IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 44, NO. 2, FEBRUARY 1997 257
2-D MOSFET Modeling Including
Surface Effects and Impact Ionization by
Self-Consistent Solution of the Boltzmann,
Poisson, and Hole-Continuity Equations
Wenchao Liang, Neil Goldsman, Isaak Mayergoyz, Fellow, IEEE, and Phil J. Oldiges
Abstract—We present a new two-dimensional (2-D) MOSFET
simulation method achieved by directly solving the Boltzmann
Transport equation for electrons, the Hole-Current Continuity
equation, and the Poisson equation self-consistently. The spher-
ical harmonic method is used for the solution of the Boltzmann
equation. The solution directly gives the electron distribution
function, electrostatic potential, and the hole concentration for
the entire 2-D MOSFET. Average quantities such as electron con-
centration and electron temperature are obtained directly from
the integration of the distribution function. The collision integral
is formulated to arbitrarily high spherical harmonic order, and
new collision terms are included that incorporate effects of sur-
face scattering and electron-hole pair recombination/generation.
I–V characteristics, which agree with experiment, are calculated
directly from the distribution function for an LDD submicron
MOSFET. Electron-hole pair generation due to impact ionization
is also included by direct application of the collision integral.
The calculations are efficient enough for day-to-day engineering
design on workstation-type computers.
I. INTRODUCTION
IN THIS paper, we present a new efficient two-dimensional(2-D) MOSFET simulation tool which is based on the
self-consistent solution of the Boltzmann, Poisson, and Hole-
Continuity equations.
To facilitate the design of deep-submicron devices, new
modeling approaches, which are based on the solution of the
Boltzmann Transport equation (BTE) [1]–[5], are being sought
to complement existing techniques. The spherical harmonic
(SH) method for solving the BTE represents a new “middle
of the road” device design option. It gives more information
than the Hydrodynamic model [6]–[11], while requiring only
slightly more computation time. It provides information which
is similar to spherical-band Monte Carlo calculations, and
is computationally many times faster. The SH method can
provide the momentum distribution function, using spherical
band models [12], [13], as well as average quantities, for an
Manuscript received April 2, 1996; revised August 5, 1996. The review
of this paper was arranged by Editor A. H. Marshak. This work was
supported by the Semiconductor Research Corporation and the National
Science Foundation.
W. Liang, N. Goldsman, and I. Mayergoyz are with the Department of
Electrical Engineering, University of Maryland, College Park, MD 20742
USA.
P. J. Oldiges is with Digital Equipment Corporation, Hudson, MA 01749
USA.
Publisher Item Identifier S 0018-9383(97)00898-8.
entire device. It is therefore a promising approach for mod-
eling hot-electron phenomena, including MOSFET reliability,
EPROM programming and impact-ionization, which depend
on the shape of the distribution function. It has been shown
to agree with Monte Carlo simulations which employ the
same input parameters [14]–[16], while requiring orders of
magnitude less CPU time to evaluate, and without statistical
noise. The SH method has been employed to solve the BTE in
one-dimensional (1-D) for BJT’s as a post-processor [17], and
self-consistently with the Poisson equation for simple 1-D test
structures [18]. It has also been used as a post-processor in
a first step to achieving 2-D MOSFET simulation [19], [20].
In addition, a generalized spherical harmonic approach has
been achieved for the left-hand side (LHS) of the Boltzmann
equation, in 1-D and 2-D, which allows the incorporation of
an arbitrary number of spherical harmonics [20], [21].
Here, we further advance the SH approach, and demonstrate
the viability of the method, by developing a stand-alone sim-
ulator for deterministic, self-consistent MOSFET modeling.
We begin by introducing the device model which consists
of the BTE for electrons, the Poisson equation, and Hole-
Current Continuity equation. We then transform the LHS
of the BTE into a tractable form using the generalized SH
approach given in [20]. Next, we extend this approach to
also express the collision integral to arbitrarily high spherical
harmonic order in a general way. We also introduce new
collision terms into the model to account for surface scattering
in MOSFET inversion layers, Shockley–Read–Hall (SHR)
electron-hole pair recombination-generation, and electron-hole
pair generation resulting from impact ionization. To facil-
itate obtaining a numerical solution, we developed a new
Scharfetter–Gummel-type discretization for the BTE. We also
develop an iterative method to self-consistently solve the entire
nonlinear device model, while simultaneously overcoming
computational restrictions imposed by the three-dimensional
(3-D) BTE. The simulation results directly provide the electron
distribution function, electric potential and hole concentration
for the entire MOSFET. To our knowledge, this is the first 2-D
MOSFET self-consistent simulation using the SH approach
that has been reported.
To help demonstrate the accuracy of the distribution func-
tion obtained by solving the BTE, we use it to simulate
0018–9383/97$10.00  1997 IEEE