An examination of field evaporation theory

SURFACE

SCIENCE 29 (1972) 37-59 0 North-Holland

AN EXAMINATION

OF FIELD DAVID

Publishing Co.

EVAPORATION

THEORY*

McKINSTRY

Physics Department, University of Nevada, Reno, Nevada 89507, U.S.A. Received 19 May 1971; revised manuscript

received 14 October 1971

Contrary to opinions expressed by earlier investigators, thorough examination of the existing models for field evaporation indicates that they are capable of predicting changes in the relative abundance of ionic species as temperature is changed and evaporation rate is held constant, without assuming atomic tunneling or slowness of electronic transition. However, this need not imply that atomic tunneling and slowness of electronic transition do not exist. Furthermore, the inclusion of energy level shifts in field evaporation theory results in the prediction of higher charge states than would be otherwise expected. This is in agreement with the observations of various investigators on copper and other metals. The examination further reveals that very significant differences in the predicted variation of evaporation rate with field exist between the image-hump and intersection models. This suggests a means by which one can attempt to determine experimentally the “correct” model to apply in a given situation. Furthermore, rigorous interpretation of the intersection model in conjunction with the experimental data of Tsong and Miiller can result in significantly different values for the distance of the equilibrium position of the surface atom from the metal surface and the polarizability of the surface atom, than obtained using linear approximations.

1. Introduction Field evaporation has typically been described by two slightly different potential energy models. Both models assume that an atom bound to a surface may have its bonding described by a one-dimensional atomic potential well. The first model, originally proposed by MiAler192), will herein be referred to as the image-hump model rather than the image-force model, as it has commonly been called, since an image-force term is also present in the equation for the ionic potential in the second field evaporation model. The second model, which was first used by Gomers) to describe field evaporation, will herein be designated the intersection model. Many questions concerning these models and the field evaporation process remain unanswered at this time. These will become apparent in the reading of this paper. However, it is not the intent of this paper to answer all of these questions. Rather, its purpose is to provide guidelines for the use of the two * Research done in partial fulfillment of the Ph.D. degree in physics at the University of Nevada, Reno. 37

38

D. MCKINSTRY

models for field evaporation in the interpretation of various experimental data on rate-sensitivity measurements 4pg) and the relative abundance of ionic specieslo-r4) obtained in recent years.

2. Background theory The following discussion [up to and including eq. (4)] of the atomic and ionic potentials applies to both evaporation models. As discussed here, the models are of the simple classical formrs) which assumes the equivalence of a force acting on a particle and the negative derivative of the particle’s potential energy. The depth of the atomic potential well n (see fig. la) is the sublimation energy required to free the atom from the surface. An additional energy c I, is needed to ionize an atom already in the gas phase,

0

f

vi

"i

a

(a)

(b)

Fig. 1. Potential energy diagrams for (a) a surface atom, (b) a surface ion, and (c) a surface ion in the presence of a field. Note the difference in zero energy level for the atom and ion.

where ne is the charge on the ion and I,, is the nth ionization potential. As the electrons removed from the atom are returned to the metal surface, an energy nc$ is recovered, where $ is the difference between the potential of the electron at the metal’s Fermi surface and the potential of the same electron free in vacuo16). Thus, the zero level for the ionic potential energy curve (see fig. 1b) lies c I,, - n4 above the zero level of the atomic potential energy curve. The energy difference QOn between the zero level fat the ionic potential and the bottom of the atomic potential well is

(1) The ionic potentialrv) consists an attractive image-force term.

of a short-range repulsive term V,,(x) and The latter is taken to be caused by the

FIELD

rearrangement

EVAPORATION

39

THEORY

of surface charge on the metal due to the presence

of the ion

and is given by -(ne)‘/4x, where e is the electronic charge, and x is the distance of the ion from the metal surface. In the presence of a positive applied electric field, the ion experiences a repelling force neF, where F is the applied field. The potential -neFx, and the ionic potential

associated with the electric field is therefore (see fig. lc) is

V.1(x 3 F) = I/r” (x) - neFx - (?f 4x

.

It is usually assumedI’) that the ionization process occurs at a sufficiently great distance from the atoms in the surface that the repulsive force on the ion formed may be neglected. Thus, the ionic potential is approximated by

Examination of this equation reveals a maximum value (referred Schottky or image hump) at the position X,,,, given by X,, = (ne/4F)*.

to as the

(4)

Superposition of the atomic and ionic curves creates three possible models for evaporationl7). The first model (fig. 2a) applies when c In-n4 is small enough that no intersection occurs between the atomic and ionic curves. For most metals c 1,-n+ is sufficiently largel’) that at expected evaporation fields intersection must occur and either the image-hump model (fig. 2b) or the intersection model (fig. 2c) applies. For this reason, only these latter two models are considered in this paper. When intersection occurs between

Fig. 2. Three possible evaporation models. (a) Simple ionic bounding, (b) evaporation over the image hump after transition from the atomic to ionic state at xn, and (c) ionization at xn followed by immediate evaporation.

40

D. MCKINSTRY

the atomic and ionic curves it is expected that ionization can occurrs-so). The image-hump model applies when intersection (ionization) occurs in a region where the forces on the ion formed are such as to draw it back toward the interior of the metal. Hence, the ion must have sufficient kinetic energy to reach beyond the image-hump maximum (where the forces on it are zero) before evaporation can occur. Otherwise, the ion will revert to the atomic state as it returns across the intersection, and probably “oscillate” between the two states without ever being freed from the surface. Thus, the energy the atom (ion) must have to reach X,, defines the activation energy. This energy is merely the difference between the ionic potential at the image-hump maximum and the minimum of the atomic potential. The activation energy is thus given by Q n = Qon - n&X

m”

-

!fe)Z =Qon _(n3e3$‘)t. 4XnVl

At the evaporation field F,,,, the activation energy is zero. For a given ionic species the activation energy is strictly a function of the electric field, and no more knowledge of the atomic potential is required in computing F,, from eq. (5) with Q, equal to zero than the sublimation energy [see eq. (l)]. Furthermore, no more need be known about the point of intersection than that it occurs at a smaller value of x than X,,. On the other hand, the intersection model applies when the intersection of the two potential curves (and hence ionization) occurs at a position where the forces on the ion repel it from the surface, and evaporation spontaneously follows ionization. Thus, the energy required for the atom to reach the point of ionization defines the activation energy for the intersection model. So, the activation energy is given by Q, = Q,, - n&x ” - peJ

4x,

’

(6)

where x,, is the distance of the point of intersection from the metal surface. The evaporation field F,, is that field for which the intersection occurs at the equilibrium position x0 of the atom, and hence the activation energy is zero. F,, may be predicted for the intersection model only if x0 as well as n (see fig. la) is known. Furthermore, the variation of activation energy with field can be determined only if specific knowledge of the atomic surface potential is available. Whichever model applies it is assumed that the evaporation rate k,, for a given species, may be related to the activation energy by the Arrhenius equation33 6, 21), namely k, = v, exp (-

Q,IkT) ,

(7)

FIELD EVAPORATION

where k is Boltzmann’s

constant,

41

THEORY

and the Arrhenius

coefficient

v, may be

regarded as the evaporation rate at the evaporation field F,,. Thus, if one has knowledge of the Arrhenius coefficient and is able to calculate the activation energy at a given field, then it is also possible to determine the evaporation rate. Because of its simplicity and lack of need for specific knowledge of the surface binding forces, the image-hump model is the more attractive of the two models. However, it is expected that at least in some cases (to be discussed later) the intersection model is physically more realistic. Previous investigators have assumed values of x:,21) which are quite near to those determined for X,, (see table 1). Therefore, both models predict approxiTABLE

1

Position of image hump calculated from eq. (4)

3.0 4.0

5.0 6.0

1.10 0.95 0.85 0.78

1.55 1.34 1.20 1.10

1.90

2.19

1.64 1.47 1.34

1.90 1.70 1.55

mately the same evaporation fields. Either model may be applicable to various metals with an appropriate set of evaporation field F and evaporation parameters characterized by the metal and the ionic species evaporating. For any such set of conditions, the use of one model assumes that the other is not applicable for a given evaporating atom. This, of course, does not imply that evaporation cannot be governed by both models at the same time for different atoms evaporating from a single field-ion specimen, or even different ionic species evaporating from identical lattice sites. Several additional effects, which are not included in eqs. (5) and (6), are considered to have an influence on the evaporation process. One of these is the shift in the energy of the atom relative to that of the ion due to the atom’s polarization by the high electric field. The ionic polarizability is assumed to be negligible by comparison with the atomic polarizability22), and higher order polarization terms are usually neglected5). Letting c1 be the atomic polarizability, the shift in energy is equal to +xF2. This has the same effect on the models as would assuming larger values of Qon for a given field. However, the polarizability term is left separate from Qon because of the obviously implied quadratic dependence of the activation energy on the electric field. Additional effects result because of the non-ideal character of the metal

42

D. MCKINSTRY

surface. It is classically assumed that the metal surface is a mathematical plane. Of course, this assumption is adequate for large distances from the surface. In reality, the surface is composed of an “electron jellium”4) which is poorly defined, and by no means “solid.” For this reason, the electric field penetrates a finite distance 2 which is taken to be the impurity screening distancess~~~), into the surface. This lowers the ionic potential curve by an amount neF1. A second correction must be included to account for the failure of the classical image potential at small distances. Obviously, one would not physically expect the existence of a singularity in the image potential at x=0. There have been several attempts at providing an image potential which approximates physical reality for small distances from the surface. Comer and Swanson20) have given the image potential as V$ (x) = - + (lie)“/(x An alternative the form

to this potential

vgx) =-

+ A).

has been given by Cutler

(yJ{l-exp(-:)i,

(8) and Davises)

in

(9)

Both of these approximations are equal to -~~e)‘~4~ in the limit as x approaches zero, and converge with the classical image potential for large x. Using the linearized Thomas-Fermi approximation, Newnsss) has recently derived an equation for the image potential which takes close range effects into consideration. His equation may be expressed as26)

This expression for the image potential is that most rigorously derived thus far. Therefore, it can probably be looked upon as that which most nearly approximates physical reality. Shown in fig. 3 is a plot of eq. (10) obtained by numerical integration. As can be seen, both eq. (8) and eq. (9) give considerably better agreement with the results of Newmae) than does the classical image potential. Of course, one may obtain a better approximation to eq. (10) using numerical techniques27), but further discussion herein either uses Fir(x) for the image potential of Newnses) or assumes the use of the Gomer-Swanson approximationso) for simple examples. Indeed, as can be seen in fig. 3, the Gomer-Swanson approximation provides excellent agreement with the results of Newnsss) for all moderate and large values of X.

FIELD

0

.2

EVAPORATION

.4

43

THEORY

1.o

.a

.6

X/A Fig. 3. Image potentials. The upper curve is the classical image potential. The middle solid curve is the image potential derived by Newns, and the lower solid curve is the Gomer-Swanson approximation. The broken curve is the Cutler-Davis modification for the image potential.

Inclusion of polarization and field penetration and (6) be replaced respectively by

effects demands

Q, = Q,, - neF (X,,,, + A) + T/if (Xm,J + $aF2, for the image-hump

(11)

model and

Q, = Qon - neF (x, + A) + Vi, (x”) + @F2, for the intersection

that eqs. (5)

model.

If one uses the approximation

(12) of eq. (8) for the

image potential, then (X,,+A) may be looked upon as the distance of the image-hump maximum from the “effective mirror surface”. Similarly, (xn+ A) is then the distance of the point of intersection from the “effective mirror surface”. Since the term +aF2 is not a function of x (for a uniform field), the position of the image hump may still be defined by eq. (4) merely by substitution of (X,,,,+ A) for X,,. Then, Q, may be written as Q,, = QOn- (n3e3F)* + &F2

(13)

for the image-hump model. Several other effects are expected to influence the field evaporation process, although they have never been discussed more than qualitatively for tungsten. These are broadening effects and energy level shifts. Gurneyss) and Gomer

44

D. MCKINSTRY

and SwansonzO) have discussed broadening effects, and Gadzukeg,sO) has provided detailed analysis and calculations for both energy level shifts and broadening effects for cesium on tungsten and potassium on platinum. Energy level shiftszg) Al?(x) are the result of the close proximity of the surface atom to the effective mirror surface. Thus situated, the combined effects of the image potentials of the ion core of the surface atom and the electron(s) about it cause an increase in the energy of the electron(s) bound to the atom. This results in an ionization potential which is less for the surface atom than for the free atom, for which the ionization potential is determined both theoretically and experimentally. Broadening occurs when the originally sharp energy level of the valence electron is influenced by the presence of the metal surface. This may be viewed as a situation of resonance tunneling2g) between the atomic and metallic states. The uncertainty principle demands that an uncertainty in the energy of the valence electron must be associated with the finite lifetime z of the states. Thus, T(x)= h/z is the amount to which the energy level is broadened. The complete expressions for these effects are too lengthy to warrant inclusion here, but an approximation for energy shifts valid at large distances from the metal surface will be discussed later. Another effect which has been only briefly mentioned previouslys), is the “partial denuding” of each surface atom of about 0.5 e of electronic charge at a field of 5 V/A. This will be ignored for now, since in terms of image potentials, this amount of charge produces only & of the effect of the image potential [see eqs. @-(IO)] for a triply charged species, such as tungsten. Furthermore, those metals which evaporate with lesser charge usually do so at lower fields17). Hence, according to Gauss’ law, the degree of “denuding” is less for them. Since the high electric fields associated with field evaporation may be expected to result in some separation of charge of the atom (polarization), in reality, AE(x) and T(X) are also functions of field, polarizability, and penetration depth of the field. For further discussion herein, both energy level shifts and broadening effects will be combined in a term representing the relative energies of the zero level for the ionic potential and the bottom of the atomic potential well. Thus, Qon in all the previous equations is replaced by Qons (x,, a, 4 F) = Qon - AE (x,, a, A, F) - ir (x,, a, 4

F) .

(14)

One should note that QOns is written as a function of x, for both the imagehump model and the intersection model, since once ionization is complete, these effects are no longer “active”, and it is the energy QOns at the time of ionization which determines the remaining energy of the ion.

FIELD EVAPORATION

TABLE

45

THEORY

2

Distance of an adatom from different crystal planes calculated on basis of the bulk crystal lattice for bee tungsten Crystal direction

110

001

112

111

556

332

221

552

x0’(A)

2.24

1.58

1.29

0.91

0.34

0.67

0.53

0.43

3. Choice of an evaporation model One might initially assume that it is reasonably simple to determine which evaporation model should be used in a given situation. The most straight forward approach is to define the “atomic surface” as that surface determined by the locations of the cores of the exposed atoms. For body-centered cubic (bee) crystal structure, it can be seen that the distance of an ad-atom from this surface is less for the high-index planes than for the low-index planes (see table 2). For the former, these values are typically less than the X,,,,,‘s, placing the intersection of the atomic and ionic curves nearer to the surface than the image hump. On the low-index planes, the reverse is true and the intersection model must surely be used. In addition, eq. (4) indicates that X,,,, is least for metals evaporating at high field strengths with lower charge (see table l), and this must be taken into consideration when attempting to choose an evaporation model. Furthermore, recent calculations by Tsong5), using the modified image potential of eq. (9) indicate that an image hump may be expected to exist only for higher charge states and/or lower evaporation fields. Using the form of the image potential given by Gomer and SwansonzO) [eq. (S)] provides an alternate interpretation for the elimination of the image hump. As already noted, the use of eq. (8) results in the acquisition of (Xm,+ 2) rather than X,,,, from eq. (4) and table 1. For those cases in which ,I is greater than (X,, + ;1), X,, would have to be negative, physically placing the image hump below the metal surface. One would never expect an evaporating ion to be “found” below the metal surface, nor would one feel justified in extending the Gomer-Swanson approximation into such a region. So, in such a case (most probable for lower charge states and higher evaporation fields), the intersection model must be applied. On the other hand, if the material in question is expected to evaporate in higher charge states at relatively low evaporation fields, so that the image hump lies well above the metal surface, then the Gomer-Swanson approximation provides excellent agreement with the results obtained by Newns26). Although the above approach is the most straight forward, and a similar approach might be used for kink-site atoms, it has been pointed out that the

D. MCKINSTRY

46

‘ ‘jellium “4,5) of free electrons at the metal surface may cause the effective mirror surface to be at a significant distance above the atomic cores. Thus, it can be argued that although the simplest approach demands the use of the intersection model for the low-index planes in (bee) crystal structure, it is possible that the mirror surface lies sufficiently far above the atomic surface that x, is less than X,, (even though xb is greater) and the imagehump model is applicable. However, there is one more argument that further confuses the choice of models. As has been pointed out by several investigatorsJ,aa), it may well be that the repulsive term in the ionic potential may be by no means negligible. The retention of the repulsive term requires that the activation energies for the image-hump and intersection models respectively be written as

and x,,cl, Q, = K(xn)- va(xo) = Qons(

tl,F) + +aF2+ I/i(x,,,F), (16)

where Vi(x, F) is defined by Vi (x, F) = Vri,,(x) - neF (x + A) + Vi,(x).

(17)

Whether the repulsive term is included or not, the intersection model requires knowledge of the atomic potential and the simultaneous solution of both sides of eq. (16) to determine x, and Q,. Such is not the case for the imagehump model until the effects included in QOns(x,, a, 2, F) and/or V,,(X,,) enter into the theory. For the former term, the value of x, may be found in the same manner as for the intersection model. Furthermore, the position of the image hump with the repulsive q’ (x,,,

term included

must satisfy the equation

F) = 0 = Vr’,,(Xmn)- neF +

vi;(X,,,J,

(18)

where for brevity, a prime on any energy term in this paper indicates differentiation with respect to x (partial differentiation with respect to x if the energy is a function of more than one variable) evaluated at the argument. Thus, for any given field F, X,, may no longer be calculated with simple knowledge of F [as with eq. (4)]. Instead, X,, is shifted toward the metal surface by an amount (the horizontal distance between the maxima labeled A’ and A in fig. 4a), which is dependent upon the repulsive force - V,‘,,(X,,). If the magnitude of the negative quantity V:,(x) is greater than Vi;(x) - nrF for all positive x, i.e. the total ionic force is repulsive at all distances from the surface, then ionic bonding cannot occur (see fig. 4b) and the intersection model must be used. Both the displacement of the image hump toward the

FIELD

EVAPORATION

47

THEORY

surface and the possible elimination of the hump altogether provide strong arguments in favor of the intersection model, even for the high-index planes where it is expected that x0 is less than the unperturbed value of X,,,,. As already mentioned, the value of Q,, and field penetration depth can influence the choice of evaporation model. Also, polarizability and both

X-

(a)

r

A

Fig. 4. The effect of the repulsive term in the ionic potential on the hypothetical image hump. The lower curves represent the usual image hump. (a) For moderate repulsion, the image-hump maximum is shifted toward the metal surface. (b) For greater repulsion, the ~ximum is eliminated altogether and any ion formed can evaporate spontaneously.

energy level shifts and broadening effects can shift the relative energies of the atomic and ionic states, and hence (as in fig. 2) influence the choice of evaporation model. Presently, there are too many unknowns associated with all of these effects to enable the certain theoretical prediction of which model best applies to a given material. Nevertheless, there are many indications that the intersection model is the most likely choice for a material such as tungsten. This, of course, is not necessarily the case for other metals. 4. Rate-sensitivity

predictions

Examination of the Arrhenius equation, i.e. eq. (7), and eq. (16), or eq. (15) for the image-hump model, reveals two obvious ways by which an investigator can vary the rate of field evaporation. Raising the temperature, while keeping the electric field and corresponding activation energy constant, increases the evaporation rate. Likewise, increasing the field, and thus decreasing the activation energy, while the temperature is held constant, also increases the evaporation rate. The sensitivity of the evaporation rate

48

D.MCKINSTRY

to the field for constant temperature is known as field sensitivitys), whereas the sensitivity of the evaporation rate to temperature for constant field is temperature sensitivity. Both are classified as rate sensitivities. An alternate experimental procedure6) involves the variation of both temperature and field at the same time in such a manner as to keep the evaporation rate constant. Since it can be shown that the mathematics of this latter procedure is directly related to that of the first two sensitivities, the experimental value obtained from such a procedure may be also considered a rate sensitivity. One may determine theoretical values for the various kinds of rate sensitivity to compare with the experimental work by differentiating the Arrhenius equation. The field sensitivity ST* is found to be

=dln(kn)

s

’

F dQn

d ln (kn)

F

din(F)

T=

dF

T

(19)

kT dF

The temperature sensitivity S, may be found by differentiating the Arrhenius equation with respect to ln( T). It is found to be equivalent to

SF= Similarly,

d ln (k,) I

Q,

OF= kT

din(T)

the third sensitivity

=

(20)

ln (v,lkJ .

S, is given as

=c!!m

s

k

din(T)

(21)

k,’

Of course, all three of these sensitivities are defined in the same way for both evaporation models. From the Arrhenius equation, one may write the expression for the activation energy as Q, = kT In (v,/k,). (22) If one assumes that v, is a slowly varying function of field and temperature compared to the exponential dependence of k, on these parameters21), one may approximate V, by a constant so that differentiation yields

dQn

d In(T)

~

= kT In (v,/k,) __ d In (F) d h(F)

Solving

this for ln(v,//r,),

I (23) it.’

one obtains In (v,/k,) = - S,S,.

Measurements

of evaporation

rate as a function

(24) of temperature

have been

* The subscript indicates the variable that is held constant during the experiment.

FIELD EVAPORATION

made by several investigators

49

THEORY

in order to test the evaporation

models

and

determine other information about the field evaporation process. A first attempt at testing the image-hump model using rate-sensitivity measurements was made by Brandone). More recently, Tsong and Mtiller4* 5, have extended rate measurements over 9 orders of magnitude of evaporation rate in an attempt to determine CIand (x0 + /2), and Taylor and Southon’, s, have made comprehensive rate measurements under a variety of conditions. Of course, other investigators g, are working in this field, but it is the experimental work just specified with which the author is most familiar. Using eq. (24) and his experimental values for ST and S,, Brandon’j) obtained a value for v, * of about 5 x 10” layers/set for tungsten. This value is compatible with the atomic vibration frequency (10’2-1013/sec), if one considers that the number of atoms on the surface which are exposed to an evaporating field may be as much as three orders of magnitude less than the number of atoms making up the surface4). Brandon, however, did not take that into consideration. Using the same approach with data on tungsten obtained by Tsong and Mtiller 4), one obtains a value of about 6 x 10’ layers/ sec. This merely indicates that there is a great deal of experimental error, and that more experimental work must be conducted before the value of the Arrhenius coefficient is pinned down for tungsten, or any other material. Brandon obtained a value for v, of about 6 x IO7 layers/set for platinum, and values ranging between 7 x IO3 and lo6 layers/set under various conditions for molybdenum. He attributed the extremely low values for molybdenum to elastic deformation. The fact that the values for platinum and tungsten were less than expected he tentatively explained as being a consequence of partial tunneling of the ions through the activation energy barrier. Later, Tsong21) rejected this hypothesis and pointed out that slowness of electronic transition could account for the reduction in the Arrhenius coefficient. It should be noted that one can determine v, directly using temperaturesensitivity measurements and eq. (20). Of course, the greatest difficulty in conducting such an experiment lies in maintaining a constant field as the specimen is dulled. This might be overcome to some extent by working with a specimen which is already relatively dull. Also, the introduction of tracer quantities of an inert gas with a known “threshold ionization field” might be used in conjunction with a mass spectrometer arrangementis) as a means of monitoring the field. The development of such techniques for the measurement of SF would enable one to check further the consistency of values obtained for v,. * At the time of his experiments, Brandons) believed n to be 2 for tungsten. Recent studies by Miillerz3), however, indicate that the third ionic species is dominant for tungsten.

50

D. MCKINSTRY

If it is assumed that the image-hump model is valid for a given set of conditions, the following approach to the determination of the field sensitivity may be used. The complete expression for the activation energy is given by eq. (15). Using this, one finds the derivative of the activation energy with respect to field to be

(25) where, of course, the total derivative of QOns includes the partial derivative with respect to F, and a term consisting of Q& times the derivative of x, with respect to F. The latter part of the term may be evaluated using techniques to be described for the intersection model. One may use eq. (18) to eliminate Vi (X,,, F) from eq. (25), and finds that if the broadening effects and energy level shifts included in Qons are neglected, and that any variation in X,, is assumed to be small compared to X,,, the derivative of activation energy with respect to field is linear in F. More generally, however, the field sensitivity, as defined by eq. (19) must be written as = neF (X,,,” + A) - aF2 - F (dQ,,,/dF)

s T

~_____

kT

(26)

Ignoring the repulsive term of the ionic potential, and energy level shifts and broadening effects, the above equation reduces to the result ST =

3 (n3e3F)* - aF* - ~~ -~kT

(27)

similar (except for the aF* term) to that obtained by BrandonIT). If one assumes this form to be accurate, then polarizability is the only factor responsible for any observed reduction in field sensitivity for the image-hump model. However, if the repulsive term is retained, S, must be determined using eq. (26), and some additional reduction may be expected because of the lesser value of X,, as determined by eq. (18). If (X,,,,,+ A) is comparable in magnitude to its own variation, then the variation of Q, with F is not linear. However, this is unlikely for those cases in which the image-hump model applies. On the other hand, results may be confused considerably by neglecting (dQ,,,/dF). Better quantitative knowledge of energy level shifts and broadening effects is needed before a prediction of the way in which they alter these results can be made with certainty. In the event that the intersection model applies in a given situation, the activation energy is given by eq. (16). To determine the field sensitivity, one

FIELD

must first differentiate

EVAPORATION

51

THEORY

eq. (16) with respect to F to obtain

dQn

=V;(~~)~$=~F-ne(x.+~)f~ dF +

aeons

{vi’(xmF) + Qb,, (x,, ~(24 6) i?,

where Vi’(x, F) is given by

V/(x, F) = Vr\(x) + Vi;(x) - neF .

(29)

As with finding x,,, the value of dx,ldF must be such that both sides of eq. (28) are simultaneously satisfied. Solving for this value and substituting back into eq. (28), and eq. (19), one obtains the expression s

= T

(IzeF(x” + A) - aF2 - F (ziQ,,,,/S’)} V,: kT (Va’ - V; - Q,;,)

(30)

Since the slope of the ionic potential is negative for any case in which the intersection model applies, if one assumes that QOnsis equal to Q,,, then the term in the parentheses of the denominator of eq. (30) is greater than the slope of the atomic potential in the region of interest. As an example of the effect this may have on the value of ST, one can consider the difference between the predicted variation of the activation energy with respect to the field for two cases in which intersection is assumed to occur very near to the image-hump maximum. It can be seen that the derivative of Q, with respect to In(F), which is here called QF for brevity, may be found for the imagehump and intersection models respectively by multiplying eqs. (26) and (30) by kT. Also, eq. (30) degenerates to eq. (26) in the case where x,=X,,,, since at that position, the slope of the ionic potential is zero. For this example, the repulsive term of the ionic potential, as well as broadening effects and energy level shifts, are ignored. One may take tungsten, which field evaporates as a triply charged species at fields of about 6 V/A as a likely example. For lack of any better information about polarizability of the surface atoms, it is assumed that a=0.32 eV-A2/V2 (equal to 4.6 A”) as given by Tsongs). Assuming that intersection occurs at or before the image-hump maximum, so that the image-hump model holds, and assuming that the Gomer-Swanson approximationzO) may be used for the image potential, one can take the value of (Xm,+Iz) from table 1, and obtain QF = neF (X,, + 1) - aF2 = 12.6 eV. If intersection

occurs at a small distance

A beyond

the image-hump

maxi-

D. MCKINSTRY

52

mum, then the intersection

model must be used and QE’ is given by

QF = {neF(X,, + A +

A) - aF2}Vi

-r/,l-py where Vi is given by Vi’ = - neF +

(3’ ~~ 4(xm”+i+A)2’

Since the slope of the ionic potential is zero at X,,,,, neF is equal to (ne)2/4(X,,, + I)l. Th ere f ore, it can be shown that to a first approximation VL is just Vi g - 2neFA/(X,,,, + 2)’ = - 2.6 eVjA for A =O.l A. For low temperature field evaporation, one does not expect Vi significantly to exceed 1 eV/A21). Therefore, in the case where intersection occurs only 0.1 8, beyond the image-hump maximum, QF is calculated to be QF = 14.4 eV x (l/3.6)

= 4 eV.

A reduction of QF (and hence field sensitivity) to a value less than 3 that predicted by the image-hump model is quite significant. This emphasizes the importance of knowing which model applies to a given experimental situation before attempting to interpret field sensitivity data, such as that obtained by Brandons*17), and by Tsong and Miiller4, 5). In analyzing eq. (30), it is apparent that the force on the atom near its equilibrium position, i.e. the negative of the slope of the atomic potential, may be expected to vary considerably, even over the relatively small range of activation energies associated with low-temperature field evaporation studies. In fact, since the slope of the atomic potential is zero at the equilibrium position, eq. (30) predicts that if the slope of the ionic potential is negative, as it must be for the intersection model, and Q& is either negative or less than the magnitude of Vl, as the evaporation field F,, is reached and the point of intersection occurs at x0, S, becomes zero. Since S, is expected to approach zero as the activation energy approaches zero only under very special circumstances for the image-hump model, a significant difference, at least in this respect, is seen to exist between the two models. This suggests the possibility of devising an experimental means of determining which model may be applied in a given situation. First of all, it is necessary to determine the Arrhenius coefficient for the given material and ionic species from experimental data and eqs. (20) and (24), hopefully with much better accuracy than presently attained. This, of course, requires improvements in both experimental techniques and equipment. One can then carry out evaporation studies at very high rates to determine the evaporation

FIELD

EVAPORATION

THEORY

53

rate at which ST becomes zero. This may require better pulser techniques than are presently availables). If this evaporation rate matches the Arrhenius coefficient within the experimental error, it is a good indication that, at least process is governed by the near the evaporation field Fob,, the evaporation intersection model. On the other hand, if the evaporation rate at which S, becomes zero is significantly less than the Arrhenius coefficient, then the image-hump model dominates the evaporation process, and/or c( is sufficiently large that polarizability is suppressing the evaporation process for higher fields 27). As is apparent from eq. (30) Sr may be very non-linear in F, especially as the point of intersection approaches the atomic equilibrium position and the activation energy approaches zero. Therefore, even though linear approximations for S, may be perfectly valid for the image-hump model, any analysis of rate-sensitivity data, which assumes the intersection model and uses linear approximations5) without prior assumptions about the atomic and ionic potentials, must be considered with extreme caution. Using such approximations, Tsong 5, obtains values for the effective polarizabilities of tungsten kink-site atoms and ad-atoms respectively of 4.6+_0.6 A3 and 6.8+_ 1.0 A3. The values of (x0 +A) he obtains are about 0.7 A and 1.3 A respectively, which are much less than those values assumed by previous investigators4). Although the values obtained for the effective polarizability, and its implied variation over the different evaporation sites, are quite reasonable, and the values of (x0 +A) are not inconceivable (although the author feels them to be very improbable), no final conclusion can be made about their accuracy without first conducting an analysis of the data which takes the potentials of the evaporating atoms and ions into consideration. Furthermore, if energy level shifts and broadening effects are found to make a significant contribution, they must also be included.

5. Relative abundance of ionic species and energy level shifts Utilizing slowness of electronic transition and atomic tunneling, Tsong2r) has been able to obtain qualitative agreement with the experimental observation of Barofsky and Mtillerls), Vanselow and Schmidtll), and Richter and Schmidtie) by which it was found that atoms of certain metals which evaporate primarily as doubly charged ions at low temperatures, instead evaporate as singly charged ions at higher temperatures. Furthermore, Tsong suggested [as did Barofsky and Mtillerio)] that qualitative agreement could not be obtained if it is assumed that field evaporation is a purely thermally-activated process. In at least one caselo), changes in the relative abundance of ionic species were noted with increase in temperature, while the electric field was

54

D. MCKINSTRY

being simultaneously Tsong’s changes

reduced

to keep the total evaporation

paper21) did not treat the effects of this reduction in the relative abundance of ionic species.

rate constant. in field on the

It is apparent from eq. (26) and eq. (30), that the field sensitivity of the evaporation rate is strongly dependent on the charges of the evaporating ions for either evaporation model. Obviously, as the electric field is reduced, the activation energy for the more highly charged species increases more rapidly than for those species of lesser charge. Thus, if at a given field a particular ionic species is found to be dominant, as the field is decreased it is expected that a field may be reached at which the next lower ionic species dominates. In general, it may be stated that the lower the electric field, is, the lower is the charge on dominating species of evaporating ions. Thus, if some mechanism, such as raising the temperature, is utilized so that evaporation can be observed for higher activation energies and lower fields, it is expected that the charge of the evaporating ion becomes less. This becomes most obvious in the extreme case for which the temperature is sufficiently high that some purely thermal evaporation occurs and the evaporating particles are atoms rather than ions. Thus, there is no limit as to how little charge is associated with the evaporating species, if sufficient thermal energy is supplied to the system. On the other hand, however, the lower temperature limit of 0°K does provide an expected maximum charge that can be obtained for an activation energy of zero. The expected charges of the evaporating ions predicted by both of the existing models have been found generally to agree with the observed speciesi724, st). However, there have been several cases at either very high evaporation rates, or very low temperatures for which the observed species is more highly charged than predicted. An example of this is copperlO), which is observed to evaporate as doubly charged ions at low temperatures. Also, tungsten has been observed to evaporate as a triply charged species 2s), in spite of the fact that it is expected to be doubly chargedl7). The question of why this occurs arises. Computations by both Chambers g, and Taylors) indicate that it is very improbable that a mechanism of “post ionization” is responsible for a change in ionic species, so this process is not considered herein. Of course, the transition temperature1as2i) for copper is observed to be so low that atomic tunneling is highly probable, and is most likely making some contribution to the evaporation process. However, at the evaporation field F,,, for which the ionic species are predictedi7), the temperature dependent phenomenon of tunneling 21) can no longer be expected to influence the choice of species. Earlier computer studies by the author279 32) have indicated that inclusion of the repulsive term in the ionic potential of copper is capable of changing the species predicted from singly to doubly

FIELD

EVAPORATION

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55

charged by repressing the evaporation of the singly charged species. However, it is expected that the repulsive term for higher ionic states is sufficiently 10~27) that it plays virtually no part in the appearance of triply and quadruply charged tungsten, or the higher charge states of other metals. Therefore, the only other effects believed to influence the field evaporation process must be considered. As may be noted from Gadzuk’s discussion on broadening sg,sO), it is reasonable to assume that, as more highly ionized states are achieved, the energy of the valence electron(s) in the surface atom, relative to the zero of energy at the bottom of the conduction band in the metal, becomes less, and hence the tunneling probability and the corresponding broadening becomes less. When the ionization potential is such that the energy of the valence electron drops below this zero level of the conduction band, then broadening effects are probably negligibless). Such is the case for the second and higher ionization potentials for tungsten, and in general one can expect the trend to be such that broadening effects become progressively less for more highly ionized states. Therefore, if broadening is not negligible, it should result in a shift toward an increase in the relative abundance of the lower ionic states, rather than the other way around.

Fig. 5.

Classical picture of a non-alkali atom and the image charges it induces in the metal.

Therefore, the effects of energy level shifts should also be investigated. If one wishes to extend the concept of energy level shifts to the non-alkali metals, which have two s electrons in their outer orbitals, one may use an approach analogous to that used by Gadzukss>sO) for the alkali metals. For the non-alkali metals, it may be assumed that the charge on the ion core (see fig. 5) is +2e, rather than +e as assumed for the alkali metals. For the configuration shown in fig. 5, the redistribution of surface charge due to the close proximity of the surface atom is such that the perturbation associated

56

with the resulting just

D. MCKINSTRY

image potential Vma1

for the electron

2eZ =

Making the same approximations this becomes 2e2 v mat 2Sf z1

z1 is

e2

e2

- .

~~~ Rl

at the z-coordinate

R2

4(s+ ~~~~~

-

(31)

ZJ

as Gadzuk29,ss)

has for the alkali metals,

e2

e2

2s+z,+z,

4(s+z,)’

(32)

where s is the distance of the atom from the effective mirror surface. To determine the energy level shift rigorously, it is necessary to conduct an integration over both z1 and z2. However, that is not warranted for this paper. Therefore, it is assumed that the atom is sufficiently far from the surface that the differences between S, s+z, and sfz, may be neglected. If this is done, then eq. (32) reduces to 2e2

e2

e2

e2

Vmat = ~~ 2s

2s

4s

4s’

(33)

This, of course, agrees with the large distance approximation for the alkali metalsso). Thus, at large distances from the surface, no distinction need be made as to the kind of atom in order to determine the energy level shift on the first ionization potential, although the “size” of the atom does determine the distance from the surface at which this approximation becomes realistic. Since it is clear that the configuration of the electrons about the atom may be ignored if the atom is sufficiently far from the surface, one may look upon the energy level shift for the nth ionization potential as being equivalent to the work done by the image charges during the ionization process. During this process, the force on the electron, located at a distance z from the atom, is just (see fig. 6)

2

e2

F;=&---

(34)

4(s + z)2’

where ne is the charge remaining on the ion after ionization is completed. Integration of this from z=O to z= co gives the approximate energy level shift on the nth ionization potential as

&;_;c

(2n

- 1) e2 4s

.

(35)

The total shift in the energy needed to bring the atom to its nth ionic state is therefore *+A&=;,$

(Z&1)=7. I

1

W2

(36)

FIELD

If one further

assumes

effects may be ignored,

EVAPORATION

57

THEORY

that at large distances

from the surface,

then QOns may be approximated

QOns= Qon- (ne)'/Js.

broadening

by (37)

Thus, it takes progressively less energy to bring an atom situated near a metal surface to higher and higher ionic states than it takes for the same atom in

Fig. 6.

Classical picture of the image charges induced in the metal surface for an ion being raised to its next higher ionic state.

free vacuuo. This is true even if the above equations prove to be gross overestimates of the actual energy level shifts near the metal surface, since the general trend is nevertheless toward increasing perturbation of the charge distribution on the metal surface as the charge on the ion above the surface increases. Therefore, it is apparent that the inclusion of energy level shifts in field evaporation theory results in the prediction of the existence of more highly ionized states than would be otherwise expected. It is highly probable that it is because the energy level shifts have not been included in previous calculations that the existing theory has failed to predict the observed presence of the more highly ionized states of tungsten, copper and other metals.

6. Conclusions From the discussion in this paper, it is apparent that the ionic species which dominates the evaporation process is strongly dependent upon the field at which evaporation occurs, as well as upon the temperature. Furthermore, the species which dominates at a given field and temperature is dependent on the repulsive part of the ionic potential and energy level shifts,

58

D. MCKINSTRY

as well as those factors already considered by previous investigators. The repulsive part of the ionic potential influences the evaporation process by repressing the lower charge states, for which the repulsive term is expected to be stronger than for the higher charge statess7). However, although this may play a role in the appearance of doubly charged copper ions, it is unlikely that it has any significant effect on the appearance of the observed triply and quadruply charged ions of various metals. On the other hand, as indicated by eq. (37), the decrease in Qon becomes greater for the higher charged states, and the activation energy for these states becomes less by a greater amount than for the lower charged states. Thus, the inclusion ef energy level shifts in field evaporation theory results in the prediction of more highly ionized states than would be otherwise expected. As shown in the section on rate sensitivity, it is important that investigators know which evaporation model best applies to a given experimental situation before any attempts are made to interpret field-sensitivity data. However, the author has also shown a means by which field-sensitivity data can be used to determine which model is most likely applicable in a given situation. This probably requires some improvements in pulser techniques 5), however. Also, it is necessary either to incorporate into the theory in a quantitative manner any effects on the evaporation process caused by the transfer of thermal energy to the surface atoms from the imaging gas62 7,17), or the field adsorption34) of imaging gas atoms on the surface. Furthermore, if the intersection model is found to apply, the use of linear approximations5) in the interpretation of field sensitivity data must be considered with extreme caution. Indeed, it may prove necessary to use an actual approximation of ss,ss) in conjunction with the experimental the atomic and ionic potentials data495) in order to determine the desired parameters cxand (x0 + A). In summation, it may be said that although there is presently a good qualitative basis for field evaporation theory, this theory is far from obtaining accurate quantitative predictions of many aspects of the evaporation process. There remains a great deal of both experimental and theoretical work to be done in the field of field evaporation and ionization. Acknowledgements The cooperation of the computing center at the University of Nevada, Reno, is gratefully acknowledged. Special thanks is extended to Dr. George Barnes for many discussions with him and his extensive aid in writing the paper. Also, helpful discussions with Drs. P. C. Bettler and P. L. Altick are greatly appreciated.

FIELD

EVAPORATION

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59

References 1) E. W. Mtiller, Naturwissenschaften 29 (1941) 533. 2) E. W. Mtiller, Phys. Rev. 102 (1956) 618. 3) R. Gomer, J. Chem. Phys. 31 (1959) 341; in: Field Emission and Field Ionization (Harvard Univ. Press, Cambridge, Mass., 1961). 4) T. T. Tsong and E. W. Mtiller, in: 16th Field Emission Symp., Pittsburg, Pa., 1969; Phys. Status Solidi (a) 1 (1970) 513. 5) T. T. Tsong, J. Chem. Phys. 54 (1971) 4205. 6) D. G. Brandon, Phil. Mag. 14 (1966) 803. 7) D. M. Taylor and M. J. Southon, in: 17th Field Emission Symp., Yale University, 1970. 8) D. M. Taylor, Ph.D. Dissertation, Univ. of Cambridge (1970). 9) R. S. Chambers, G. Ehrlich and M. Vesely, in: 17th Field Emission Symp., Yale Univ., 1970. 10) D. F. Barofsky and E. W. Mtiller, Surface Sci. 10 (1968) 177. 11) R. Vanselow and W. A. Schmidt, in: 13th Field Emission Symp., Cornell Univ., 1966 12) E. L. Richter and W. A. Schmidt, in: 14th Field Emission Symp., National Bureau of Standards, 1967. 13) E. W. Mtiller, S. V. Krishnaswamy and S. B. McLane, Surface Sci. 23 (1970) 112. 14) E. W. Mtiller, in: 17th Field Emission Symp., Yale Univ., 1970. 15) H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950). 16) M. J. Southon, Ph.D. Thesis, University of Cambridge (1963). 17) D. G. Brandon, in: Field-Zon Microscopy, Eds. J. J. Hren and S.Ranganathan(Plenum. New York, 1968) ch. 3. 18) C. Zener, Proc. Roy. Sot. (London) A 140 660 (1933). 19) D. R. Bates, Atomic and Molecular Processes (Academic Press, New York, 1962) p. 608. 20) R. Gomer and L. W. Swanson, J. Chem. Phys. 38 (1963) 1613; L. W. Swanson and R. Gomer, J. Chem. Phys. 39 (1963) 2813. 21) T. T. Tsong, Surface Sci. 10 (1968) 102. 22) D. G. Brandon, Surface Sci. 3 (1965) 1. 23) T. T. Tsong and E. W. Mtiller, Phys. Rev. 181 (1969) 530. 24) J. Friedel, Advan. Phys. 3 (1954) 446. 25) P. H. Cutler and J. C. Davis, Surface Sci. 1 (1964) 194. 26) D. M. Newns, J. Chem. Phys. 50 (1969) 4572. 27) D. M. McKinstry, Ph.D. Dissertation, University of Nevada, Reno (1971). 28) R. W. Gurney, Phys. Rev. 47 (1935) 479. 29) J. W. Gadzuk, Surface Sci. 6 (1967) 133. 30) J. W. Gadzuk, Phys. Rev. B 1 (1970) 2110. 31) E. W. Miiller and T. T. Tsong, Field-Ion Microscopy: Principles and Applications (Elsevier, Amsterdam, 1969). 32) D. M. McKinstry, in: 17th Field Emission Symp., Yale Univ., 1970. 33) J. W. Gadzuk, private communication (1971). 34) T. T. Tsong and E. W. Miiller, Phys. Rev. Letters 25 (1970) 911. 35) R. Ftirth, Proc. Roy. Sot. (London) A 183 (1945) 87. 36) A. Mtiller and M. Drechsler, Surface Sci. 13 (1969) 471.