Armenian Journal of Physics, 2014, vol. 7, issue 3, pp. 136-146
CHARGE CARRIER’S DISTRIBUTION
IN THE INVERSION CHANNEL OF NANOSIZED FETs
F.V. GASPARYAN
Yerevan State University, 1 Alex Manoogian St., 0025 Yerevan, Armenia
e-mail: fgaspar@ysu.am
Received 7 May 2014
Abstract – The charge carrier’s distribution in the inversion layer of the nanowire based
field-effect transistor is analyzed using classical and quantum-mechanical evaluation of
carrier distribution. It is shown that size quantization essentially changes carrier transport
through the channel.
Keywords: nanosized FETs, inversion channel, size distribution
1. Introduction
Semiconductor nanosize devices on the base of nanowires (NWs) and nanotubes (NTs) have been
the subjects of comprehensive research in recent years due to their unique and specificelectrophysical, optical, magnetic and mechanical properties. NWs and NTs are promising candidates for
application in a variety of fields, such as engineering, electronics, optoelectronics, biophysics,
biomedicine [1-5]. Recently a large progress has been made in fabrication, device physics, modeling
and simulation of electrical propertiesof NW and NT field-effect transistors (FETs) [6,7]. Among
other electronic devices silicon nanosized FETs play a principal role in modern electronics due to
their compatibility to CMOS process. The small NW cross-section offers a significant gate control
over the drain current, therefore NW FETs represent ultimate building blocks for nanoelectronics.
However, for development of ultrasensitive devices, such as bio-chemical and gas sensors,
deepunderstanding of the transport mechanisms in FET structures is of crucial importance. Currently
the unified charge control model is generally accepted for metal-oxide-semiconductor FETs [8-10].
Gasparyan || Armenian Journal of Physics, 2014, vol. 7, issue 3
Gate voltage can be applied both: as usual metal electrodeand through the reference electrode using
electrolyte medium.Liquid-gated FETs are widely used in bio-chemical sensors [11, 12]. In case of
liquid-gated NW FETs there are still a lot of open questions. For example, nowadays, two main
concepts are considered for optimization sensitivity and selectivity of Si NW sensors: using the subthreshold mode or above-threshold mode [12,13].
The distribution of the mobile charge carriers determines the static and dynamic electrophysical and optical behavior of the transistor. Indeed, the distribution of the carriers in the NW
FETs channel may differ from its classical counterpart due to quantum confinement. It may lead to
overestimation of different kinds of FET parameters during fitting. Quantization may result in the
shift of location of the mobile carrier’smaximum density from the front gate interface. This can be
considered to be equivalent to increasing of the effective tunneling distance to the trap(s) located in
the front oxide layer. Note that usually, in case of very small channel cross sections (
10
10
cm ), the only single trap may determine the channel current behavior.
The influence of size quantization on the current transport mechanisms into the nanosize
channel of liquid-gated FETs has not yet been reported.
In this paper we study charge carrier’s distribution and conditioned by themunique properties
of liquid-gated NW FETs.It is shown the importance of considering of the quantization effect in the
inversion layer.
The schematic picture of theinvestigated structure presented in Fig.1.Here RE is the
reference electrode,
is the applied gate voltage,
is the source-drain current, ,
of the NW, by FOX signify front oxide layer, and byBOX - buried oxide layer.
137 and
are sizes
CHARGE CARRIER’s DISTRIBUTION || Armenian Journal of Physics, 2014, vol. 7, issue 3 2. Distribution of the charge carriers in the inversion channel
2.1. General considerations
The statistical and dynamical behavior of the source-drain current is defined unambiguously by
the distribution of the mobile charge carrier’s concentration over the conducting channel. We
consider the case of inversion n-channel liquid-gated FET andtherefore majority of processes in the
structure is determined by the electrons. Obviously, concentration of mobile carriers in the channel
depends on the coordinate
(see Fig. 1) and applied gate voltage. At the same time FOX-NW
surface concentration is only gate voltage dependent. Hence overall concentration can be presented
as follows:
,
Here
,
.
is the electron surface concentrationper unit area at the FOX interface and
(1)
,
in
[cm-1] is the some characteristic function, which describes the charge carrier distribution in the
plane of the channel (see Fig.1).
Vg
z
RE
Electrolyte
F O X FOX y
W
n+-S
t
0
p-Si NW
Vsd
n+-D
Isd
BOX
p-Si substrate
L
Vbg
x
Fig. 1. Schematic of the p-Si NW channel between FOX
and BOX layers, source (S) and drain (D).
138 Gasparyan || Armenian Journal of Physics, 2014, vol. 7, issue 3
The surface concentration can be described using the unified charge control model. For the
investigated structure it can be defined from the following expression [13]:
ln
,
Here
isthethreshold voltage,
is the electron charge,
.
,
(2)
is the capacitance of FOX layer,ns,t is
the surface density of electrons per unit area at the threshold voltage (
at
,
), and is
the factor of the transistor non-ideality:
1
1
,
is the capacitance of silicon depletion layer;
space and silicon, correspondingly;
Si-NW;
,
and
are the dielectric permittivities of free
is the doping acceptor concentration in both Si-substrate and
is the thermal voltage.
value, which depends on
It should be noted that influence of electrolyte is included into
potentials of the semiconductor, BOX, FOX, NW layers and electrolyte and can be presented as
follows [11, 14-18]:
2
,
(3)
;
,
0;
2
Here
is the flat-band voltage,
0;
,
,
;
;
;
,
;
,
and
.
,
are the electric
is the surface potential of the NW-FOX
is the intrinsic carrier concentration in bulk silicon;
139 ln
is the Fermi potential;
potentials of the bulk solution and bulk silicon substrate;
interface;
2
is the capacitance of BOX
CHARGE CARRIER’s DISTRIBUTION || Armenian Journal of Physics, 2014, vol. 7, issue 3 layer;Φ and Φ
are the work functions of the silicon and oxide layer, correspondingly;
is the molar
are the dielectric permittivities of water and electrolyte, correspondingly;
is the molar concentration of hydrogen ions at oxide
concentration of cations in the electrolyte,
surface;
and
is the solution molar concentration;
,
and
,
are the trap surface concentrations
and
per unit area in the FOX-NW and BOX-NW interfaces, correspondingly;
are the
charges of front and buried oxide layers, correspondingly.
Concentration
,
can be expressed as
.
,
(4)
For the further calculations it is necessary to define surface potential of the NW-FOX interface
. It
can be calculated using Eq. (3) and density of minority carriers per unit area:
ln
ln ln 1
.
in terms of
Note that Eq. (2) has no analytical solution for
(5)
. The following approximate
solution is suitable for the strong inversion and sub-threshold regimes [17]:
2
After determining
,
ln 1
, we calculate
,
.
(6)
according to classical and quantum-mechanical
approaches in order to evaluate the influence of peculiarities in carrier distribution for both cases on
the physical processes taking place in the channel.
2.2. Classical approach
In order to find the function
classical dependence of
,
for the case of classical approach we use the following
[18]:
.
140 (7)
Gasparyan || Armenian Journal of Physics, 2014, vol. 7, issue 3
Here
is the density of states in the conduction band of a semiconductor,
potential at the FOX-NW interface. To determine
is the contact
we have to solve the Poisson equation:
,
where
(8)
is the space charge density for the fully ionized acceptor (boron in Si) centers:
1
Here ,
and
,
.
(9)
are the carrier’s non-equilibrium and equilibrium concentrations, respectively.
We can use the following boundary conditions for solving Eq. (8) (see Fig.1):
∞
0,
0
.
(10)
Using Eqs. (9) and (10), one can obtain the following solution of Eq. (8):
1
,
(11)
where
⁄
,
,
is the Debye’s screening length.
Then using expression for
,
ln
from Eq. (5), finally we have
ln ln 1
141 .(12)
CHARGE CARRIER’s DISTRIBUTION || Armenian Journal of Physics, 2014, vol. 7, issue 3 Concentration of electrons in the channel,n(x), cm-3
2.5x1019 1019 4V
7.5x1018 3V
5x1018 2V
2.5x1018 1V
0
10
30
40
50
20
Coordinate of x direction in the channel from FOX surface, nm
calculated for the several
Fig. 2.The carrier concentration
,
values of gate voltages at 300 K.
,
Figure 2 shows classical dependencies of
calculated using Eq. (12) and parameters
described below. For numerical computation we use the following values, which correspond to
sample
geometry
0.026 eV
50 nm,
80,
and
typical
300 K ,
8.85
of
materials
200 nm,
10
F⁄cm,
for
investigated
0.015 mol⁄L,
10 cm ,
100 nm,
78,
parameters
9 nm,
0.26
structure:
0.001 mol⁄L,
500 nm,
11.6,
3.9,
.
2.3. Quantum-mechanical approach
The quantum-mechanical (QM) distribution of mobile carriers within the inversion layer in
the NW FET can be obtained by solving self-consistently the Schrödinger equation and Poisson
,
equation. QM calculation gives the following result for thecharacteristic
,
| |
,
where
142 function[19,20]:
0 ,
(13)
Gasparyan || Armenian Journal of Physics, 2014, vol. 7, issue 3
| |
(14)
and
,
is the Airy function,
NW channel,
is the effective electronic mass,
is the electric field strength in the
is the quantized energy levels for electrons of the inversion channel in a triangular
0. According to Ref. [21] the
is the i-th solution of the equation
potential well [21],
surface electric fields are typically of 10
for silicon
(15)
10
V⁄cm, energy levels
0.03
0.06 eV and
is equal to2.338. Using Eqs. (1), (13), (14) and above listed parameters for investigated
structure we have computed dependences of
,
and
,
for different gate voltages. Figure
3 shows the obtained results. As one can see, the concentration of mobile charge carriers in the QM
approach differs considerably from the classical case (Fig.2). The curves have well pronounced
peaks near the interface.
Concentration of electrons in the channel, n(x), cm-3
5x1019 4V
4x1019 3x1019 3V
2x1019 1x1019 2V
1V
0 0
1
2
3
4
5
Coordinate of x direction in the channel from FOX surface, nm
calculated using
Fig.3.Dependences of carrier concentration
,
Eqs. (1), (6), (13) for several values of gate voltages atT=300 K.
143 CHARGE CARRIER’s DISTRIBUTION || Armenian Journal of Physics, 2014, vol. 7, issue 3 Such a behavior is caused by the quantization effect in the triangular potential well near the
FOX-NW interface (see Eq. (15)). Increasing of the gate voltage results in increasing of the maximal
concentration value and shift of the maxima towards the FOX-NW interface. Majority of the
electrons are located near the FOX layer and occupy the region of 1-2 nm depth in the channel.
More than 90% of the channel thickness does not contribute into the dynamical processes and is in
passive state. In QM case a conception of the uniform inversion layer approximation becomes
inappropriate. It should be emphasized that in the QM case the maximal values of concentration are
one order of magnitude higher than in the classical distribution case. After defining charge carrier
distribution for classical and quantum cases we can evaluate the drain current of the Si NW
transistor for both cases.
3. Conclusion
It is shown that inthe inversion channel of the nanosize FET are significant difference
between charge carrier distributions forms in classical and QM approaches. This difference can have
strong influence on the carrier transport through several nanosize field-effect structures. In classical
approach the charge carrier distribution has a maximum at the interface FOX-NW, whereas in the
QM approach the maximum displaces away from the interface. The value of electron concentration
increases with increasing of the gate voltage as well as its maximum relocates closer to the FOXNW interface. Majority of the electrons concentrates near the FOX surface and occupies the region
from 1 to 2 nm. This fact should be taken into account in designing submicron devices. Those
results can be useful both for deep insight and for accurate qualitative and quantitative description of
the physical processes taking place not only in the electrolyte-gated Si NW FETs but also can be
144 Gasparyan || Armenian Journal of Physics, 2014, vol. 7, issue 3
applied for the several nanosize FETs. Classical and QM models of electron charge distribution can
be applied to find appropriate description of physical phenomena which takes place.
Acknowledgment: Author greatly appreciates the support of German Academic Exchange
Service (DAAD) for the research grant.
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