J Comput Electron (2014) 13:818–825
DOI 10.1007/s10825-014-0592-x
Poisson–Nernst–Planck model for an ionic transistor based
on a semiconductor membrane
Alexey Nikolaev · Maria E. Gracheva
Published online: 22 July 2014
© Springer Science+Business Media New York 2014
Abstract In this paper we developed a Poisson–Nernst–
Planck model of an ionic current flowing through a nanopore
in a layered solid-state membrane made of a single highlydoped n-Si layer sandwiched between two thick oxide layers
which we call the ionic transistor. We studied this layered
membrane for a range of source-drain voltages while keeping the gate (the semiconductor membrane) voltage fixed
at a certain value, which was later varied too. We find that
for this ionic transistor to be effective in controling the ion
fluxes through the nanopore, the gate voltage must be kept
relatively large. Another solution could be to increase the
surface negative charge on the membrane or to replace the
outer oxide layers with the semiconductor material, such as
the p-Si material. The developed model can be applied to
study ionic filtering and separation properties of membranes
of different composition and nanopore geometries.
Keywords Nanopore · Semiconductor membrane ·
Poisson–Nernst–Planck model · Ion flux · Computations ·
Nanotechnology
1 Introduction
The Poisson–Nernst–Planck (PNP) model is a continuum
theory which was originally developed to calculate ionic
fluxes through a biological ion channel [1]. The PNP model
allows to evaluate the ionic current passing through the channel and the channel’s permeability to ions [2]. In the PNP
model, the Poisson equation is applied to describe the elecA. Nikolaev · M. E. Gracheva (B)
Department of Physics, Clarkson University, Potsdam ,
NY 13699, USA
e-mail: gracheva@clarkson.edu
123
trostatic potential in the system, with the gradient of this
potential (the electric field) serving as a driving force for
the ionic motion. In addition, the Nernst–Planck equations
describe the electrodiffusion of ions due to the gradient in
ion concentration and the gradient in electrostatic potential.
The Nernst–Planck equations are coupled with the Poisson
equation to form the complete PNP model. The PNP model
is a very well established model, which, despite it’s known
limitations, is widely used in biology, chemistry, engineering
and physics [3,4].
Recently, a considerable attention has been drawn to
the nanofluidic ionic diodes and transistors [5–13]. These
solid-state devices allow for control over the ionic and
molecular flows through a channel in the device or in the
environment in close proximity to the device due to their
nanoscale features. For the ionic flow through a nanometer diameter pore in a solid-state membrane the diode-like
and transistor-like ionic transport characteristics are usually obtained due to a non-uniform fixed surface charge
which is patterned on the channel walls, as it was done
in [5]. In a semiconductor membrane, the non-uniform electric potential in a channel or a pore can be achieved by varying the doping concentration of the semiconductor material, and as a consequence, the non-uniform distribution of
the ionic species may be controlled by the voltage applied
to the semiconductor membrane or a specific layer in this
membrane, as it was shown in [14–16]. Asymmetric shape
of the nanopore also leads to the non-linear ion currentvoltage characteristics [6–8,14–16]. Alternatively, the same
effect can be obtained by varying the pH of the solution
on either side of the membrane, in which case the resulting transistor-like ionic transport characteristics are apparent [8,12].
In the recent years, the ability of the gate-operated
nanopore in a solid-state membrane to exert influence on
J Comput Electron (2014) 13:818–825
biological molecule transversing the nanopore has become
increasingly appreciated in the view of a truly breakthrough
potential in cheap and ultrafast DNA sequencing with applications in research, national security and personal medicine [17–19]. New materials, such as ultrathin graphenebased membranes, are being discussed as the next evolution
of these devices [20–22].
Nanopore-carrying multilayered semiconductor membranes in a variety of designs have been introduced and studied by us previously [14–16,23]. In particular, we have shown
that unlike dielectric membranes that always attract cations at
the electrolyte/membrane interface whenever a negative surface charge is present, the properties of the double layer in
semiconductor membranes can be adjusted in a broad range.
We found that in the presence of a dopant charge in the
semiconductor membrane, the shape of the nanopore and the
fixed negative surface charge resulting from the pore fabrication process have competing influences on the double layer
formation [14]. Inversion of the electrolyte surface charge
from negative to positive is observed for n-Si membranes,
while no such effect occurs for dielectric and p-Si membranes.
Furthermore, we have already shown that a nanopore in a
silicon membrane can be used as an electrically tunable ion
filter [15,23]. Other structures, studied by us, include double
and triple layered n-Si and p-Si membranes [15,16]. The
rectification of ionic transport through a nanopore in a double layered p − n semiconductor nanopore was observed,
while transistor-like ionic current blocking and switching
has been characteristic of the triple layered n − p − mn
membrane [15,16].
Here, we apply the Poisson–Nernst–Planck model to calculate the ionic current through the nanopore in a singlelayer semiconductor membrane made of n-doped silicon
material. We sandwich this electrically active layer between
two thick layers of dielectric silicon dioxide material (each
of these layers is as thick as the central n-Si layer of the
membrane). This membrane is similar in operation to a
p − n − p (a triple-layered) membrane introduced and
studied by us before [16], however the present setup is
much more simpler in design. With this simplified system,
while utilizing the PNP model, we demonstrate that the
nanopore may be used to control the ionic current through
the membrane while the middle n-Si layer serves as an
electrode gating the flow of ions. We currently explore
the model developed here on more elaborate membrane
setups, designed for the ion flow and biomolecule manipulation.
In the present paper, the membrane and nanopore
geometry is described in Sect. 2, the mathematical formulation of the PNP model is given in Sect. 3, and the
obtained results are presented in Sect. 4, we conclude with
Sect. 5.
819
2 Semiconductor membrane device
The device studied here is similar in operation to other experimentally and theoretically studied solid-state transistor-like
devices [5,6,9,17,24–26]. While the material make up of
these extensively studied devices is different, they all pursue
the same goal: these devices aim to either detect or control
the electric potential inside the nanoconstriction (a nanochannel or a nanopore), and to exert influence on the dynamics
of biomolecules (i.e. DNA) and/or ions passing through the
said constriction.
The uniqueness of our device is in the explicit use of semiconductor materials in the membrane composition. Silicon
based, capacitor-like membranes are currently used by an
experimental group [25], while silicon nanowires are also
used in membrane designs developed and studied by another
experimental group [24], in both instances the nanopore
membranes are used for DNA characterization.
Our ionic transistor has the following design. The solidstate membrane is composed of three layers: SiO2 material,
n-Si material, and again SiO2 material. The electrically active
middle n-Si layer, sandwiched between dielectric SiO2 layers, is 8 nm thick, and the total membrane thickness is 26 nm.
A cylindrical nanopore with the diameter of 2 nm is formed in
this membrane, as shown in Fig. 1. Thus, we have a problem
with cylindrical symmetry. Even though we consider a cylindrical nanopore, the shape of the pore can be easily changed
to a double conical or any other shape, which introduces
additional geometrical effects on the electrical potential and
the concentrations of ions in the pore. The n-Si membrane
in the pore region is isolated from the solution by a 1-nm
thick layer of SiO2 . We apply voltage VG to the middle n-Si
layer, which acts as a gate electrode for the ions of the aqueous solution, in which the membrane is immersed. The electrolyte concentration is [KCl]0 =0.1 M. The Debye screening
Fig. 1 Triple-layered membrane: the membrane is made of n-Si material sandwiched between two layers of SiO2 material. The insert shows
the ion distribution inside the nanopore while the middle n-Si layer (the
gate electrode) is biased at VG = 0: in this situation Cl− ions are in
excess at the center of the nanopore, while K+ ions are in abundance at
the nanopore entrance and exit
123
820
length of the membrane electric potential is λ D =0.96 nm
for this solution concentration. Since the nanopore diameter is 2 nm, this allows us to use of the PNP model as the
validity of it for charged membranes with nanopore radii
down to Debye length was confined previously by Brownian
Dynamics simulations and Monte Carlo simulations [27,28]
concentration of the counter ions. For a charged channel
the paper [27] reports lower. This is the situation in our
nanopore membrane, where a moderate charge is always
present.
The surface of our membrane contains negative charge
with the density of σ = −0.16 e/nm2 . Two thick oxide layers play crucial part in the membrane setup: the negative
surface charge on the oxide-electrolyte boundary attracts
[K+ ] ions, producing (at the center of the nanopore) the
local concentration of potassium ions [K+ ] about four time
larger than the bulk concentration, and the local concentration of chlorine ions [Cl− ] approximately 0.25 of the bulk
concentration. These concentration values are obtained at
zero applied electrolyte bias. Even larger charge separation is possible, if the membrane negative surface charge is
larger, or if additional p-Si layers are added instead of SiO2
layers, as it was done in [16], however, here we develop
the PNP model for the ionic current that was not implemented in our earlier work. In other words, we develop a
more sophisticated model for a less complicated membrane
system.
Due to the presence of three distinct solid state layers in the
membrame, the electrolyte inside the pore also forms three
distinct regions, Fig. 1. The two regions adjacent to the top
and bottom silicone dioxide layers (carrying negative charge
at the surface) are similar to the heavily doped Source and
Drain regions of the mosfet. In these electolyte regions, the
concentration of potassium ions is greater than that of the
chlorine ions, [K+ ][Cl− ]. The central region is adjacent
to n-doped silicon layer, but insulated from it by a 1-nm
thin layer of oxide. In the unbiased state, [Cl− ]>[K+ ] in the
central pore region, and the whole ionic transistor is similar
to the p-channel enchancement-mode (normally off) mosfet
[29]p. 362. A non-zero voltage VG must be applied to turn the
mosfet on.
The voltage terminals that are applied at the top and bottom compartments filled with electrolyte act as the Source
and the Drain terminals. The geometry of our system has
the mirror symmetry with respect to the plane of the membrane, therefore, the difference between these two terminals
is purely nominal. We call the left terminal the “Source”,
V S , and the right is the “Drain”, V D . For all the following
calculations, we assume that our membrane (the ionic transistor) and the voltage sources form a common-source electric
circuit.
The membrane/electrolyte system is characterized by the
following variables: ϕ(
r ), the electric potential; n(
r ) and
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J Comput Electron (2014) 13:818–825
p(
r ), the concentrations of electrons and holes in the semir ) and [Cl− ](
r ), the
conductor material in membrane; [K+ ](
concentrations of potassium and chlorine ions in the electrolyte solution (KCl).
3 Method
3.1 Poisson- -Nernst–Planck model
We placed the origin of our coordinate system at the center
of the pore, whereas the x-axis in this set-up coincides witht
the pore axis. The concentration of ions varies in the pore
region, reflecting the charge distribution in the membrane
layers and screening of the electrical potential in the pore
vicinity. To obtain the electrostatic potential distribution ϕ(
r)
generated by the membrane and the electrolyte, we solve
Poisson equation [14–16,30]:
∇ · (ε(
r )∇ϕ(
r )) = −ρ(
r ),
(1)
where ρ(
r ) is the charge density, ε(
r ) = ε0 εr (
r ), with εrSi =
Si
11.7 in silicon (Si), εr = 3.9 in silicon dioxide (SiO2 ), and
εrSi = 78.0 in the electrolyte, ε0 is the permittivity of free
space.
The charge density ρ(
r ) is given by:
ρ(
r ) = e{ p(
r ) − n(
r)
+ Nd+ (
r ) − Na− (
r ) + Nsur f (
r)
r ) − [Cl − ](
r )},
+ [K + ](
(2)
where e is the elementary charge, Na− and Nd+ are the acceptor and donor densities in the semiconductor regions, and
Nsur f is the equivalent volumetric density of the fixed oxide
surface charge (σsur f ) on the membrane surface at the interface with electrolyte.
In the semiconductor regions, n and p are governed
by Fermi-Dirac statistics, and concentrations are expressed
using the Fermi-Dirac integrals:
n(
r ) = Nceff F1/2
p(
r) =
Nveff
Nceff = 2
F1/2
e(ϕ − VG ) − E gSi /2
,
kb T
−e(ϕ − VG ) − E gSi /2
2π m ∗n kb T
h2
kb T
3/2
,
Nveff = 2
(3)
, where
2π m ∗p kb T
h2
(4)
3/2
.
Above, m ∗n and m ∗p are effective masses of electrons and
holes, E gSi is the energy band gap of Si, VG is the voltage
applied to the n-Si layer. More details on the electrostatic
part of the model can be found in [30].
J Comput Electron (2014) 13:818–825
821
Local concentrations of potassium and chlorine ions,
r ) and [Cl− ](
r ), in Eq. (2) are computed from the solu[K+ ](
tion of the steady-state Nernst-Planck equations [27]:
∇ · [μi Ci ∇ϕ + z i Di ∇Ci ] = 0, Ci = [K + ], [Cl − ],
(5)
c|x=±L x = [K Cl]0 ,
( J · ĵ)| y=±L y = 0,
( J · k̂)|z=±L z = 0,
( J · n̂) = 0.
δM
where D K − (Cl + ) = 1.95(2.03) × 10−5 cm2 /s is the diffusion
coefficient, μi is the mobility, Di = μi kb T /e, z i = ±1
depending on the ionic species charge sign.
Steady state Nernst–Planck equations for ionic current
densities:
Furthermore, we set the electric potential on both sides of
the membrane, away from it. In addition, that the electric field
is zero on the other box boundaries. The boundary conditions
for the electric potential are
∇ · J = 0
ϕ|x=L x = VD + VBKICl ,
∂ϕ ∂ϕ =
= 0.
∂ y y=±L y
∂z z=±L z
Current densities are
JCl = −eμ[Cl − ]∇ϕ + eDCl − ∇[Cl − ],
JK = −eμ[K + ]∇ϕ − eD K + ∇[K + ].
And the resulting equations for [Cl− ] and [K+ ] are
∇ · (−μ[Cl − ]∇ϕ + D∇[Cl − ]) = 0,
+
(6)
+
∇ · (−μ[K ]∇ϕ − D∇[K ]) = 0.
(7)
The discretization of the Nernst–Planck equations is usually much more important then the discretization of the
Poisson’s equation. This procedure is well explained in
papers [31–34].
The total current through the pore is the sum of currents of
both ionic species, K+ and Cl− through a pore cross-section
S
I = I [Cl
−]
+ I [K
+]
=
S
JCl − · d A +
JK + · d A.
S
It is natural to choose the cross-section S so, that it is perpendicular to the axis of the pore. The specific position of the
cross-section within the pore is not important, because the
correct discretization of the Nernst–Planck equations guarantees that the total ionic current is conserved at any crosssection inside the nanopore.
3.2 Boundary Conditions
The PNP model requires boundary conditions to be solved.
These are as follows. The electrolyte concentration is maintained at [K Cl]0 . We assume there is no flux of ions on
the box boundaries. Similarly, there is no flux normal to the
membrane. Mathematically, these are formulated as follows:
ϕ|x=−L x = VS + VBKICl ,
VBKICl is the built-in potential of the electrolyte with
respect to silicon. Ultimately, it depends on the material of
the drain and source electrodes. Here, it is assumed to be
equal to 0.077 V [15].
The system of equations is solved self-consistently using
a finite difference method. Potential ϕ and concentrations n,
p, [Cl − ], [K + ] are replaced with their discrete representations on a grid with a variable grid-point size from 0.2 nm
to 0.8 nm. Gummel’s method is implemented for Poisson’s
equation to speed up its convergence. Finally, all systems of
linear equations are solved using Gauss-Seidel method.
If VS = VD ≡ VD&S , there is no ionic current across the
membrane, thus the solution of the Nernst-Planck equations
(6) and (7) is the standard Boltzmann distribution:
q(ϕ − VD&S − VBKCl
−
I )
[Cl ] = [K Cl]0 exp
(8)
kT
−q(ϕ − VD&S − VBKCl
+
I )
[K ] = [K Cl]0 exp
(9)
kT
Solution of a such simplified electrostatic problem (equations 1, 8 and 9) provides initial conditions for the full PNP
problem.
4 Results
4.1 Model validation: SiO2 membrane
We validate our computational model by testing it on a
dielectric membrane with no surface charge. We compare
computed ionic curent with current estimated analytically.
This can be done easily if two facts are considered. First,
the cylindrical nanopore in such membrane must have uniformly distributed ionic concentrations [K + ] = [Cl − ] =
[K Cl]0 (hence, no diffusion contribution to the current),
123
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J Comput Electron (2014) 13:818–825
while the ionic current is produced only by the drift component qμ[K Cl]0 ∇ϕ.
Second, the magnitude of the electric field can be estimated as follows. The current of each ion species must be
constant for each cross-section, and I ∝ E · (Area). The
size of the modeled system is 17 nm × 17 nm × 53 nm, and
the diameter of the pore is 2 nm, thus the cross-sectional area
changes from (17 nm)2 outside to π nm2 inside the pore. The
ratio of this areas is ≈92. Therefore, the electric field inside
the pore is about 100 times larger than that of the outside,
Oxide, no surface charge. Cylindrical.
150
Ohm’s law
Simulation
100
I (pA)
50
I (VDS ) = 2qμ[K Cl]0 × (VDS /L) × π R 2
(10)
Here, the factor ”2” comes from the assumption that both
ion species have the same mobility μ. We take both valences
to be |z| = 1. The membrane thickness is assumed to be
L = 26 nm.
The comparison between the simulated ionic current with
the PNP model and the calculated current obtained from
Eq. 10 is shown in Fig. 2. The simulated points fit calculated
straight line very well. Also, both used assumptions are correct: the concentration of ions indeed remains constant and
the electric potential varies linearly in the pore , while the
electric potential is constant outside (results are not shown).
4.2 Triple-layered SiO2 – n-Si – SiO2 membrane
0
4.2.1 I–V curves and the transfer characteristics
-50
-100
-150
and we can safely assume that the voltage drop happens only
inside the pore.
Using Ohm’s law, the total current is estimated as:
-0.6 -0.4 -0.2
0
0.2
0.4
0.6
VDS (V)
Fig. 2 The comparison of the ionic current flowing through a cylindrical nanopore of diameter 2 nm as found from the PNP model (squares)
and from Eq. 10 (dash line). The electrolyte concentration is set to 0.1 M
and the ”source-drain” voltage, V DS , between the ends of the modeled
system is varied. The nanopore is made in a SiO2 membrane that has
no surface charge. The simulated points are in excellent agreement with
the theory
The current–voltage characteristins (the I–V curves) are
shown in Fig. 3. For negative VDS , currents almost saturate, maintaining a small slope. A negative gate voltage
VG S is necessary to increase this saturation current. The
device behavior resembles that of the standard mosfets. The
already saturated current slowly increases with VDS almost
linearly, which is similar to the channel-length modulation
in mosfets, and the Early effect in Bipolar Junction Transistors (bjt).
As it is shown in Fig. 3 (the left panel), for VG S ≥ 0,
there is a region of high conductance for small VDS (less
than 0.1 V , the right panel), and current quickly saturates
Oxide-n-Si-oxide. Cylindrical.
Oxide-n-Si-oxide. Cylindrical.
400
20
300
10
100
I (pA)
I (pA)
200
0
0
-10
-100
VGS = 0.4V
VGS = 0.0V
VGS = -0.4V
VGS = -0.8V
-200
-300
-1
-0.8 -0.6 -0.4 -0.2 0
VDS (V)
0.2 0.4 0.6
-20
VGS = 0.4V
VGS = 0.0V
VGS = -0.4V
VGS = -0.8V
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
VDS (V)
Fig. 3 The current–voltage characteristics (I–V curves) for the oxide-n-Si-oxide membrane with the surface charge σsurf = −0.16 e/nm 2 .
The full range of studied VDS voltages on the left, and only small voltages (|V DS | < 0.4 V ) on the right
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J Comput Electron (2014) 13:818–825
823
Oxide-n-Si-oxide. Cylindrical. VDS=-0.05V.
5
0.1
4
[Cl-] / [KCl]0
φ (V)
-0.1
-0.2
-0.3
-0.4
-0.5
VGS=0.4V
VGS=0.0V
VGS=-0.4V
VGS=-0.8V
-0.6
-20
-13
VGS=0.4V
VGS=0.0V
VGS=-0.4V
VGS=-0.8V
3
2
1
-4
4
13
0
-20
20
-13
4
13
4
[Cl ] / [KCl]0
-0.1
-
-0.2
-0.3
VGS=0.4V
VGS=0.0V
VGS=-0.4V
VGS=-0.8V
-13
4
13
20
4
13
20
Oxide-n-Si-oxide. Cylindrical. VDS=-0.6V.
5
VGS=0.4V
VGS=0.0V
VGS=-0.4V
VGS=-0.8V
4
3
2
0
-20
-4
x (nm)
1
-4
0
-20
20
+
0
φ (V)
-4
[K ] / [KCl]0
0.1
-13
2
Oxide-n-Si-oxide. Cylindrical. VDS=-0.6V.
5
-0.6
-20
3
x (nm)
Oxide-n-Si-oxide. Cylindrical. VDS=-0.6V.
-0.5
VGS=0.4V
VGS=0.0V
VGS=-0.4V
VGS=-0.8V
1
x (nm)
-0.4
4
+
0
Oxide-n-Si-oxide. Cylindrical. VDS=-0.05V.
5
[K ] / [KCl]0
Oxide-n-Si-oxide. Cylindrical. VDS=-0.05V.
VGS=0.4V
VGS=0.0V
VGS=-0.4V
VGS=-0.8V
3
2
1
-13
x (nm)
-4
4
x (nm)
13
20
0
-20
-13
-4
4
13
20
x (nm)
Fig. 4 The electrical potential and the ionic concentrations (K+ and Cl− ) through the central axis of the pore for V DS = −0.05 V (top row) and
VDS = −0.6 V (bottom row)
for larger VDS . A similar nanofluidic system was shown in
[10,11], and it resembles two diodes with their n-terminals
connected (as in pnp bjt). It is shown in Fig. 4 that for larger
VDS (bottom row), there is a depletion region at x ≈ 4 nm,
while for small |VDS | < 0.1 V (top row), concentrations of
ions is relatively large, hence, the
√ conductance is higher.
The transfer characteristic I versus VG S is shown in
Fig. 5. It demonstrates the efficiency of the gate voltage, and,
is obtained for VDS
√ = −1.0 V. In mosfets [29] (p. 353),
saturation current I ∝ (VG S − VT ), where VT is called
the threshold voltage. Thus, the transfer characteristic is
expected to be linear for VG S < VT , however, this is not
the case: Assuming that VT ≈ 0 V , the curve becomes nonlinear for VG S < −0.6 V , because current does not reach
saturation for these voltages. Therefore, on the one hand, the
transfer characteristic should be obtained for stronger drainsource bias, such as VDS = −1.0 V . On another hand, we
cannot choose VDS < −0.6 V , because the excess concentration of [K + ] on the right side of the pore vanishes for strong
negative source-drain voltage VDS . Therefore, the system
requires an improvement: there should be higher concentration of the excess [K + ] in the channel. One of the possibilities
is to change the oxide surface charge from −0.16 e/nm 2 to
−0.5 e/nm 2 , which is done in the next section.
4.2.2 Modified system: the increased surface charge
σsurf = −0.5 e/nm2
In this case, the excess concentration of [K + ] is almost twice
larger (≈8 × [K Cl]0 ) than it was before, and it provides a
basis for stronger performance of the device: the differential
conductances remain flat for saturated I-V curves even for
VDS = −1.4 V, and the new transfer characteristic is linear, with the slope 21.4 pA1/2 V−1 , and the threshold voltage
VT ≈ 1.08 V. It is important to note that the modified transistor is on for zero gate-source voltage, so it is a depletionmode fet.
The current–voltage curves are shown in Fig. 6. It can be
seen clearly that for the negative VDS currents practically saturate (although, they still have a small slope). A negative gate
voltage VG S is necessary to increase this saturation current.
The transfer characteristic demonstrates the efficiency of the
gate voltage: the current begins to increase at VG S ≈ −0.1 V
with almost constant transconductance g ≈ 359 pA/V. However, the current for VG S < 0.6 V does not saturate at reasonable drain-source voltages, and it makes it difficult to estimate
the exact shape of the transfer characteristics, it is possible
that the ideal curve should be a quadratic function of VG S
[29] (p. 353), and not linear. However, in general, the device
123
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J Comput Electron (2014) 13:818–825
Oxide-n-Si-oxide. Cylindrical. VDS = -1.0V.
0
I
1/2
(pA)
1/2
-5
-10
-15
-20
Fit: (21.4 VGS - 23.2) pA
Simulation
-25
0
0.2
0.4
0.6
0.8
1
1.2
1.4
VGS (V)
Table 1 Values of model parameters utilized in the simulations
Parameter
Value
E gSi
1.124 eV
E gSi O2
9.0 eV
qχ Si
4.05 eV
qχ Si O2
0.95 eV
ε Si
11.70
ε Si O2
3.9
Nd+
Na−
2.0 × 1020 cm−3
Nsur f
−4.0 × 1020 cm−3
2.0 × 1020 cm−3
T
300 K
ε K Cl
80.0
[K Cl]0
0.1 M
Fig. 5 The transfer characteristics for the oxide-n-Si-Oxide system
with surface charge σsurf = −0.16 e/nm2
Oxide-n-Si-oxide. Cylindrical.
200
I (pA)
0
-200
-400
-600
VGS = 1.2V
VGS = 0.8V
VGS = 0.4V
VGS = 0.0V
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4
VDS (V)
Fig. 6 The current–voltage characteristics for the modified system
with a greater surface charge of σsurf = −0.5 e/nm2
behavior is very similar to a standard mosfet, where the
saturated current slowly increases with VDS almost linearly.
More precise determination of the characteristics may be
done. Nevertheless, shown results are already very promising, and the original design can be enhanced to reveal other
properties of the system, as well as to improve the performance of the device (Table 1).
5 Conclusion
In this paper we developed a Poisson–Nernst–Planck model
of an ionic current flowing through a nanopore in a layered
solid-state membrane made of a single highly-doped n-Si
layer sandwiched between two thick SiO2 layers which we
call the ionic transistor following [8]. We studied this layered
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membrane for a range of source-drain voltages while keeping the gate (the semiconductor membrane) voltage fixed at
a certain value, which was later varied too. We find that for
this ionic transistor to be effective in controling the ion fluxes
through the nanopore, the gate voltage must be kept relatively
large. Another solution could be to increase the surface negative charge on the membrane or to replace the outer oxide
layers with the semiconductor material, such as the p-doped
Si material.
The developed model is very general, and as such, can be
applied to study ionic filtering and separation properties of
membranes of different composition and nanopore geometries.
Acknowledgments This work was supported in part by NSF Grant
No. CBET-1119446 and by XSEDE Award for computational resources No. TG-PHY110023.
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