untitled
Coulomb blockade of field emission from nanoscale conductors
O. E. Raichev*
Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Prospekt Nauki 45, 03028, Kiev, Ukraine
�Received 9 February 2006�
Theoretical description of the field emission of electrons from nanoscale objects weakly coupled to the
cathode is presented. It is shown that the field-emission current increases in a steplike fashion due to single-
electron charging which leads to abrupt changes of the effective electric field responsible for the field emission.
A detailed consideration of the current-voltage characteristics is carried out for a nanocluster modeled by a
metallic spherical particle in the close vicinity of the cathode and for a cylindrical silicon nanowire grown on
the cathode surface.
PACS number�s�: 79.70.�q, 73.23.Hk, 73.40.Gk
I. INTRODUCTION
The discrete nature of electric charge reveals itself in the
transport of electrons through small conductors �nanopar-
ticles or other nanoscale objects� weakly coupled to the
source and drain electrodes �current-carrying leads� owing to
the Coulomb blockade effect. Numerous manifestations of
the charge quantization in transport properties, the most fa-
miliar of them are the Coulomb blockade oscillations of the
electric current as a function of the gate voltage and the
Coulomb staircase in the current-voltage characteristics,
have attracted considerable attention in the past years.1 Since
the fundamentals of the transport theory in the Coulomb
blockade regime have been established,2–4 the Coulomb
blockade-based physics has been applied to various issues of
electron transport in mesoscopic systems, and the field of its
applications expands in line with the advances in nanotech-
nology.
Usually, the influence of the Coulomb blockade on the
current in two-terminal devices is considered under assump-
tion that the coupling between the nanoscale object and the
leads is not sensitive to the number of electrons N determin-
ing the object charge eN. This corresponds to the introduc-
tion of ohmic �or nearly ohmic� effective resistances describ-
ing this coupling. Though this assumption often works well,
it can be violated, for example, in nanomechanical
systems,5–7 where charging of the object gives rise to its
displacement towards one of the leads thereby changing its
tunnel coupling to both leads. In this paper we study a situ-
ation when the sensitivity of the tunnel coupling to the num-
ber of electrons does not require a mechanical displacement
and is determined by the nature of tunneling. This implies a
device layout and conditions similar to those used in the
recent experiments on field emission of electrons from me-
tallic nanoclusters,8–10 silicon nanowires11–15 and
nanocones,16,17 and carbon nanotubes �see, for example,
Refs. 11 and 18–26�, when small �nanoscale� objects are
formed on the source electrode �cathode�, the latter is then
negatively biased with respect to the drain electrode �anode�
in vacuum. The current between the electrodes flows owing
to the field emission of electrons from nanoscale objects,
because the electric field F at the tips of the objects is higher
than in the other places of the device. The field-emission
current is described by the Fowler-Nordheim formula27
I = ASF2 exp�− F
�, F = 4�2m
3�e��
W3/2, �1�
where m is the free electron mass, W is the work function of
the emitting material, S is the effective emitting area, and A
is a constant expressed through the work function and Fermi
energy �F of the emitting material
�e�3��F/W
4�2 � ��F + W�
. �2�
The effective field F, which describes the tunnel coupling
between the nanoscale object and the anode, depends on the
object charge, which is induced by the applied voltage V
=V1−V2, where V1 and V2 are the cathode and anode poten-
tials, respectively. Under conditions of Coulomb blockade,
i.e., when the electric connection between the cathode and
the object is weak and the charging energy of the object
considerably exceeds the temperature T, the continuous
variation of the voltage V leads to discrete changes of the
object charge in units of e, and, consequently, to correspond-
ing discrete changes of the field F. Therefore, one may in-
troduce the field FN, which is a function of the discrete num-
ber N and continuous variable V. Next, if the current in the
device is limited by the field emission, the single-electron
tunneling processes become important. This means that, at a
fixed voltage V, the object stays mostly in the states with N
and N−1 electrons, the number N is determined by the volt-
age. In the N-electron state, no electrons can come to the
object from the cathode until an electron leaves the object by
tunneling through the barrier, see Fig. 1�a�. Then the object
appears in the N−1-electron state and returns to the
N-electron state before the next Fowler-Nordheim tunneling
event takes place. The field-emission current in these condi-
tions is given by Eq. �1� with F=FN and can be denoted as
IN. If the bias eV increases, the state with N+1 electrons
becomes more favorable, and the current changes in a step-
like fashion from IN to IN+1. This leads to staircaselike
current-voltage characteristics, which may look similar to the
usual Coulomb staircases.28–30 However, since the sensitivity
of the tunneling to the number of electrons is involved, the
staircaselike current-voltage characteristics can exist under
rather peculiar conditions, when the source-drain bias is or-
ders of magnitude larger than the charging energy.
The rest of the paper is devoted to quantitative studies
based on the physical idea outlined above. In Sec. II we give
the basic equations and calculate the current in the simplest
case of an idealized emitter shown in Fig. 1�b�. In Sec. III we
calculate the current from a nanocluster modeled by a spheri-
cal particle on the metallic cathode surface and from a semi-
conductor wire �nanowhisker� grown perpendicular to the
cathode surface. The discussion and concluding remarks are
given in the last section.
II. GENERAL CONSIDERATION
We consider the case of classical �or metallic� Coulomb
blockade, when the electron energy level separation in the
nanoscale object can be neglected in comparison to both
temperature and charging energy. Since the object is assumed
to be weakly coupled to the cathode, we study the sequential
tunneling process and not the coherent one. It is convenient
to investigate the electron transport by applying the kinetic
equation2 �Master equation� for the distribution function PN
describing the probability for the object to be in the state
with N electrons. Assuming that the electric connection be-
tween the cathode and the object is characterized by the con-
ductance G, this equation is written as
= QN+1 − QN, �3�
where
1 − exp�− �EN/T�
�PN − PN−1 exp�− �EN/T�	
+ PNIN/�e� . �4�
Here �EN= �e
2 /C��N−1/2−C2V /e	 is the difference in Cou-
lomb energies for the objects with N and N−1 electrons, C is
the total capacitance, and C2 is the capacitance of the object
with respect to the anode �the capacitance with respect to the
cathode is given by C1=C−C2�. The first term in expression
�4� has the usual form2 and corresponds to the current be-
tween the object and the cathode. It is written as a difference
of the contributions describing the departure of an electron
from the object in the N-electron state and arrival of an elec-
tron at the object in the N−1-electron state. The second term
corresponds to the field-emission current from the object in
the N-electron state. Since no electrons come to the object
from the anode, this term does not contain a contribution
describing arrival of electrons. In the stationary case, Eq. �3�
is reduced to the form QN=const, where the constant can be
chosen equal to zero. After determining PN from the equation
QN=0 with the use of the normalization condition 
NPN=1,
the total current is given by
PNIN. �5�
Under the condition GT� �e � IN, which means that the object
is in thermal equilibrium with the cathode, the stationary
solution of Eq. �3� is written as PN=Z
−1 exp�−EN /T�, where
EN= �e
2 /2C��N−C2V /e	
2 is the Coulomb energy, and Z
N exp�−EN /T� is the partition function. The current in this
case is determined by the expression
J = Z−1
IN exp�− EN/T� . �6�
Let us apply the solution �6� to the idealized model of
emitter, Fig. 1�b�, when the emission takes place from a
spherical nanoparticle of radius R, placed at a distance d
from the cathode. The distance between the cathode and an-
ode is L. The connection c-p denotes a low-transparent con-
tact �for example, tunnel barrier� between the particle and the
cathode, which does not contribute to the field-emission
properties and electrostatics of the device. Assuming d�R,
we have C=R, C2=Rd /L, and neglect the charge polariza-
tion of the particle because this polarization is small in com-
parison to the total charge eN induced by the applied voltage.
The number of electrons is estimated as N�C2V /e
=RdF0 / �e�, where F0=−V /L is the applied electric field. The
effective field for the nanoparticle with N electrons is FN
= �e �N /R2, and the partial currents IN in these conditions are
given by
IN = AS�eN/R
2�2exp�− FR2/�e�N� , �7�
where the emitting area S, in the idealized model considered
here, can be approximated by the total surface area of the
nanoparticle, S=4�R2. In Fig. 2 we plot the current-voltage
characteristics of the idealized emitter, calculated according
to Eqs. �6� and �7�, where A is given by Eq. �2� with W
=5.1 eV and �F=5.5 eV �taken for Au�, and the geometrical
parameters are chosen as R=5 nm and d=0.5 �m. The char-
acteristics look like staircases with flat regions �plateaus� be-
tween the steps, which are visible even at room temperature.
It is possible to estimate the relative heights of the steps by
calculating the ratio of the currents IN and IN−1 emitted from
the nanoparticle with N and N−1 electrons
� exp� FR2
�e�N�N − 1�
� . �8�
In spite of the fact that the charged nanoparticle typically
contains a large number of electrons, N
100, one can al-
FIG. 1. �a� The mechanism of single-electron tunneling in the
Fowler-Nordheim regime. �b� Schematic representation of the ide-
alized emitter.
ways find a regime when the ratio IN / IN−1 is not small in
comparison to unity. This necessarily implies a weak Fowler-
Nordheim tunneling, when F /F=FR2 / �e �N�1.
In the calculations described above, the applicability of
the Fowler-Nordheim formula requires R�W / �e �F, which is
rewritten as R�e2N /W, or, according to N�RdF0 / �e�, as
�e �F0�W /d, independent of the nanoparticle radius. This
condition is satisfied at high enough applied voltages. If
�e �F0=eV /L�W /d, the approximation of a triangular poten-
tial barrier is not quite good, and one should consider the
tunneling through the barrier described by the potential en-
ergy W−e2N�1/R−1/r� at r�R, where r is the distance
from the center of the spherical nanoparticle; the tunneling
through the potential barrier of this form is described in Ref.
31. Even under the condition �e �F0�W /d, which is satisfied
in the calculations shown in Fig. 2, the relative change of the
current per one step, IN / IN−1−1, appears to be significant,
because the exponent FR2 / �e �N�N−1� in Eq. �8� is estimated
as c�W / �e �F0d�
2, where the dimensionless constant c
=4/3�2me4 /�2W is noticeably larger than unity.
If the current is high enough, the field emission cannot
remain the bottleneck for the electron transfer from the cath-
ode to the anode, and a finite resistance G−1 becomes essen-
tial. The nanoparticle in these conditions is no longer in equi-
librium with the cathode. This means that the distribution PN
is established kinetically, and several states with different
charges coexist at a fixed voltage �see the inset in Fig. 2�. As
a consequence, the Coulomb blockade features are washed
out. This case requires a numerical solution of the equation
QN=0. The corresponding current-voltage characteristics of
the idealized emitter calculated by using the RC time C /G
=100 ps are also shown in Fig. 2. The degradation of the
current steps appears to be stronger with increasing voltage,
because the current increases and the nanoparticle-cathode
link becomes more important. The shape of the steps in this
case resembles the usual Coulomb staircase.
III. MORE COMPLEX EXAMPLES
After demonstrating the possibility of the Coulomb-
blockade staircase of the field emission on a model example,
it is worth to consider more complex cases. Indeed, the
model example discussed above has certain disadvantages.
First of all, it is hardly possible to connect a particle placed
far from the cathode surface by a link �c-p in Fig. 1�b�	
which does not contribute to the electrostatic properties of
the device. Second, the model of uniform charging is insuf-
ficient: the charge polarization of the nanoscale object ap-
pears to be important and should be always taken into ac-
count, see below in this section. Therefore, the model shown
in Fig. 1�b� is suitable only for the purposes of illustration of
the basic physics described by Eqs. �3�–�6�. To have a closer
approach to reality, we point out that the nanoscale objects
investigated in the above-cited experiments on field emission
can be roughly divided into two classes: the objects whose
dimensions in all directions are comparable �nanoclusters or
nanoparticles�, and the objects whose length in the direction
of the applied field is much larger than their transverse size
�nanowires or nanowhiskers�. The following consideration is
carried out for the cases of nanoclusters and nanowires of the
simplest geometries, when the electric fields FN and the ca-
pacitances C and C2 can be determined consistently by solv-
ing corresponding electrostatic problems. The current is cal-
culated according to Eq. �6�, under the assumption that the
objects are in equilibrium with the cathode.
A. Field emission from nanoclusters
Below we consider the field emission from a nanocluster
modeled by a spherical metallic particle of radius R depos-
ited on the flat cathode surface. To provide a finite capaci-
tance C, one should assume a finite separation d-R between
the particle and the metallic cathode plate �for instance, one
can imagine that the particle resides on an oxidized surface�,
see the inset to Fig. 3. Besides, this assumption provides
electrical isolation of the particle from the cathode, which is
FIG. 2. Current from the idealized emitter as a function of the
applied field F0=−V /L for the case of small C /G �nanoparticle in
thermal equilibrium with the cathode, upper curves� and for the
case of C /G=100 ps �lower curves�, at the temperatures T=77 K
�solid� and T=293 K �dashed�. The inset shows the distribution
function PN at F0=5	10
5 V/cm for the second case.
FIG. 3. Charge density per unit length in z direction for a spheri-
cal metallic nanocluster placed at the distance 0.1 R from the me-
tallic cathode. Here 
0=F0R /2. The inset shows the geometry of
the problem and the directions of the field emission �arrows�.
a necessary condition for the Coulomb blockade. The elec-
trostatics of the plane-sphere system is known, and the field
and charge distributions in this case can be found in the form
of rapidly converging infinite series arising from the poten-
tials of image point charges and point dipoles.32 Such a con-
sideration allows one to present the distribution of the elec-
trostatic potential energy near the particle in the approximate
U�r,�� � W + e��F0R�1 + 
�cos � − 1�	
− ��eN − C2V	/C�
r − R
, �9�
where r and � are the radial and azimuthal coordinates of the
spherical coordinate system with the origin at the center of
the particle, and �, 
, and � are the dimensionless constants
of the order of unity, which are to be determined from nu-
merical calculations. Such calculations also give us the ca-
pacitances C and C2.
33 Note that if the charge quantization is
neglected �so that N=C2V /e when the particle is in equilib-
rium with the cathode�, � is identified with the field enhance-
ment factor conventionally used in the physics of field emis-
sion. The expression �9� provides an excellent description of
the electrostatic potential at r−R�R /2 and at small �. It
allows one to take into account deviations of the potential
energy from the linear form W− �e �F�r−R� and, therefore, to
find corrections to the Fowler-Nordheim tunneling exponent.
Neglecting such corrections in the prefactor, we obtain the
following expression for the partial currents
IN = ASFN
2 exp�− F
�� �e�FNR
��x� =
� x2�x − 1��2 − arctan �x − 1� − x� , �10�
where A is given by Eq. �2�, the dimensionless function ��x�
describes the corrections to the tunneling exponent, and the
effective emitting area S=2�R2�FN
2 /
F�F0�
�2�R2�FN /
F� is reduced due to the angular dependence of
the radial field described by Eq. �9�. The field FN is given by
FN = �F0 + �
�e�N − C̃2F0
, �11�
where the quantity C̃2=C2L does not depend on the distance
L between the cathode and anode. Note that, since we always
assume that L is much larger than any dimension of the
nanoscale object, the capacitance C2 is always proportional
to 1 /L, and it is more convenient to replace C2 �V� by C̃2F0.
This substitution also allows us to represent the Coulomb
energy standing in Eq. �6� as
�N − C̃2F0/�e�	
2. �12�
Further calculations are done for the separation d−R
=0.1R, when C=2.16R, C̃2=1.74R
2, �=4.32, �=1.22, and

=0.66. Figure 3 shows the distribution of negative charges
on the surface of the spherical particle staying in equilibrium
with the cathode for this case ��e �N= C̃2F0 is assumed�. The
distribution of the radial field F�z� at the surface of the par-
ticle is given by the same dependence, F�z� /F0=
�z� /
The field-emission current from the nanocluster described
above has been calculated according to Eqs. �6� and �10�–
�12� at R=5 nm. The results of the calculations shown in Fig.
4 demonstrate the staircaselike behavior caused by the Cou-
lomb blockade. However, in contrast to the staircases shown
in Fig. 2, the current continues to increase between the steps.
This occurs because of electrostatic polarization of the nano-
particle. According to Eq. �11�, when the particle charge is
constant, the increase in the applied field F0 leads to an in-
crease in the effective field FN because the factor �
−�C̃2 /CR is positive. For the chosen particle radius, the
steps of the current are clearly visible at liquid nitrogen tem-
perature but poorly visible at room temperature. Neverthe-
less, the Coulomb blockade features at room temperature be-
come quite distinct in the plots of the derivative of the
current, as shown in the inset to Fig. 4.
B. Field emission from nanowires
Let us consider the field emission from a small semicon-
ductor wire modelled by a cylinder of radius R and length d,
which ends with a hemispherical tip of the same radius, see
the inset to Fig. 5. The cathode substrate upon which the
wire is grown is assumed to be a metal �or a heavily doped
semiconductor� so that one can use the method of image
charges instead of solving the electrostatic problem in the
whole space. The electric isolation of the wire from the cath-
ode in this case takes place in a natural way, because a
Schottky barrier is formed between the wire and the metallic
cathode �in the case of semiconducting cathode there can be
a heterobarrier or an interband p-n barrier�. In other words,
FIG. 4. Current from the spherical nanocluster of radius R
=5 nm as a function of the applied field F0=−V /L at T=4.2 K
�solid� and 77 K �dashed�. The inset shows the derivative of the
current at T=293 K.
the wire region adjacent to the cathode becomes depleted of
electrons and positively charged because of the presence
of donors �we assume that the wire is uniformly doped
with bulk donor density nD�. When a bias eV is applied
between the cathode and anode, the wire acquires a con-
siderable negative charge because of tunneling or thermi-
onic emission of electrons from the cathode through the
barrier. When the field emission from the wire of nan-
oscale radius becomes essential, the density of induced
negative charges per unit length of the wire appears to be
much larger than the equilibrium charge density 
=�R2 �e �nD even if nD is of the order of 10
18 cm−3. For
this reason, one can use the “metallic” approximation
assuming that the charges in the wire are placed mostly on its
surface. This means that the electron density distribution
n�� ,z�, which depends on the radial coordinate � of the cy-
lindrical coordinate system connected with the wire, is given
by n�� ,z�= �2�� �e � �−1���−R�
�z�+nD for z�d and n�� ,z�
= �2�� �e � �−1���−�R2− �z−d�2	
�z�+nD for d�z�d+R,
where 
�z� is the density of negative charges on the surface
per unit length. Since this approximation is based on the
assumption that the screening length is small in comparison
with the wire radius, it works better for wider wires. For
silicon wires, whose field-emission properties are currently
the subject of investigations,11–15 the metallic approximation
remains suitable even for the radius of several nanometers,
because, owing to the large effective masses and six-valley
degeneracy, the density of electron states in n-Si appears to
be high enough to provide the Thomas-Fermi screening
length less than one nanometer for Fermi energies �F
�0.01 eV. The metallic approximation, of course, fails to
describe the region of the wire in the close vicinity of the
cathode, where the depletion occurs. Nevertheless, since this
region is a small part of the whole wire, see the charge dis-
tribution in Fig. 5, its presence cannot considerably modify
the parameters calculated as described below.
According to the discussion given here, we search for the
charge distribution 
�z� satisfying the integral equation
U�z� = U0 − �e�F0z + �
dz�K�z,z��
�z�� , �13�
where U�z� is the potential energy counted from the Fermi
level in the cathode material, U0 is the barrier height, and
K�z ,z�� is the potential of interaction between the electrons
in the points z and z� of the wire surface in the presence of
the cathode plate, see the Appendix. Equation �13� is accom-
panied with additional requirements: U�z�=0 at z�z0 and
�z�=−
D at z�z0, where z0 is the depletion edge coordi-
nate, which is to be found self-consistently. The first of these
requirements corresponds to a full screening of the bare po-
tential U0− �e �F0z by the induced charges of the wire, while
the second one models the presence of the positive charges in
the depletion region z�z0. Once the distribution 
�z� is
found, the total charge of the wire, −�0
d+Rdz
�z�, as well as
the distribution of electric field around the wire, can be cal-
culated. To find the capacitance C and describe modification
of the effective field under single-electron charging, one may
calculate the variation of the total charge and the field at the
tip �at z=d+R� with respect to a small variation of U0. Equa-
tion �13� is solved numerically by using the method of itera-
tions. The dependence of the effective field FN on F0 and N
can be represented in the form similar to Eq. �11�
FN = ��F0�F0 + �
�e��N + B� − C̃2F0
C�F0�R
, �14�
while the Coulomb energy is written as
2C�F0�
�N + B − C̃2F0/�e�	
2. �15�
These equations take into account a finite �though weak�
dependence of the capacitance C and field enhancement fac-
tor � on the applied field F0. The dependence of the param-
eters C̃2 and � on F0 appears to be much weaker and can be
neglected. The positive dimensionless constant B reflects the
fact that the average number of induced charges is smaller
than C̃2F0 / �e�. These features appear because the system un-
der consideration is not entirely metallic and contains a
depletion region whose length changes with F0.
The numerical calculations leading to the results pre-
sented below are done for U0=0.7 eV, which approximately
corresponds to the Schottky barrier height for n-Si in contact
with Al.34 The chosen donor density is nD=2	10
18 cm−3.
The parameters standing in Eqs. �14� and �15�, however, are
not sensitive to nD, except for the capacitance C, which
changes within 10% when nD varies from 10
18 cm−3 to 2
	1018 cm−3. Figure 5 shows the charge density distribu-
tion for the wire of radius R=5 nm and length d=0.1 �m
at F0=10
6 and 2	106 V/cm. The charge density shows
a nearly linear growth through the main part of the wire
and a sharp enhancement at the hemispherical tip from
which the field emission occurs. The dependence of the
field enhancement factor and capacitance on the applied
electric field is shown in Fig. 6, and the other calcu-
lated parameters are C̃2=2.44 dR, �=0.414, and B=12.14.
FIG. 5. Charge density per unit length for the cylindrical wire
whose geometry is shown in the inset �see parameters in the text�.
The plots of the field-emission current calculated with the
use of the parameters listed here are given in Fig. 7. The
calculations are done according to Eqs. �6�, �14�, and �15�,
and the Fowler-Nordheim formula for the partial current,
IN=ASFN
2 exp�−F /FN�. Since the calculated radial electric
field in the region of the tip weakly depends on z �in contrast
to the case of the nanocluster studied above� and sharply
decreases in the region of transition to the cylindrical part of
the wire, the effective emitting area S is estimated by the
total area of the hemispherical tip, S=2�R2. The work func-
tion is taken for silicon, W=4.2 eV. Next, the Fermi energy
standing in the expression for A, see Eq. �2�, is estimated
from the equation �F��e �FinrTF, where Fin��F0 /� is the
field inside the semiconductor near the end of the tip, rTF is
the Thomas-Fermi screening length, and � is the dielectric
constant of the semiconductor. Such an estimate, carried out
for n-Si, leads to �F�0.1 eV at F0�2	10
6 V/cm. The pic-
ture of Coulomb staircase shown in Fig. 7 is basically the
same as that in Fig. 4. Again, the increase of the current with
the applied field is determined by the increase of the effec-
tive field �14� due to both single-electron charging �steps�
and charge polarization under a constant charge �regions be-
tween the steps�. The main difference is that the interval of
the applied field needed for addition of one electron to the
wire is considerably reduced, owing to the larger capacitance
C2, and appears to be of about 1.2 V/�m �further reduction
of this interval takes place with the increase of the wire
length, see below�. Next, since the capacitance C increases
considerably in comparison to the case of nanocluster of the
same radius, the Coulomb blockade features at room tem-
perature are poorly visible even in the derivative plot, see the
inset. Nevertheless, these features remain pronounced at T
=77 K.
With the increase of the wire length d, the parameters
entering Eqs. �14� and �15� are modified as shown in Fig. 8.
The field enhancement factor and the capacitances increase
nearly in a linear way, while the parameter �, which charac-
terizes relative contribution of charging to the effective field,
slightly decreases �for comparison, the idealized emitter con-
sidered in the previous section is described by the parameters
�=d /R, �=1, C=R, and C̃2=dR, where d is the distance
from the cathode to the emitting sphere�. The increase of the
total capacitance C makes it difficult to observe the Coulomb
staircase in long wires. For example, at d=1 �m one should
have temperatures considerably lower than 77 K. The inter-
val of the applied field corresponding to the addition of one
electron is inversely proportional to C̃2. This interval de-
creases very fast with the increase of d and becomes equal to
2.5	102 V/cm at d=1 �m.
FIG. 6. Field dependence of the enhancement factor and capaci-
tance for the cylindrical wire with R=5 nm and d=0.1 �m.
FIG. 7. Current from the cylindrical wire of radius R=5 nm and
length 0.1 �m as a function of the applied field F0=−V /L at T
=4.2 K �solid� and 77 K �dashed�. The inset shows the derivative of
the current at T=293 K.
FIG. 8. Dependence of the parameters �, �, C, and C̃2 on the
length of the wire for R=5 nm and F0=5	10
5 V/cm.
IV. CONCLUSIONS
The key point of the presented theoretical study is the
possibility of noticeable modification of the effective electric
field causing the field emission from a nanoscale conductor
by addition of just one electron to this conductor. Formally,
this modification is described by introducing the effective
field FN, which determines the partial current IN, and by
evaluating the dependence of this field on the bias applied
between the cathode and anode, see Eqs. �11� and �14�. As a
result of this effect, the current-voltage characteristics of the
field emission show steps in the Coulomb blockade regime.
In other words, the steplike current-voltage characteristics
related to single-electron charging �Coulomb staircases� may
exist even under the conditions of field-emission experi-
ments, when the applied bias is orders of magnitude larger
than the charging energy. The steps on the current-voltage
characteristics can be visible at 77 K in the case of field
emission from nanoclusters and nanowires of 10 nm diam-
eter and submicron length. In the regions between the steps,
where the total charge of the nanoscale object is constant, the
current increases with the increase of the applied bias owing
to charge polarization.
The staircases described in this work are similar to the
usual Coulomb staircases obtained in the transport through
small metallic islands28–30 or quantum dots �see Ref. 35 for
review� with strong asymmetry in the barriers. In both cases,
each step of the current is associated with addition of an
electron to the nanoscale object, and the applied source-drain
voltage drops mostly across the low-transparent barrier �the
barrier between the object and the drain�. Therefore, the pe-
riodicity of the steps in both cases is determined by the
object-drain capacitance C2. However, the steps in the sec-
ond case are formed due to shifts of effective �N-dependent�
electrochemical potential of the object with respect to elec-
trochemical potentials of the source and drain. For this rea-
son, the usual Coulomb staircase shows well-defined steps
when C2 is greater than the object-source capacitance C1,
while in the opposite situation, C1�C2, the steps are sup-
pressed and the current-voltage characteristic approaches to a
linear dependence.29,30 In contrast, in the case described in
this work the steps are formed due to changes in the prob-
ability of Fowler-Nordheim tunneling from the object to the
drain �anode�. That is why the steps are clearly visible under
the condition C1�C2, which is imposed by the field-
emission layout considered in this paper. To summarize, the
sensitivity of the field emission to the number of electrons in
the nanoscale object makes it possible to obtain the Coulomb
staircases under the conditions when such staircases cannot
be observed in the transport through small metallic islands or
quantum dots.
The quantitative consideration has been applied here to
some simple models of the nanoscale objects, whose electro-
static properties necessary for description of the field en-
hancement and charging have been determined consistently.
Consequently, the number of geometrical parameters charac-
terizing the objects has been minimized. For example, the
nanowire has been characterized only by its length d and
radius R. In reality, the geometrical structure of objects is
more complicated. For example, their tips may contain sharp
regions which provide a more efficient field emission. In
fact, high field-emission currents from nanoscale objects are
typically observed at the applied fields of the order of
105 V/cm, which requires the field enhancement factors
much larger than those calculated in this paper. On the other
hand, the presence of sharp tips cannot strongly modify the
capacitances of the objects. The general picture of the single-
electron tunneling under the field-emission regime also re-
mains valid. For possible application to experiments, the
field enhancement due to charging can be described by equa-
tions of the kind of Eqs. �11� and �14�, where � and � should
be considered as parameters to be determined experimen-
tally.
At the present time, there is no experimental evidence of
the Coulomb staircase phenomenon under the field emission.
Though the current-voltage characteristics sometimes show
steplike features, see, for example, Ref. 11, these features are
not regular and, most probably, should be attributed to insta-
bilities of the emission process and burning out of the emit-
ting material. There are numerous reasons which make ob-
servation of the phenomena considered in this paper difficult.
First of all, in most cases the nanoscale objects on the cath-
ode surface form dense arrays. This means that the field
emission takes place from a macroscopic number of objects
which are electrostatically coupled. The charging and field-
emission properties appear to be considerably different36
from those of individual objects. The Coulomb blockade
phenomena in this case should be dramatically suppressed by
the size dispersion of the objects and by the effects of mutual
screening. Investigation of field emission from individual ob-
jects is possible in the cases of metallic nanoclusters8–10 and
carbon nanotubes.26 However, there exists the problem of
electric isolation of these objects from the cathode, which is
one of the necessary conditions for Coulomb blockade. No
special attempts to achieve such an isolation in the field-
emission experiments have been undertaken so far, except
for the nanomechanical system investigated in Ref. 7, where
the electron emission from an isolated Au island to a
submicron-sized electrode has been observed. Most of the
experiments on field emission are carried out at room tem-
perature, though existing experimental techniques also allow
measurements at liquid nitrogen temperature. This means
that the Coulomb blockade phenomena can be observed only
for small-sized objects whose capacitances are low enough
�see the results of Sec. III�. Besides, the interval of the ap-
plied field corresponding to addition of one electron strongly
decreases in the case of emission from long nanowires,
which requires high resolution with respect to field. In sum-
mary, a search for the Coulomb blockade features in the
field-emission current would require a special planning of
experiment. The author hopes that the presented theoretical
study will stimulate experimental investigations in this direc-
tion.
ACKNOWLEDGMENTS
The author is grateful to A. I. Klimovskaya for stimulat-
ing discussions.
APPENDIX: KERNEL OF EQUATION (13)
If z�d and z��d, K�z ,z��=K0�z ,z��−K0�−z ,z��, where
K0�z,z�� = �
��z − z��2 + 2R2�1 − cos ��
. �A1�
If z�d and z��d, K�z ,z��=K0�z ,z��−K0�−z ,z��, where
K0�z,z�� = �
��d − z�2 + 2R�d − z�cos �� + 2R2�1 − sin �� cos ��
. �A2�
If z�d and z��d, K�z ,z��=K0�z ,z��−K0�z ,−z��, where
K0�z,z�� = �
��d − z��2 + 2R�d − z��cos � + 2R2�1 − sin � cos ��
. �A3�
Finally, if z�d and z��d,
K�z,z�� = �
� �e��2R2�1 − cos � cos �� − sin � sin �� cos ����
�4d2 + 4dR�cos � + cos ��� + 2R2�1 + cos � cos �� − sin � sin �� cos ��� . �A4�
In Eqs. �A2�–�A4�, cos �= �z−d� /R and cos ��= �z�−d� /R, so � and �� are the azimuthal angles. The integrals are taken over
the polar angle �. The function K�z ,z�� is also representable in the form of full elliptic integrals.
*Electronic address: raichev@isp.kiev.ua
1 Single Charge Tunneling, edited by H. Grabert and M. H. De-
voret, NATO ASI Series B 294 �Plenum Press, New York, 1992�.
2 I. O. Kulik and R. I. Shekhter, Zh. Eksp. Teor. Fiz. 68, 623
�1975� �Sov. Phys. JETP 41, 308 �1975�	.
3 D. V. Averin and K. K. Likharev, J. Low Temp. Phys. 62, 345
�1986�; and in Mesoscopic Phenomena in Solids, edited by B. L.
Altshuler, P. A. Lee, and R. A. Webb �Elsevier, Amsterdam,
1991�.
4 C. W. J. Beenakker, Phys. Rev. B 44, 1646 �1991�.
5 L. Y. Gorelik, A. Isacsson, M. V. Voinova, B. Kasemo, R. I.
Shekhter, and M. Jonson, Phys. Rev. Lett. 80, 4526 �1998�.
6 A. Erbe, C. Weiss, W. Zwerger, and R. H. Blick, Phys. Rev. Lett.
87, 096106 �2001�.
7 D. V. Scheible, C. Weiss, J. P. Kotthaus, and R. H. Blick, Phys.
Rev. Lett. 93, 186801 �2004�.
8 M. E. Lin, R. P. Andres, and R. Reifenberger, Phys. Rev. Lett. 67,
477 �1991�.
9 M. E. Lin, R. Reifenberger, and R. P. Andres, Phys. Rev. B 46,
15490 �1992�.
10 M. E. Lin, R. Reifenberger, A. Ramachandra, and R. P. Andres,
Phys. Rev. B 46, 15498 �1992�.
11 C. S. Chang, S. Chattopadhyay, L. C. Chen, K. H. Chen, C. W.
Chen, Y. F. Chen, R. Collazo, and Z. Sitar, Phys. Rev. B 68,
125322 �2003�.
12 S. Johnson, A. Markwitz, M. Rudolphi, H. Baumann, S. P. Oei,
K. B. K. Teo, and W. I. Milne, Appl. Phys. Lett. 85, 3277
�2004�.
13 N. N. Kulkarni, J. Bae, C.-K. Shih, S. K. Stanley, S. S. Coffee,
and J. G. Ekerdt, Appl. Phys. Lett. 87, 213115 �2005�.
14 J. C. She, K. Zhao, S. Z. Deng, J. Chen, and N. S. Xu, Appl.
Phys. Lett. 87, 052105 �2005�.
15 J. C. She, S. Z. Deng, N. S. Xu, R. H. Yao, and J. Chen, Appl.
Phys. Lett. 88, 013112 �2006�.
16 Q. Wang, J. J. Li, Y. J. Ma, Z. L. Wang, P. Xu, C. Y. Shi, B. G.
Quan, S. L. Yue, and C. Z. Gu, Nanotechnology 16, 2919
�2005�.
17 Y. L. Chueh, L. J. Chou, S. L. Cheng, J. H. He, W. W. Wu, and L.
J. Chen, Appl. Phys. Lett. 86, 133112 �2005�.
18 A. G. Rinzler, J. H. Hafner, P. Nikolaev, L. Lou, S. G. Kim, D.
Tomanek, P. Nordlander, D. T. Colbert, and R. E. Smalley, Sci-
ence 269, 1550 �1995�.
19 P. G. Collins and A. Zettl, Appl. Phys. Lett. 69, 1969 �1996�.
20 Q. H. Wang, A. A. Setlur, J. M. Lauerhaas, J. Y. Dai, E. W. Seelig,
and R. P. H. Chang, Appl. Phys. Lett. 72, 2912 �1998�.
21 S. Fan, M. G. Chapline, N. R. Franklin, T. W. Tombler, A. M.
Cassell, and H. Dai, Science 283, 512 �1999�.
22 R. H. Baughman, A. A. Zakhidov, and W. A. de Heer, Science
297, 787 �2002�.
23 S. H. Jo, Y. Tu, Z. P. Huang, D. L. Carnahan, J. Y. Huang, D. Z.
Wang, and Z. F. Ren, Appl. Phys. Lett. 84, 413 �2004�.
24 M. Mauger, V. T. Binh, A. Levesque, and D. Guillot, Appl. Phys.
Lett. 85, 305 �2004�.
25 N. de Jonge, M. Allioux, M. Doytcheva, M. Kaiser, K. B. K. Teo,
R. G. Lacerda, and W. I. Milne, Appl. Phys. Lett. 85, 1607
�2004�.
26 Z. Xu, X. D. Bai, E. G. Wang, and Z. L. Wang, Appl. Phys. Lett.
87, 163106 �2005�.
27 R. H. Fowler and L. W. Nordheim, Proc. R. Soc. London, Ser. A
119, 173 �1928�.
28 J. B. Barner and S. T. Ruggiero, Phys. Rev. Lett. 59, 807 �1987�.
29 K. Mullen, E. Ben-Jacob, R. C. Jaklevic, and Z. Schuss, Phys.
Rev. B 37, 98 �1988�.
30 R. Wilkins, E. Ben-Jacob, and R. C. Jaklevic, Phys. Rev. Lett. 63,
801 �1989�.
31 L. D. Landau and E. M. Lifshitz, Quantum Mechanics �Perga-
mon, Oxford, 1977�.
32 W. R. Smythe, Static and Dynamic Electricity �McGraw-Hill,
New York, 1968�.
33 If d=R, the parameters are obtained analytically: �=7��3� /2
�4.21, 
=93��5� /56��3�−3/4, �=�2 /8, and C2= ��
2 /6�R2 /L,
but C is equal to infinity �C diverges in a logarithmic way as the
separation d-R goes to zero�.
34 Metal-Semiconductor Interfaces, edited by A. Hiraki �IOS Press,
Amsterdam, 1995�.
35 L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R.
M. Westervelt, and N. S. Wingreen, Electron Transport in Quan-
tum Dots, in Proceedings of the Advanced Study Institute on
Mesoscopic Electron Transport, edited by L. L. Sohn, L. P. Kou-
wenhoven, and G. Schön �Kluwer, 1997�.
36 T. A. Sedrakyan, E. G. Mishchenko, and M. E. Raikh, cond-mat/
0504042 �unpublished�.

  Theoretical description of the field emission of electrons from nanoscale
objects weakly coupled to the cathode is presented. It is shown that the field-
emission current increases in a step-like fashion due to single-electron
charging which leads to abrupt changes of the effective electric field
responsible for the field emission. A detailed consideration of the
current-voltage characteristics is carried out for a nanocluster modelled by a
metallic spherical particle in the close vicinity of the cathode and for a
cylindrical silicon nanowire grown on the cathode surface.

untitled
Coulomb blockade of field emission from nanoscale conductors
O. E. Raichev*
Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Prospekt Nauki 45, 03028, Kiev, Ukraine
�Received 9 February 2006�
Theoretical description of the field emission of electrons from nanoscale objects weakly coupled to the
cathode is presented. It is shown that the field-emission current increases in a steplike fashion due to single-
electron charging which leads to abrupt changes of the effective electric field responsible for the field emission.
A detailed consideration of the current-voltage characteristics is carried out for a nanocluster modeled by a
metallic spherical particle in the close vicinity of the cathode and for a cylindrical silicon nanowire grown on
the cathode surface.
PACS number�s�: 79.70.�q, 73.23.Hk, 73.40.Gk
I. INTRODUCTION
The discrete nature of electric charge reveals itself in the
transport of electrons through small conductors �nanopar-
ticles or other nanoscale objects� weakly coupled to the
source and drain electrodes �current-carrying leads� owing to
the Coulomb blockade effect. Numerous manifestations of
the charge quantization in transport properties, the most fa-
miliar of them are the Coulomb blockade oscillations of the
electric current as a function of the gate voltage and the
Coulomb staircase in the current-voltage characteristics,
have attracted considerable attention in the past years.1 Since
the fundamentals of the transport theory in the Coulomb
blockade regime have been established,2–4 the Coulomb
blockade-based physics has been applied to various issues of
electron transport in mesoscopic systems, and the field of its
applications expands in line with the advances in nanotech-
nology.
Usually, the influence of the Coulomb blockade on the
current in two-terminal devices is considered under assump-
tion that the coupling between the nanoscale object and the
leads is not sensitive to the number of electrons N determin-
ing the object charge eN. This corresponds to the introduc-
tion of ohmic �or nearly ohmic� effective resistances describ-
ing this coupling. Though this assumption often works well,
it can be violated, for example, in nanomechanical
systems,5–7 where charging of the object gives rise to its
displacement towards one of the leads thereby changing its
tunnel coupling to both leads. In this paper we study a situ-
ation when the sensitivity of the tunnel coupling to the num-
ber of electrons does not require a mechanical displacement
and is determined by the nature of tunneling. This implies a
device layout and conditions similar to those used in the
recent experiments on field emission of electrons from me-
tallic nanoclusters,8–10 silicon nanowires11–15 and
nanocones,16,17 and carbon nanotubes �see, for example,
Refs. 11 and 18–26�, when small �nanoscale� objects are
formed on the source electrode �cathode�, the latter is then
negatively biased with respect to the drain electrode �anode�
in vacuum. The current between the electrodes flows owing
to the field emission of electrons from nanoscale objects,
because the electric field F at the tips of the objects is higher
than in the other places of the device. The field-emission
current is described by the Fowler-Nordheim formula27
I = ASF2 exp�− F
�, F = 4�2m
3�e��
W3/2, �1�
where m is the free electron mass, W is the work function of
the emitting material, S is the effective emitting area, and A
is a constant expressed through the work function and Fermi
energy �F of the emitting material
�e�3��F/W
4�2 � ��F + W�
. �2�
The effective field F, which describes the tunnel coupling
between the nanoscale object and the anode, depends on the
object charge, which is induced by the applied voltage V
=V1−V2, where V1 and V2 are the cathode and anode poten-
tials, respectively. Under conditions of Coulomb blockade,
i.e., when the electric connection between the cathode and
the object is weak and the charging energy of the object
considerably exceeds the temperature T, the continuous
variation of the voltage V leads to discrete changes of the
object charge in units of e, and, consequently, to correspond-
ing discrete changes of the field F. Therefore, one may in-
troduce the field FN, which is a function of the discrete num-
ber N and continuous variable V. Next, if the current in the
device is limited by the field emission, the single-electron
tunneling processes become important. This means that, at a
fixed voltage V, the object stays mostly in the states with N
and N−1 electrons, the number N is determined by the volt-
age. In the N-electron state, no electrons can come to the
object from the cathode until an electron leaves the object by
tunneling through the barrier, see Fig. 1�a�. Then the object
appears in the N−1-electron state and returns to the
N-electron state before the next Fowler-Nordheim tunneling
event takes place. The field-emission current in these condi-
tions is given by Eq. �1� with F=FN and can be denoted as
IN. If the bias eV increases, the state with N+1 electrons
becomes more favorable, and the current changes in a step-
like fashion from IN to IN+1. This leads to staircaselike
current-voltage characteristics, which may look similar to the
usual Coulomb staircases.28–30 However, since the sensitivity
of the tunneling to the number of electrons is involved, the
staircaselike current-voltage characteristics can exist under
rather peculiar conditions, when the source-drain bias is or-
ders of magnitude larger than the charging energy.
The rest of the paper is devoted to quantitative studies
based on the physical idea outlined above. In Sec. II we give
the basic equations and calculate the current in the simplest
case of an idealized emitter shown in Fig. 1�b�. In Sec. III we
calculate the current from a nanocluster modeled by a spheri-
cal particle on the metallic cathode surface and from a semi-
conductor wire �nanowhisker� grown perpendicular to the
cathode surface. The discussion and concluding remarks are
given in the last section.
II. GENERAL CONSIDERATION
We consider the case of classical �or metallic� Coulomb
blockade, when the electron energy level separation in the
nanoscale object can be neglected in comparison to both
temperature and charging energy. Since the object is assumed
to be weakly coupled to the cathode, we study the sequential
tunneling process and not the coherent one. It is convenient
to investigate the electron transport by applying the kinetic
equation2 �Master equation� for the distribution function PN
describing the probability for the object to be in the state
with N electrons. Assuming that the electric connection be-
tween the cathode and the object is characterized by the con-
ductance G, this equation is written as
= QN+1 − QN, �3�
where
1 − exp�− �EN/T�
�PN − PN−1 exp�− �EN/T�	
+ PNIN/�e� . �4�
Here �EN= �e
2 /C��N−1/2−C2V /e	 is the difference in Cou-
lomb energies for the objects with N and N−1 electrons, C is
the total capacitance, and C2 is the capacitance of the object
with respect to the anode �the capacitance with respect to the
cathode is given by C1=C−C2�. The first term in expression
�4� has the usual form2 and corresponds to the current be-
tween the object and the cathode. It is written as a difference
of the contributions describing the departure of an electron
from the object in the N-electron state and arrival of an elec-
tron at the object in the N−1-electron state. The second term
corresponds to the field-emission current from the object in
the N-electron state. Since no electrons come to the object
from the anode, this term does not contain a contribution
describing arrival of electrons. In the stationary case, Eq. �3�
is reduced to the form QN=const, where the constant can be
chosen equal to zero. After determining PN from the equation
QN=0 with the use of the normalization condition 
NPN=1,
the total current is given by
PNIN. �5�
Under the condition GT� �e � IN, which means that the object
is in thermal equilibrium with the cathode, the stationary
solution of Eq. �3� is written as PN=Z
−1 exp�−EN /T�, where
EN= �e
2 /2C��N−C2V /e	
2 is the Coulomb energy, and Z
N exp�−EN /T� is the partition function. The current in this
case is determined by the expression
J = Z−1
IN exp�− EN/T� . �6�
Let us apply the solution �6� to the idealized model of
emitter, Fig. 1�b�, when the emission takes place from a
spherical nanoparticle of radius R, placed at a distance d
from the cathode. The distance between the cathode and an-
ode is L. The connection c-p denotes a low-transparent con-
tact �for example, tunnel barrier� between the particle and the
cathode, which does not contribute to the field-emission
properties and electrostatics of the device. Assuming d�R,
we have C=R, C2=Rd /L, and neglect the charge polariza-
tion of the particle because this polarization is small in com-
parison to the total charge eN induced by the applied voltage.
The number of electrons is estimated as N�C2V /e
=RdF0 / �e�, where F0=−V /L is the applied electric field. The
effective field for the nanoparticle with N electrons is FN
= �e �N /R2, and the partial currents IN in these conditions are
given by
IN = AS�eN/R
2�2exp�− FR2/�e�N� , �7�
where the emitting area S, in the idealized model considered
here, can be approximated by the total surface area of the
nanoparticle, S=4�R2. In Fig. 2 we plot the current-voltage
characteristics of the idealized emitter, calculated according
to Eqs. �6� and �7�, where A is given by Eq. �2� with W
=5.1 eV and �F=5.5 eV �taken for Au�, and the geometrical
parameters are chosen as R=5 nm and d=0.5 �m. The char-
acteristics look like staircases with flat regions �plateaus� be-
tween the steps, which are visible even at room temperature.
It is possible to estimate the relative heights of the steps by
calculating the ratio of the currents IN and IN−1 emitted from
the nanoparticle with N and N−1 electrons
� exp� FR2
�e�N�N − 1�
� . �8�
In spite of the fact that the charged nanoparticle typically
contains a large number of electrons, N
100, one can al-
FIG. 1. �a� The mechanism of single-electron tunneling in the
Fowler-Nordheim regime. �b� Schematic representation of the ide-
alized emitter.
ways find a regime when the ratio IN / IN−1 is not small in
comparison to unity. This necessarily implies a weak Fowler-
Nordheim tunneling, when F /F=FR2 / �e �N�1.
In the calculations described above, the applicability of
the Fowler-Nordheim formula requires R�W / �e �F, which is
rewritten as R�e2N /W, or, according to N�RdF0 / �e�, as
�e �F0�W /d, independent of the nanoparticle radius. This
condition is satisfied at high enough applied voltages. If
�e �F0=eV /L�W /d, the approximation of a triangular poten-
tial barrier is not quite good, and one should consider the
tunneling through the barrier described by the potential en-
ergy W−e2N�1/R−1/r� at r�R, where r is the distance
from the center of the spherical nanoparticle; the tunneling
through the potential barrier of this form is described in Ref.
31. Even under the condition �e �F0�W /d, which is satisfied
in the calculations shown in Fig. 2, the relative change of the
current per one step, IN / IN−1−1, appears to be significant,
because the exponent FR2 / �e �N�N−1� in Eq. �8� is estimated
as c�W / �e �F0d�
2, where the dimensionless constant c
=4/3�2me4 /�2W is noticeably larger than unity.
If the current is high enough, the field emission cannot
remain the bottleneck for the electron transfer from the cath-
ode to the anode, and a finite resistance G−1 becomes essen-
tial. The nanoparticle in these conditions is no longer in equi-
librium with the cathode. This means that the distribution PN
is established kinetically, and several states with different
charges coexist at a fixed voltage �see the inset in Fig. 2�. As
a consequence, the Coulomb blockade features are washed
out. This case requires a numerical solution of the equation
QN=0. The corresponding current-voltage characteristics of
the idealized emitter calculated by using the RC time C /G
=100 ps are also shown in Fig. 2. The degradation of the
current steps appears to be stronger with increasing voltage,
because the current increases and the nanoparticle-cathode
link becomes more important. The shape of the steps in this
case resembles the usual Coulomb staircase.
III. MORE COMPLEX EXAMPLES
After demonstrating the possibility of the Coulomb-
blockade staircase of the field emission on a model example,
it is worth to consider more complex cases. Indeed, the
model example discussed above has certain disadvantages.
First of all, it is hardly possible to connect a particle placed
far from the cathode surface by a link �c-p in Fig. 1�b�	
which does not contribute to the electrostatic properties of
the device. Second, the model of uniform charging is insuf-
ficient: the charge polarization of the nanoscale object ap-
pears to be important and should be always taken into ac-
count, see below in this section. Therefore, the model shown
in Fig. 1�b� is suitable only for the purposes of illustration of
the basic physics described by Eqs. �3�–�6�. To have a closer
approach to reality, we point out that the nanoscale objects
investigated in the above-cited experiments on field emission
can be roughly divided into two classes: the objects whose
dimensions in all directions are comparable �nanoclusters or
nanoparticles�, and the objects whose length in the direction
of the applied field is much larger than their transverse size
�nanowires or nanowhiskers�. The following consideration is
carried out for the cases of nanoclusters and nanowires of the
simplest geometries, when the electric fields FN and the ca-
pacitances C and C2 can be determined consistently by solv-
ing corresponding electrostatic problems. The current is cal-
culated according to Eq. �6�, under the assumption that the
objects are in equilibrium with the cathode.
A. Field emission from nanoclusters
Below we consider the field emission from a nanocluster
modeled by a spherical metallic particle of radius R depos-
ited on the flat cathode surface. To provide a finite capaci-
tance C, one should assume a finite separation d-R between
the particle and the metallic cathode plate �for instance, one
can imagine that the particle resides on an oxidized surface�,
see the inset to Fig. 3. Besides, this assumption provides
electrical isolation of the particle from the cathode, which is
FIG. 2. Current from the idealized emitter as a function of the
applied field F0=−V /L for the case of small C /G �nanoparticle in
thermal equilibrium with the cathode, upper curves� and for the
case of C /G=100 ps �lower curves�, at the temperatures T=77 K
�solid� and T=293 K �dashed�. The inset shows the distribution
function PN at F0=5	10
5 V/cm for the second case.
FIG. 3. Charge density per unit length in z direction for a spheri-
cal metallic nanocluster placed at the distance 0.1 R from the me-
tallic cathode. Here 
0=F0R /2. The inset shows the geometry of
the problem and the directions of the field emission �arrows�.
a necessary condition for the Coulomb blockade. The elec-
trostatics of the plane-sphere system is known, and the field
and charge distributions in this case can be found in the form
of rapidly converging infinite series arising from the poten-
tials of image point charges and point dipoles.32 Such a con-
sideration allows one to present the distribution of the elec-
trostatic potential energy near the particle in the approximate
U�r,�� � W + e��F0R�1 + 
�cos � − 1�	
− ��eN − C2V	/C�
r − R
, �9�
where r and � are the radial and azimuthal coordinates of the
spherical coordinate system with the origin at the center of
the particle, and �, 
, and � are the dimensionless constants
of the order of unity, which are to be determined from nu-
merical calculations. Such calculations also give us the ca-
pacitances C and C2.
33 Note that if the charge quantization is
neglected �so that N=C2V /e when the particle is in equilib-
rium with the cathode�, � is identified with the field enhance-
ment factor conventionally used in the physics of field emis-
sion. The expression �9� provides an excellent description of
the electrostatic potential at r−R�R /2 and at small �. It
allows one to take into account deviations of the potential
energy from the linear form W− �e �F�r−R� and, therefore, to
find corrections to the Fowler-Nordheim tunneling exponent.
Neglecting such corrections in the prefactor, we obtain the
following expression for the partial currents
IN = ASFN
2 exp�− F
�� �e�FNR
��x� =
� x2�x − 1��2 − arctan �x − 1� − x� , �10�
where A is given by Eq. �2�, the dimensionless function ��x�
describes the corrections to the tunneling exponent, and the
effective emitting area S=2�R2�FN
2 /
F�F0�
�2�R2�FN /
F� is reduced due to the angular dependence of
the radial field described by Eq. �9�. The field FN is given by
FN = �F0 + �
�e�N − C̃2F0
, �11�
where the quantity C̃2=C2L does not depend on the distance
L between the cathode and anode. Note that, since we always
assume that L is much larger than any dimension of the
nanoscale object, the capacitance C2 is always proportional
to 1 /L, and it is more convenient to replace C2 �V� by C̃2F0.
This substitution also allows us to represent the Coulomb
energy standing in Eq. �6� as
�N − C̃2F0/�e�	
2. �12�
Further calculations are done for the separation d−R
=0.1R, when C=2.16R, C̃2=1.74R
2, �=4.32, �=1.22, and

=0.66. Figure 3 shows the distribution of negative charges
on the surface of the spherical particle staying in equilibrium
with the cathode for this case ��e �N= C̃2F0 is assumed�. The
distribution of the radial field F�z� at the surface of the par-
ticle is given by the same dependence, F�z� /F0=
�z� /
The field-emission current from the nanocluster described
above has been calculated according to Eqs. �6� and �10�–
�12� at R=5 nm. The results of the calculations shown in Fig.
4 demonstrate the staircaselike behavior caused by the Cou-
lomb blockade. However, in contrast to the staircases shown
in Fig. 2, the current continues to increase between the steps.
This occurs because of electrostatic polarization of the nano-
particle. According to Eq. �11�, when the particle charge is
constant, the increase in the applied field F0 leads to an in-
crease in the effective field FN because the factor �
−�C̃2 /CR is positive. For the chosen particle radius, the
steps of the current are clearly visible at liquid nitrogen tem-
perature but poorly visible at room temperature. Neverthe-
less, the Coulomb blockade features at room temperature be-
come quite distinct in the plots of the derivative of the
current, as shown in the inset to Fig. 4.
B. Field emission from nanowires
Let us consider the field emission from a small semicon-
ductor wire modelled by a cylinder of radius R and length d,
which ends with a hemispherical tip of the same radius, see
the inset to Fig. 5. The cathode substrate upon which the
wire is grown is assumed to be a metal �or a heavily doped
semiconductor� so that one can use the method of image
charges instead of solving the electrostatic problem in the
whole space. The electric isolation of the wire from the cath-
ode in this case takes place in a natural way, because a
Schottky barrier is formed between the wire and the metallic
cathode �in the case of semiconducting cathode there can be
a heterobarrier or an interband p-n barrier�. In other words,
FIG. 4. Current from the spherical nanocluster of radius R
=5 nm as a function of the applied field F0=−V /L at T=4.2 K
�solid� and 77 K �dashed�. The inset shows the derivative of the
current at T=293 K.
the wire region adjacent to the cathode becomes depleted of
electrons and positively charged because of the presence
of donors �we assume that the wire is uniformly doped
with bulk donor density nD�. When a bias eV is applied
between the cathode and anode, the wire acquires a con-
siderable negative charge because of tunneling or thermi-
onic emission of electrons from the cathode through the
barrier. When the field emission from the wire of nan-
oscale radius becomes essential, the density of induced
negative charges per unit length of the wire appears to be
much larger than the equilibrium charge density 
=�R2 �e �nD even if nD is of the order of 10
18 cm−3. For
this reason, one can use the “metallic” approximation
assuming that the charges in the wire are placed mostly on its
surface. This means that the electron density distribution
n�� ,z�, which depends on the radial coordinate � of the cy-
lindrical coordinate system connected with the wire, is given
by n�� ,z�= �2�� �e � �−1���−R�
�z�+nD for z�d and n�� ,z�
= �2�� �e � �−1���−�R2− �z−d�2	
�z�+nD for d�z�d+R,
where 
�z� is the density of negative charges on the surface
per unit length. Since this approximation is based on the
assumption that the screening length is small in comparison
with the wire radius, it works better for wider wires. For
silicon wires, whose field-emission properties are currently
the subject of investigations,11–15 the metallic approximation
remains suitable even for the radius of several nanometers,
because, owing to the large effective masses and six-valley
degeneracy, the density of electron states in n-Si appears to
be high enough to provide the Thomas-Fermi screening
length less than one nanometer for Fermi energies �F
�0.01 eV. The metallic approximation, of course, fails to
describe the region of the wire in the close vicinity of the
cathode, where the depletion occurs. Nevertheless, since this
region is a small part of the whole wire, see the charge dis-
tribution in Fig. 5, its presence cannot considerably modify
the parameters calculated as described below.
According to the discussion given here, we search for the
charge distribution 
�z� satisfying the integral equation
U�z� = U0 − �e�F0z + �
dz�K�z,z��
�z�� , �13�
where U�z� is the potential energy counted from the Fermi
level in the cathode material, U0 is the barrier height, and
K�z ,z�� is the potential of interaction between the electrons
in the points z and z� of the wire surface in the presence of
the cathode plate, see the Appendix. Equation �13� is accom-
panied with additional requirements: U�z�=0 at z�z0 and
�z�=−
D at z�z0, where z0 is the depletion edge coordi-
nate, which is to be found self-consistently. The first of these
requirements corresponds to a full screening of the bare po-
tential U0− �e �F0z by the induced charges of the wire, while
the second one models the presence of the positive charges in
the depletion region z�z0. Once the distribution 
�z� is
found, the total charge of the wire, −�0
d+Rdz
�z�, as well as
the distribution of electric field around the wire, can be cal-
culated. To find the capacitance C and describe modification
of the effective field under single-electron charging, one may
calculate the variation of the total charge and the field at the
tip �at z=d+R� with respect to a small variation of U0. Equa-
tion �13� is solved numerically by using the method of itera-
tions. The dependence of the effective field FN on F0 and N
can be represented in the form similar to Eq. �11�
FN = ��F0�F0 + �
�e��N + B� − C̃2F0
C�F0�R
, �14�
while the Coulomb energy is written as
2C�F0�
�N + B − C̃2F0/�e�	
2. �15�
These equations take into account a finite �though weak�
dependence of the capacitance C and field enhancement fac-
tor � on the applied field F0. The dependence of the param-
eters C̃2 and � on F0 appears to be much weaker and can be
neglected. The positive dimensionless constant B reflects the
fact that the average number of induced charges is smaller
than C̃2F0 / �e�. These features appear because the system un-
der consideration is not entirely metallic and contains a
depletion region whose length changes with F0.
The numerical calculations leading to the results pre-
sented below are done for U0=0.7 eV, which approximately
corresponds to the Schottky barrier height for n-Si in contact
with Al.34 The chosen donor density is nD=2	10
18 cm−3.
The parameters standing in Eqs. �14� and �15�, however, are
not sensitive to nD, except for the capacitance C, which
changes within 10% when nD varies from 10
18 cm−3 to 2
	1018 cm−3. Figure 5 shows the charge density distribu-
tion for the wire of radius R=5 nm and length d=0.1 �m
at F0=10
6 and 2	106 V/cm. The charge density shows
a nearly linear growth through the main part of the wire
and a sharp enhancement at the hemispherical tip from
which the field emission occurs. The dependence of the
field enhancement factor and capacitance on the applied
electric field is shown in Fig. 6, and the other calcu-
lated parameters are C̃2=2.44 dR, �=0.414, and B=12.14.
FIG. 5. Charge density per unit length for the cylindrical wire
whose geometry is shown in the inset �see parameters in the text�.
The plots of the field-emission current calculated with the
use of the parameters listed here are given in Fig. 7. The
calculations are done according to Eqs. �6�, �14�, and �15�,
and the Fowler-Nordheim formula for the partial current,
IN=ASFN
2 exp�−F /FN�. Since the calculated radial electric
field in the region of the tip weakly depends on z �in contrast
to the case of the nanocluster studied above� and sharply
decreases in the region of transition to the cylindrical part of
the wire, the effective emitting area S is estimated by the
total area of the hemispherical tip, S=2�R2. The work func-
tion is taken for silicon, W=4.2 eV. Next, the Fermi energy
standing in the expression for A, see Eq. �2�, is estimated
from the equation �F��e �FinrTF, where Fin��F0 /� is the
field inside the semiconductor near the end of the tip, rTF is
the Thomas-Fermi screening length, and � is the dielectric
constant of the semiconductor. Such an estimate, carried out
for n-Si, leads to �F�0.1 eV at F0�2	10
6 V/cm. The pic-
ture of Coulomb staircase shown in Fig. 7 is basically the
same as that in Fig. 4. Again, the increase of the current with
the applied field is determined by the increase of the effec-
tive field �14� due to both single-electron charging �steps�
and charge polarization under a constant charge �regions be-
tween the steps�. The main difference is that the interval of
the applied field needed for addition of one electron to the
wire is considerably reduced, owing to the larger capacitance
C2, and appears to be of about 1.2 V/�m �further reduction
of this interval takes place with the increase of the wire
length, see below�. Next, since the capacitance C increases
considerably in comparison to the case of nanocluster of the
same radius, the Coulomb blockade features at room tem-
perature are poorly visible even in the derivative plot, see the
inset. Nevertheless, these features remain pronounced at T
=77 K.
With the increase of the wire length d, the parameters
entering Eqs. �14� and �15� are modified as shown in Fig. 8.
The field enhancement factor and the capacitances increase
nearly in a linear way, while the parameter �, which charac-
terizes relative contribution of charging to the effective field,
slightly decreases �for comparison, the idealized emitter con-
sidered in the previous section is described by the parameters
�=d /R, �=1, C=R, and C̃2=dR, where d is the distance
from the cathode to the emitting sphere�. The increase of the
total capacitance C makes it difficult to observe the Coulomb
staircase in long wires. For example, at d=1 �m one should
have temperatures considerably lower than 77 K. The inter-
val of the applied field corresponding to the addition of one
electron is inversely proportional to C̃2. This interval de-
creases very fast with the increase of d and becomes equal to
2.5	102 V/cm at d=1 �m.
FIG. 6. Field dependence of the enhancement factor and capaci-
tance for the cylindrical wire with R=5 nm and d=0.1 �m.
FIG. 7. Current from the cylindrical wire of radius R=5 nm and
length 0.1 �m as a function of the applied field F0=−V /L at T
=4.2 K �solid� and 77 K �dashed�. The inset shows the derivative of
the current at T=293 K.
FIG. 8. Dependence of the parameters �, �, C, and C̃2 on the
length of the wire for R=5 nm and F0=5	10
5 V/cm.
IV. CONCLUSIONS
The key point of the presented theoretical study is the
possibility of noticeable modification of the effective electric
field causing the field emission from a nanoscale conductor
by addition of just one electron to this conductor. Formally,
this modification is described by introducing the effective
field FN, which determines the partial current IN, and by
evaluating the dependence of this field on the bias applied
between the cathode and anode, see Eqs. �11� and �14�. As a
result of this effect, the current-voltage characteristics of the
field emission show steps in the Coulomb blockade regime.
In other words, the steplike current-voltage characteristics
related to single-electron charging �Coulomb staircases� may
exist even under the conditions of field-emission experi-
ments, when the applied bias is orders of magnitude larger
than the charging energy. The steps on the current-voltage
characteristics can be visible at 77 K in the case of field
emission from nanoclusters and nanowires of 10 nm diam-
eter and submicron length. In the regions between the steps,
where the total charge of the nanoscale object is constant, the
current increases with the increase of the applied bias owing
to charge polarization.
The staircases described in this work are similar to the
usual Coulomb staircases obtained in the transport through
small metallic islands28–30 or quantum dots �see Ref. 35 for
review� with strong asymmetry in the barriers. In both cases,
each step of the current is associated with addition of an
electron to the nanoscale object, and the applied source-drain
voltage drops mostly across the low-transparent barrier �the
barrier between the object and the drain�. Therefore, the pe-
riodicity of the steps in both cases is determined by the
object-drain capacitance C2. However, the steps in the sec-
ond case are formed due to shifts of effective �N-dependent�
electrochemical potential of the object with respect to elec-
trochemical potentials of the source and drain. For this rea-
son, the usual Coulomb staircase shows well-defined steps
when C2 is greater than the object-source capacitance C1,
while in the opposite situation, C1�C2, the steps are sup-
pressed and the current-voltage characteristic approaches to a
linear dependence.29,30 In contrast, in the case described in
this work the steps are formed due to changes in the prob-
ability of Fowler-Nordheim tunneling from the object to the
drain �anode�. That is why the steps are clearly visible under
the condition C1�C2, which is imposed by the field-
emission layout considered in this paper. To summarize, the
sensitivity of the field emission to the number of electrons in
the nanoscale object makes it possible to obtain the Coulomb
staircases under the conditions when such staircases cannot
be observed in the transport through small metallic islands or
quantum dots.
The quantitative consideration has been applied here to
some simple models of the nanoscale objects, whose electro-
static properties necessary for description of the field en-
hancement and charging have been determined consistently.
Consequently, the number of geometrical parameters charac-
terizing the objects has been minimized. For example, the
nanowire has been characterized only by its length d and
radius R. In reality, the geometrical structure of objects is
more complicated. For example, their tips may contain sharp
regions which provide a more efficient field emission. In
fact, high field-emission currents from nanoscale objects are
typically observed at the applied fields of the order of
105 V/cm, which requires the field enhancement factors
much larger than those calculated in this paper. On the other
hand, the presence of sharp tips cannot strongly modify the
capacitances of the objects. The general picture of the single-
electron tunneling under the field-emission regime also re-
mains valid. For possible application to experiments, the
field enhancement due to charging can be described by equa-
tions of the kind of Eqs. �11� and �14�, where � and � should
be considered as parameters to be determined experimen-
tally.
At the present time, there is no experimental evidence of
the Coulomb staircase phenomenon under the field emission.
Though the current-voltage characteristics sometimes show
steplike features, see, for example, Ref. 11, these features are
not regular and, most probably, should be attributed to insta-
bilities of the emission process and burning out of the emit-
ting material. There are numerous reasons which make ob-
servation of the phenomena considered in this paper difficult.
First of all, in most cases the nanoscale objects on the cath-
ode surface form dense arrays. This means that the field
emission takes place from a macroscopic number of objects
which are electrostatically coupled. The charging and field-
emission properties appear to be considerably different36
from those of individual objects. The Coulomb blockade
phenomena in this case should be dramatically suppressed by
the size dispersion of the objects and by the effects of mutual
screening. Investigation of field emission from individual ob-
jects is possible in the cases of metallic nanoclusters8–10 and
carbon nanotubes.26 However, there exists the problem of
electric isolation of these objects from the cathode, which is
one of the necessary conditions for Coulomb blockade. No
special attempts to achieve such an isolation in the field-
emission experiments have been undertaken so far, except
for the nanomechanical system investigated in Ref. 7, where
the electron emission from an isolated Au island to a
submicron-sized electrode has been observed. Most of the
experiments on field emission are carried out at room tem-
perature, though existing experimental techniques also allow
measurements at liquid nitrogen temperature. This means
that the Coulomb blockade phenomena can be observed only
for small-sized objects whose capacitances are low enough
�see the results of Sec. III�. Besides, the interval of the ap-
plied field corresponding to addition of one electron strongly
decreases in the case of emission from long nanowires,
which requires high resolution with respect to field. In sum-
mary, a search for the Coulomb blockade features in the
field-emission current would require a special planning of
experiment. The author hopes that the presented theoretical
study will stimulate experimental investigations in this direc-
tion.
ACKNOWLEDGMENTS
The author is grateful to A. I. Klimovskaya for stimulat-
ing discussions.
APPENDIX: KERNEL OF EQUATION (13)
If z�d and z��d, K�z ,z��=K0�z ,z��−K0�−z ,z��, where
K0�z,z�� = �
��z − z��2 + 2R2�1 − cos ��
. �A1�
If z�d and z��d, K�z ,z��=K0�z ,z��−K0�−z ,z��, where
K0�z,z�� = �
��d − z�2 + 2R�d − z�cos �� + 2R2�1 − sin �� cos ��
. �A2�
If z�d and z��d, K�z ,z��=K0�z ,z��−K0�z ,−z��, where
K0�z,z�� = �
��d − z��2 + 2R�d − z��cos � + 2R2�1 − sin � cos ��
. �A3�
Finally, if z�d and z��d,
K�z,z�� = �
� �e��2R2�1 − cos � cos �� − sin � sin �� cos ����
�4d2 + 4dR�cos � + cos ��� + 2R2�1 + cos � cos �� − sin � sin �� cos ��� . �A4�
In Eqs. �A2�–�A4�, cos �= �z−d� /R and cos ��= �z�−d� /R, so � and �� are the azimuthal angles. The integrals are taken over
the polar angle �. The function K�z ,z�� is also representable in the form of full elliptic integrals.
*Electronic address: raichev@isp.kiev.ua
1 Single Charge Tunneling, edited by H. Grabert and M. H. De-
voret, NATO ASI Series B 294 �Plenum Press, New York, 1992�.
2 I. O. Kulik and R. I. Shekhter, Zh. Eksp. Teor. Fiz. 68, 623
�1975� �Sov. Phys. JETP 41, 308 �1975�	.
3 D. V. Averin and K. K. Likharev, J. Low Temp. Phys. 62, 345
�1986�; and in Mesoscopic Phenomena in Solids, edited by B. L.
Altshuler, P. A. Lee, and R. A. Webb �Elsevier, Amsterdam,
1991�.
4 C. W. J. Beenakker, Phys. Rev. B 44, 1646 �1991�.
5 L. Y. Gorelik, A. Isacsson, M. V. Voinova, B. Kasemo, R. I.
Shekhter, and M. Jonson, Phys. Rev. Lett. 80, 4526 �1998�.
6 A. Erbe, C. Weiss, W. Zwerger, and R. H. Blick, Phys. Rev. Lett.
87, 096106 �2001�.
7 D. V. Scheible, C. Weiss, J. P. Kotthaus, and R. H. Blick, Phys.
Rev. Lett. 93, 186801 �2004�.
8 M. E. Lin, R. P. Andres, and R. Reifenberger, Phys. Rev. Lett. 67,
477 �1991�.
9 M. E. Lin, R. Reifenberger, and R. P. Andres, Phys. Rev. B 46,
15490 �1992�.
10 M. E. Lin, R. Reifenberger, A. Ramachandra, and R. P. Andres,
Phys. Rev. B 46, 15498 �1992�.
11 C. S. Chang, S. Chattopadhyay, L. C. Chen, K. H. Chen, C. W.
Chen, Y. F. Chen, R. Collazo, and Z. Sitar, Phys. Rev. B 68,
125322 �2003�.
12 S. Johnson, A. Markwitz, M. Rudolphi, H. Baumann, S. P. Oei,
K. B. K. Teo, and W. I. Milne, Appl. Phys. Lett. 85, 3277
�2004�.
13 N. N. Kulkarni, J. Bae, C.-K. Shih, S. K. Stanley, S. S. Coffee,
and J. G. Ekerdt, Appl. Phys. Lett. 87, 213115 �2005�.
14 J. C. She, K. Zhao, S. Z. Deng, J. Chen, and N. S. Xu, Appl.
Phys. Lett. 87, 052105 �2005�.
15 J. C. She, S. Z. Deng, N. S. Xu, R. H. Yao, and J. Chen, Appl.
Phys. Lett. 88, 013112 �2006�.
16 Q. Wang, J. J. Li, Y. J. Ma, Z. L. Wang, P. Xu, C. Y. Shi, B. G.
Quan, S. L. Yue, and C. Z. Gu, Nanotechnology 16, 2919
�2005�.
17 Y. L. Chueh, L. J. Chou, S. L. Cheng, J. H. He, W. W. Wu, and L.
J. Chen, Appl. Phys. Lett. 86, 133112 �2005�.
18 A. G. Rinzler, J. H. Hafner, P. Nikolaev, L. Lou, S. G. Kim, D.
Tomanek, P. Nordlander, D. T. Colbert, and R. E. Smalley, Sci-
ence 269, 1550 �1995�.
19 P. G. Collins and A. Zettl, Appl. Phys. Lett. 69, 1969 �1996�.
20 Q. H. Wang, A. A. Setlur, J. M. Lauerhaas, J. Y. Dai, E. W. Seelig,
and R. P. H. Chang, Appl. Phys. Lett. 72, 2912 �1998�.
21 S. Fan, M. G. Chapline, N. R. Franklin, T. W. Tombler, A. M.
Cassell, and H. Dai, Science 283, 512 �1999�.
22 R. H. Baughman, A. A. Zakhidov, and W. A. de Heer, Science
297, 787 �2002�.
23 S. H. Jo, Y. Tu, Z. P. Huang, D. L. Carnahan, J. Y. Huang, D. Z.
Wang, and Z. F. Ren, Appl. Phys. Lett. 84, 413 �2004�.
24 M. Mauger, V. T. Binh, A. Levesque, and D. Guillot, Appl. Phys.
Lett. 85, 305 �2004�.
25 N. de Jonge, M. Allioux, M. Doytcheva, M. Kaiser, K. B. K. Teo,
R. G. Lacerda, and W. I. Milne, Appl. Phys. Lett. 85, 1607
�2004�.
26 Z. Xu, X. D. Bai, E. G. Wang, and Z. L. Wang, Appl. Phys. Lett.
87, 163106 �2005�.
27 R. H. Fowler and L. W. Nordheim, Proc. R. Soc. London, Ser. A
119, 173 �1928�.
28 J. B. Barner and S. T. Ruggiero, Phys. Rev. Lett. 59, 807 �1987�.
29 K. Mullen, E. Ben-Jacob, R. C. Jaklevic, and Z. Schuss, Phys.
Rev. B 37, 98 �1988�.
30 R. Wilkins, E. Ben-Jacob, and R. C. Jaklevic, Phys. Rev. Lett. 63,
801 �1989�.
31 L. D. Landau and E. M. Lifshitz, Quantum Mechanics �Perga-
mon, Oxford, 1977�.
32 W. R. Smythe, Static and Dynamic Electricity �McGraw-Hill,
New York, 1968�.
33 If d=R, the parameters are obtained analytically: �=7��3� /2
�4.21, 
=93��5� /56��3�−3/4, �=�2 /8, and C2= ��
2 /6�R2 /L,
but C is equal to infinity �C diverges in a logarithmic way as the
separation d-R goes to zero�.
34 Metal-Semiconductor Interfaces, edited by A. Hiraki �IOS Press,
Amsterdam, 1995�.
35 L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R.
M. Westervelt, and N. S. Wingreen, Electron Transport in Quan-
tum Dots, in Proceedings of the Advanced Study Institute on
Mesoscopic Electron Transport, edited by L. L. Sohn, L. P. Kou-
wenhoven, and G. Schön �Kluwer, 1997�.
36 T. A. Sedrakyan, E. G. Mishchenko, and M. E. Raikh, cond-mat/
0504042 �unpublished�.
