Image charge method for electrostatic calculations in field-emission diodes
G. Mesa, E. Dobado-Fuentes,a) and J. J. Sáenzb)
Departamento de Fı́sica de la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid,
Spain
~Received 11 July 1995; accepted for publication 5 September 1995!
We present a method to calculate the electrostatic field between a metallic tip of arbitrary shape and
a sample surface. The basic idea is to replace the electrodes by a set of ‘‘image’’ charges. These
charges are adjusted in order to fit the boundary conditions on the surfaces. As an application of the
method, we describe the field characteristics of a field-emission diode as a function of the gap
between electrodes for different tip shapes. A comparison between numerical and analytical results
is presented. The results do not depend on the overall tip geometry only for gap distances smaller
than '1/2 the tip radius. The field enhancement factor due to the presence of small protrusions on
the tip apex is calculated and their influence in near-field-emission scanning tunneling microscopy
is also discussed. We show that the electron-field emission from the sample is stable against
tip-shape changes due to adsorbate diffusion or atomic rearrangements. © 1996 American Institute
of Physics. @S0021-8979~95!02524-8#
I. INTRODUCTION
The electric field generated by sharp tips is the most
critical parameter governing electron-field emission and
closely related phenomena. Field electron and ion
microscopes,1,2 field-emission electron guns,3 thin-film fieldemission cathodes,4 and many devices based on vacuum field
emission are found throughout the sciences. The advances in
engineering of ultrasharp tips at atomic scale5,6 generated
renewed interest in this field. The remarkable properties of
the electron beam generated from these nanometer-sized
sources depends critically on the electric field around the tip
apex.7–11
Additional motivation for our work arises from the combination of field-emission devices with scanning tunneling
microscope ~STM! technology.12–16 It has been recently
shown that ‘‘near-field-emission STM’’17 provides a direct,
noninvasive approach for investigating at nanometer scale.
An early example of such approach was Young’s
‘‘topografiner.’’18 When a STM is operated in the near-fieldemission regime ~like a topografiner!, the electric-field
strength at the emitter surface determines the I vs V characteristics through the Fowler–Nordheim equation.2,19 Operating the instrument in the constant current mode implies an
approximately constant field at the emitter surface. By solving the Laplace’s equation we can then calculate the field
strength everywhere between the tip and the sample, and
thus determine the relationship between the tip–sample distance S and the emitter voltage V.18
Most of the predictions of the electric-field shape and
amplitude in the tip–sample gap used so far are based on
simple analytical models or sophisticated numerical calculations. Moreover, theory an experiments on field-emission
systems have been mainly focused in the far field regime,
i.e., when a tip cathode is placed at a macroscopic distance
a!
Also at Instituto de Ciencia de Materiales, Sede B, C.S.I.C., Facultad de
Ciencias C-III, Madrid, Spain.
b!
Author to whom correspondence should be addressed; Electronic mail:
juanjo@dune.fmc.uam.es
J. Appl. Phys. 79 (1), 1 January 1996
from the sample ~anode! surface. The aim of this work is to
present a method to calculate the electrostatic field between a
metallic tip of arbitrary ~axial symmetric! shape and a
sample surface for any tip–surface distance. The basic idea is
to replace the electrodes by a set of ‘‘image’’ charges. These
charges are adjusted by means of a standard least-squares
method in order to fit the boundary conditions on the surfaces. The method is much simpler and faster than standard
methods used to solve the Laplace equation. It is also rather
flexible and can be applied to many different problems ranging from the design and characterization of field-emission
guns to the study of field effects in STM experiments.20,21 As
a particular application, in this article we focus on the field
characteristics of a field-emission diode as a function of the
gap between electrodes for different tip shapes. The field
enhancement effect due to the presence of a small protrusion
on the tip apex is also analyzed. We show that some conclusions obtained from the standard analytical approximations
are wrong and a careful calculation of the electric field is
needed to interpret correctly the experimental data.
In Sec. II we present our image-charge method and discuss the details of the field calculation. In Sec. III we present
the numerical results for smooth tips together with a comparison with the most common analytical approaches. The
presence of a small protusion on the tip apex modifies drastically the field characteristics both in the far-field and in the
near-field emission regime. This is discussed in Sec. IV. In
Sec. V we study the influence of tip-shape changes in the
performance of a near-field-emission STM.17 The concluding
remarks are given in Sec. VI.
II. IMAGE-CHARGE METHOD
The calculation of the electric field in a field-emission
device is far from being a trivial problem. One of the main
difficulties in the computation is to handle accurately the
large change in geometrical scale between the tip radius and
the tip to sample spacing. The situation is even worse when
the tip has not a smooth shape, for example, due to the presence of small nanometer-scale protrusions.
0021-8979/96/79(1)/39/6/$6.00
© 1996 American Institute of Physics
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V cn ~ r ,z;z n ! 5
V sm ~ r ,z;z m !
5
FIG. 1. Scheme of the ‘‘image’’ charge method. The tip is replaced by a set
of points and segments of charge. Each charge has its corresponding image
on the other side of the flat conducting surface at V50. The tip surface is
described by a set of discret points. The charges are adjusted to fit the
potential V5V 0 at these points. The inset shows the equipotential lines
around a small bump of radius R b placed on the tip apex.
qn
4pe0
1
Ar 2 1 ~ z2z n ! 2
,
S
~2!
D
1
qm
~ z2z m 1L m ! 1 Ar 2 1 ~ z2z m 1L m ! 2
ln
.
4 p e 0 2L m
~ z2z m 2L m ! 1 Ar 2 1 ~ z2z m 2L m ! 2
The position z n and strength q n of each point charge and the
length 2L m , the position of the center z m , and the total
charge, q m of each segment are the unknowns of the problem. These parameters are adjusted in order to fit the boundary condition on the tip surface, V5V 0 @notice that the condition on the flat surface, V50, is already included in Eq.
~1!#.
To describe the geometry of the tip we discretize its
surface taking a sample of P constant potential points,
r j 5( r j ,z j ) ( j51, P@N,M ), so that we have a higher density in those places with lower curvature radius. In this way
we avoid a deformation of equipotential curves on the apex
of the tip. The best values for L i , z i , and q i are obtained
after minimization of
P
In the past decades many methods for calculating the
electrostatic field in field-emission systems have been
developed.3 Among these, effective charge-density
methods22–24 are especially appropriate for the calculation of
capacitances or the computation of trajectories in electrostatic focusing.22–24,10 With a clever choice of the charge
distribution on the surface it is even possible to simulate the
electric field at atomic scale.9
From a general point of view, the charge-density methods emphasize the superposition principle and the selfconsistency of the boundary conditions,24 e.g., the charges on
the conductor are calculated by requiring that the potential be
a constant on each conductor surface, in contrast with the
finite-difference/-element approach in which the potential at
each point is obtained as the average of the potentials at the
adjacent points. Following the same basic ideas, we propose
a new method specially suitable to calculate the electrostatic
field between a metallic tip of arbitrary shape and a sample
surface for any tip–surface distance. Making use of the classical electrostatic image method, we replace the electrodes
by a set of image charges. The charge strength and position
are then adjusted in order to fit the boundary conditions on
the surfaces.
For simplicity, let us discuss in some detail the case of a
sharp tip with axial symmetry in front of a grounded ~V50!
flat conducting surface ~see Fig. 1!. First we place, along the
symmetry axis, a set of segments and discrete charges as
well as their electrostatical images. The electrostatic potential at a point ~r,z! would be
x 25
( @ V ~ p j ,z j ! 2V 0 # 2 .
j51
~3!
We use a nonlinear least-squares minimization routine25 to
get the best ensemble of parameters that adjust the potential
to the shape of the tip. After convergence, the equipotential
fits almost perfectly the original surface ~see Fig. 1!.
The main advantage of the method is that it is possible to
describe the potential over very different scales ~from the
nanometer to the centimeter range! with a small number of
parameters.26 In the case of a sharp tip with a small protrusion, 3– 4 charges and 5–10 segments are enough to get an
almost perfect fit for any tip–surface distance.
III. FIELD CHARACTERISTICS FOR SMOOTH TIPS
For standard field emitters the tip acquires a smooth,
almost hemispheroidal shape. The electric field F at the tip
apex is usually written as F5V/(kR) where R is the radius
of curvature of the tip and k is a geometric factor2 @it is also
common to define a b factor, b51/(kR)#. This k factor not
only depends on the tip shape, but also on the overall geometry of the system. In particular, for a tip in front of a flat
conducting surface, k also depends on the tip–surface distance.
We have applied the image-charge method to study the
k-factor behavior for different tip geometries as a function of
the gap distance between electrodes. First, however, we give
a brief presentation of the results of standard analytical approximations.
N
V ~ r ,z ! 5
(
n51
A. Analytical approximation
@ V cn ~ r ,z;z n ! 2V cn ~ r ,z;2z n !#
M
1
(
m51
@ V sm ~ r ,z;z m ! 2V sm ~ r ,z;2z m !# ,
~1!
where the sums runs over N charges and M segments and
40
J. Appl. Phys., Vol. 79, No. 1, 1 January 1996
Probably the most simple approach to find the k factor is
to consider the tip as an hyperboloid in front of a flat surface
and use the prolate spheroidal coordinate system ~h,j,w! to
obtain solutions of the Laplace equation.27 In this system,
surfaces of constant h represent equipotential surfaces and
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FIG. 2. k t factor vs the gap distance in units of the tip radius of curvature
S/R for different tip geometries. Since k t is proportional to the applied
voltage, this curves also correspond to the voltage-distance characteristics in
a constant current ~field! near-field-emission experiment ~electrons are emitted from the tip!.
lines of constant j and w represent electric field lines. The tip
and sample surfaces are an hyperboloid h5h0 and a flat
surface h50, respectively. The radius of curvature of the tip
apex R and the tip–sample distance S determine the choice
of h0 , h05@S/(S1R)#1/2. The electrostatic potential, satisfying the boundary condition V5V 0 at the surface of the hyperboloid and V50 at the flat surface, is given by
V ~ h , j , w ! 5V ~ h ! 5V 0
ln@~ 11 h ! / ~ 12 h !#
.
ln@~ 11 h 0 ! / ~ 12 h 0 !#
~4!
The field at the tip apex is then given by
F t5
F
1
2h0
ln
S DG
11 h 0
12 h 0
21
S1R
V0 .
SR
~5!
These equations give directly the dependence of the k factors
with the tip–sample distance S. The results for k t 5V 0 /(F t R)
as a function of S/R are shown in Fig. 2. Since k is proportional to the applied voltage, the k vs S plot gives the dependence of the applied voltage with the tip–sample distance in
a constant current ~constant field! near-field-emission experiment. k t (S) would correspond to the case of negative tip
polarity, i.e., electron field emission from the tip.
In the far field regime (S@R) the k t factor takes the
simple form2
k t'
S D
4S
1
ln
.
2
R
~6!
This would give a straight line in an experimental V 0 – S plot
when the tip–sample distance is plotted logarithmically in
units of R. The straight line could then be extrapolated to
zero emitter voltage and the corresponding distance would
be 0.25R.28 On the other hand, when the tip–sample disJ. Appl. Phys., Vol. 79, No. 1, 1 January 1996
tance S is a small fraction of the radius, the emitter voltage is
a linear function of S and the slope of the V 0 – S plot gives
directly the value of the electric field.
This nice method to determine both the field strength
and the tip radius was in fact proposed by Young and
co-workers18 in relation with the topografiner; however, as
we see, it turns out that some of the results obtained from
this approach are wrong. In particular, for S.R, the exact
results for real hyperboloids strongly deviate from the analytical approximation. The reason for that behavior can be
found in the construction of hyperboloidal surfaces from the
prolate spheroidal coordinate system. In this system, once
the distance S and the curvature R are fixed, they completely
determine the opening angle u of the hyperboloid. This implies that for a given R, a change in S changes the global
shape of the tip. Although this is not very important for small
distances, it leads to clear deviations for larger ones.
In Fig. 2 we have also plotted the results for the sphere–
plane system. Notice that this approach gives essentially the
same behavior as the hyperboloid for distances S,0.5R.
This indicates that in this range the electric field does not
depend on the overall shape of the tip ~this is confirmed by
the exact calculations below!. Of course, as the distance increases the k t factor for a sphere saturates to 1 as expected
from its definition.
B. Numerical results
Although the analytical approximations are currently
used in model calculations of field-emission characteristics,
it is not clear up to what extend these approaches give a good
description of the electric fields. In Fig. 2 we present calculated k factors for different tip shapes together with the analytical results discussed above. For simplicity we have included only two general geometries: hyperboloid and a
truncated cone with an spherical apex ~see Fig. 2!. In these
simple cases the tip geometry is characterized by the curvature radius R at the tip apex and the asymptotic half-angle of
the macroscopic tip u. For a given curvature radius and for
tip–sample distances S,0.5R the field on the tip apex does
not depend on the shape of the tip. It is then possible to use
the simplest approaches to study near-field-emission problems; however, as the tip sample distance is in the order or
larger than the tip radius the behaviour of the field depends
critically on the tip geometry. This has considerable interest
since, as discussed above, it is believed you can get information on the tip radius from the slope of the V 0 – S plot at large
distances.18 Our results ~Fig. 2! show that this is not an easy
task and care must be taken in analyzing these curves. As an
example, in Fig. 2 we show the range of estimated radius
obtained from different tip shapes having angles between 10°
and 20°. They differ in more than a 50% for tips having the
same R.
IV. FIELD ENHANCEMENT EFFECTS OF SMALL
PROTRUSIONS
The presence of a small protrusion on the tip apex leads
to a local field enhancement ~lower k t ! due to the distortion
and compression of equipotentials in its vicinity.29,7,9 If
Mesa, Dobado-Fuentes, and Sáenz
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FIG. 3. k t factor for a hyperbolic tip ~u510°! with a small protrusion placed
on the tip apex. Results for different ratios R b /R as well as the results
without protrusion are also shown. The inset shows the field enhancement
F b /F t due to the protrusions as a function of the gap distance.
present on the tip apex, most of the field-emission current
would arise from such protrusions. The method of producing
such protruding ~‘‘teton’’! tips in a controlled way as well as
the study of their peculiar ~far-field! emission characteristics
have been discussed in the last few years.6 –11 On the other
hand, atomic-size bumps can also appear spontaneously as a
consequence of adsorption or diffusion processes.
For a hemispherical bump of radius R b on the apex of a
blunt tip (R b !R) the k factor can be shown29 to be 1/3 that
of the blunt tip (F b 53F t ); however, the field enhancement
factor depends, in general, on the tip-protrusion geometry.9
Moreover, it must also depend on the gap distance between
electrodes. In Fig. 3 we have plotted k t vs S for a hyperboloid ~u510°! with different hemispherical bumps ~radius
R b 50.1R and 0.05R! on the apex. As expected, the presence of any small bump on the tip surface drastically modify
the field characteristics at any tip–sample distance. The effect of the different ratios R b /R on the field enhancement,
F b /F t , is shown in the inset of Fig. 3. As it can be seen, only
for distances larger than R, F b /F t takes an approximately
constant value which approaches 3 for very small ratios
R b /R. It is worth noticing that an extrapolation to zero emitter voltage in the k t – S plot is independent on the presence of
the protrusion on the tip apex.
V. TIP GEOMETRY EFFECTS IN NEAR-FIELDEMISSION STM
In this section we discuss the influence of the tip geometry in near-field-emission STM.17 Attempts of using the
field emitted current from a tip as a parameter for generating
images were done by Young and co-workers;18 however, the
field emission of electrons from a sharp tip is not exempt
from drawbacks. The high energy of the electrons impinging
the sample could modify it.30 In addition, adsorbates that
42
J. Appl. Phys., Vol. 79, No. 1, 1 January 1996
FIG. 4. k factor for a hyperbolic tip ~u510°! corresponding to the emission
from the tip (k t ) and from sample (k s ). Continuous lines are the results of
the analytical hyperboloidal approximation. Solid symbols correspond to a
smooth tip. Open symbols reflect the changes in k due to the appearance of
a small hemisferical protrusion (R b 50.05R) on the tip apex. The inset
shows the influence of a bump in the equipotential lines for a distance of
0.6R. Thin lines correspond to the smooth tip. As it can be seen, the equipotentials close to the sample are almost unaffected by the change of the tip
apex geometry.
may appear on the tip would produce instabilities in the
emission characteristics. This is due to changes in the work
function and, as we have seen, in the field close to the protrusion; however, the changes in the field decay very fast
with the distance from the protrusion ~at least in the far-field
regime29,7!. It has been argued17 that they have a negligible
effect on the field at the sample surface. As we see, the
electric-field calculations show that this is indeed the case.
The instabilities can then be avoided if the polarity is reversed, i.e., if the electrons are field emitted from the sample.
As we have discussed above the k vs S plot gives the
dependence of the applied voltage with the tip–sample distance in a constant current experiment. When the electrons
are emitted from the sample we can define a k s factor,
k s 5V 0 /(F s R), where F s is the field on the sample surface
~just below the tip apex!. In the analytical approximation, F s
is simply given by
F s5
R
F ,
S1R t
~7!
where F t is given by Eq. ~5!. We have calculated k s for the
same geometries as in Fig. 2. As a general result we find that,
for distances S,R, there is no difference between the different geometries and the analytical approach gives a good description of the field. As an example, in Fig. 4 we present a
comparison between the k t ~solid triangles! and k s ~solid
circles! factors for an hyperboloidal tip ~u510°! as a function of S. The results of the analytical approach are also
shown ~solid lines!. For distances S!R the voltage drops
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linearly within the tip–sample gap and both k factors are
linear with S. At distances of the order of '0.2R, k already
deviates from the linear behavior.
The presence of a small bump on the tip ~due to adsorbate diffusion, atomic rearrangements, etc.! can produce a
dramatic change in the field characteristics. In Fig. 4 we
present the changes in k due to the sudden appearance of a
small protrusion of radius R b 50.05R. This has mainly two
effects. First, the distance between the tip and sample is
slightly reduced leading to a small increase in the field. At
the same time, the field is strongly enhanced in the vicinity
of the bump while it is almost unaffected the sample surface
~see inset in Fig. 4!. This effects have important consequences in the performance of the STM in the near-fieldemission regime. In a STM experiment in the constant current ~field! mode at fixed applied voltage, the feedback loop
would try to keep the field constant by changing the tip–
sample distance. Let us assume a typical STM tip radius of,
let say, '10 nm and a working distance of '3 nm.17 When
electrons are field emitted from the tip, the appearance of an
atomic-size bump ~R b '0.5 nm! on the tip apex would lead
the tip to retract more than 10 nm in order to keep constant
field and voltage ~i.e., constant k t !. On the other hand, the
emission from the sample is almost unaffected.
VI. CONCLUSIONS
We have presented an image-charge method to calculate
the electrostatic field between a metallic tip of arbitrary
shape and a sample surface. As an application of the method,
we have described the field characteristics of a field-emission
diode as a function of the gap between electrodes for different tip shapes. We have shown that the electrostatic field
generated by a tip close to a flat surface only depends on the
radius of curvature of the tip apex. In particular, for small
distances, the analytical hyperboloidal approximation is in
agreement with the numerical results. However, for tip–
surface distances S larger than '0.5R the results depend on
the overall shape of the tip and the analytical approach can
not longer be applied. From the extrapolation to zero voltage
of the experimental V –log(S) plots at constant current ~field!
it is possible to estimate the tip radius of curvature but with
an uncertainty of the order of '50%.
We have also studied the effects due to the presence of a
small protrusion on the tip apex. Interestingly, we find that
they do not modify the extrapolated radius. We have shown
that the protrusion-induced field enhancement increases with
the tip–surface distance up to S'R. For larger distances the
field enhancement remains approximately constant.
The influence of small bumps in near-field-emission
STM characteristics have been also discussed. We have
shown that the field on the sample surface is almost unaffected by the presence of a small protrusion on the tip apex.
As a consequence the electron emission from the sample is
stable against tip–shape changes due to adsorbate diffusion
or atomic rearrangements.
The image-charge method is rather flexible and it can be
applied to study many different problems such as analysis of
electron or ion trajectories, calculation of effective tunneling
barriers, field-induced diffusion in STM experiments, etc.
J. Appl. Phys., Vol. 79, No. 1, 1 January 1996
Together with a surface charge approach9 it could be used to
simulate the electric field for ‘‘real’’ tips including both macroscopic and atomic scale geometries. The application of the
method to study capacitances and electrostatic forces in scanning probe microscopy is in progress.
ACKNOWLEDGMENTS
We gratefully acknowledge stimulating discussions with
R. Garcı́a, J. Gómez, J. Mendez, and J. M. Soler. We also
gratefully acknowledge support from the DGICYT under
Contract No. PB92-0081.
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See, for example, W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.
Flannery, Numerical Recipes. The Art of Scientific Computing ~Cambridge
University Press, Cambridge, 1992!, Chap. 15.
26
For a given tip geometry, we have found that the positions and lengths of
the effective charges are almost independent on the tip–surface distance.
Since the potential is a linear function of the charges, after a first nonlinear
minimization ~or after an appropriate guess of positions and lengths! the
problem becomes a simple linear-least-squares fitting. In this case, the
potential can be obtained with a neglegible computational effort.
27
E. W. Müller, Z. Phys. 108, 668 ~1938!; R. Haefer, ibid. 116, 604 ~1940!;
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1
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28
This idea was already proposed by Young and co-workers @Rev. Sci. Inst.
43, 999 ~1972!#. They found a value of 0.33R instead of the right one of
0.25R. The reason for that discrepancy is that they did not extrapolate
from the real straight line portion of the curve as can be seen by comparing their Fig. 15 with our Fig. 2.
44
J. Appl. Phys., Vol. 79, No. 1, 1 January 1996
29
30
D. J. Rose, J. Appl. Phys. 27, 215 ~1956!.
Atomic-scale desorption induced by electrons field emitted from a STM
tip has been reported recently by T.-C. Shen, C. Wang, G. C. Abeln, J. R.
Tucker, J. W. Lyding, Ph. Avouris, and R. E. Walkup, Science 268, 1590
~1995!.
Mesa, Dobado-Fuentes, and Sáenz
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