Local field and potential barrier in tunneling processes

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Applied Surface Science 94/95 (1996) 68-72

Local field and potential barrier in tunneling processes M.M. Mollicone, L.C.O. Dacal, C.M.C. de Castilho * Instituto de F[sica, Universidade Federal da Bahia, Campus Universit6rio da Federaq8o, 40210-340 Salvador, Bahia, Brazil Received 6 August 1995; accepted 5 September 1995

Abstract Ionisation rate-constants in conditions of imaging processes in field ion microscopy are calculated by considering a local electric field that varies along the potential barrier. The results are then compared with previous calculations where the field along the barrier is taken as constant.

1. Introduction The determination of tunnelling probability rates of electrons under the influence of strong electric fields is important when interpreting phenomena related to field emission [1], field ion microscopy [2] and neutralization of highly charged ions when approaching a surface [3]. An electron, with total energy E, can tunnel through a barrier confined between two regions of positive kinetic energy with finite probability. This probability is strongly dependent on the integral of [ V - E] 1/2 (which is a positive quantity inside the barrier) along the barrier, the limits of the integral being the points where the kinetic energy changes sign, usually called the "classical turning points". Frequently the shape of the barrier is simplified, in order to make the necessary calculations easier. However, the model for the potential barrier must reflect the physical situation properly since it has been shown [4] that the form of

the barrier, even for one-dimensional models, strongly affects tunnelling probabilities. In the field ion microscope (FIM) the tunnelling electron is transferred from an atom (usually a noble gas atom) to a positively charged sample. Most of the ionisation of the imaging atoms occurs at distances a few angstroms from the sample. To determine the ionization rate constant (ionization probability per unit of time) as a function of the atom's position, an analytical expression can be obtained or a numerical calculation can be performed, with more approximations in the former case. However, numerical calculations [5-8] usually assume a constant field along the potential barrier. This is clearly unsatisfactory, at least when the ionization takes place a few ~mgstrSms from the surface, since the field intensity varies significantly in this region. This constant field assumed is either the field value at the atomic nucleus position [5-7] or an estimated average field between the classical turning points [8]. On an atomic scale a surface presents protrusions (kinks, steps, adsorbed atoms, etc.) of the order of about 2 A, whereas important tunnelling processes occur within distances of about two or three times o

* Corresponding author. Tel.: +55 71 247 2033; fax: +55 71 235 5592; e-mail: [email protected].

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M.M. Mollicone et al. /Applied Surface Science 94 / 95 (1996) 68-72

69

this value. The distance between the classical tuming points of the potential barrier is of much the same size. So, if we model the field dependence on position in order to represent, even roughly, the tip protrusions, it seems to be important to take into account the field variation along the barrier. The aim of this work is to evaluate how important the inclusion of a variable field along the potential barrier is, with particular reference to the tunnelling conditions in the FIM. To do this it is essential to properly model the surface geometry.

2. Calculation We have performed numerical calculations for the ionization rate constant adopting the same procedure presented in Ref. [5], where the tunnelling rate-constant p was given, in atomic units, by

p=A2ufz=ff/2 4,=0 o = o

Xexp(-23/2fr'(°'¢)[V(r,.,ro(O,~o) 0 ,

qg)-E]

× s i n 0 dO d~o,

1/2 d r )

(1)

where A2v is a prefactor, V(r, O, q~) is the potential energy between ro( O, q~) and r~( O, q~) which are the classical turning points. In Ref. [5] the potential energy, V(r, O, q~), was given by:

V( r, O, q~) = - ( Z / r ) - Fr

cos 0,

(2)

where F is the electric field, considered as constant along the potential barrier and taken as its value at the nucleus position, Z is an effective charge and 0 = 7r is the field direction. The difference between the present calculation and the ones in Refs. [5-7] is that here the value of the electric field is variable along the potential bartier or, in other words, V(r, 0, q~) is taken as its " t r u e " value for the assumed geometry of the surface, whereas previously the field along the barrier was kept constant for a fixed position of the atom or ion. The local enhancement of the electric field is considered as the most important factor in explaining the resolution and contrast of the FIM. This has been stressed from interpretation of experimental data [9] and also from theoretical calculations [8,10].

\

Fig. 1. Different possibilities for the geometry of the sample: (a) hyperboloid with a superimposed hemisphere; (b) detail of (a); (c) hemisphere superimposed on a plane.

For modelling the surface and, as a consequence, the field dependence with position, we have considered two aspects: (i) the general shape of the plane or curved surface, and (ii) the presence or absence of atomic size protrusions on the surface. We combine these two aspects in order to get four cases. The adopted overall curved surface is a hyperboloidal one. The electric potential for a positively charged surface, to which a zero potential is associated, is given by [1 1]:

V= -Fo{(Rt/2 )

ln[(R t + 2 r cos 05)Rt] },

(3)

where F 0 is the field at a point infinitely close to the top of the surface apex of the hyperbola, R t is the apex radius, and r and 05 are as shown in Fig. 1, (a) and (b). The electric potential for a plane surface positively charged is obviously given by:

V= -FoZ,

(4)

which is also the limit of Eq. (3) when R t tends to infinity. Modelling a small protrusion by considering a hemisphere superimposed on a smooth plane surface leads to:

V= -Fo[r

cos 0 5 - 0 3 cos

05/r 2]

(5)

for a half sphere of radius a on a plane and 05 as shown in Fig. l(c). When a hemisphere is superimposed on the very top of an hyperboloid, and adopting a nomenclature

70

M.M. Mollicone et al. / Applied Surface Science 94 / 95 (1996) 68-72 1.80E-7 --

compatible with the previous equations, the potential is given by [12]: V = -Fo{(Rt/2

) l n [ ( R t + 2 r cos ~b)/Rt]

1,20E-7 -

- a 3 cos (9/r2}.

(6)

AS is well known, ionisation of the imaging gas atom in the FIM occurs beyond a minimum distance from the sample. This distance, known as the critical distance [2], is sometimes related to other parameters of the imaging atom and sample by: eFx c = 1 - d p - e2/16ZreoXc + 0 . 5 F 2 ( ofa - -

iJ 4.IX]E-.8 -

ai), (7)

where I is the ionisation potential, qb is the workfunction, e is the elementary positive charge, F (assumed as constant) is the electric field and a s are, respectively, the atomic and ionic polarizabilities. Eq. (7) is frequently simplified to: eFx c = I - qb.

~.

(9)

3. Results We have performed a numerical integration of Eq. (1) using different expressions for the potential energy V ( r , O, q~), corresponding to Eqs. (3), (4), (5) and (6). In this work we have considered helium as the imaging gas, which has an ionization potential of 24.58 eV and an electric polarizability of 2.29 × 10 -41 J m 2 V -2 (1.545 a.u.). The tip apex radius, in the case of a hyperboloidal sample, was taken as 600 and the hemisphere radius, when used, as 1.58 A. The work-function was taken as 4.47 eV. The prefactor A2v [13,14] was taken as: A2~, = 1 / [ 1 2 7 r e 2 ] .

~

I

I

1

I

'

I

D i l t ~ x ~ From C,eemr of l-lilf Spi~efe (A) F i g . 2. l o n i s a t i o n r a t e - c o n s t a n t in t h e c a s e o f t h e s a m p l e h a v i n g a smooth hyperboloidal

shape and the field along the barrier consid-

as: (a) constant; (b) variable. The tip apex radius is 600 and the ionisation rate is in (atomic unit of time)- ~.

ered

(8)

In our calculation, coherently with the importance we have given to the local field, the critical distance was determined from considering the adequate equipotential surface compatible with the energy level for the electron transition to the Fermi level inside the sample, i.e. [XCeF( x ' ) d x ' = I ~0

0.~=,.0

(10)

This value does not strongly depend on field [14] and is frequently taken as 1.0 [15]. This is not an important difference but just a scale factor for a fixed species of the imaging gas atom. Fig. 2 shows the ionisation rate per unit of time (ionisation rate-constant) as a function of distance from the tip apex in the case of a smooth hyperboloid, showing the two cases for the field intensity along the barrier: (a) constant field and (b) variable field. As a result of the relatively slow variation of the field intensity with distance from the tip, the difference between the two cases is relatively small. Obviously, there is no difference when we consider the sample as a plane surface, since in this case the field intensity is constant everywhere. The presence of a protrusion, even on a plane, causes a significant change in the situation, as compared with the previous one. This is shown in Fig. 3. In Fig. 4 the general shape of the tip is hyperboloidal and a half-sphere is superimposed on it. What is observed in Fig. 3 is now even more pronounced. We can define a width, w, for the peak of the ionisation rate curve as a function of distance, as the distance between points where p is a maximum and half of this value (see Fig. 5). Table 1 shows values of w calculated for the cases of Figs. 2-4. The increase in the values of w, when we compare values

M.M. Mollicone et al. / Applied Surface Science 9 4 / 9 5 (1996) 68-72

71

6.(X)E-8 --

i

I I

4,00E..8

s I

I

I

!

b

I

w

2.O0E..0 -

Dist=~ ~

tip ~

(1)

Fig. 5. Defmition of a width for the peak as the distance between points where the ionisation rate-constant P is maximum and half-maximum. 0.00E+O

I 10

'

[ 20

'

I 30

'

I 40

Di=tllCe From Center of HJf Spllefe (k)

6.011

Fig. 3. lonisation rate constant in the case of a planar sample's shape plus a hemisphere superimposed on it. The field along the barrier taken as: (a) constant; (b) variable. The hemisphere radius is 1.58 A and the ionisation rate is in (atomic unit of time) ~.

for constant field with the ones for a variable field, results from the way the critical distance was calculated. In both cases the critical distance was calculated using Eq. (9) and consideration of a constant

|a O 4.00

5.00E-6

3.00

4.00E-6

~

3,s 1;; 8

i

4.0

I

~

I

~

4.5 s.o Local F.JoMcField (VI.~)

F

5.5

Fig. 6. Critical distance, in Itngstr6ms, as a function of F 0 (in V/,~) and three different criteria: (a) as given by Eq. (7), without the polarizability term; (b) as given by Eq. (8); (c) as given by Eq. (9).

3.00E-6

O

| 2.0OE-6

field, which would be coherent with curves a in Figs. 2 - 4 , affects even strongly the values of w. This can be inferred from Fig. 6 where the variation of the

1.00E~

O.00E+O

I ' I I I 10 20 30 40 Distance From Cente¢ of Haft Sphere (~)

i 50

Fig. 4. lonisation rate constant in the case of the sample having a smooth hyperboloidal shape plus a hemisphere superimposed on its apex. The field along the barrier was considered as: (a) constant (b) variable. The tip apex radius is 600 A,, the half-sphere radius is 1.58 /~ and the ionisation rate is in (atomic unit of time)- t.

Table 1 Model for the sample

Smooth hyperboloid Plane + half-sphere Hyperboloid + half-sphere

Width of the peak (A) Constant field

Variable field

11.5 0.36 0.34

11.6 0.46 0.44

72

M.M. Mollicone et al./ Applied Surface Science 9 4 / 9 5 (1996) 68-72

critical distance (calculated by different procedures) with the value of F 0 is shown.

Acknowledgements Financial support from CNPq, CADCT (Brazil) is gratefully acknowledged.

4. Discussion and conclusion The importance of the local field on the imaging process of the FIM has been pointed out in the literature by several authors [2,7-9]. One aspect that perhaps is not yet clear is whether the local enhancement is mainly a result of the geometry and electronic structure of the sample or whether it is necessarily related to imaging gas adsorbed atoms. Previous numerical calculations and analytical expressions for the ionization rate-constant assume a constant field along the potential barrier. Previous numerical results have, at most, included the local variation of the field in so far as it is related to the atom's (or ion's) nucleus position. The field variation along the barrier, as far as we known, was not included in any previous determination of the ionisation rate-constant. We have shown that this significantly affects the ionization rate-constant very close to the emitter. As a direct consequence of changing the ionisation rate-constant, other probabilities [7] related to ionisation and post-ionisation of atoms or ions in the FIM and AP-FIM are also affected.

References [1] R.H. Fowler and L. Nordheim, Proc. R. Soc. London A 119 (1928) 173. [2] E.W. Miiller and T.T. Tsong, in: Field Ion Microscopy, Principles and Applications (Elsevier, New York, 1969). [3] R. Morgenstem and J. Das, Europhys. News 25 (1994) 3. [4] R. Haydock and D.R. Kingham, J. Phys. B (At. Mol. Phys.) 14 (1981) 385. [5] R. Haydock and D.R. Kingham, Surf. Sci. 103 (1981) 239. [6] S.C. Lain and R.J. Needs, Surf. Sci. 277 (1992) 359. [7] C.M.C. de Castilho and D.R. Kingham, Surf, Sci. 173 (1986) 75. [8] R.G. Forbes, J. Phys. D (Appl. Phys.) 18 (1985) 973. [9] Y. Suchorski, W.A. Schmidt and J.H. Block, Appl. Surf. Sci. 76/77 (1994) 101. [10] C.M.C. de Castilho, PhD Thesis, University of Cambridge (1986). [11] P.J. Birdseye and D.A. Smith, Surf. Sci. 23 (1970) 198. [12] C.M.C. de Castilho and D.R. Kingham, J. de Phys. (Paris) C2, Suppl. 3, 47 (1986) 23. [13] D.R. Kingham, Surf. Sci. 116 (1982) 273. [14] S.C. Lam and R.J. Needs, Surf. Sci. 177 (1992) 359. [15] S.C. Larn and R.J. Needs, Appl. Surf. Sci. 76/77 (1994) 61.