its lattice, and X(E) is a function of the incident ion energy, the atomic density, and the masses and atomic numbers of the collision partners. Using Rol’s theory, one can compute the total number of sputtered ions and atoms per incident ion, with good agreement with experiment. Up to this point, we can summarize the general features and assumptions of the proposed ion sputtering model. (a> Ion sputtering as discussed herein is a “one particle at a time” phenomenon. That is, in most analytics ion sputtering situations, the primary beam current density is so low that successive ions have essentially no effect on each other. It is certainly true at the experimentally employed current density of 30 mA/cm2. (b) Sputtered particles, ions or neutral atoms, originate only from the surface layer. (c) Almost all emitted atoms initially leave the surface in the ionized state. A very small number of these (less than 1%) survive neutralization on passage through a region, a few-lattice spacings thick, above the metal surface. (d) The total sputtering yieId, (ions plus neutrals) is adequately explained by Rol’s theory (eq. (1)).
H.G. Prival /Model
(e) The observed sputtered which survive neutralization.
of ion sputtering process
ion yield consists of just those sputtered
of the sputtering
With these assumptions, the sputtered ion yield of target metal X, $ sputtered ions per incident primary ion, can be expressed as:
(2) where S, is the total yield of sputtered atoms plus ions per primary ion, and P,, is the probability of an emitted ion of metal X being neutralized as it leaves the surface. Since we assume that S, can be computed adequately using Rol’s theory, all that remains is to compute the neutralization probability P,,. It is postulated that the mechanism by which neutralization takes place is the following: Consider fig. 1 which is a qualitative representation of the situation. As a surface ion leaves its normal lattice site (fig. la) and travels a certain short distance from its nominal position, the emitted ion causes a potential well to appear in front of the metal surface, as shown in fig. lb. This distance, X,tn (fig. lb), is obviously larger than the peak amplitude of the lattice vibration at the temperature in question, and smaller than the lattice spacing. In the proposed model, it is taken to be a certain fraction of the nearest neighbor distance of the lattice, and is a measure of the electrostatic “looseness” of an atom in its lattice. As the atom travels further away from the surface, the potential at the surface, which was originally, Emin, below the Fermi level, at the distance x,in, steadily increases until at the distance xb (see fig. lc) the potential due to the receding ion equals the Fermi level. For distances greater than xb, a potential barrier begins to reappear (fig. Id), and increases until for large distances from the surface, (fig. le), the barrier recovers to the value of the work function. In the range of distances, X,ln to xb, the potential barrier normally present at the metal surface is completely cancelled, resulting in a conduction electron being emitted from the metal with very high probability. It is this electron which neutralizes the departing.ion. Obviously this is a time varying situation, where the probability of an electron being emitted under the influence of the potential of the departing ion changes continuously with time. To make the problem mathematically soluble, we adopt a quasi-static approach, which is reasonable since the electron velocities are much larger than the ionic velocities involved. However, first we make the observation that we know ion neutralization occurs with very high probability. Second, we observe that in general, those quantum mechanical processes which are classically forbidden occur with very low probability, except for certain resonance processes. Hence, for positions of the ion greater than xb (where the potential barrier begins to reappear) and less than X,in (where the well first appears) the probability of electron emission is negligible.
of ion spuiren.n~ process
(e) Fig. 1. Potential in the vicinity of the surface as an ion leaves. (a) Ion in its lattice site. (b) Ion a distance xmin fram the metal surface. (c) Ion a distance xb from the surface, SomeWhat larger than Xmin, net surface potential zero. (d) Ion still further from surface; surface barrier begins to reappear. (e) Ion considerable distance from the surface, surface barrier fully developed again.
Therefore, we focus attention only on the interval x,jn - xb. For distances greater than xb, electrons can only leave the surface by tunnelling through a barrier. While resonance tunneiling can occur with high probability, this
H. G. Prival /Model
of ion sputtering process
phenomenon only occurs for certain very narrow ranges of distance and electron energy, and since the conduction electrons have an essentially continuous energy distribution (up to the Fermi level) the contribution to the neutralization probability of resonance tunnelling is small. We treat this problem as a time (or distance) varying transmission and reflection problem, where the transmission and reffection is caused by the potential distribution in the vicinity of the surface. We consider the problem to be a quasi-static one where the inst~taneous position of the ion, x,, is simply a parameter in the scattering of an electron by a stationary potential. The problem is treated in the same manner as Nordheim and Fowler  by matching the wavefunction and its derivative inside the metal to that outside, and from the continuity equations, determining the reflection and transmission coefficents. However, even for relatively simple potentials, the mathematical complexity of even a one dimensional solution is considerable. 3.1. Calculation of the neutralization probability In order to use the time independent theory for a really time varying problem, we must convert the problem form one having a time variation to one which is only (explicitly) distance varying. To do this, we divide the interval, xb -x,rn, within which the probability of neutralization is non-zero, into M equal subintervals. The time independent probability of neutralization of an ion in the kth subinterval, pk, is found by computing the transmission coefficient, T(E, k), for an electron under the influence of the field of an ion located in the kth position subinterval, multiplying this transmission coefficient by the probability (density) that an electron having energy between E and Et AE encounters the metal surface within a certain region in the vicinity of the ion, and within the time interval the ion spends in the kfh subinterval; and then integrating over the range of energies within which an emitted electron can neutralize the ion. The electron energy distribution is obtained from the simple free electron model as: for
Using a simple kinetic theory argument, we can estimate the flux density of electrons with energy between E and E t dE that impinges on a plane normal to the x axis. If u(E) is the velocity of an electron, then: (4) = 5 (~~/~~~3) =0
for E,< Ef forE>&.
H.G. Privai/Model of ion sputtering process
Since the total distance xb - X,in within which electron capture takes place is less than a few 8, we define a capture cross section such that electrons encountering the surface within a disk of radius XI, where XI is the free ion radius, centered on the point at which an ion leaves the surface, have a finite probability of neutralizing the ion. Conversely, those encountering the Surface outside of this disk have zero probability of participating in ion neutralization. Therefore, the total number of ebctrons per second imp~~ng on the capture area which have energy between E and E t dE is simply: iv(@ dE = ~~~~~3~~3) E cl.&)
and the total number of electrons per second which penetrate the surface and which are capable of neutralizing the departing ion is: Fe = sN(E) T(E) d.E’,
where T(E) is the transmission coefficient. Now the time interval that an ion spends in each distance subinterval, xk, can be estimated from a knowledge of the average ion emission energy, and consequent average ion emission velocity, u,x, which is a constant determined by measurement. The average time spent by the ion in the kth subinterval, tk, is estimated as tk = (xb -
Further, the distribution in time of those electrons which penetrate the surface is random, subject only to the fact that, on the average, Fe electrons per second penetrate the surface within the capture area. Therefore, the probability, pk, that one or more electrons penetrate the capture area in the time interval, fk, iS Simply pk =
1 - eXp(--F&k) .
&, therefore, is the neutralization probability for an ion in the kth position interval. 3.2. Cuhlation of the transmission coefficient To compute the probability pk we must first compute the trans~ssion coefficient T(E). We.do this, as mentioned earlier, following a procedure similar to that of Nor~eim and Fowler. For this approach, we require a model of the potential due to an isolated ion. We model the potential using the f~i~ar approach of a coulomb potential modified by a screening fiction, i.e.
where f(x, Z) must have the property that f(0, 2) = Z, the atomic number, and
H.G. F’rival/Model of ion sputtering process
2) = 1. We use here a variation on one due to Tietz  Zb f(x,
where x1 is the ionic radius, and b is a constant. A value of b of l/3 seems to be most satisfactory. A plot of this potential for a silver ion (2 = 47) for several values of b appears in fig. 2. However, rather than work with the screened Coulomb potential of eqs. (9) and (10) directly, with its attendant mathematical difficulties, for purposes of solving the Schrodinger equation, we replace this potential by an equivalent piece-wise constant potential, shown in fig. 3. The potential Vr is simply the value of the actual screened potential of eq. (9) at a distance equal to the distance of the ion from the surface, and V2 is found as the squared mean square root of the central portion of the actual potential. The method used to compute the transmission coefficient T(E) is outlined in Appendix A. Since the mathematics involved is straightforward but tedious, a computer program was written to compute T(E). Once T(E) is available, the neutralization probability for an ion in the kth position subinterval, pk, is readily found from eqs. (6), (7) and (8).
60 r d =-50 2 540 K z
Fig. 2. Screened
, I 1 2.0 3.0 DISTANCE FROM ION - A
of eq. (9) for a silver ion.
H.G. Primalj Modef of ion sputtering process
Fig. 3. Stepwiseapproximation to the potential due to an ion at x = ~0. To compute the total probability of neutralization of an ion as it traverses the interal, Xb-&&,, we must add up the probability that it is neutralized in each of the M subint~~als. Now these probabilities must be the probabilities of mutually exclusive events in order for the probabilities to be simply additive. Consequently, we require the probability, &, that an ion is neutralized in exactly the kth subinterval. This is the probability that an ion is nor neutralized in the first k -- 1 intervals and is neutralized in the kth interval. In symbols: Pk=fl
Hence the total probability, PN, of an ion being neutralized in traversing the interval Xb-xmin is:
and consequently, the probabiiity of survival, f,, is simply: P,=l
H.G. Prival /Model
of ion sputtering process
Table 3 Computed and measured values of the relative sputtered ion yields (ions per sputtered atom) Element
Computed ion yield (ion/ sputtered atom)
Normalized computed ion yields (Si = 1)
Measured relative ion yields by the author (Si = 1)
Measured relative ion yields by Satkiewicz [ I$] (Si = 1)
0.03 0,147 0.168 0.023 0.000166 0.0000124 0.000003
0.178 0.875 1.0 0.136 0.00084 0.000074 0.000018
0.105 0.371 1.0 0.04 0.0025 0.0015 0.00016
0.22 6.5 1.0 0.097 0.016 0.002 1 0.00006 I
Al Si Cr CU AS AU
The computer program which computes 7’(E) also performs the above computations and gives the survival probability, given the following physical constants of the metal: the work function, (b, the Fermi energy, Er, the ionic radius, XI, the m~imum distance at which the ion field is effective, x,in, the distance at which the surface barrier re-appears, xb, the average ion emission energy, J&r, the ionization energy, El, the atomic number, 2, and the principal mass number,M. Values of the probability computed using this program are given in table 3 for a number of metals for which these constants are available. Unfortunately, a full set of metal constants is available for only a relatively few of the 32 elements listed in table 2. Appendix B gives the sources of the numerical data used. For comparison, values of the measured relative ion sputtering yields are given along with the computed values, both normalized so that the value for silicon is unity. In addition, sputtering yields measured by another worker are also given.
4. Discussion So far we have considered only mono-valent metals, that is, those whose ions have a single positive charge. However, the same model is applicable to poly-valent metals, with essentially no change. The reason for this is that the potential energy of an electron due to the electrostatic field of an ion is proportional to the charge. Hence, at a given position, a doubly charged ion produces a potential well twice as deep as a singly charged ion. Consequently, the probability of a doubly charged ion capturing an electron and becoming singly charged is much higher than the probability of a singly charged ion being neutralized. However, this latter probability, we have seen, is itself almost unity. Therefore, for purposes of computing the sputtered ion yield, we can consider even poly-valent metals as if they contributed only mono-valent ions. That this is true can be readily seen by exa~n~g the three mass
KG. R&al /Model
of ion sputtering process
spectrograms of fig. 4, which are for the metals magnesium (divalent), aluminum (trivalent), and silicon (quadrivalent) respectively. In all three cases, the height of the peak representing the singly charged metal is one to two orders of magnitude
MASS NWBER Mass Spectrum of Magnesium
MASS NUMBER Mass Spectrum of Aluminum Sample
of ion sputtekg process
Mass Spectrum of Silicon Sample
Fig. 4. Mass spectra of (a) magnesium (divalent), (quadrivalent).
(b) aluminum (trivalent),
and (cl silicon
higher than those representing multiply charged states. The theory given in this paper does not show any explicit variation with bombardment direction relative to the crystallographic axes. Experimentally, it is known that such variations do take place. However, work by Yurasova and Karpuzov  has shown that the variations of the ion sputtering yield and total yield with angle are essentially simiIar. Hence, this variation with angle of the ion yield is attributable to the variation in total yield. While the version of Rol’s theory given in eq. (1) does not show an angular distribution for purposes of simplicity, the original work does give a definite angular variation. The only anticipated source of error from this assumption that all directional ion sputtering effects are related to those of the total sputtering yield, is that due to the variation with direction of the Fermi level. In summary, the model given in this paper seems to be able to explain most of observed features of ion sputtering from metals. The model predicts the very large variations in ion sputtering yields from different metals observed experimentally even though the values of the material constants change by relatively small amounts from metal to metal. The greatly enhanced ion yield observed when sputtering from oxide films on metals which are strong oxide formers, is also explainable. The dipole electric field resulting from a broken molecular bond and the simultaneous ejection of two oppositely charged ions is both initially weaker and decreases with distance faster than that due to the emission of a single metal ion. By similar reasoning, the observed very high sensitivity of the sputtered ion yield to surface conditions and contaminants would not be unexpected. However, these topics are not treated in this paper, nor is the yield of negative ions, which is explainable in terms of the same theory.
H.G. Prival /Model
of ion sputtering process
The observed very weak dependence of sputtered ion yields on macroscopic target temperature is also explained by the theory. In the version presented herein, there is no explicit temperature variation. The greatest effect of target macroscopic temperature on sputtered ion yield is through the temperature dependence of metal electron energy distribution (the 0 K distribution is given in eq. (3)). The basic hypothesis of the model described herein is that sputtered particles are initially emitted from a metal surface in the same charge state as they exist in the undisturbed metal, i.e., as ions, and that they are largely neutralized as a result of quantum mechanical transmission and reflection phenomena occurring within short distances (say 5 A) of the metal surface. Further, the number of ions surviving neutralization on leaving the surface is a small percentage of the total number of sputtered atoms. The total ion sputtering yield is obtained as the product of the total sputtering yield (neutrals plus ions) and the neutralization survival probability. The total sputtering yield is computed using Rol’s theory, and the neutralization survival probability is computed using the procedure previously outlined. While the dependence of ion sputtering yields on the material constants is quite complicated, in general one can say that those ions which encounter an electron transmission peak at some point in their travel on leaving the metal, have a high probability of neutralization and hence a low sputtering yield. Those that do not, have relatively strong sputtered ion yields. Because of the extremely complicated way in which the material constants interact to determine the ion sputtering yield, one cannot make any definitive statements regarding the dependence of sputtering yield on any one or two constants. For example, it would be very useful if one could say that in general ion sputtering yield goes down with increasing 2, or has some particular dependence on Er. However, reference to tables 1 and 2 shows numerous counterexamples. The values given in table 3 shows general agreement between theoretical and experimental data, and gives absolute values which appear to be reasonable in view of the limited experimental data available in the literature. For example, the computed ion sputtering yield for silicon (ions per incident primary ion) is about 0.05 which is of the same order of magnitude as values given in the literature. The agreement with experiment for the more strongly sputtering metals seems to be within experimental error. For the weaker sputtering metals, the ratio of experimental to theoretical yields is larger. This may be because of a magnified sensitivity to impurities in these materials. It can be seen by examining ion sputtering spectrograms of pure metals, that the surfaces of metals are almost invariably contaminated with sodium, aluminum, silicon, and the usual elemental constituents of atmospheric dust, all of which have very strong ion sputtering yields. Further, at a pressure of only about 10e6 Torr, at which most of the measurements were made, the measured sputtering yields of many materials, particular strong oxide former% would be enhanced in value over what would be obtained with “harder” vacuums, because of the presence of oxygen. &other possible explanation of some of the discrepancies between theory and
H.G. I+-ival/Model of ion sputten.ng process
experiment is the difficulties involved in obtaining good values for the material constants required in the theory. In order to use the proposed theory for the analysis of unknown materials, it is necessary to have an understanding of the manner in which some of the material properties vary with composition. Some work in this direction has already been done, and some data on the variation of work function with composition of alloys is available, as is crystallographic data on alloys, and some information on the Fermi level of alloys. However, the utility of the proposed theory for analysis is limited to the extent material constants remain poorly known. The theory presented herein has been limited to the consideration of positive ion neutralization. The same theory can be used to treat negatve ion production. However, this will not be discussed in this paper. It is hoped, however, that the model presented here will help in gaining the necessary insight into the ion sputtering process to eventually allow the routine use of secondary ion mass spectrometry for precise quantitative analysis.
The author wishes to thank Dr. K. Heinrich of the Surface Microanalysis Section, Analytical Chemistry Division of the National Bureau of Standards for making the facilities of his section and of the Bureau of Standards available to the author as a guest worker. Thanks are also due Mr. R. Myklebust, Mr. C. Fiori, Dr. D. Nebury and many others in the Surface Microanalysis Section and throughout the Bureau of Standards without whose help the work reported herein could not have been carried out. Appreciation must also be expressed to Dr. F. Satkiewicz of the Johns Hopkins University, Applied Physics Laboratory for furnishing certain material contants and experimental data. Finally, indebtedness to Dr. C. Feldman of the Johns Hopkins University, Applied Physics Laboratory for many stimulating discussions, much helpful advice, and continuing encouragement, is gratefully acknowledged.
of the transmission coefficient
The method used to find the transmission probability T(E) is the following: With the piecewise constant potential model of fig. 3, an emitted electron traverses a number of regions in each of which the potential is a constant. In the jth such
H.G. Primal j Model
region, the Schrodinger
of ion sputtering
--- A2 d2~~+(Jrj_E)$i=0
2m, d.x2 where Vi is the potential [email protected]
) = Ai exp(i$x)
in the jth region. The solution to thisequation
+ Bj exp(-ik+)
k, = (2rn~~‘/‘/~ ,
k, = [2m,(E + VI)] “‘/fi ,
k3 = [2m,(E + V,)] “‘/Fi ,
k, = kl ,
and with the condition that B5 = 0. Using the condition that at the boundaries between regions the wavefunctions, given by eqs. (A.2), and their derivatives are continuous, one can successively solve the resultant pairs of equations until one finally has a relation between A I and B1. The reflection coefficient at the metal surface is defined as R=iBr/‘A,12, and the transmission
T(E) is simply
Determination of material constants One of the difficult aspects of this work has been the probIem of finding a set of relatively consistent values for the material constants required in the ions sputtering model. Some of the nine required constants such as the atomic number, 2, and the principal mass number, M, are known precisely. Others, such as the isotopic ratios of the various isotopes of the elements considered, and the ioni~tion energies, EI, are known to at least three figure accuracy. However, for other quantities there is considerable uncertainty. The values of the work function, #J, used herein and given in table 2 were obtained from Volume III Part 1 of “Techniques of Metals Research” 1231. This publication gives a compilation of values of the work functions of the eiements as determined by various workers over the period from 1926 to 1970. The
H.G. Frival /Model of ion sputtering process
principal methods used depended on the observation of the photoelectric effect, thermionic emission, or a contact potential difference. The values of different workers differed considerably. For example, the values given for tungsten vary between 4.34 and 5.25 V. For copper the values varied between 4.4 and 5.61 V, etc. The particular value chosen and which appears in table 2 was obtained as an average of the more recent values given in ref.  where the values averaged were only those applicable to lower temperatures. Obviously for those elements for which only one value was given, or only one recent value, that value was used. Following this procedure, at least some value was obtained for the work function for all 32 elements studied. For the most part, it is believed that the values of table 2 are at least within 0.1 to 0.2 V of the correct value. However, the work function for crystalline materials is found to be a function of the orientation of the surface with respect to the crystallographic axes. The “average” values of table 2 should thus apply to poly-crystalline materials. If single crystals of known orientation are being observed, it may be important to use the work function corresponding to this orientation. If single crystals are being observed and the orientation is not known, or poly-crystalline materials are considered where the individual crystallites are comparable in size to (or larger than) the primary beam diamater, then the best that one can do is to use the tabulated values and accept the resulting error which may be up to OS V. The situation for values of the Fermi energy, Ef, is considerably worse. The values that appear in table 2 were obtained from electronic density of states data, largely from NBS Special Publication 323 , and from private conversations with Dr. A. McAlister and Dr. L. Bennett of the NBS Institute for Materials Research. The width of the conduction band of a metal can be estimated from the spectrum of photoelectron emission from metals irradiated by “soft” X-rays. The values given are probably no better than 20.5 eV. A few values of the Fermi level were obtained from a work by Cracknell and Wong . If one assumes that the metal is isotropic and that the Sommerfeld model holds, the Fermi level of a metal can be computed from its valence, s, and atomic density, Ne, as follows: Ef = (tz2/2m,)(3n2N,&2’3
As a matter of interest, these calculations were carried out for- the 32 elements investigated, and the results tabulated in table 2, column 15. Since many metals exhibits more than one valence, the value used in computing Ef was the one given by the metal’s electronic structure, except for the case of palladium, which has a filled outer shell and hence a valence of 0. For palladium, the chemical valence of 2 was used. To obtain a value for the radius of a free ion xr, several approaches were considered. The first was the use of the crystal ionic radius, tabulated values of which can be found in the “Handbook of Chemistry and Physics” . However, the radii desired are those of the free ion and not those of an ion bound in a solid,
H.G. Prival /Model
of ion sputtering process
which are measured by X-ray diffraction techniques. Another method considered was the use of the radial distance corresponding to the maximum ,of the radial location of an electron in an ion. However, this too was rejected since we are interested in the net external electrical properties of the ion and not its charge distribution. Choosing a dimension for an ion is a somewhat ambiguous process. A single ion may be considered one of the smallest or one of the largest objects in the universe, since both its radial wavefunction and electric field extend to inanity. The method finally chosen to define the ionic radius used an analogy with the hydrogen ion. The hydrogen atom (for which the screening function is unity) has a radius of the Bohr radius, 0.53 A, i.e., that radius which corresponds to the first eigen-value of the radial wave function. The potential at a distance of 0.53 A from a proton is 27.3 V. We define the ion radius, x1, for any ion as that distance at which the potential as determined by the potential model of eq. (9) is 27.3 V. From eq. (9), for x =x1, we have: 27.3 = --%4aeexr
Zb + 1 1+ 1
or xr = 0.53
zb + 1 zb-‘t
These values are given in table 2. The barrier distance xb is that distance from the surface at which the potential due to an ion, at this distance, has risen to the Fermi level. Since the potential as calculated from eq. (9) is with reference to the vacuum level, and the work function is defined as the difference between the vacuum and Fermi levels, the barrier distance, xb, is computed from the equation:
zb + (x,,/x,)* zb- ’ + (x,,/x,)*’
eq, (B.4) for xb gives a cubic equation in xb, which when solved gives the vahies of xb tabulated in table 2. The distance, x,in is obtained as a number proportional to the nearest neighbor distance. We find the proportionality constant which gives best agreement with experiment to be 0.25. The crystal lattice data appearing in table 2 were obtained from works by Taylor and eagle (271 and from Wyckoff (281. The values obtained by various workers appear to agree to within 0.02 a. Probably the material constant given in table 2 whose value is most questionable is the average ion emission energy, E,r. Fortunately, the sputtering yield as computed using the model of section 3 is not strongly dependent on this constant. Many of the values for the average ion emission energy were obtained from experiments performed by F. Satkiewicz of the Applied Physics Laboratory of Johns Hopkins University. The values shown in table 2 were obtained from him in a
private communication. A few ion emission spectra obtained from pure metals appear in a work by McDonald . The values of E,t for aluminum, maganese, and copper obtained from this paper seem to roughly agree with the values provided by Satkiewicz [ 1.51. Some values, shown in parentheses in table 2, were obtained from other sources, notably Thompson , and Stuart et al. . While this data was, in some cases, measured at energies other than the 20 keV used throughout this work, it is possible to extrapolate the given data to the energy used herein, since studies have been made of the variation of energy spectra with bombarding energy. See Stuart [311* References [l] Ya. M. Fogel, Usp. Fiz. Nauk 91 (1967) 75.
[ 21 J.M. Schrooer, Surface Sci. 34 (1973) 571.     
2. Jurela, Intern. J. Mass Specry. Ion Phys. 12 (1973) 33. C.A. Anderson, and J.R. Hinthorne, Science 175 (1972) 853. 2. Sroubek, Surface Sci. 44 (1974) 47. P. Joyes, J. Phys. (Paris) 30 (1969) 365. A. Benninghoven, Z. Physik 220 (1969) 159. [ 81 A. Benninghoven, D. Jaspers and W. Sichtermann, Appl. Phys. 11 (1976) 35.  K.F.J. Heinrich and D.E. Newbury, Eds., NBS Special Publication 427 (US Dept. of Commerce, Washington, DC, 1975). [lo] R. Castaing, Thesis, University of Paris (1951). [l l] P. Rol, J. Fluit and J. Kistemaker, Physica 26 (1960) 1009. [ 121 P. Rol, D. Onderdelinden and J. Kistemaker, in: Proc. Third Intern. Congr. on Vacuum Techniques, Stuttgart, 1965, Vol. 1, p. 75.  H. Oechsner, Appl. Phys. 8 (1975) 185.  P. Sigmund, Phys. Rev. 184 (1969) 383. [ 151 F. Satkiewicz, private communication (1976). [ 161 0. Almen and G. Bruce, Nucl. Instr. Methods 11 (1961) 257.  A. Sommerfeld, Naturwissenschaften 41 (1927) 825. [ 181 L. Landau, Physik. Z. Soviet Union 2 (1932) 46.  C. Zener, Proc. Roy. Sot. (London) Al37 (1932) 696.  R. Fowler and L. Nordheim, Proc. Roy. Sot. (London) All9 (1928) 173.  T. Tietz, J. Chem. Phys. 22 (1954) 2094.  V. Yurasova and D. Karpuzov, Soviet phys.-Solid State 9 (1968) 197.  R. Bunshah, Gen. Ed., Techniques of Metals Research Vol. VI, Part 1, Measurement of Physical Properties, Some Special Properties, Ed. E. Passaglia (Wiley-Interscience, New York, 1972).  L. Bennet, Ed., Electronic Density of States, NBS Specieal Publ. 323 (US Dept. of Commerce, Washington, DC, 1971).  A. Cracknell and K. Wong, The Fermi Surface (Clarendon, Oxford, 1973).  R.C. Weast, Ed., Handbook of Chemistry and Physics, 52nd ed. (Chemical Rubber Publ. Co., Cleveland, Ohio, 1971). [ 27) A. Taylor and B. Kagle, Crystallographic Data on Metal and Alloy Structures (Dover, New York, 1963).  R.G. Wyckoff, Crystal Structures (Interscience, New York, 1960).  R.J. MacDonald, Advan. Phys. 19 (1970) 457. 1301 M.W. Thompson, Phil. Mag. 18 (1968) 377.  R. Stuart, G. Wehner and G. Anderson, J. Appl. Phys. 40 (1969) 803.