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Field ionization theory: A new, analytic, formalism
Surface Science 103 (1981) 239-247 0 North-Holland Publishing Company
FIELD IONIZATION THEORY:
A NEW, ANALYTIC,
FORMALISM
Roger HAYDOCK and David R. KINGHAM Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, UK
Received 14 May 1980; accepted for publication 20 August 1980
A new formalism for the calculation of field ionization rate-constants near a model metal surface is derived. Approximate analytic formulae are given for the ionization rate-constant of an atom as a function of distance from the metal surface, for the total probability of ionization on a single pass through the ionization zone and for the width of the ionization zone. Theoretical estimates of best image fields are given for gas atoms of interest in field ion microscopy. The clarity of approach and computational ease of the method is shown to be compatible with the production of acceptably accurate results.
1. Introduction
In this paper we consider the process of field ionization as it occurs in the field ion microscope [l] where gas atoms drift towards the metal tip under the influence of the applied electric field. As the atom approaches the metal tip it may be ionized by an electron tunnelling into the metal. A theoretical explanation of this was first given by Mtiller and Bahadur [2] who achieved qualitative agreement between their theoretical calculations and the observed experimental results. Since then many other authors have investigated field ionization near a metal surface theoretically [3-91. We present here a derivation of an analytic formula for the electron transition rateconstant for field ionization and for the ionization zone width. The advantage of an analytic method is that the dependence of electron transition rateconstant and ionization zone width on factors such as field strength, ionization potential of the gas atom and metal surface work-function is obtained explicitly. The slight loss of absolute accuracy inherent in the method is a small price to pay for this.
2. Model potential
The analytic approach relies on the choice of model potential for the system, our choice is a uniform depth potential well for the metal with sides of height $J above the Fermi level of the metal where 4 is the zero field local work-function, 239
R. Haydock, D.N. Kinghant / Field ionization theory
240
and a uniform electric field potential with no penetration into the metal plus a hydrogenic atomic potential centred on the position of the gas atom. That is (in S.I.) Vmodel = -.@/4neor
-
eFr cos % .
(11
with an abrupt cut-off at the model metal surface (fig. 1). In eq. (1) e is the electron charge, E@the permittivity of free space, F the applied electric field strength at the position of the nucleus of the gas atom and Ze is the effective nuclear charge seen by the tunnelling electron, Yis measured from the nucleus of the gas atom and 0 = 71 is the field direction. With such a model potential an accurate three-dimensional calculational method [ 101 may be used. We note that a critical distance exists for field ionization [Z] because the electron may only tunnel to empty states at or above the Fermi level of the metal. This distance is given approximately by eFz,=B-@
1
(9
where B is the binding energy of the electron in the atom and z, is measured from the model potential metal surface. The model potential neglects the effects of any field adsorbed gas atoms which may be present and assumes a featureless planar metal surface. The “image” potential due to the interaction of the gas atom and tunnelling electron with the metal, and the one-electron potential of the gas atom are both represented by the term -Ze2/47re0r where 2 is a variable parameter of the model which must be restricted to physically reasonable values. The calculational method is quasi-classical in that it treats the motion of the gas atom classically. The tunnelling electron is treated quantum-mechanically but no account is taken of any reflected electron waves from the metal surface, as such reflection is only a small effect for ionization close to the critical distance. The above approximations inevitably mean that the model
Fig. 1. A onedimensional representation of the model potential used in the calculation of field ionization rate. # is the work function, i?F the Fermi level of the metal, zo the distance of the atom from the metal surface and B the zero field ionization potentiat of the atom.
R. Haydock, D.R. Kingham / Field ionization theory
241
system does not include all the physics of the real system, but it does include the essentials of electron tunnelling through a three-dimensional, field-dependent, potential barrier and thus the results of the calculation are relevant to the real problem of field ionization.
3. Derivation of analytic formulae The Schroedinger
equation
for the model system is then
v*- (B+ T/model>1 d’= 0 >
(3)
where B is taken to be equal to the ionization potential of the atom, that is small corrections to B are ignored, though this constraint may be relaxed without affecting the formalism. An essentially timedependent problem has been approximated here by a time-independent problem; this is valid so long as the ionization rate is small on an atomic time scale, as it is in this case. The time dependence is recovered by allowing only an outgoing wave solution to the equation. Eq. (3) is not separable in spherical polar coordinates but an approximate outgoing solution is
where A and r. are constants k(r, 19)~ =$
(B - Ze2/4neor
and - eFr cos 0) ,
(5)
outside the metal and k(r, 0)” is a negative constant inside the metal. This solution, analogous in three dimensions to the JWKB solution in one dimension, may be shown to be valid to a good approximation in the region where 0 is small, that is the region in which $(r, 0) is largest and which gives the major contribution to the tunnelling probability [lo]. The ionization rate-constant, I, then is equal to the electron probability flux through a surface, S, perpendicular to the field direction, in the classically allowed region of the model potential which corresponds to the interior of the metal. So I= /I$@, s
e)l* u, dS >
(6)
where v, is the component of the electron velocity normal to the metal surface. Now inside the metal k(r, 0) is imaginary and the exponent in eq. (4) has constant real part. Thus inside the metal A2 I$(r,O)l*=
r*Ik(r, e)l
k(r, 0) h ro
1 ,
(7)
242
R. Haydock, D.R. Kingham /Field
ionization theory
where z,/cos 0 is the outer classical turning point, that is the distance from the atomic nucleus to the model surface, in a direction 0 to the normal from nucleus to surface. For small 0 the kinetic energy hZ lk(r, 0)1*/2m N imu:. Changing now, for the sake of clarity, to atomic units in which h = m = e = 1, so that the unit of energy is the Hartree (27.2 eV), the unit of field strength is 514.2 V nm-’ and the unit of length is the Bohr radius (5.29 X IO-* nm) we have n/2
2n
ZQ/COSH
A*r-* exp P23’2
I= s.i
O=O *o
[
J^
ro
(B - Z/r ~ Fr cos 0) I’* dr r* sin 0 d0 d@ . 1 (8)
Now as the atomic wavefunctions for atoms in a strong electric field are not well known a reasonable approach is to evaluate the tunnelling integral over the whole potential barrier and to introduce an electron frequency factor v to represent the rate of tunnelling attempts. The inner integral in eq. (8) is evaluated in terms of complete elliptic integrals for which an approximation valid for F < B* (as is the case) is used. Translating from atomic units the F < B* condition is equivalent to F < 128 V nm-’ for hydrogen and to F Q 420 V nm-’ for helium. The outer integral may then be evaluated noting that the tunnelling only occurs in the region where 0 is small, i.e. where the barrier is shortest. The result is &)
3rrA’uF = ‘~~~~ z312 [B3’2 ~ (B - Fz,,)~‘~]
qz/B?‘*
p/2
X exp
3~
[B3’* ~ (B -~ Fz,)~‘*] j ,
(9)
for the ionization rate-constant of a stationary atom as a function of distance, zo, from the metal surface. The above derivation and hence eq. (9) is valid for (B - 4)/F < z. B/F it is readily shown that the terms in (B - Fz,) in eq. (9) must be discarded and the dependence of I on z. disappears as we are now considering free space field ionization. If we take [lo] A* = (1 2rrvhe2)-1 ,
(10)
where v,, is the electron frequency in hydrogen then this formula gives the asymptotically correct result as F -+ 0 for the free-space field ionization rate-constant of hydrogen, where B = $, which is [ 111 in atomic units I = 4F’
exp(-2/3F)
.
(11)
The exponential dependence of 1 on z. in eq. (9) shows that a zone will exist close to the critical distance z, in which the ionization rate is much greater than elsewhere. The electric field, F, will vary as a function of distance from the surface. This variation may be modelled by postulating an inverse square field strength
R. Huydock, D.R. Kingham / Field ionization theory
243
dependence near the critical surface. If Fe is the field strength at the critical distance and R the radius of curvature of the equipotentials at z, we have FcR2 = F&(R + 6)’ ,
(12)
where S is the distance beyond zC and F8 is the field at distance S . From eqs, (9) and (12) we have, to a good approbation for S
exp(-PS)
,
that is
(13)
where exp and /3 = 23f2[#1f2 + 4(BY2 - #3f2)/3FcR]
.
(15)
The probab~ity of an atom being ionized as it moves a distance dz with component of velocity normal to the surface u(z) is p(z) da = Z(z) u(z)-’ dz ) so the probability W> = r z
I1 -
(16)
of the atom being ionized as it approaches the metal is
P(E)1P(E)di
07)
=l-ev[-~dBldt], z
(18)
and the total probab~ty PT, is
of ionization
on a single pass through the ionization
[ zc 1.
PT = 1 - exp - 7
only occurs close to z, and we take u(z, + 6) = v{zC) = u,
6
P(z, t 6) = 1 - exp - 1 [
PT =1-
expf-l(zc)@u,)-* ]
I(z,) vi’ exp(-PO
d$
1,
.
Defining the width, S,, of the ionization P(zc + S,) = $PT )
(19)
p(l) de
If Pr is small ionization giving
zone,
a!) (21)
zone by (22)
244
R. Haydock, D.R. Kingham /Field ionization theory
we find
and for P, Q 1 S, = In 2//3 .
(24)
The velocity of the atom is given by 1,rrUJ~= ;C@.
(2%
neglecting thermal energy, on its first pass through the critical surface, where a: is the atomic polarizability [12], but the atom is then slowed down by thermal accommodation as it bounces on the surface so that its velocity tends to become U, ~ fkTtip/m)“2
where Ttip is the tip temperature.
4. Numerical results and discussion Results are given in table 1 for the probability of ionization of a helium atom on a single normal pass through an ionization zone, that is P, calculated from eqs. (14), (15) and (21), when the field strength at the critical distance is the best image field for helium (45 V nm-*) and at 99% and 80% of this field. Probabilities are quoted for the case when the atom has polarization energy and for when thermal accommodation has occurred to give effective gas temperatures of 200, 80, 20, 4 and 1 K. The work-function is taken as 4.5 eV and Z as 1 S, where this value is estimated to account approximately for both the one-electron potential of the atom and the image potential. The numbers given in table 1 are dependent on the details Table 1 The probability of ionization of a helium atom on a single pass through an ionization best image field (BIF) and at 99% and 80% of BIF for various atom kinetic energies __
zone at
Atom energy
PT @IF)
PT (99% of RII;)
PT (80% Of BIF)
Potarization energy Thermal energy (200 Kf ( 8OK) ( 20 K)
1.1 x to-2
9.6 x 10-S
2.5 x 104
4.8 7.4 1.4 2.9 5.0
4.2 6.3 1.2 2.5 4.4
8.5 1.3 2.7 6.0 1.2
( (
4K) lK1
x x x x x
10-Z 10-z 10-l 10-l lo-’
x x x x x
10-Z 10-2 10-l 10-l 10-l
x x X X x
10-4 10-J 1O-3 10-j 10-2
R. Haydock, D.R. Kingham / Field ionization theory
245
Table 2 Theoretical estimates of best image fields from this work for rare gases based on an assumed value of 45 V nm-’ for helium, compared with the experimental data of Chamberlain [ 131 and the theoretical estimates of Mtiller and Tsong [ 141 Gas
Ne
Ar Kr Xe
Best image field (V nm-‘) This work
Experiment ref. [ 13 J
Theory ref. (141
35.0 18.6 14.3 10.3
35.0 19.0 14.5-15.0 11.0-13.0
35.0 19.4 15.4 12.5
of the model potential and on the value chosen for 2 and are thus not intended to show absolute values of ionization probability but rather to demonstrate variations in this probability. Calc~ations are made for other gases using the same model potential and if we assume that these calculated probabilities will be the same for all gases at their respective best image fields, above the same surface of work-function 4.5 eV, we obtain the theoretical estimates of such fields given in table 2. If a different model potential (with 2 = 2) is used for helium and the other gases calculated probabilities may again be compared. The best image fields found using this model are almost identical with the previous case indicat~g a very weak model dependence of the results in table 2. These theoretical estimates are compared with the experimental values given by Chamberlain [13] and the theoretical estimates of Muller and Tsong D41. The half width of the ionization zone, 6,, calculated from eqs. (15) and (24) is given in table 3 for all rare gases at best image field above surfaces of work-function 4.5 and 5.5 eV. Values of R much less than the radius of the tip are taken to represent the varying field strength above a protruding atom. The numbers given in table
Table 3 The half width of the ionization zone, 6,, for rare gases at BIF above surfaces of work function 4.5 eV and 5.5 eV for various local surface radii of curvature, R R (nm)
1.5 2.5 5.0 10.0
6,
(nm)
0 = 4.5 eV
@=5.5 eV
0.016 0.020 0.024 0.029
0.014 0.018 0.022 0.027
246
R. Ifaydock, D.R. Kingham /Field iorlization theor),
3 are only weakly dependent on the details of the model potential, but do depend on the field variation above the surface. From table 1 we see that the probability of ionization on a single pass through an ionization zone increases as the energy (velocity) of the gas atom decreases from its initial polarization energy (equivalent to a temperature of about 4000 K). This is due simply to the increase in residence time of the atom in the ionization zone. It is important to realize that the atom would have to bounce about 300 times on the tip [14] before being fully accommodated to the tip temperature, but that on each bounce it will not necessarily pass through an ionization zone. At best image field the ionization zones occur only in the enhanced field regions above protruding surface atoms and so a gas atem will only pass through an ionization zone once in about 10 bounces. Table 1 shows that PT. is about 0.01 while the atom still has most of its polarization energy, which is reasonable as it indicates that significant though not necessarily complete thermal accommodation [15] may occur before the atom is ionized. Table 1 also shows how ionization probability on a single pass through the ionization zone changes with varying field strength. It is not yet clear whether field-dependent changes in ionization probability are the major cause of the contrast in the field ion microscope or whether field-dependent gas concentration effects [IS] are also important. Either or both of these effects could explain a field ion micrograph as being a representation of field strength variations above the surface. Alternative explanations, in terms of extended surface electron states [16,17] Fermi surface structure [7], or apex adsorbed atoms [18,19], have also been proposed, though, in the present authors’ view, these are of secondary importance to field strength variations. Table 2 shows that this work is in good agreement with experiment [ 131 and the simple estimates of Mtiller and Tsong [14] except that our value for the best image field of Xe seems rather low. There is no reason to suspect that these calculations are less accurate for Xe than for other gases and the cause of this discrepancy may well deserve further investigation. The results in table 3 are in good agreement with experimental values of ionization zone widths [3] of about 0.02 nm. These calculated zone widths are much less than previous JWKB calculations [3]. The improvement in accuracy is due partly to a better, three-dimensional, version of the JWKB method and partly to the inclusion of field strength variation with- distance from the surface. The predicted changes in ionization zone width with changing work-function or changing field strength (eqs. (15) and (24)) are small but may be experimentally observable.
5. Conclusion Analytic formulae have been derived for the ionization rate-constant of an atom above a metal surface (eq. (9)) for the total probability of ionization on a single pass through the ionization zone (eqs. (14), (15) and (21)) and for the width of
R. Haydock, DR. Kingham /Field
ionization theory
241
the ionization zone (eqs. (IS) and (24)). The explicit dependence of these quantities on factors such as field strength, ionization potential of the gas atom and work function of the metal has thus been demonstrated. We believe that this paper presents the first calculations for field ionization above a metal surface which give both a reasonable estimate of ionization probab~ty and good agreement with experimental values for ionization zone width.
Acknowledgements
We are grateful to Dr. Richard G. Forbes for helpful and constructive comments, One of us (D-RX.) acknowledges the financial support of the Science Research council.
References [l] [2] f3] [4]
E.W. Mtiller, 2. Physik 131 (1951) 136. E.W. Miiller and K. Bahadur, Phys. Rev. 102 (1956) 624. T.T. Tsong and E.W. MiiBer, J. Chem. Phys. 41 (1964) 3279. D.S. Boudreaux and P.H. Cutler, Solid State Commun. 3 (1965) 219; Phys. Rev. 149 (1966) 170;Surface Sci. 5 (1966) 230. [5] M.E. Alferieff and C.B. Duke, J. Chem. Phys. 46 (1966) 938. [6] A.J. Jason, Phys. Rev. 156 (1967) 266. [7] S.P. Sharma, S.J. Fonash and G.L. Schrenk, Surface Sci. 23 (1970) 30. [8] I.V. Goldenfield, E.N. Karol and V.A. Pokrovsky, Intern. J. Mass. Spectrom. Ion Phys. 5 (1970) 337. [9] J.A. Appelbaum and E.G. McRae, Surface Sci. 47 (1975) 445. ]lO] R. Haydock and D.R. Kingham, J. Phys. B, to be published. [ 111 L.D. Landau and E.M. Lifshitz, Qu~tum Mechanics (Pergamon, Oxford, 1965) section [12] EG. Forbes, Surface Sci. 64 (1977) 367. [ 131 P. Chamberlain, thesis, Univ. of Cambridge (1971). [ 141 E.W. Mtiller and T.T. Tsong, Field Ion Microscopy (Elsevier, New York, 1969). [15] J. Duffel and R.G. Forbes, J. Phys. D (Appl. Phys.) 11 (1978) L123. [16] Z. Knor and E.W. MiilJer, Surface Sci. 10 (1968) 21. [ 171 2. Knor, J. Vacuum Sci. Technol. 8 (1971) 57. [18] D.A. Nolan and R.M. Herman, Phys. Rev. 38 (1973) 4099. [ 191 H. Iwasaki and S. Nakamura, Surface Sci. 49 (1975) 664.
FIELD IONIZATION THEORY:
A NEW, ANALYTIC,
FORMALISM
Roger HAYDOCK and David R. KINGHAM Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, UK
Received 14 May 1980; accepted for publication 20 August 1980
A new formalism for the calculation of field ionization rate-constants near a model metal surface is derived. Approximate analytic formulae are given for the ionization rate-constant of an atom as a function of distance from the metal surface, for the total probability of ionization on a single pass through the ionization zone and for the width of the ionization zone. Theoretical estimates of best image fields are given for gas atoms of interest in field ion microscopy. The clarity of approach and computational ease of the method is shown to be compatible with the production of acceptably accurate results.
1. Introduction
In this paper we consider the process of field ionization as it occurs in the field ion microscope [l] where gas atoms drift towards the metal tip under the influence of the applied electric field. As the atom approaches the metal tip it may be ionized by an electron tunnelling into the metal. A theoretical explanation of this was first given by Mtiller and Bahadur [2] who achieved qualitative agreement between their theoretical calculations and the observed experimental results. Since then many other authors have investigated field ionization near a metal surface theoretically [3-91. We present here a derivation of an analytic formula for the electron transition rateconstant for field ionization and for the ionization zone width. The advantage of an analytic method is that the dependence of electron transition rateconstant and ionization zone width on factors such as field strength, ionization potential of the gas atom and metal surface work-function is obtained explicitly. The slight loss of absolute accuracy inherent in the method is a small price to pay for this.
2. Model potential
The analytic approach relies on the choice of model potential for the system, our choice is a uniform depth potential well for the metal with sides of height $J above the Fermi level of the metal where 4 is the zero field local work-function, 239
R. Haydock, D.N. Kinghant / Field ionization theory
240
and a uniform electric field potential with no penetration into the metal plus a hydrogenic atomic potential centred on the position of the gas atom. That is (in S.I.) Vmodel = -.@/4neor
-
eFr cos % .
(11
with an abrupt cut-off at the model metal surface (fig. 1). In eq. (1) e is the electron charge, E@the permittivity of free space, F the applied electric field strength at the position of the nucleus of the gas atom and Ze is the effective nuclear charge seen by the tunnelling electron, Yis measured from the nucleus of the gas atom and 0 = 71 is the field direction. With such a model potential an accurate three-dimensional calculational method [ 101 may be used. We note that a critical distance exists for field ionization [Z] because the electron may only tunnel to empty states at or above the Fermi level of the metal. This distance is given approximately by eFz,=B-@
1
(9
where B is the binding energy of the electron in the atom and z, is measured from the model potential metal surface. The model potential neglects the effects of any field adsorbed gas atoms which may be present and assumes a featureless planar metal surface. The “image” potential due to the interaction of the gas atom and tunnelling electron with the metal, and the one-electron potential of the gas atom are both represented by the term -Ze2/47re0r where 2 is a variable parameter of the model which must be restricted to physically reasonable values. The calculational method is quasi-classical in that it treats the motion of the gas atom classically. The tunnelling electron is treated quantum-mechanically but no account is taken of any reflected electron waves from the metal surface, as such reflection is only a small effect for ionization close to the critical distance. The above approximations inevitably mean that the model
Fig. 1. A onedimensional representation of the model potential used in the calculation of field ionization rate. # is the work function, i?F the Fermi level of the metal, zo the distance of the atom from the metal surface and B the zero field ionization potentiat of the atom.
R. Haydock, D.R. Kingham / Field ionization theory
241
system does not include all the physics of the real system, but it does include the essentials of electron tunnelling through a three-dimensional, field-dependent, potential barrier and thus the results of the calculation are relevant to the real problem of field ionization.
3. Derivation of analytic formulae The Schroedinger
equation
for the model system is then
v*- (B+ T/model>1 d’= 0 >
(3)
where B is taken to be equal to the ionization potential of the atom, that is small corrections to B are ignored, though this constraint may be relaxed without affecting the formalism. An essentially timedependent problem has been approximated here by a time-independent problem; this is valid so long as the ionization rate is small on an atomic time scale, as it is in this case. The time dependence is recovered by allowing only an outgoing wave solution to the equation. Eq. (3) is not separable in spherical polar coordinates but an approximate outgoing solution is
where A and r. are constants k(r, 19)~ =$
(B - Ze2/4neor
and - eFr cos 0) ,
(5)
outside the metal and k(r, 0)” is a negative constant inside the metal. This solution, analogous in three dimensions to the JWKB solution in one dimension, may be shown to be valid to a good approximation in the region where 0 is small, that is the region in which $(r, 0) is largest and which gives the major contribution to the tunnelling probability [lo]. The ionization rate-constant, I, then is equal to the electron probability flux through a surface, S, perpendicular to the field direction, in the classically allowed region of the model potential which corresponds to the interior of the metal. So I= /I$@, s
e)l* u, dS >
(6)
where v, is the component of the electron velocity normal to the metal surface. Now inside the metal k(r, 0) is imaginary and the exponent in eq. (4) has constant real part. Thus inside the metal A2 I$(r,O)l*=
r*Ik(r, e)l
k(r, 0) h ro
1 ,
(7)
242
R. Haydock, D.R. Kingham /Field
ionization theory
where z,/cos 0 is the outer classical turning point, that is the distance from the atomic nucleus to the model surface, in a direction 0 to the normal from nucleus to surface. For small 0 the kinetic energy hZ lk(r, 0)1*/2m N imu:. Changing now, for the sake of clarity, to atomic units in which h = m = e = 1, so that the unit of energy is the Hartree (27.2 eV), the unit of field strength is 514.2 V nm-’ and the unit of length is the Bohr radius (5.29 X IO-* nm) we have n/2
2n
ZQ/COSH
A*r-* exp P23’2
I= s.i
O=O *o
[
J^
ro
(B - Z/r ~ Fr cos 0) I’* dr r* sin 0 d0 d@ . 1 (8)
Now as the atomic wavefunctions for atoms in a strong electric field are not well known a reasonable approach is to evaluate the tunnelling integral over the whole potential barrier and to introduce an electron frequency factor v to represent the rate of tunnelling attempts. The inner integral in eq. (8) is evaluated in terms of complete elliptic integrals for which an approximation valid for F < B* (as is the case) is used. Translating from atomic units the F < B* condition is equivalent to F < 128 V nm-’ for hydrogen and to F Q 420 V nm-’ for helium. The outer integral may then be evaluated noting that the tunnelling only occurs in the region where 0 is small, i.e. where the barrier is shortest. The result is &)
3rrA’uF = ‘~~~~ z312 [B3’2 ~ (B - Fz,,)~‘~]
qz/B?‘*
p/2
X exp
3~
[B3’* ~ (B -~ Fz,)~‘*] j ,
(9)
for the ionization rate-constant of a stationary atom as a function of distance, zo, from the metal surface. The above derivation and hence eq. (9) is valid for (B - 4)/F < z. B/F it is readily shown that the terms in (B - Fz,) in eq. (9) must be discarded and the dependence of I on z. disappears as we are now considering free space field ionization. If we take [lo] A* = (1 2rrvhe2)-1 ,
(10)
where v,, is the electron frequency in hydrogen then this formula gives the asymptotically correct result as F -+ 0 for the free-space field ionization rate-constant of hydrogen, where B = $, which is [ 111 in atomic units I = 4F’
exp(-2/3F)
.
(11)
The exponential dependence of 1 on z. in eq. (9) shows that a zone will exist close to the critical distance z, in which the ionization rate is much greater than elsewhere. The electric field, F, will vary as a function of distance from the surface. This variation may be modelled by postulating an inverse square field strength
R. Huydock, D.R. Kingham / Field ionization theory
243
dependence near the critical surface. If Fe is the field strength at the critical distance and R the radius of curvature of the equipotentials at z, we have FcR2 = F&(R + 6)’ ,
(12)
where S is the distance beyond zC and F8 is the field at distance S . From eqs, (9) and (12) we have, to a good approbation for S
exp(-PS)
,
that is
(13)
where exp and /3 = 23f2[#1f2 + 4(BY2 - #3f2)/3FcR]
.
(15)
The probab~ity of an atom being ionized as it moves a distance dz with component of velocity normal to the surface u(z) is p(z) da = Z(z) u(z)-’ dz ) so the probability W> = r z
I1 -
(16)
of the atom being ionized as it approaches the metal is
P(E)1P(E)di
07)
=l-ev[-~dBldt], z
(18)
and the total probab~ty PT, is
of ionization
on a single pass through the ionization
[ zc 1.
PT = 1 - exp - 7
only occurs close to z, and we take u(z, + 6) = v{zC) = u,
6
P(z, t 6) = 1 - exp - 1 [
PT =1-
expf-l(zc)@u,)-* ]
I(z,) vi’ exp(-PO
d$
1,
.
Defining the width, S,, of the ionization P(zc + S,) = $PT )
(19)
p(l) de
If Pr is small ionization giving
zone,
a!) (21)
zone by (22)
244
R. Haydock, D.R. Kingham /Field ionization theory
we find
and for P, Q 1 S, = In 2//3 .
(24)
The velocity of the atom is given by 1,rrUJ~= ;C@.
(2%
neglecting thermal energy, on its first pass through the critical surface, where a: is the atomic polarizability [12], but the atom is then slowed down by thermal accommodation as it bounces on the surface so that its velocity tends to become U, ~ fkTtip/m)“2
where Ttip is the tip temperature.
4. Numerical results and discussion Results are given in table 1 for the probability of ionization of a helium atom on a single normal pass through an ionization zone, that is P, calculated from eqs. (14), (15) and (21), when the field strength at the critical distance is the best image field for helium (45 V nm-*) and at 99% and 80% of this field. Probabilities are quoted for the case when the atom has polarization energy and for when thermal accommodation has occurred to give effective gas temperatures of 200, 80, 20, 4 and 1 K. The work-function is taken as 4.5 eV and Z as 1 S, where this value is estimated to account approximately for both the one-electron potential of the atom and the image potential. The numbers given in table 1 are dependent on the details Table 1 The probability of ionization of a helium atom on a single pass through an ionization best image field (BIF) and at 99% and 80% of BIF for various atom kinetic energies __
zone at
Atom energy
PT @IF)
PT (99% of RII;)
PT (80% Of BIF)
Potarization energy Thermal energy (200 Kf ( 8OK) ( 20 K)
1.1 x to-2
9.6 x 10-S
2.5 x 104
4.8 7.4 1.4 2.9 5.0
4.2 6.3 1.2 2.5 4.4
8.5 1.3 2.7 6.0 1.2
( (
4K) lK1
x x x x x
10-Z 10-z 10-l 10-l lo-’
x x x x x
10-Z 10-2 10-l 10-l 10-l
x x X X x
10-4 10-J 1O-3 10-j 10-2
R. Haydock, D.R. Kingham / Field ionization theory
245
Table 2 Theoretical estimates of best image fields from this work for rare gases based on an assumed value of 45 V nm-’ for helium, compared with the experimental data of Chamberlain [ 131 and the theoretical estimates of Mtiller and Tsong [ 141 Gas
Ne
Ar Kr Xe
Best image field (V nm-‘) This work
Experiment ref. [ 13 J
Theory ref. (141
35.0 18.6 14.3 10.3
35.0 19.0 14.5-15.0 11.0-13.0
35.0 19.4 15.4 12.5
of the model potential and on the value chosen for 2 and are thus not intended to show absolute values of ionization probability but rather to demonstrate variations in this probability. Calc~ations are made for other gases using the same model potential and if we assume that these calculated probabilities will be the same for all gases at their respective best image fields, above the same surface of work-function 4.5 eV, we obtain the theoretical estimates of such fields given in table 2. If a different model potential (with 2 = 2) is used for helium and the other gases calculated probabilities may again be compared. The best image fields found using this model are almost identical with the previous case indicat~g a very weak model dependence of the results in table 2. These theoretical estimates are compared with the experimental values given by Chamberlain [13] and the theoretical estimates of Muller and Tsong D41. The half width of the ionization zone, 6,, calculated from eqs. (15) and (24) is given in table 3 for all rare gases at best image field above surfaces of work-function 4.5 and 5.5 eV. Values of R much less than the radius of the tip are taken to represent the varying field strength above a protruding atom. The numbers given in table
Table 3 The half width of the ionization zone, 6,, for rare gases at BIF above surfaces of work function 4.5 eV and 5.5 eV for various local surface radii of curvature, R R (nm)
1.5 2.5 5.0 10.0
6,
(nm)
0 = 4.5 eV
@=5.5 eV
0.016 0.020 0.024 0.029
0.014 0.018 0.022 0.027
246
R. Ifaydock, D.R. Kingham /Field iorlization theor),
3 are only weakly dependent on the details of the model potential, but do depend on the field variation above the surface. From table 1 we see that the probability of ionization on a single pass through an ionization zone increases as the energy (velocity) of the gas atom decreases from its initial polarization energy (equivalent to a temperature of about 4000 K). This is due simply to the increase in residence time of the atom in the ionization zone. It is important to realize that the atom would have to bounce about 300 times on the tip [14] before being fully accommodated to the tip temperature, but that on each bounce it will not necessarily pass through an ionization zone. At best image field the ionization zones occur only in the enhanced field regions above protruding surface atoms and so a gas atem will only pass through an ionization zone once in about 10 bounces. Table 1 shows that PT. is about 0.01 while the atom still has most of its polarization energy, which is reasonable as it indicates that significant though not necessarily complete thermal accommodation [15] may occur before the atom is ionized. Table 1 also shows how ionization probability on a single pass through the ionization zone changes with varying field strength. It is not yet clear whether field-dependent changes in ionization probability are the major cause of the contrast in the field ion microscope or whether field-dependent gas concentration effects [IS] are also important. Either or both of these effects could explain a field ion micrograph as being a representation of field strength variations above the surface. Alternative explanations, in terms of extended surface electron states [16,17] Fermi surface structure [7], or apex adsorbed atoms [18,19], have also been proposed, though, in the present authors’ view, these are of secondary importance to field strength variations. Table 2 shows that this work is in good agreement with experiment [ 131 and the simple estimates of Mtiller and Tsong [14] except that our value for the best image field of Xe seems rather low. There is no reason to suspect that these calculations are less accurate for Xe than for other gases and the cause of this discrepancy may well deserve further investigation. The results in table 3 are in good agreement with experimental values of ionization zone widths [3] of about 0.02 nm. These calculated zone widths are much less than previous JWKB calculations [3]. The improvement in accuracy is due partly to a better, three-dimensional, version of the JWKB method and partly to the inclusion of field strength variation with- distance from the surface. The predicted changes in ionization zone width with changing work-function or changing field strength (eqs. (15) and (24)) are small but may be experimentally observable.
5. Conclusion Analytic formulae have been derived for the ionization rate-constant of an atom above a metal surface (eq. (9)) for the total probability of ionization on a single pass through the ionization zone (eqs. (14), (15) and (21)) and for the width of
R. Haydock, DR. Kingham /Field
ionization theory
241
the ionization zone (eqs. (IS) and (24)). The explicit dependence of these quantities on factors such as field strength, ionization potential of the gas atom and work function of the metal has thus been demonstrated. We believe that this paper presents the first calculations for field ionization above a metal surface which give both a reasonable estimate of ionization probab~ty and good agreement with experimental values for ionization zone width.
Acknowledgements
We are grateful to Dr. Richard G. Forbes for helpful and constructive comments, One of us (D-RX.) acknowledges the financial support of the Science Research council.
References [l] [2] f3] [4]
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