Quantum Mechanical Effects Correction Models for
Inversion Charge and Current-Voltage (I-V) Characteristics of the MOSFET Device
H. Abebe* and E. Cumberbatch**
Nanotech2003 1
*
University of Southern California, Information Sciences Institute, MOSIS Service, 4676 Admiralty Way,
Marina del Rey, California 90292, USA, abebeh@mosis.org
**
Claremont Graduate University, Department of Mathematics, Claremont, California 91711, USA,
ellis.cumberbatch@cgu.edu
ABSTRACT
Analytic 1-D quantum mechanical effects correction
formulae for the MOSFET inversion charge and I-V
characteristics are derived from the density gradient (DG)
model using matched asymptotic expansion techniques.
Results for the classical drift-diffusion (DD) equations
using asymptotic techniques have been achieved by Please
[1], Ward [2] and Markowich [3]. Ward's results were
improved in [4] to achieve explicit formulae for MOSFET
I-V characteristics, which are accurate over the range of
device voltages.
Ancona, [6], introduced the DG theory to model
quantum effects in electron and hole transport equations.
The numerical simulation results of the I-V and the
capacitance-voltage (C-V) characteristics using the DG
model showed good comparison with data, see [7,8]. This
numerical approach has been useful at the device
simulation level. However, for circuit analysis applications
a simple analytic model is required.
Keywords: Device modeling, MOSFET, Quantum effects,
SPICE.
1
INTRODUCTION
When the dielectric thickness of the MOSFET device is
reduced below 4 nm, quantum mechanical (QM) effects
near the silicon/silicon-oxide interface become significant,
and the accuracy of the current SPICE quasi-physical
models deteriorates rapidly [6-8].
QM tunneling and confinement affects the profile of the
inversion charge in the direction perpendicular to the
interface. The laws of quantum mechanics require that the
electron density vanish at the interface, but the classical
theory based on the drift-diffusion model gives maximum
charge density at the interface. This is the main discrepancy
between the classical and quantum models.
Several models have received attention. Full solutions
of the Schrodinger and Gauss equations, and various other
1
approximations, all require high level numerical
simulations. Accurate modeling of QM effects in
MOSFETs requires a multi-dimensional solution of the
Schrodinger and Gauss equations. This multi-dimensional
microscopic solution is very difficult for practical circuit
application. The DG theory is a macroscopic approach of
modeling QM effects that can be generated directly from
QM by taking the lowest order solutions. The quantum
mechanical model followed here was introduced by
Ancona, [6-8], and this is being solved for variety of
devices since it has shown good accuracy compared with
more complete models. The DG model has been found to
adequately account for QM effects for applications of
interest in semiconductors [8-10].
The usual DD current density model is obtained by
assuming that the internal energy densities of electron and
hole gases have a logarithmic dependence on the charge
densities. A more general series expansion of the density
using the kinetic theory of gases gives that the energy
depends not only on the density but also on the gradient of
gas density [6,8]. This is the central assumption of the DG
theory and it is derived from electron and hole kinetics by
applying hydrodynamic theory. The model has been
compared with quantum microscopic solutions for
tunneling and confinement effects; see [6], p. 7964, [9], p.
1228, and [10], p. 9537. These comparisons show good
agreement with the quantum solutions.
The results of the DG model show that the charge
density is reduced significantly in a small layer close to the
silicon/silicon-oxide interface, but its behavior outside this
layer is similar to the non-quantum, classical solution. This
is known as a boundary layer phenomenon [11] in other
areas of physics such as fluid or solid mechanics. Boundary
layers most often occur when a small parameter multiplies
the highest order derivatives. This is evident in equation (3)
2
in which the term ( Bn ) multiplying the derivative on the
−9
L.H.S is O (10 ) . As a consequence of this we approach
the solution of equation (3) as a boundary layer problem
and we present the results of that solution in what follows.
This work is given more completely in [12]. Since the
The 2003 Nanotech Conference Proceedings, Feb 23-27, San Francisco, U.S.A.
governing equations for electron and hole current flow are
similar, our work here is concentrated only on the NMOS
device, in which electrons are the dominant carriers.
The scaling introduced in (2) yields O (1) changes in
the scaled potential over O (1) changes in the scaled
distance x that represent the depletion depth. The factor
Bn2 λ /(lnλ ) 2 is O(10 −4 ) for λ = 10 7 ( N = 1017 ) .
2
MODEL EQUATIONS
In the DG theory, the classical electrostatic potential V
has a quantum correction Vqn [7, equation (5)],
Vqn = φ + Vth ln( n / ni ) − V
where Vqn = 2bn (∇
2
(1)
n ) / n , bn = h 2 /(12 me q ) .
This indicates that the quantum correction term is
significant in a layer much smaller than the depletion length
and a fraction of the length scale of the inversion layer. The
correction term may be disregarded outside this narrow
quantum layer. This is a typical boundary layer
phenomenon. Numerical solutions confirm this behavior
showing that the QM effect on the electron density is
substantial in a narrow layer close to the oxide interface,
reducing it from high values to zero at the interface. (See
Figure 1, numerical data provided by Asenov et al [5]). The
density is zero at the oxide interface with the inversion
0
This representation of Vqn in terms of n is obtained by the
inclusion of the density gradient dependence. ( h is Plank
constant divided by 2π , m e effective electron mass, q
electron charge.) Equation (1) is the DG approximation of
the Schrodinger equation with wave function n , where
n is the carrier charge density, Vth is the thermal voltage,
0
charge peak at 5A to 15A . The solution of (3) in this
narrow layer is called the inner solution. Outside the
quantum layer the electron density resembles a classical
profile, though not the classical profile at the same
parameter values. The outer solution for the quantum
problem is a classical solution shifted by the effect of the
quantum layer. (The outer/inner terminology is that used in
asymptotic theory.) Over the inner quantum layer the
electrostatic potential changes much less dramatically than
the electron density.
and ni is the intrinsic density. An alternative interpretation
is that the right-hand side of (1) represents the Boltzmann
statistics for electrons and the left-hand side is a quantum
mechanical correction to the Boltzmann statistics.
The boundary layer method is facilitated by the use of
approximate solutions valid in different regions of the x1
domain, of size dependent on parameter scalings of the
dependent and independent variables. Here we apply
Ward's parameter scaling that is used in [2] for the coordinate perpendicular to the channel, electrostatic
potential, quasi-Fermi potential
parameters, and
drain/source voltage, respectively:
_
_
x1 = xLd ln λ / λ , (V , φ ) = ( w, φ )Vth ln λ , Vds = V ds Vth (2)
where λ = max
N ( x1 )
, N ( x1 ) is the channel doping
ni
density, and Ld is the Debye length.
Rewriting equation (1) using the scaling in (2) gives,
Figure 1: Electron density profile perpendicular to the
inverted channel. Relative effective mass of electron in
density gradient is 0.2, effective gate voltage 2V, substrate
17
−3
doping 5 X 10 cm , and oxide thickness 4nm.
Bn2 λ
d 2 n _ ln( n / n i )
=φ+
−w
2
ln λ
n dx
(ln λ ) 2
where Bn = 2 bn /Vth Ld .
2
2
(3)
In this work we make the assumption that the electrostatic
potential in the inner quantum layer is a constant. The
composite solution is given as sum of the inner and outer
expansions. This sum is then corrected by subtracting the
common part. In our case the common part is found to be
the peak quantum charge density, n0 . Thus
n( x ) = nout − (n 0 − nin )
(4)
x = ( Bn λ / 2 ln λ ) ∫
0
ds
s n 0 − s + s log( s / n 0 )
device channel length of 42 nm, the aspect ratio ε w is
about 1. Hence we do not expect solutions (5) and (6) to be
valid for device channel lengths smaller than 42 nm (for
The validity of DG model in (1) also requires ε a to be
small [10, equation (3.13)], where
*
_
( wqs −φ ) ln λ
This restriction is applied to the validity of the
solutions in (5) and (6). Equation (7) indicates that for
λ ~ 10 7 .)
is the solution for nin (x ) , given in inverse form, and
where n0 = ni e
(7)
*
where
nin
ε *w = ( Ld / L ) ln λ / λ (see Ward [2]).
is both the asymptotic form of
nin as it merges into the outer solution and the value of the
outer solution at x = 0 . The parameter w qs is the surface
potential as seen by the outer solution, and its magnitude is
less than the magnitude of the classical surface potential by
some amount ∆ ws . In this work the value of ∆ ws is
determined by fitting with numerical data. However, it is
possible to get an expression for w qs from the interface
ε a* = h 2 /(8 Kb Tme L2 )
(8)
If we consider a channel length of 10 nm, ε a at room
*
temperature is about 0.02.
The channel current with quantum correction is
compared with the classical and numerical results of [7].
The comparisons show that the fit with the numerical result
for 80 nm is excellent (see Figure 2 below).
boundary condition of the outer solution by considering the
quantum layer thickness as part of the oxide thickness. That
work is in progress.
The quantum correction of inversion charge is
determined from (4) by integrating the second term in the
bracket. This gives
QC = − n0 R1 Bn
(5)
where R1 = 1.187 .
The quantum correction for the channel current at strong
inversion with mobility µ n is derived from
_
I dsc = − K bTµ n Ld
V ds
ln λ
_
W
ln( λ ) ∫ QC d φ .
L
0
(6)
Figure 2: Channel current for 80 nm gate length and
35A0gate oxide MOSFET, gate voltage Vgs =1.8, 1.5, 1.2V
from top to bottom, and µ n = 1500 cm / Vs . DG
numerical data is taken from Biegel et al [7].
2
K b is the Boltzmann constant, T is the
temperature, W is the device width, and L is the channel
where
length. The outer solution for the current, which (6)
modifies, is derived by the method used in [4].
3
RESULTS AND COMPARISON
The quasi 1-D solution is valid if ε w is small, where
*
ε w* is defined by
In Figure 2, ∆ ws is assumed to have a linear relation
with the gate voltage and coefficients of the linear equation
are determined by fitting with data (see [12]). Hence the
quantum correction introduced here adds two parameters to
be found from data. Percentage reductions of the current
due to the quantum effect for different oxide thickness and
gate bias are shown in Figure 3.
This work has presented results obtained by the
boundary layer technique for the 1-dimensional DG model;
quantum corrections for the inversion charge and channel
current are achieved. These analytical models are
appropriate for SPICE application.
Figure 3: Normalized channel current reduction versus
oxide thickness, from [12].
Since QM effects have a profound impact on the charge
induced by the gate at the surface of the channel, the C-V
characteristics at the gate are also affected. The numerical
solution of Ancona in [8, Figure 4] shows gate capacitance
reduction in the accumulation and inversion regions. Work
in that area is in progress, along with investigations using
the asymptotic approach of quantum effects in other
devices.
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