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Ion current generation in the field ion microscope
Surface Science 0 North-Holland
52 (1975) 588.. 596 Publishing Company
ION CURRENT GENERATION
IN THE FIELD ION MICROSCOPE
I. Dynamic approach Hiroshi IWASAKI and Shogo NAKAMURA Institute of Scientific and Industrial Research, Osaka University, Suita, Osaka, Japan 56.5 Received
18 March
1975; revised manuscript
received
11 August
1975
The numbers of gas particles arriving at unit tip surface in unit time from a field free region are derived as functions of velocity components for a spherical tip. It is shown that a considerable fraction of the gas particles arrives at the tip having large tangential velocities. The simple model of collision of a particle with a metal surface is used and the trajectories and rebounds of particles are tracked. The principal method to calculate the total ion current is shown. The capture probability of particles by the dipole attraction potential is shown to increase when the tip temperature is lowered, the field strength is increased, the mass ratio of the gas atom to the metal atom increases and the gas temperature is lowered.
1. Introduction The ion current generation in the field ion microscope (FIM) has been studied by Miiller [ 11, Comer [2], and Southon [3] dynamically. In the treatises, the total gas supply to the field-ion emitter surface is found. The rate constants for ionization and escaping from the tip region without ionization are calculated and finally the total ion current is obtained. Realistically, the supply of captured atoms must be used instead of the total supply as Southon [3] has indicated. Both rate constants and the probability of capture [3] are quantities averaged over the velocity distribution of the particles at the tip region in equilibrium and so are the functionals of the distribution function [4]. In this paper, instead of calculating the averaged quantities such as effective ionization rate on the basis of an assumed, somewhat ambiguous velocity distribution, it is shown that the ion current can be calculated thoroughly in a dynamical situation. The incident and rebound trajectories of all particles are tracked and the ionized fractions generated in passing through the ionization zone are summed up. The simplification that the particles scattered from an emitter surface have an average velocity is employed. It enabled us to easily get information on the influence of the many variables such as the tip temperature and the gas temperature, in the field
H, Iwasaki, S. Nakamura/Ion
current generation in FM.
range where particles are ionized in a few hops. In the succeeding paper [4], the velocity distribution region is derived by the quasi-static approach.
I
589
of the particles at the tip
2. The supply function In analyzing the hopping process of gas particles, it will be assumed for simplicity that the emitter is spherical. Then, the magnitude of the electric field F is given by F(r) = F,(R JG2,
(1)
where Ft is the electric field at the tip surface, r is the distance to the center of the tip and R, the tip radius. A particle approaching the tip with u, its velocity when very far from the tip, and p, the distance of closest approach to the tip center if the electric field were zero, has radial and tangential velocities urr(r) and q(r). These are found by assuming the conse~ation of energy and angular momentum as follows: u,(r) = u[ 1 - (j~+‘r)~+ (u~/u)~(R~/~)~]“~,
@a)
q’)
@)
= up/r,
where mu;/2
= Ep = oF32,
(2c)
and LJ is dipole attraction velocity, Ep is the polarization energy of a gas atom, ar is its po Parizability and m its mass. The number of gas particles that pass through a certain plane far from the tip in unit time with u between u and utdu and p between p and p+dp is called N(v,p) du dp.
Fig. 1. A trajectory of a gas particle from a field free region to the tip surface; un and ut are the velocity components at the tip surface.
H. Iwasaki, S. Nakamurallon
590
As the velocity distribution
current generation
in FIM I
is Maxwellian far from the tip, N(u,p) du dp is given by
N(u,p) du dp = n(m/27rkTJ
3/2 u3 exp(-mu2/2kT,J
27~~dp du,
(3)
where n and Tg are the density and the temperature of the ambient gas, respectively, and k is the Boltzmann constant, From eqs. (2a) and (2b) P dp = -(R,/v)~
(4a)
u, du,,
(4b)
P dp = (R,/u)2 ut dut,
where u, and ut are equal to un(Rt) and u,(Rt), respectively. The numbers of particles that approach the tip from a given direction with u between u and u+du and hit the tip surface in unit time with un between u, and u,tdu, and with I+ between ut and ut+du, are called N(u,u,) du du, and N(u,uJ du du,, respectively. We obtain from eq. (3) and eqs. (4a) and (4b) du du, = n(m/2nkTg)3’2 u exp(--mu2/2kTg)
2rrRF un dudu,,
(5a)
N(u,ut) du du, = n(m/2nkTg) 3’2 u exp (-mu2/2kTg)
21rRfu, du du,.
(5b)
N(u,u,)
For a given value of u, a maximum value pm,(u) of p, at which the particle will reach the tip, exists [5]. So, there are minimum values urnin and urnin of u at which the particle will hit the tip surface with a given u, and ut, respectively: Umin(un) =
up - un
for u,
(LIZ- LJy
forun>up,
L
(64
for ut
for ut >JzuP.
(6b)
The numbers of gas particles that hit the unit tip surface in unit time with u, between u, and u,+du, and with ut between ut and q+dq are called N,(u,) du, and N&u,) dut, respectively. We obtain from eqs. (5a) and (Sb) and eqs. (6a) and (6b) N&J
= sdfl
= So(m/kTg)
duN(u,u,)/(4nR:)
7 Urnin X
u, exp [-m(u,
- u,)2/2kT,J
exp (Ep/kTg) un exp (-mui/2kTg)
N#J,) = jdaj
for un < up, for un > up;
(7a)
dWv,)/(4~~,2)
Umin(ut) = So(m/kTg)
X
ut exp [-Wf/2)2/(4~pkTg)1 for ut
ut exp (-muf/2kTg)
for ut > fiu,;
(7b)
H. Iwasaki, 5’. Nakamurajlon
current generation
in FM.
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591
VELOCITYCOMPONENT V, , Vt
Fig. 2. Distributions of supplied particles, Nr,&,) and N&t) for helium, at a field strength of 4.5 V/A. The value of up eq. (2c), is 0.38 eV . The ratios of the maximum values Nn(300”)/Nn(200) and Nt(300”)/Nt(20°) are 0.017 and 0.034, respectively. Values of the thermal velocity, uth are shown by the arrows.
where So = n(kTg/2nm)V2,
(7c)
and So is the supply function in the absence of the electric field and a is the solid angle. The distributions, Nn(un) and N,(u,), for helium, at Ft = 4.5 V/A are depicted in fig. 2. The thermal velocity, uth = (~~g~m)~2 is also indicated in fig. 2. The numerical value of velocity is expressed by the value of the reduced velocity defined as (,t1~/2)l/~ in eV112. The incident particle characterized by p near pm,(u) will be accelerated by the dipole attraction force to have a large value of ut. As can be seen from fig. 2, a large fraction of particles arrive at the tip surface with large tangential velocities. Differentiating Nt(ut) with respect to ut, we obtain the most probable tangential velocity, lIti: utm = (2/m)“2 (kTgEp)v4.
(8)
We obtain the supply function S by integration as follows.
S= ~du~‘“’ 0
0
= SoI(nE,/kTg)v2
dpW,p)
= 1 du,N,(v,) 0
erf [(Ep/kTg)1’2] + exp (-Ep/kTg)}.
This is the formula for S as derived by Southon [3].
(9)
592
H. Iwasaki,
S. Nakarnurallorl
current generation
in FM.
I
3. Ion current generation Now we introduce the collision matrix following Van Eckelen [5] : particles that have hit the surface with velocity (uA,u;) rebound with a velocity distribution W(u,,vJ = b(u,,t+,u~,u;). The collision matrix is derived on the basis of classical “hard cube model” [6] (see Appendix). A particle that left the surface with velocity (u,,uJ has a radial kinetic energy E(r) at distance r from the center of the tip as follows: E(r) = mu;(r)/2
= ,u;
+ (mu;/2)
[ 1 - (I$/#]
~~Ep [ 1 - (I$/Y)~].
(10)
A particle will escape if its kinetic energy E(r) is positive for any I and otherwise it will return to the tip and hit the surface again. There exists the minimum value u,, of u, at which E(r) has no zeros and so a particle can escape: U nc
=
u&l - mL#4Ep).
(11)
Then, an incident particle with initial velocity u will be eventually tra ped if it loses kinetic energy, by collision with the surface, more than mu212 (mu,rl/2)*/4Ep instead of mu2/2. When a particle, which has velocity (un, ut) at the surface, passes through the ionization zone, it is ionized with the probability Q(u,,uJ. The probability Q is given by Q(u,J+)
=
1 - exp [-~(u~,~J/~I,
(12)
where Rt+Zc+d Qpq
=
J Rt+Zc
d,. Iu,(a)l
’
(13)
and T is the ionization liFetime of a particle in the ionization zone which is Z, above the surface and whose depth is d. In eq. (13) the u,(v) are given by eq. (2a) and by eq. (10) for a newly arriving particle from a field free region and for a rebounding particle, respectively. We use the formula for 7 given by Gomer [2] : 7 = v-l exp [0.68 (I -~ @)(I - 7.6Fi(2)1/2/Ft],
(14)
with the ionization energy I and the work function Cpin eV, and Ft in V/A; v is the orbital frequency of the tunneling electron in the gas atom. Particles which are captured at first impact may leave the tip region during their many hops by either thermal activation or field ionization. The procedure is repeated by a computer till the number of particles N(u,p) converges to a very small fraction for each incidence characterized by u and p. The total ion current is obtained by summing up the numbers of generated ions. It is impracticable to perform the whole process discussed above by a computer and we simplify the rebound process as follows. It is assumed that a fraction P, of rebounding particles that have hit the surface with velocity (uk,u;) will escape with
H. Iwasaki, S. Nakamurajlon
current generation
in FIM. I
593
velocity (LJne,ut) and the remainder of them go on trajectories returning to the surface with velocity (u,,Q, where P,, une and u, are defined as follows. Pe =
s
~(U,,>~Jt,+J;)
Usa)
du,,
Unc U
ne
=
s
u, b(vn,vt,Uh,v;>dvnlP,,
(15b)
“nc
“nc
urn =
_I v, b(v,,v,,u;,u;>dv,/(l
-P,>.
(15c)
0
4. Results and discussion The total ion currents of helium from tungsten are calculated by repeating the hopping 200 times for each incidence on a NEAC 700 computer, where R, = 5008, a=0.205Aj,Z, =3.5&d=0.38,~=2.4X 1016sec-1,1= 24.6eV and@=4.5eV Except for a tip at a temperature higher than 300 K, almost the same are assumed. results are obtained for different choices of the collision matrix, b and bLs (see Appendix). The tip temperature TS has effects on the total ion current through the collision matrix. Only a fraction of lop7 of the particles are supplied to the hopping states that do not reach the ionization zone, after 65 hops when Ft = 4.5 V/A and TS = 80K. For TS = 20 K, the fraction increases to 10e3 and only 45 hops are needed to decrease the hopping height below Z,. The dynamic equilibrium between the gas ionization probability and the probability of escape without ionization from the tip region by thermal activation may play an important role in determining the amount of the ion current [7]. The number of particles that had velocity u, when very far from the tip and arrive at a unit tip surface in unit time, N(v, T.J is given by N(v,T& = Jda
Tax(“)
dpN(u,p),(4nR:).
(16)
0
The fraction of N(v, T ), which escape without ionization after the first impact or in a few hops, is called R!e(u, TS). The temperature dependence of both quantities may be noticed. Some results of K,(v, Ts) for the collision matrix b, together with ZV(u,T ), are depicted in figs. 3a, 3b and 3c. Figs. 3a and 3b show that incident par. larger initial velocity than about 0.20 eVY* can almost escape from the ticles a avmg tip region after a few collisions. It is also shown that the probability of escape, K, decreases as field strength increases (from fig. 3b) and the mass ratio of the gas particle to the metal atom, ~1increases (see fig. 3~). It can be seen from fig. 3c that the trapped fraction of the total incident particles increase as the temperature of the gas in the field free region becomes low. This
594
H. iwasaki, S. Nakamura/lon
current generation in FIM. I
1.0
0.1
0.2 VELOCITYV (.V""
0.3 1
(a)
80 K
I
I
I
I
0.2
0‘1
VELOCITYV (eV"'
0.3 f
(b)
0.1
0.2
0.3
VELOCITY V
(eV 1'2 )
CC)
0.4
OS5
H. Iwasaki,
S. Nakamurallon
current generation
in FM
I
595
shows clearly the large dependence of the ion current on the gas temperature and the importance of keeping the average gas temperature low by the metal electrode in contact with the cold finger of the conventional FIM design [ 1,8]. The purely dynamical approach for calculating the total ion current which is shown in section 3 may be performed by, e.g., the Monte Carlo method. However, the simplification employed by using eqs. (15a), (15 b) and (1%) ullderest~nates the probability of escape of particles after multiple collisions with the surface. So, the calculation of the ion current by the simplified procedure is valid in relatively high field region and high tip temperature region. The behaviour of the ion current for the entire field and temperature range will be studied in the succeeding paper.
Appendix
The scattering of a gas atom from a solid surface is called thermal scattering when the kinetic energy of the gas atom is a few tenths of an eV [9] and this is the case in the usual condition of FIM. This scattering is well explained by the simple classical model which assume the solid atom as a cube and so the velocity component parallel to the surface is conserved [6]. We base the collision matrix on this model. The relation of the normal velocity components of the gas atom before and after the collision may be obtained from the velocity distribution of the free metal particles, assuming conservation of energy and momentum [5] : b(u,,U,,U:,,U;, = cqu,-u;,(u,+u’,) X exp i-M[(l
+dy,
-
(1-
p)vkl 2/(~kTs)l,
where M is the mass of the metal atom and C is the normal~ation mined by the following normalization condition: m
(A.11 constant
deter-
s s
du, b(u,,u,,u~,u;) = 1. 64.2) d”t -ca 0 When a gas atom is considered to have collided with the metal atom of effective collision speed as Logan and Stickney (61 supposed, the collision matrix is given by bL&J”‘utJ;,Q
= CLS S(br - u;)
x 12 Ph/ ](I + P)U, - (1 - /&I X exp I-Mbk](L
P2
+ P)U, - (1 - ~)u~l/(4k7’Sy)I
(A.3)
Fig. 3. The escape probability K, versus velocity of an incident particle when very far from the tip, U, for helium with a tungsten tip, (a) at a field strength of 4.5 V/A, (b) at a surface temperature of 80 K, (c) at F = 4.5 V/A and T, = 80 K, for P equal to ordinary value of 0.02177 and 2.5, together with N(u,7’&
H, Iwasaki, S. Nakamurajlon
596
current generation
ir? FIM. I
Both collision matrices give the average u, av as lJn av = #-
~)/(l +P) + ~WJJ/ tM~jl(l+ v)].
(A.4)
Logan and Stickney called the collision characterized by (A.4) a representative collision. As bLS(un,ut, uh,u;) shows very sharp peak at u,, = u,,, the results obtained by using this matrix are almost equal to those based on the representative collision. The average accommodation coefficient obtained from eq. (A-4) resembles that of a two-dimensional classical hard sphere collision.
References [ 1) E.W. Miiller, Advan.
Electron.
Electron
Phys. 13 (1960) (Harvard
[ 21 R. Gomer, Field Emission and Field Ionization
83. Univ. Press, Cambridge,
Mass., 1961). [ 31 M.J. Southon, Ph. D. Thesis, Univ. Cambridge (1963). [4] H. Iwasaki and S. Nakamura, Surface Sci. 52 (197.5) 597. IS] H.A.M. van Eekelen, Surface Sci. 21 (1970) 21. [6] R.M. Logan and R.E. Stickney, J. Chem. Phys. 44 (1966) 195. 171 T.T. Tsong and E.W. Miiller, J. Appl. Phys. 37 (1966) 3065. [8] F.W. Miiller, S. Nakamura, 0. Nishikawa and S.B. McLane, J. Appl. Phys. 36 (1965) 2496. (91 I:.O. Goodman, Surface Sci. 26 (1971) 364.
52 (1975) 588.. 596 Publishing Company
ION CURRENT GENERATION
IN THE FIELD ION MICROSCOPE
I. Dynamic approach Hiroshi IWASAKI and Shogo NAKAMURA Institute of Scientific and Industrial Research, Osaka University, Suita, Osaka, Japan 56.5 Received
18 March
1975; revised manuscript
received
11 August
1975
The numbers of gas particles arriving at unit tip surface in unit time from a field free region are derived as functions of velocity components for a spherical tip. It is shown that a considerable fraction of the gas particles arrives at the tip having large tangential velocities. The simple model of collision of a particle with a metal surface is used and the trajectories and rebounds of particles are tracked. The principal method to calculate the total ion current is shown. The capture probability of particles by the dipole attraction potential is shown to increase when the tip temperature is lowered, the field strength is increased, the mass ratio of the gas atom to the metal atom increases and the gas temperature is lowered.
1. Introduction The ion current generation in the field ion microscope (FIM) has been studied by Miiller [ 11, Comer [2], and Southon [3] dynamically. In the treatises, the total gas supply to the field-ion emitter surface is found. The rate constants for ionization and escaping from the tip region without ionization are calculated and finally the total ion current is obtained. Realistically, the supply of captured atoms must be used instead of the total supply as Southon [3] has indicated. Both rate constants and the probability of capture [3] are quantities averaged over the velocity distribution of the particles at the tip region in equilibrium and so are the functionals of the distribution function [4]. In this paper, instead of calculating the averaged quantities such as effective ionization rate on the basis of an assumed, somewhat ambiguous velocity distribution, it is shown that the ion current can be calculated thoroughly in a dynamical situation. The incident and rebound trajectories of all particles are tracked and the ionized fractions generated in passing through the ionization zone are summed up. The simplification that the particles scattered from an emitter surface have an average velocity is employed. It enabled us to easily get information on the influence of the many variables such as the tip temperature and the gas temperature, in the field
H, Iwasaki, S. Nakamura/Ion
current generation in FM.
range where particles are ionized in a few hops. In the succeeding paper [4], the velocity distribution region is derived by the quasi-static approach.
I
589
of the particles at the tip
2. The supply function In analyzing the hopping process of gas particles, it will be assumed for simplicity that the emitter is spherical. Then, the magnitude of the electric field F is given by F(r) = F,(R JG2,
(1)
where Ft is the electric field at the tip surface, r is the distance to the center of the tip and R, the tip radius. A particle approaching the tip with u, its velocity when very far from the tip, and p, the distance of closest approach to the tip center if the electric field were zero, has radial and tangential velocities urr(r) and q(r). These are found by assuming the conse~ation of energy and angular momentum as follows: u,(r) = u[ 1 - (j~+‘r)~+ (u~/u)~(R~/~)~]“~,
@a)
q’)
@)
= up/r,
where mu;/2
= Ep = oF32,
(2c)
and LJ is dipole attraction velocity, Ep is the polarization energy of a gas atom, ar is its po Parizability and m its mass. The number of gas particles that pass through a certain plane far from the tip in unit time with u between u and utdu and p between p and p+dp is called N(v,p) du dp.
Fig. 1. A trajectory of a gas particle from a field free region to the tip surface; un and ut are the velocity components at the tip surface.
H. Iwasaki, S. Nakamurallon
590
As the velocity distribution
current generation
in FIM I
is Maxwellian far from the tip, N(u,p) du dp is given by
N(u,p) du dp = n(m/27rkTJ
3/2 u3 exp(-mu2/2kT,J
27~~dp du,
(3)
where n and Tg are the density and the temperature of the ambient gas, respectively, and k is the Boltzmann constant, From eqs. (2a) and (2b) P dp = -(R,/v)~
(4a)
u, du,,
(4b)
P dp = (R,/u)2 ut dut,
where u, and ut are equal to un(Rt) and u,(Rt), respectively. The numbers of particles that approach the tip from a given direction with u between u and u+du and hit the tip surface in unit time with un between u, and u,tdu, and with I+ between ut and ut+du, are called N(u,u,) du du, and N(u,uJ du du,, respectively. We obtain from eq. (3) and eqs. (4a) and (4b) du du, = n(m/2nkTg)3’2 u exp(--mu2/2kTg)
2rrRF un dudu,,
(5a)
N(u,ut) du du, = n(m/2nkTg) 3’2 u exp (-mu2/2kTg)
21rRfu, du du,.
(5b)
N(u,u,)
For a given value of u, a maximum value pm,(u) of p, at which the particle will reach the tip, exists [5]. So, there are minimum values urnin and urnin of u at which the particle will hit the tip surface with a given u, and ut, respectively: Umin(un) =
up - un
for u,
(LIZ- LJy
forun>up,
L
(64
for ut
for ut >JzuP.
(6b)
The numbers of gas particles that hit the unit tip surface in unit time with u, between u, and u,+du, and with ut between ut and q+dq are called N,(u,) du, and N&u,) dut, respectively. We obtain from eqs. (5a) and (Sb) and eqs. (6a) and (6b) N&J
= sdfl
= So(m/kTg)
duN(u,u,)/(4nR:)
7 Urnin X
u, exp [-m(u,
- u,)2/2kT,J
exp (Ep/kTg) un exp (-mui/2kTg)
N#J,) = jdaj
for un < up, for un > up;
(7a)
dWv,)/(4~~,2)
Umin(ut) = So(m/kTg)
X
ut exp [-Wf/2)2/(4~pkTg)1 for ut
ut exp (-muf/2kTg)
for ut > fiu,;
(7b)
H. Iwasaki, 5’. Nakamurajlon
current generation
in FM.
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591
VELOCITYCOMPONENT V, , Vt
Fig. 2. Distributions of supplied particles, Nr,&,) and N&t) for helium, at a field strength of 4.5 V/A. The value of up eq. (2c), is 0.38 eV . The ratios of the maximum values Nn(300”)/Nn(200) and Nt(300”)/Nt(20°) are 0.017 and 0.034, respectively. Values of the thermal velocity, uth are shown by the arrows.
where So = n(kTg/2nm)V2,
(7c)
and So is the supply function in the absence of the electric field and a is the solid angle. The distributions, Nn(un) and N,(u,), for helium, at Ft = 4.5 V/A are depicted in fig. 2. The thermal velocity, uth = (~~g~m)~2 is also indicated in fig. 2. The numerical value of velocity is expressed by the value of the reduced velocity defined as (,t1~/2)l/~ in eV112. The incident particle characterized by p near pm,(u) will be accelerated by the dipole attraction force to have a large value of ut. As can be seen from fig. 2, a large fraction of particles arrive at the tip surface with large tangential velocities. Differentiating Nt(ut) with respect to ut, we obtain the most probable tangential velocity, lIti: utm = (2/m)“2 (kTgEp)v4.
(8)
We obtain the supply function S by integration as follows.
S= ~du~‘“’ 0
0
= SoI(nE,/kTg)v2
dpW,p)
= 1 du,N,(v,) 0
erf [(Ep/kTg)1’2] + exp (-Ep/kTg)}.
This is the formula for S as derived by Southon [3].
(9)
592
H. Iwasaki,
S. Nakarnurallorl
current generation
in FM.
I
3. Ion current generation Now we introduce the collision matrix following Van Eckelen [5] : particles that have hit the surface with velocity (uA,u;) rebound with a velocity distribution W(u,,vJ = b(u,,t+,u~,u;). The collision matrix is derived on the basis of classical “hard cube model” [6] (see Appendix). A particle that left the surface with velocity (u,,uJ has a radial kinetic energy E(r) at distance r from the center of the tip as follows: E(r) = mu;(r)/2
= ,u;
+ (mu;/2)
[ 1 - (I$/#]
~~Ep [ 1 - (I$/Y)~].
(10)
A particle will escape if its kinetic energy E(r) is positive for any I and otherwise it will return to the tip and hit the surface again. There exists the minimum value u,, of u, at which E(r) has no zeros and so a particle can escape: U nc
=
u&l - mL#4Ep).
(11)
Then, an incident particle with initial velocity u will be eventually tra ped if it loses kinetic energy, by collision with the surface, more than mu212 (mu,rl/2)*/4Ep instead of mu2/2. When a particle, which has velocity (un, ut) at the surface, passes through the ionization zone, it is ionized with the probability Q(u,,uJ. The probability Q is given by Q(u,J+)
=
1 - exp [-~(u~,~J/~I,
(12)
where Rt+Zc+d Qpq
=
J Rt+Zc
d,. Iu,(a)l
’
(13)
and T is the ionization liFetime of a particle in the ionization zone which is Z, above the surface and whose depth is d. In eq. (13) the u,(v) are given by eq. (2a) and by eq. (10) for a newly arriving particle from a field free region and for a rebounding particle, respectively. We use the formula for 7 given by Gomer [2] : 7 = v-l exp [0.68 (I -~ @)(I - 7.6Fi(2)1/2/Ft],
(14)
with the ionization energy I and the work function Cpin eV, and Ft in V/A; v is the orbital frequency of the tunneling electron in the gas atom. Particles which are captured at first impact may leave the tip region during their many hops by either thermal activation or field ionization. The procedure is repeated by a computer till the number of particles N(u,p) converges to a very small fraction for each incidence characterized by u and p. The total ion current is obtained by summing up the numbers of generated ions. It is impracticable to perform the whole process discussed above by a computer and we simplify the rebound process as follows. It is assumed that a fraction P, of rebounding particles that have hit the surface with velocity (uk,u;) will escape with
H. Iwasaki, S. Nakamurajlon
current generation
in FIM. I
593
velocity (LJne,ut) and the remainder of them go on trajectories returning to the surface with velocity (u,,Q, where P,, une and u, are defined as follows. Pe =
s
~(U,,>~Jt,+J;)
Usa)
du,,
Unc U
ne
=
s
u, b(vn,vt,Uh,v;>dvnlP,,
(15b)
“nc
“nc
urn =
_I v, b(v,,v,,u;,u;>dv,/(l
-P,>.
(15c)
0
4. Results and discussion The total ion currents of helium from tungsten are calculated by repeating the hopping 200 times for each incidence on a NEAC 700 computer, where R, = 5008, a=0.205Aj,Z, =3.5&d=0.38,~=2.4X 1016sec-1,1= 24.6eV and@=4.5eV Except for a tip at a temperature higher than 300 K, almost the same are assumed. results are obtained for different choices of the collision matrix, b and bLs (see Appendix). The tip temperature TS has effects on the total ion current through the collision matrix. Only a fraction of lop7 of the particles are supplied to the hopping states that do not reach the ionization zone, after 65 hops when Ft = 4.5 V/A and TS = 80K. For TS = 20 K, the fraction increases to 10e3 and only 45 hops are needed to decrease the hopping height below Z,. The dynamic equilibrium between the gas ionization probability and the probability of escape without ionization from the tip region by thermal activation may play an important role in determining the amount of the ion current [7]. The number of particles that had velocity u, when very far from the tip and arrive at a unit tip surface in unit time, N(v, T.J is given by N(v,T& = Jda
Tax(“)
dpN(u,p),(4nR:).
(16)
0
The fraction of N(v, T ), which escape without ionization after the first impact or in a few hops, is called R!e(u, TS). The temperature dependence of both quantities may be noticed. Some results of K,(v, Ts) for the collision matrix b, together with ZV(u,T ), are depicted in figs. 3a, 3b and 3c. Figs. 3a and 3b show that incident par. larger initial velocity than about 0.20 eVY* can almost escape from the ticles a avmg tip region after a few collisions. It is also shown that the probability of escape, K, decreases as field strength increases (from fig. 3b) and the mass ratio of the gas particle to the metal atom, ~1increases (see fig. 3~). It can be seen from fig. 3c that the trapped fraction of the total incident particles increase as the temperature of the gas in the field free region becomes low. This
594
H. iwasaki, S. Nakamura/lon
current generation in FIM. I
1.0
0.1
0.2 VELOCITYV (.V""
0.3 1
(a)
80 K
I
I
I
I
0.2
0‘1
VELOCITYV (eV"'
0.3 f
(b)
0.1
0.2
0.3
VELOCITY V
(eV 1'2 )
CC)
0.4
OS5
H. Iwasaki,
S. Nakamurallon
current generation
in FM
I
595
shows clearly the large dependence of the ion current on the gas temperature and the importance of keeping the average gas temperature low by the metal electrode in contact with the cold finger of the conventional FIM design [ 1,8]. The purely dynamical approach for calculating the total ion current which is shown in section 3 may be performed by, e.g., the Monte Carlo method. However, the simplification employed by using eqs. (15a), (15 b) and (1%) ullderest~nates the probability of escape of particles after multiple collisions with the surface. So, the calculation of the ion current by the simplified procedure is valid in relatively high field region and high tip temperature region. The behaviour of the ion current for the entire field and temperature range will be studied in the succeeding paper.
Appendix
The scattering of a gas atom from a solid surface is called thermal scattering when the kinetic energy of the gas atom is a few tenths of an eV [9] and this is the case in the usual condition of FIM. This scattering is well explained by the simple classical model which assume the solid atom as a cube and so the velocity component parallel to the surface is conserved [6]. We base the collision matrix on this model. The relation of the normal velocity components of the gas atom before and after the collision may be obtained from the velocity distribution of the free metal particles, assuming conservation of energy and momentum [5] : b(u,,U,,U:,,U;, = cqu,-u;,(u,+u’,) X exp i-M[(l
+dy,
-
(1-
p)vkl 2/(~kTs)l,
where M is the mass of the metal atom and C is the normal~ation mined by the following normalization condition: m
(A.11 constant
deter-
s s
du, b(u,,u,,u~,u;) = 1. 64.2) d”t -ca 0 When a gas atom is considered to have collided with the metal atom of effective collision speed as Logan and Stickney (61 supposed, the collision matrix is given by bL&J”‘utJ;,Q
= CLS S(br - u;)
x 12 Ph/ ](I + P)U, - (1 - /&I X exp I-Mbk](L
P2
+ P)U, - (1 - ~)u~l/(4k7’Sy)I
(A.3)
Fig. 3. The escape probability K, versus velocity of an incident particle when very far from the tip, U, for helium with a tungsten tip, (a) at a field strength of 4.5 V/A, (b) at a surface temperature of 80 K, (c) at F = 4.5 V/A and T, = 80 K, for P equal to ordinary value of 0.02177 and 2.5, together with N(u,7’&
H, Iwasaki, S. Nakamurajlon
596
current generation
ir? FIM. I
Both collision matrices give the average u, av as lJn av = #-
~)/(l +P) + ~WJJ/ tM~jl(l+ v)].
(A.4)
Logan and Stickney called the collision characterized by (A.4) a representative collision. As bLS(un,ut, uh,u;) shows very sharp peak at u,, = u,,, the results obtained by using this matrix are almost equal to those based on the representative collision. The average accommodation coefficient obtained from eq. (A-4) resembles that of a two-dimensional classical hard sphere collision.
References [ 1) E.W. Miiller, Advan.
Electron.
Electron
Phys. 13 (1960) (Harvard
[ 21 R. Gomer, Field Emission and Field Ionization
83. Univ. Press, Cambridge,
Mass., 1961). [ 31 M.J. Southon, Ph. D. Thesis, Univ. Cambridge (1963). [4] H. Iwasaki and S. Nakamura, Surface Sci. 52 (197.5) 597. IS] H.A.M. van Eekelen, Surface Sci. 21 (1970) 21. [6] R.M. Logan and R.E. Stickney, J. Chem. Phys. 44 (1966) 195. 171 T.T. Tsong and E.W. Miiller, J. Appl. Phys. 37 (1966) 3065. [8] F.W. Miiller, S. Nakamura, 0. Nishikawa and S.B. McLane, J. Appl. Phys. 36 (1965) 2496. (91 I:.O. Goodman, Surface Sci. 26 (1971) 364.