Quantum Mechanical Compact Modeling of Inversion Charge in Double-Gate
MOSFETs Unifying Symmetric and Asymmetric Operation Modes
Ze Yuan, Meng Li, and Zhiping Yu
Institute of Microelectronics, Tsinghua University, Beijing 100084, China
e-mail: yuzhip@tsinghua.edu.cn
Abstract - In this paper, a physics-based analytical
model for both symmetrically and asymmetrically
biased Double-Gate
quantum
(DG) MOSFETs considering
mechanical
effects
is
proposed
and
demonstrated. Schrödinger and Poisson’s equations are
solved simultaneously using a variational approach.
The
inversion
distributions
charge
and
perpendicular
to
electrical
the
potential
channel
are
expressed in closed forms. The calculated electron
distributions and electrical potential, verified by results
obtained from numerical simulation, provide good
A. Device Structure and Potential Definition
Fig. 1 shows the schematic of a generic DG-MOSFET.
x-axis is the transverse direction to the channel with its
origin located in the middle of the silicon film. Note that
two oxide layers do not necessarily have the same
thickness and two gates can be biased arbitrarily.
B. The Variational Approach for the Calculation of
Inversion Charge Considering QM Effects
In order to model the inversion charge density, Poisson
and
Schrödinger’s
equations
should
be
solved
physical insight on the quantization caused by quantum
simultaneously, which is usually done by numerical
confinement under arbitrary gate bias.
simulation. For analytical QM modeling, a variational
approach can be adopted, whose validity is guided by the
I. Introduction
The compact modeling of DG-MOSFET has been
under intensive research for quite a few years, with more
attention turned to the quantum mechanical (QM)
modeling
recently
[1].
However,
while
quantum
mechanical effects have been considered for symmetric
cases [2], they are largely neglected so far in aDG.
modeling [3].
For the suppression of short-channel effects, a thin
silicon film tsi with thickness less than 20nm is often used
and the quantum confinement effects can no longer be
Fig.1. Schematic of a generic DG-MOSFET.
form of trial wavefunction.
ignored. It has been shown that QM effects play a crucial
For rectangle potential well defined by the front and
role for device operation when the channel thickness is
back-gate oxide barriers, the wavefunctions of different
smaller than 20nm [4], [5].
subbands can be solved analytically as [4]
In this paper, a QM model for DG-MOSFETs, unifying
symmetric and asymmetric modes, is presented. The
i  x  

 x 1 
2
sin  i  1     
tsi
 tsi 2  

(1)
variational method [6] has been applied to solve Poisson
and
Schrödinger’s
equation
simultaneously.
The
model-predicted results agree well with those from 2D
device simulation.
II. Modeling Process
It is shown that, for potential well in a constant
electrical field, the trail function can be further set as [1]

 x 1   i
 i  x   i  ai  sin  i  1      e tsi

 tsi 2  
ax
(2)
where ai are undetermined parameter related to the
boundary conditions,
degree of asymmetry, and i  ai  are normalization
constants. At zeros field, ai
 ox
0 , the wavefunctions are
Vg1  V fb  S1
tox1
  si
d
dx
(5)
x 
tsi
2
identical to Eq. (1).
In order to unify symmetric and asymmetric conditions,
 ox
both the position of the peak of the electron distribution
and the degree of asymmetry must be taken into
considerations. Hence, we introduce three parameters ai ,
bi and ci to describe the generalized trial wavefunctions

 x 1  
 ax
 b x 
 i  x   i sin  i  1     ci exp   i   exp  i  
 tsi 2   
 tsi 
 tsi  

(3)
where i=0, 1, 2,…, and i  ai , bi , ci  is normalization
coefficient so that
tsi
2
t
 si
2

tox 2
  si
d
dx
(6)
x
tsi
2
the potential   x  can be obtained in closed form. The
effectiveness of this approximation has been proven in [2].
IV. Summary
A compact model unifying symmetric and asymmetric
DG-MOSFET
considering
quantum
confinement
in
transverse direction is presented. The variational approach
is used to calculate energy level and wavefunction. 1D
Schrödinger
 i 2  x dx  1 .
Vg 2  V fb  S 2
and
Poisson’s
equation
are
solved
simultaneously. Comparison between the model predicted
data and the numerical device simulation data further
For symmetric biased conditions, ci
1 and the trail
validates the accuracy of our model.
wavefunctions are identical to Eq. (3). For extreme
asymmetric cases, ai
bi , hence have the same form as
Eq. (2). Furthermore, it guarantees a smooth transition
References
[1]
on the Threshold Voltage of Undoped Double-Gate
between the two extreme cases.
MOSFETs,” IEEE Electron Device Letters, vol. 26, no. 8,
By employing the gradual channel approximation
pp. 579-582, Aug. 2005.
(GCA), the 1D Poisson’s equation in the direction
[2]
perpendicular to the channel has the general form of
d 2  x 
dx 2
V. Trivedi, and J. Fossum, “Quantum-Mechanical Effects
L.
Ge,
and
J.
Fossum,
“Analytical
Modeling of
Quantization and Volume Inversion in Thin Si-Film DG

q
q
 N  2  x   N A  (4)
 n  x   N A  
 si 
 si  inv
where   x  is the electrostatic potential, n( x) is the
MOSFETs,” IEEE Trans. Electron Devices, vol. 49, no. 2,
pp. 287-294, Feb. 2002.
[3]
Symmetric and Asymmetric DG MOSFETs,” IEEE Trans.
electron volume density, Ninv is the total electron density,
  x  is the overall wavefunction, and N A is doping
H. Lu, and Y. Taur, “An Analytic Potential Model for
Electron Devices, vol. 53, no. 5, pp. 1161-1168, May, 2006.
[4]
G. Baccarani, and S. Reggiani, “A Compact Double-Gate
concentration respectively. We consider electrons in a
MOSFET Model Comprising quantum-mechanical and
channel with <100> crystal direction and treat with
nonstatic effects,” IEEE Trans. Electron Devices, vol. 46,
different
effective
mass,
mx  0.92m0
and
my  mz  0.19m0 explicitly. The total electron density is
no. 8, pp. 1656-1666, Aug. 1999.
[5]
Y. Taur, et al., “CMOS Scaling into the Nanometer
obtained from the superposition of contribution from six
Regime,” Proc. of the IEEE, vol. 85, no. 4, pp. 486-504,
energy valleys in the reciprocal space. The overall
Apr. 1997.
wavefunction is the composite of different energy levels,
defined as
 2  x    Ni i2  x  Ninv .
Ni
[6]
Layers,” Phys. Rev. B, vol 5, pp. 4891-4899, 1972.
is the
i
inversion charge density in the ith subband. In this work,
the first five energy levels are calculated.
By integrating Ninv 02  x   N A twice and apply the
F. Stern, “Self-Consistent Results for n-Type Si Inversion
[7]
Taurus-Device User Guide, 2004 Edition, Synopsys.