Download Modeling of field-assisted emission from a negative electron affinity

Modeling of field-assisted emission from a negative electron
affinity cathode
A. Mayera兲
Laboratoire de Physique du Solide, Facultés Universitaires Notre-Dame de la Paix,
Rue de Bruxelles 61, B-5000 Namur, Belgium
M. S. Chung
Department of Physics, University of Ulsan, Ulsan 680-749, Korea
N. Kumar
UHV Technologies, Inc., 450 South Freeway, Fort Worth, Texas 76104-3503
B. L. Weiss
Department of Physics, 130 CAC, The Pennsylvania State University, Altoona, Pennsylvania 16601
N. M. Miskovsky and P. H. Cutler
Department of Physics, 104 Davey Laboratory, The Pennsylvania State University,
University Park, Pennsylvania 16802
共Received 3 March 2006; accepted 8 November 2006; published 4 January 2007兲
A device has been reported recently 关M. W. Geis et al., Appl. Phys. Lett. 87, 192115 共2005兲兴 in
which electrons transit through the image states of a negative electron affinity glass substrate before
being emitted to the vacuum. The external field required for this emission may be as low as
10 V / cm, which is up to three orders of magnitude lower than the fields encountered with other
materials and configurations. The authors present a description and analysis of this device, which
includes the glass substrate and the metallic contacts. The model also accounts for the ionization of
cesium in the glass. Two stages are considered: 共i兲 the activation step in which a 200 V bias is
established through the glass and 共ii兲 the operational state in which the transfer of electrons from the
metallic contacts to the glass surface is enhanced by positive ions trapped in the glass. The results
provide insight into the role of the positive charges trapped in the glass. It is shown that the
conditions are encountered, both in the activation step of the device and in its subsequent unbiased
working mode, to account for the emission reported experimentally. © 2007 American Vacuum
Society. 关DOI: 10.1116/1.2404683兴
I. INTRODUCTION
Field emission from thin films has been studied since the
pioneering work of Malter in 1936.1 Other contributions include those of Dittmer,2 Boziak et al.,3 Araki and Hanawa,4
and Fedorovich et al.5 More recently, Geis et al.6 presented a
device in which electrons transit through the image states7–9
of a Cs2Si4O9 glass surface before being emitted to the
vacuum. Since this material exhibits a negative electron affinity property,10,11 the location of the image states is in the
forbidden energy gap region of the glass. Hence, the electrons occupying these states are quantum-mechanically forbidden to enter the material. This results in a substantial
increase of the lifetime of these states,12–20 which enables the
electrons to reach regions of the surface from where they are
eventually emitted to the vacuum. As demonstrated in a previous paper,21 the electrons that fill the image states are only
0.4 eV below the vacuum level so their extraction is easily
achieved 共the emission is actually mostly thermal field emission over the barrier兲. Experimentally, such field-assisted
emission to the vacuum has been achieved with external
fields as low as 10 V / cm, which is up to three orders of
a兲
Author to whom correspondence should be addressed; electronic mail:
[email protected]
109
J. Vac. Sci. Technol. B 25„1…, Jan/Feb 2007
magnitude lower than values encountered with other materials and more conventional configurations 共typically
⬎1 V / ␮m兲.22,23
The complete device is depicted in Fig. 1. The glass supports metallic contacts, which are the actual sources of electrons. The mechanism that leads to the filling of the image
states may be a triple-junction emission at the metallic
contacts24–27 or a regular field emission from protrusions of
these contacts.2,5,28,29 In the activation stage of the device,
this supply of electrons is induced by an internal bias of the
order of 200 V, which is established through the glass. In the
activated state of the device, the potential associated with
positive charges trapped in the glass30 replaces the need for
this internal bias.6,21
It is the objective of this article to provide a more complete qualitative and quantitative explanation of the operational conditions of this device by presenting a description
that includes the glass and the metallic contacts. The description, which is developed in Sec. II, also accounts for the
ionization of the glass. We consider two stages of the process: 共i兲 the activation step in which a 200 V bias is established through the glass 共Sec. III兲 and 共ii兲 the operational
state in which the transfer of electrons from the metallic
contact to the glass surface is enhanced by positive ions
1071-1023/2007/25„1…/109/6/$23.00
©2007 American Vacuum Society
109
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Mayer et al.: Modeling of field-assisted emission
FIG. 1. Schematic depicting the device of Geis et al. The Cs2Si4O9 glass has
typically width of 2 ␮m and a length of 10 ␮m. The metallic contacts have
a width of a few hundreds of nanometers and a length of the order of 1 ␮m.
In the activation stage of the device, an internal bias of 200 V is applied
through the glass. This bias is removed once a sufficient amount of positive
charges has been trapped in the glass. An external field is applied between
this cathode and a facing anode in order to extract electrons from the image
states of the glass.
trapped in the glass 共Sec. IV兲. The results provide insight
into the role of the positive charges trapped in the glass.
It is shown that the local fields are sufficient, both in the
activation step of the device and in its subsequent unbiased
working mode, to account for the emission reported
experimentally.
II. METHODOLOGY
We consider a two-dimensional model for the cross section of the device of Geis et al., which includes a 2 ␮m thick
glass, the two metallic contacts, and the underlying Si sub-
Vi,j =
110
strate. The two contacts consist of two parts: 共i兲 a first part
made of Al with a width of 0.4 ␮m and a height of 70 nm
and 共ii兲 a second part made of W with a width of 1 ␮m and
a height of 300 nm. The width of the whole device is 10 ␮m.
The Si support and the right contact are grounded. Depending on the working conditions, the left contact is either at a
potential of 200 V 共activation stage兲 or 0 V 共operational
state兲. In both cases, a macroscopic field of 10 V / cm is applied to the device.
In order to determine the electric field at the surface of the
glass and assess whether conditions for emission to the
vacuum are encountered, it is necessary to solve Poisson’s
equation in this representation of the device. We do so by
considering a two-dimensional discretization of the cross
section of the device and by integrating this equation on the
elementary cell associated with each point of the grid. This
transforms Poisson’s equation ⵜ · D = ␳, where D is the electric displacement and ␳ is the charge density, into its integral
form 兰兰D · dS = Qin, where the integral is performed on the
surface of each discretization cell. In this last equation, dS is
an outward-oriented surface vector and Qin refers to the net
charge contained in the discretization cell. Expressing all
quantities in terms of finite differences, the potential Vi,j of
each cell turns out to be related to that of its neighbors by the
relation
共⑀i+1,jVi+1,j兲/⌬x2 + 共⑀i−1,jVi−1,j兲/⌬x2 + 共⑀i,j+1Vi,j+1兲/⌬z2 + 共⑀i,j−1Vi,j−1兲/⌬z2 + ␳i,j
,
共⑀i+1,j + ⑀i−1,j兲/⌬x2 + 共⑀i,j+1 + ⑀i,j−1兲/⌬z2
where the subscripts i and j refer to values of the dielectric
constant ⑀, of the electric potential V, and of the charge density ␳ at the position 共i · ⌬x , j · ⌬z兲 in the cross section of the
device 共⌬x and ⌬z are, respectively, the horizontal and vertical discretization steps兲.31 We assume that the device is
invariant in the transverse y direction.
When the boundary values of the potential are specified,
one can determine the values in the whole system by iterating this equation. The physical parameters of this model are
the values of the dielectric constant ⑀, which is intrinsic to
each material, and the values of the charge density ␳, which
accounts for positive ions trapped in the glass. The charge
density ␳ is calculated during the iteration process according
to the local values of the electric field and using an additional
equation given in the Appendix.
III. CHARACTERIZATION OF THE DEVICE
IN ITS ACTIVATION STAGE
We first consider the activation step of the device of Geis
et al. in which the left metallic contact is at a potential of
200 V and the right contact and the metallic support are
grounded. The device is also subject to an external field of
J. Vac. Sci. Technol. B, Vol. 25, No. 1, Jan/Feb 2007
共1兲
10 V / cm, whose role is to extract the electrons from the
surface of the glass and guide them to the facing anode. This
situation as well as the physical processes discussed in this
section are represented in Fig. 2.
The 200 V bias is responsible for two effects: 共i兲 the production of positive ions in the glass and 共ii兲 the extraction of
electrons from the right metallic contact into the image states
of the glass surface. These electrons are extracted in a subsequent step to the vacuum because of the external field. The
ionization of the glass plays an important role in the situation
FIG. 2. Schematic depicting the device of Geis et al. in its activation stage.
An internal bias of 200 V is applied through the glass, which results in an
accumulation of positive charges on the left side of the glass. Electronic
emission originates from the right metallic contact and transit through the
image states of the glass before being emitted to the vacuum because of the
external field.
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Mayer et al.: Modeling of field-assisted emission
111
FIG. 3. Schematic depicting the device of Geis et al. in its operational stage.
Because of the positive charges in the glass, electrons are attracted from the
left metallic contact into the image states of the glass surface. They transit
through these states before being emitted to the vacuum because of the
external field.
presented in the next section by enabling the transfer of electrons from the left metallic contact to the glass surface without the need of the 200 V bias 共see Fig. 3兲. We consider field
ionization as the main source of positive ions in the glass,
although there is experimental evidence that ions coming
from the anode may contribute as well.32
In order to get more insight into the working conditions of
the device, it is necessary to look at the electric potential in
the device as well as at the normal component of the electric
field at the surface of the glass. These two quantities are
represented in Figs. 4 and 5. The last quantity is especially
important as it determines the emission from the image states
of the glass surface.
Because of the sign of the polarization, the electronic flow
must start from the right metallic contact. The mechanism for
this emission may be either 共i兲 a triple-junction fieldemission process taking place at the corner where the glass,
the metallic contact, and the vacuum meet along the same
line,24–27 or 共ii兲 a regular field-emission process from one or
several protrusions of the metallic contacts, provided the local fields are sufficient.2,5,28,29 This emission results in the
filling of the image states of the glass surface and to a subsequent emission to the vacuum because of the external field.
The electric field along the line where a triple-junction
emission may occur is of the order of 1.5 V / ␮m in the x
direction and 1 V / ␮m in the z direction. These values are
obtained from the resolution of Poisson’s equation presented
in Sec. II. They account for the polarization of the glass but
FIG. 4. Electric potential in the device during its activation stage. A 200 V
potential is applied at the left metallic contact, while the right contact and
the metallic support are grounded. An external field of 10 V / cm is applied.
The equipotentials are separated by increments of 5 V.
JVST B - Microelectronics and Nanometer Structures
FIG. 5. Electric field 共normal component兲 at the surface of the glass. A
200 V bias is established through the glass and an external field of 10 V / cm
is applied. The solid line stands for the result obtained considering ionized
cesium. The dashed line stands for the result obtained without the ions.
Finally, the dot-dashed line stands for the result obtained when the glass is
removed.
not for the image fields at this stage. Although relatively
weak 共but of the order of fields used in conventional techniques兲, these values may be sufficient to fill the image states
of the glass substrate.26,27 The second possibility, a regular
emission from the protrusions of the metallic contacts, is
supported by the fact that this emission would be enhanced
by the proximity of the metallic protrusions to the glass surface, the accumulation of positive charges in the glass, and
multiple image interactions of the emitted electrons with
both the metallic protrusions and the glass. Whether a triplejunction mechanism or this emission depicting is responsible
for the filling of the image states depends on the exact geometrical characteristics of the device.
In order to explain the emission from the image states of
the glass surface, it is necessary to look at the normal component of the electric field on the glass surface 共Fig. 5兲. The
average value of this field, for x ranging between 2 and
8 ␮m, is 35 V / ␮m. This value is essentially determined by
the 200 V bias and is negligibly dependent on the 10 V / cm
external field and on the ions trapped in the glass. According
to our recent publication,21 this field is sufficient to justify
the emission reported by Geis et al.6 provided the filling of
the image states of the glass surface is appropriate 共Fermi
level pinned at the ground state of the image potential or
within a few tenths of eV from this level兲. Our previous
model21 shows indeed that a 10 nA emission is achieved at
that 35 V / ␮m field value provided the Fermi level at the
surface of the glass is 0.44 eV below the ground state of the
image potential 共the transverse dimension of the actual device is 2 mm.32兲 This corresponds to a density of image
states as low as 5 ⫻ 105 e cm−2. In these conditions, the emission is essentially determined by the thermal excitation of
electrons to states propagating ballistically above the surface
barrier 共above the barrier emission兲. The factor e−⌬E/kT, with
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Mayer et al.: Modeling of field-assisted emission
⌬E the energy difference between the apex of the surface
barrier and the Fermi level, k the Boltzmann constant, and
T = 300 K the temperature, takes here a value of 8 ⫻ 10−12.
It can be noted that the electric field just below the left
metallic contact is seven times larger than the one obtained
in a situation where the glass is replaced by vacuum. This
field enhancement is a consequence of the reduction
of the field in the glass 共field enhancement due to
dielectrics is similarly encountered in the triple-junction
configuration26,27兲. The maximal value of this field,
1.3 V / nm, is so strong that all the image states of the glass
substrate belong to the continuous part of the spectrum. In
contrast with the remaining part of the glass surface, where
the Fermi level is expected to be close to the ground state of
the image potential 共reflecting the filling of this state兲, the
disappearance of this ground state just below the left metallic
contact may result in a Fermi level exhibiting stronger variations. The variations of the Fermi level are indeed related to
the electronic current density and to the carrier density. Because of the strong electric field and the negative electron
affinity of the glass surface, one can expect this current density to be significant. One can therefore conjecture that a
significant current may flow in this region in the very first
moments that follow the application of the 200 V bias 共depending on the supply of electrons in this part of the device兲.
This transitory state is followed by a charge redistribution,
which enables the device to reach an equilibrium. The
buildup of positive charges in the glass substrate, which
partly counterbalances the field associated with the 200 V
bias, is one part of this process. The second part of this
process is related to the filling of the image states, which in
turn depends on the position of the Fermi level at each point
of the glass surface.33 This filling as well as the mechanisms
that determine it constitute therefore an important issue for
reaching a deeper understanding of the device.
112
FIG. 6. Density of ionized cesium in the glass substrate, as determined
during the activation step of the device 共200 V applied through the glass
substrate兲. The maximal density is 4 ⫻ 1023 e m−3 共left part of the figure兲.
Adjacent contour lines indicate a reduction of the charge density by a factor
of 100. The lowest contour corresponds to a density of 1 e m−3.
metallic contact. In the current simulation, the net ionization
charge reaches the value of 105e, where e is the absolute
value of the electronic charge. This particular value depends
on the density of cesium, on its ionization probability, and on
the geometrical parameters of the device.
As in the previous case, an important information in order
to determine the electronic currents is the normal component
of the electric field on the surface of the glass substrate. This
quantity is represented in Fig. 7. The figure reveals that the
electric field below the left metallic contact 共around
0.2 V / nm兲 is sufficient to extract electrons from that contact.
As stated earlier, the mechanism for that emission may be
either 共i兲 a triple-junction emission process or 共ii兲 a regular
field emission from one or several protrusions of the metallic
contact. These electrons are then injected into the image
states of the glass surface.
In order to determine whether conditions are met for these
electrons to be subsequently emitted to the vacuum, it is
IV. CHARACTERIZATION OF THE DEVICE
IN ITS OPERATIONAL STAGE
When the 200 V bias is removed, the device starts working in a different mode. The electrons are attracted from the
metallic contacts to the glass surface, not by the 200 V bias
as previously, but presumably by the positive charges
trapped in the glass. Since these charges are essentially below the left contact, it is from that contact that the emission
originally starts. These electrons transit again through the
image states of the glass surface before being emitted to the
vacuum because of the external field. This second situation
as well as the physical processes discussed in this section are
represented in Fig. 3.
To gain some insight into this second working condition,
we represented in Fig. 6 the density of positive charges that
are induced during the activation stage of the device 共as explained in the Appendix, we consider field-ionized cesium as
the main source of these positive charges although oxygen,
silicium, and ion bombardment from the anode may contribute as well兲. The density of these positive charges reaches its
maximal value below the corners and asperities of the left
J. Vac. Sci. Technol. B, Vol. 25, No. 1, Jan/Feb 2007
FIG. 7. Electric field 共normal component兲 at the surface of the glass. The
glass contains a net positive charge of 105e because of the ionized cesium.
An external field of 10 V / cm is applied.
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Mayer et al.: Modeling of field-assisted emission
necessary to look at the potential barrier and electric field in
the remaining part of the surface. It turns out that the normal
component of the electric field in this remaining part of the
surface keeps positive values 共i.e., the local field points outwards of the surface兲. The field associated with positive ions
in the glass dominates indeed the 10 V / cm external field
共which points towards the surface兲. The emission from the
surface states of the glass is therefore not accounted for by
the local values of the electric field on this surface but rather
by the ability of electrons in these surface states to be thermally exited to states propagating ballistically above the surface potential barrier 共this potential barrier is calculated using the method described in Sec. 2兲. There is a turning point
between the surface of the glass and the anode, where the
external field finally dominates the field associated with the
positive ions in the glass. Electronic emission only occurs
when electrons are thermally excited to states for which ballistic transport above this turning point is possible.
Using the model developed in our previous paper,21 we
can compute the current emitted from each point of the glass
surface. Besides the potential and field values calculated up
to this point, the result also depends on the position of the
Fermi level on the glass surface. In order to distinguish between the current that leaves the surface and the one that
actually reaches the anode, it would also be necessary to
compute the electron trajectories. If we assume for simplicity
that the emitted electrons follow straight vertical trajectories
between the surface of the glass and the anode, we obtain the
10 nA current observed experimentally32 provided the Fermi
level on the surface of the glass is 0.22 eV below the ground
state of the image potential 共which is 0.4 eV below the
vacuum level兲. This corresponds to a surface charge density
of 2 ⫻ 109 e cm−2. The fact that the surface charge density is
higher than in the previous section is consistent with the fact
that the field at the contact from which the current originally
starts is higher 共0.2 V / nm instead of 1 V / ␮m previously兲.
Figure 8 finally shows the current emitted from each point of
the glass surface. It reveals that the emission from the surface of the glass is centered on the right part of that surface.
The factor e−⌬E/kT that quantifies the ability of electrons in
the surface states of the glass surface to be thermally excited
to states propagating above the surface barrier has a peak
value of 3 ⫻ 10−11 on that part of the surface.
V. CONCLUSIONS
In conclusion, we propose the following model for the
operation of the device reported by Geis et al. After an activation step that leads to the accumulation of positive ions in
the glass, electrons are emitted from the left metallic contact
to the ground image state of the glass substrate. Since these
electrons are quantum-mechanically forbidden to enter the
glass and neutralize these ions, they flow towards the right
part of the glass surface, from which they are finally emitted
to the vacuum because of the external fields.
These simulations reveal that the conditions required in
order to achieve the emission reported by Geis et al. are
actually reached, both in the activation step of the device and
JVST B - Microelectronics and Nanometer Structures
113
FIG. 8. Electric emission from the surface of the glass. The glass contains a
net positive charge of 105e because of the ionized cesium and an external
field of 10 V / cm is applied. Straight vertical trajectories between the surface of the glass and the anode are assumed.
in its subsequent unbiased working mode. It also appears that
the fields below the metallic electrodes are sufficient to justify an emission from these electrodes to the glass surface.
Further analysis should include the mechanisms that determine the filling of the surface states. The emission associated
with this filling may come from either the protrusions of the
metallic contacts or from the triple junction where the glass,
the right metallic contact, and the glass meet along the same
line.
ACKNOWLEDGMENTS
This work is partially supported by a SBIR Phase I grant
from the US Department of Energy under Contract No. DEFG02-05ER84342. One of the authors 共A.M.兲 was supported
by the National Fund for Scientific Research 共FNRS兲 of Belgium. The authors acknowledge the use of the Interuniversity Scientific Computing Facility 共ISCF兲 and the Belgian State Interuniversity Research Program on “Quantum
size effects in nanostructured materials” 共PAI/IUPP5/01兲.
The authors are grateful to M.W. Geis, P. Lerner, and G.
Mahan for useful discussions.
APPENDIX: MODEL FOR THE FIELD IONIZATION
OF CESIUM
We describe in this appendix the model used to compute
the field ionization of the glass. For this Cs2Si4O9 material,
the ionization density is dominated by cesium. The electron
affinity of cesium is indeed smaller than that of silicium and
oxygen. These two elements may however constitute, after
cesium, an additional and non-negligible source of charge
entrapment. Besides field effects, ion bombardment from the
anode may contribute as well to the ionization process.32 The
current analysis will, however, focus on the first effect. In
particular, we will consider that the effect of the local field F
is to reduce the potential barrier that otherwise keeps the
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Mayer et al.: Modeling of field-assisted emission
114
electrons bound to their atom. Field ionization then occurs
either by thermally enhanced excitation above this field reduced barrier or by tunneling through it.
The potential energy along the direction of the field is
given by
V共r兲 = −
1 Zeffe2
− eFr,
4␲⑀0 r
共A1兲
where the effective nucleus charge Zeff is that relevant to the
electron in the highest occupied atomic level and r is the
distance to the atomic nucleus. The lowering of this barrier is
then given by
⌬V = − 2e
冑
Zeffe
冑F.
4␲⑀0
共A2兲
In the conditions of this article, we found the ionization
probability to be dominated by thermally enhanced excitation above this field reduced barrier. The density of ionized
atoms was then approximated by34
␳ionized ⯝ ␳neutrale−共IE+⌬V兲/kT Pevac ,
共A3兲
where IE is the first ionization energy and Pevac the probability for emitted electrons to be evacuated from the glass. This
equation corresponds actually to a Boltzmann approximation, which is widely used in plasma physics. The parameters
relevant to cesium are, respectively, Zeff = 13.65 and IE
= 3.9 eV.34 The density ␳neutral of cesium atoms was given a
value of 6 ⫻ 1021 cm−3, which corresponds to its concentration in the Cs2Si4O9 compound.
The parameter Pevac of Eq. 共A3兲 depends on the ability of
the emitted electrons to be evacuated from the glass. This
evacuation may proceed from a regular emission from the
surface or proceed through the metallic contacts. In both
cases, a propagation of these electrons through gap states,
local defects, or the conduction band of the glass appears as
a necessity. The probability associated with these mechanisms is described empirically by the parameter Pevac, which
is given a value of 10−4. Considering lower values reduces
the density of ionized atoms in the glass, which essentially
affects the surface electric field just below the left metallic
contact. As discussed in the text, the surface emission essentially depends on the apex value of the potential barrier between the surface of the glass and the anode. It turns out that
the calculated emission does not depend critically on the
particular value of Pevac 共a reduction of this factor by two
orders of magnitude results in an increase of the surface
emission by a factor 3 only兲.
J. Vac. Sci. Technol. B, Vol. 25, No. 1, Jan/Feb 2007
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