Thickness Dependence of the Effective Masses in
a Strained Thin Silicon Film
Viktor Sverdlov
1
, Oskar Baumgartner
1
, Thomas Windbacher
1
, Franz Schanovsky
2
, and Siegfried Selberherr
1
1
Institute for Microelectronics
2
Christian Doppler Laboratory for TCAD at the Institute for Microelectronics
TU Wien
Gußhausstraße 27-29/E360
A-1040 Wien, Austria
{sverdlov|baumgartner|windbacher|schanovsky|selberherr}@iue.tuwien.ac.at
Abstract— By comparing results obtained with the density-
functional method, empirical pseudo-potential method, and
empirical tight-binding method it is demonstrated that the
conduction band structure is accurately described by the two-
band k·p model. The later model is used to investigate the
subband structure in ultra-thin (001) silicon films. It is
demonstrated for the first time that the unprimed subbands with
the same quantum number are not equivalent in ultra-thin films
and develop different effective masses along [110] and [-110]
directions. Using the two-band k·p model the dependence of the
subband effective masses on strain and thickness is calculated. It
is shown that the mass along tensile stress in [110] direction
decreases with strain guaranteeing current enhancement in thin
films. Shear strain also introduces large splitting between the
unprimed subbands with the same n. Finally, the dependence of
the effective masses in primed subbands is calculated and found
to agree well with recent pseudopotential calculations.
Keywords - two-band k·p model, shear strain, subband splitting,
strain and thickness dependent effective masses
I. INTRODUCTION
The rapid increase in computational power and speed of
integrated circuits is supported by the continuing size reduction
of semiconductor devices’ feature size [1]. With scaling
apparently approaching its fundamental limits, the
semiconductor industry is facing critical challenges and new
engineering solutions are required to improve CMOS device
performance. Strain-induced current enhancement is one of the
most attractive solutions to increase the device speed and will
maintain its key position among possible technology
innovations. In addition, a multi-gate MOSFET architecture is
expected to be introduced for the 22nm technology node.
Combined with a high-k dielectric/metal gate technology and
strain engineering, a multi-gate MOSFET appears to be the
ultimate device for high-speed operation with excellent channel
control, reduced leakage currents, and low power budget.
Uniaxial [110] strain modifies the transversal effective
masses of the two (001) valleys [2]. The mass change is
accurately described by the two-band kp model of the
conduction band [3]. The two-band kp model [2-4] provides a
general framework to compute the subband structure, in
particular the dependence of the subband effective masses on
shear strain. In case of a square potential well with infinite
walls, which is a good approximation for the confining
potential in ultra-thin Si films, the dispersion equations were
obtained [5], however, an accurate analysis of the electron
subband structure is still missing.
Here we describe the subband structure and the dependence
of the subband effective masses on strain and thickness t by
solving the two-band kp model in a thin (001) film under
[110] tensile strain numerically. This allows an analysis of
subband energies and effective masses for both primed and
unprimed subbands in (001) silicon films.
II. THE TWO-BAND KP MODEL
The two-band kp model was suggested first in [2] to
explain experimentally observed strain dependence of the
cyclotron effective mass on the orientation of the external
magnetic field. It can be written as
Figure 1. Comparison of the conduction band of silicon computed with
DFT, EPM, sp
3
d
5
s
*
, and the k p method. The k p model is accurate up to
an energy of 0.5eV.
978-1-4244-3947-8/09/$25.00 ©2009 IEEE 51