Si/Si12 x Gex heterostructures: Electron transport and field-effect transistor
operation using Monte Carlo simulation
Philippe Dollfusa)
Institut d’Électronique Fondamentale, CNRS URA22, Université Paris-Sud, Bâtiment 220, F-91405 Orsay,
CEDEX, France
~Received 20 March 1997; accepted for publication 24 June 1997!
Using an ensemble Monte Carlo simulation, we study the electron transport properties in tensile
strained Si grown on a Si12x Gex substrate and in an n-channel Si/Si12x Gex modulation-doped
field-effect transistor ~MODFET!. Owing to the strain-induced modification of the conduction-band
structure, the in-plane drift mobility in undoped material reaches 3250 cm2/V s at 300 K ~for x
> 0.2! and 31000 cm2/V s at 77 K ~for x>0.05!. The two-dimensional Monte Carlo modeling of
0.18 mm gate Si/SiGe MODFETs reveals a high velocity overshoot of 2.53107 cm/s related to a
significant ballistic transport through the gated part of the strained Si channel. For a 0.18 mm gate
and a Ge mole fraction x50.3, intrinsic transconductance and transition frequency as high as 460
mS/mm and 85 GHz are obtained. © 1997 American Institute of Physics.
@S0021-8979~97!02819-3#
I. INTRODUCTION
There have been large and successful efforts to introduce
strained Si or SiGe epilayers in silicon device technology to
take advantage of increased mobilities and of band-gap engineering techniques currently used in III–V technology.
Significant results have been first obtained with strained
SiGe layers grown on a Si substrate, which leads to a type I
band alignment1 suitable for an npn heterojunction-bipolar
transistor and p-channel field-effect transistor ~FET!. To the
contrary, a type II band alignment, convenient for an
n-channel FET, is obtained using a tensile strained Si layer
grown on a relaxed SiGe substrate.1 This configuration leads
to very high electron Hall mobility values2,3 and to high performance
modulation-doped
field-effect
transistors
~MODFETs!.4,5 At room temperature, mobilities exceeding
2800 cm2/V s and transconductances as high as 390 mS/mm
have been reported. One of the key parameters leading to this
performance is the strained-induced splitting energy DE S between D valleys that enhances the number of electrons having a transverse effective mass in the plane of growth. As a
function of the Ge mole fraction x in the Si12x Gex pseudosubstrate, the splitting energy is given by DE S 50.6
3eV. 6 The n-MODFET operation is also controlled by the
conduction-band offset DE C at the Si/Si12x Gex interface,
whose value seems still controversial. For x50.3, experimental determinations gave DE C 50.18 eV, 7 as well as
DE C 50.29 eV. 2
We have used an ensemble Monte Carlo technique to
study the electron transport properties in tensile strained Si,
and to investigate the performance of an n-channel
Si/Si12x Gex MODFET. The strained Si transport model is
basically the same as the one used for bulk Si. They only
differ in the splitting energy between ellipsoidal D valleys.
The bulk Si and SiGe model is detailed in the Appendix. As
in many other models,8–11 the D–D intervalley scattering is
feasible by the exchange of three f and three g phonons. In
a!
Electronic mail: philippe.dollfus@ief.u-psud.fr
J. Appl. Phys. 82 (8), 15 October 1997
these models, zero-order deformation potentials are generally
used, even for the three transitions ~f -transverse-acoustic,
g-transverse-acoustic, and g-longitudinal-acoustic, usually
denoted, respectively f 1, g1, and g2!, which are forbidden
by scattering selection rules.12 Ferry et al. pointed out that
deformation potentials may be of a higher order in the phonon wave vector,13 and a Si model including two zero-order
high-energy phonon transitions, and two first-order lowenergy phonon transitions has been proposed in agreement
with selection rules.14 The present model takes into account
three zero-order and three first-order D–D intervalley transitions, as detailed in the Appendix. The band structure and
transport parameters of strained Si have been then introduced
in our two-dimensional ~2D! Monte Carlo device model to
study 0.18 mm n-channel Si/Si12x Gex MODFETs. The results presented in this paper emphasize the nonstationary
character of electron transport in this type of device.
The analysis of electron transport in tensile strained Si is
presented in Sec. II. The MODFET simulation results are
described and discussed in Sec. III. Conclusions are given in
Sec. IV.
II. ELECTRON TRANSPORT IN Si: EFFECT OF
TENSILE STRAIN
The Monte Carlo model of electron transport in strained
Si is derived from the model presented in the Appendix for
bulk Si. The effective masses and the scattering parameters
are assumed to be unmodified by strain. The effect of strain
is included in the splitting energy DE S between the twofold
degenerate D valleys ~hereafter, noted normal valleys! having the longitudinal axis normal to the plane of growth and
the fourfold degenerate D valleys ~hereafter, called parallel
valleys! having the longitudinal axis in this plane. The splitting energy DE S is given as a function of Ge mole fraction x
in the pseudosubstrate by DE S 50.63eV. 6 In the case of
tensile strain, the normal valleys shift down in energy, while
the parallel valleys shift up. This results in a repopulation of
electrons favoring the normal valleys. This is illustrated in
0021-8979/97/82(8)/3911/6/$10.00
© 1997 American Institute of Physics
3911
FIG. 1. Proportion of electrons in ‘‘normal’’ D valleys in strained Si as a
function of the electric field applied parallel to the plane of growth. The
parameter x is the Ge mole fraction in the SiGe pseudosubstrate on which Si
is strained.
Fig. 1, which is a plot of the percentage of electrons residing
in the normal valleys, as a function of the electric field applied parallel to a D direction in the growth plane. The results obtained at room temperature are shown for several
mole fractions in the substrate, ranging from x50 ~without
strain; DE S 50, dashed line! to x50.3 (DE S 50.18 eV). Under a low in-plane electric field (E i ,1 kV/cm), a Ge fraction of 0.2 in the substrate, i.e., DE S 50.12 eV, is enough for
most of the electrons to reside in normal valleys. The electron heating under a higher electric field tends to distribute
electrons among all D valleys, as in bulk Si. This behavior is
reflected on the corresponding drift velocity field characteristics ~Fig. 2!. For x<0.2, the low-field velocity increases as
x increases, since ever more electrons experience a transverse effective mass in traveling in the plane of growth. As
soon as all the electrons reside in normal valleys, i.e., for x
>0.2, increasing the Ge fraction x in the substrate has no
more effect on the velocity. The maximum resulting drift
mobility ~Fig. 3, circles! is then about 3250 cm2/V s ~for x
>0.2!, which is more than twice as high as in unstrained Si.
This value is 14% higher than the best experimental Hall
mobility obtained for x50.3.3 These results indicate clearly
that for an n-FET operating at room temperature, the only,
but not negligible interest in using Ge mole fractions higher
FIG. 2. Velocity-field characteristics of electrons in strained Si for different
Ge mole fractions in the pseudosubstrate.
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J. Appl. Phys., Vol. 82, No. 8, 15 October 1997
FIG. 3. Electron in-plane drift mobility in strained Si as a function of the Ge
mole fraction in the pseudosubstrate at T5300 K ~circles! and T577 K
~squares!; solid lines correspond to undoped material, and dashed lines to an
ionized impurity concentration equal to 1014 cm23.
than 0.2, is to get a higher conduction-band discontinuity at
the Si/SiGe interface and then to improve the electron confinement in the strained Si channel layer.
At 77 K, the computed drift mobility is 20 500 cm2/V s
in unstrained and undoped Si and reaches a maximum value
of 31 000 cm2/V s as soon as x is greater than 0.05 ~Fig. 3,
squares, solid line!. Indeed, due to the lower thermal energy
3k B T/2510 meV a Ge fraction x50.05, i.e., a splitting energy DE S 530 meV, is enough to fill the normal valleys and
then to fully benefit from a lower effective mass in the plane
of growth and from a lower intervalley scattering rate. It
should be noted that this high mobility has been computed
without impurity scattering. However, at 77 K, a residual
ionized impurity concentration as low as 1014 cm23 influences significantly the mobility. In such conditions, we obtain a maximum mobility of 21 500 cm2/V s ~Fig. 3, squares,
dashed line!, which is more consistent with the experimental
value of 18 000 cm2/V s reported for x50.3.3 Whatever the
Ge mole fraction may be in the pseudosubstrate, the computed saturation velocity in Si is 9.23106 cm/s at 300 K
~Fig. 2!, and 1.33107 cm/s at 77 K, which is in excellent
agreement with the measurements.3
III. DEVICE SIMULATION
The above described transport model has been introduced in MONACO, our Monte Carlo device simulator. A
three-dimensional version of this tool is available,15 but the
2D version has been used in the present work. Using a finitedifference method, the 2D Poisson’s equation is solved in a
rectangular meshing at each time step, typically, equal to 1
fs. About 22 000 particles were initially implanted in the
devices simulated in this work. The charge neutrality is
maintained at each Ohmic contact surface, which is the only
condition of carrier injection into the device. The characteristics of an injected particle are randomly chosen from a
Maxwellian distribution function weighted by the velocity
component perpendicular to the contact surface. A particle
multiplication technique11 is used to statistically enhance the
number of rare events in pre-defined crucial regions, e.g., in
the channel of FETs at low gate voltage.
We have simulated 0.18 mm gate Si/Si12x Gex MODFETs at 300 K, with x50.2 and x50.3. The basic device
Philippe Dollfus
FIG. 5. Drain current versus drain voltage at three different gate voltages for
a Si/Si0.7Ge0.3 device with DE C 50.18 eV.
FIG. 4. Schematic cross section of simulated devices.
structure is shown in Fig. 4. The Si12x Gex pseudosubstrate
and the tensile strained Si channel are doped to N D 0
51015 cm23. The channel is 10 nm thick and separated from
the top SiGe layer by a 3 nm thick Si12x Gex spacer
~t c 510 nm, t s 53 nm). The gate is deposited on a top 12 nm
Si12x Gex layer doped to N D 5331018 cm23. The gate
Schottky barrier height F B is assumed to be 0.6 eV.
As pointed out in the introduction, the value of the
conduction-band discontinuity DE C at the Si/SiGe interface
is still debated. Assuming a Vegard’s law for the variation of
DE C as a function of x, and on the basis of the experimental
value DE C 50.18 eV reported by Stern and Laux7 for x
50.3, one obtains
DE C 50.63eV.
~1!
Assuming DE C 50.29 eV for x50.3, the x dependence
of DE C becomes
2
DE C 50.973eV.
~2!
Most results presented in this work have been obtained
using the pessimistic rule ~1!.
channel becomes as high as in the access zones that tend to
be resistive enough to introduce a non-negligible series resistance. This phenomenon contributes also to the drain current limitation, yielding a fall-off of transconductance at high
gate bias.
The increase in DE C with x causes a shift in V T , which
enlarges the range of the gate voltage operation, and a higher
confinement the in Si channel. Both effects have repercussions on the current and the transconductance. The maximum
transconductance is 295 mS/mm for x50.2, and 350 mS/mm
for x50.3. The total gate capacitance C GS 1 C GD can be
calculated by the derivation of the total charge present in the
device, with respect to the gate voltage at a given drain voltage. The transition frequency f T may be then calculated as
g m /2p (C GS 1C GD ), which yields a maximum f T value of 56
GHz for x50.2, 66 GHz for x50.3 ~Fig. 6, dashed lines!.
For comparison, we computed the I–V characteristics
for the Si/Si0.7Ge0.3 device using the conduction-band discontinuity DE C 50.29 eV, according to the optimistic rule
~2!. The resulting increase in the channel electron sheet density leads to g m max5460 mS/mm and f T max585 GHz.
B. Electron transport in the channel
A. Electrical characteristics
Two devices differing from one another in the Ge mole
fraction of the pseudosubstrate have been studied: x50.2,
i.e., DE C 5DE S 50.12 eV, and x50.3, i.e., DE C 5DE S
50.18 eV. The output current versus drain voltage characteristics obtained for x50.3 are plotted in Fig. 5. They exhibit a good saturation behavior, with limited short-channel
effects. Taking into account the Schottky barrier height of
0.6 eV and the characteristics of the gated region, the theoretical flatband voltage is 20.08 V. This is consistent with
the determination of the threshold voltage V T by linear extrapolation of T D -V GS characteristic at 0.1 V drain voltage
that leads to V T '20.05 V.
Figure 6 shows, for both devices, the transconductance
~in solid lines! as a function of gate voltage, for a drain–
source bias of 1 V. Whatever the device we consider, the
drain current is limited at high gate voltage, i.e., at V GS
> F B 2 0.2 V, by the appearance of a parasitic channel in
the SiGe top layer and of a gate leakage current. Furthermore, the electron sheet density in the gated region of the
J. Appl. Phys., Vol. 82, No. 8, 15 October 1997
The studied devices have a short enough active zone to
yield a significant nonstationary transport. Furthermore,
while most electrons are retained in the normal valleys of the
FIG. 6. Transconductance ~solid lines! and transition frequency ~dashed
lines! versus gate voltage at V DS 51 V for Si/Si12x Gex devices differing in
x; close circles, x50.3; open circles, x50.2. DE C is chosen according to
the rule DE C 50.63eV.
Philippe Dollfus
3913
FIG. 7. Electron density ~solid lines! and velocity ~dashed line! along the
channel. The bias conditions are V GS 50.2 V, V DS 51 V. The vertical
dashed lines represent the gate limits.
strained channel, the intervalley scattering frequency is reduced in comparison with the case of unstrained Si. This
reinforced the probability of observing ballistic electrons, or
at least, out of equilibrium electrons.
In what follows, we mainly consider the device with x
50.3 and DE C 50.18 eV, according to Eq. ~1!, to show the
nonstationary nature of electron transport. The chosen bias
point, i.e., V GS 50.2 V and V DS 51 V, corresponds to the
maximum transconductance. In Fig. 7, the electron densities
in normal and parallel valleys are plotted along the channel
~solid lines!. Over a large part of the channel, most electrons
remain in the normal valleys, to the benefit of their velocity.
A significant transfer into the parallel valleys appears only at
the drain side of the channel ~X.150 nm in Fig. 7!. The
resulting average electron velocity reaches the maximum
value of 2.53107 cm/s ~Fig. 7, dashed line!, which is much
higher than the stationary saturation velocity equal to 9.2
3106 cm/s ~Fig. 2!.
The highly nonstationary nature of the electron transport
in this device is confirmed in Fig. 8, where we plot the distribution of electron velocities ~component parallel to the
source–drain direction! located in the last cells of the gatecontrolled zone of the channel, i.e., between X5175 nm and
X5180 nm in Fig. 7. A part of this distribution is well fitted
FIG. 9. Potential-energy e p and electron total-energy e t along the channel.
D e p is the drop of the potential energy across the gated region. The hachured rectangle symbolizes the zone in which the distribution of electron
velocities ~plotted in Fig. 8! is calculated. The bias conditions are V GS
50.2 V and V DS 51 V.
by a Maxwellian function centered on the saturation velocity
~dashed line!; it corresponds to the equilibrium electron
population. The distribution exhibits also a well-defined peak
of out-of-equilibrium electrons centered on a peak velocity
v p 5 6.8 3 107 cm/s. Considering electrons having a transverse effective mass along X, and a nonparabolicity coefficient of 0.5, this peak velocity corresponds to a kinetic energy of 0.42 eV that is exactly the drop of potential energy
D e p , i.e., of the bottom of the conduction band, across the
gated part of the channel ~Fig. 9!. A significant part of the
peak may be, thus, attributed to purely ballistic electrons.
This is consistent with the profile of electron total energy e t ,
i.e., the sum of potential energy e p and electron kinetic energy e k , in the channel ~Fig. 9!; this profile is nearly flat on
a large portion of the gated region.
These results ~Figs. 8 and 9! of electron transport in a
Si/Si12x Gex device obtained with x50.3 are very similar to
those obtained with x50.2, even from a quantitative viewpoint. The main difference originates obviously from the
lower conduction-band discontinuity DE C at the Si/Si0.8Ge0.2
interface that facilitates the real-space transfer towards the
SiGe layers. For x50.3 (DE C 50.18 eV), 62% of the
source–drain current still flows inside the Si channel at the
gate end, i.e., at X5180 nm in Figs. 7 and 8. Due to a less
efficient confinement, this proportion is limited to 42% for
x50.2 (DE C 50.12 eV).
IV. CONCLUSION
FIG. 8. Distribution of electron velocities parallel to the source–drain direction in the last cells of the gated part of the Si channel ~between X
5175 nm and X5180 nm!. The dashed line represents a Maxwellian distribution centered on v sat. The bias conditions are V GS 50.2 V and V DS
51 V.
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J. Appl. Phys., Vol. 82, No. 8, 15 October 1997
Using an ensemble Monte Carlo technique, we have
studied the electron transport in tensilely strained Si and investigated 0.18 mm Si/Si12x Gex MODFETs. The straininduced lowering of normal D valleys is shown to enhance
the electron in-plane drift mobility up to 3250 cm2/V s at 300
K, in fair agreement with Hall measurements in Si/SiGe heterostructures. A mobility of 31 000 cm2/V s is obtained at 77
K in the undoped material. This enhancement originates
from the fact that, in traveling in the plane of growth at low
energy, most electrons experience a small effective mass, the
transverse mass, with reduced intervalley scattering rates.
This behavior has repercussions on 0.18 mm gate MODFET
operation through a peak velocity overshoot of 2.5
Philippe Dollfus
3107 cm/s. We observe that a significant amount of electrons cross the gated region of the channel in ballistic flight.
Without specific optimization of device structure, and depending on the conduction-band discontinuity introduced to
model the Si/Si12x Gex interface, high transconductance and
transition frequency may be achieved ~at least 460 mS/mm
and 85 GHz. This confirms the great potential of such a
device for future high-speed applications, despite the necessity of building a SiGe pseudosubstrate that complicates the
epitaxial process for device fabrication.
APPENDIX: MONTE CARLO MODEL OF ELECTRON
TRANSPORT IN BULK Si AND Si12 x Gex
In this work, the conduction-band structure used to
model the electron transport in bulk and strained Si and SiGe
materials is based on the classical band structure of bulk Si
that consists of six ellipsoidal nonparabolic D valleys located
along the @100# directions at 85% of the Brillouin zone edge.
The longitudinal relative mass, the transverse relative mass,
and the nonparabolicity coefficient are assumed to be, respectively, m l 50.916 m 0 , m t 50.19 m0 , and a50.5. The
scattering mechanisms included in the model are acoustic
intravalley phonon scattering, three f and three g intervalley
phonon scatterings, impurity scattering, and, for SiGe, alloy
scattering.
The acoustic intravalley phonon scattering is treated as
an elastic process with the usual value of deformation potential D ac59 eV. The intervalley D–D transitions are treated
by considering both f and g processes. Using zeroth-order
transition matrix, the selection rules allow f scattering with
longitudinal-acoustic ~f -LA or f 2! and transverse-optical
~f -TO or f 3! phonons and g scattering with longitudinaloptical ~g-LO or g3! phonons. However, it has been shown
that some forbidden transitions with low-energy phonons
must be included to fit the experimental transport data on a
large range of temperatures. The f -TA ( f 1), g-TA (g1),
and g-LA (g2) phonons are then introduced in most Monte
Carlo models in contravention of the selection rules. To remedy this inconsistency, Ferry13 proposed treating the forbidden transitions by expanding the transition matrix to the first
order in the phonon wave vector. The experimental transport
data may be then fitted by theoretical calculation in agree-
FIG. 10. Velocity-field characteristics in undoped bulk Si at 300 and 77 K.
The circles and crosses represent simulation results ~this work! for a field
applied parallel to, respectively, the ^100& and ^111& directions. The dashed
and solid lines are experimental results from Ref. 8.
ment with selection rules. Using this approach, the present
model takes into account three zero-order ~g-LO, f -LA,
f -TO! and three first-order ~g-TA, g-LA, f -TA! D–D intervalley transitions. For first-order processes, we employ the
scattering rate expression calculated by Yamada et al.14 by
taking the isotropic approximation. The characteristics of intervalley transitions we have used are summarized in Table I.
The values of phonon energy are those proposed by Asche
and Sarbei,16 and the associated deformation potentials have
been considered as fitting parameters. The choice of these
parameters is not unique and discrepancies exist between the
values proposed in the literature.8–11,14,17,18 For instance, the
zeroth-order deformation potential associated with the g-LO
phonon lies between 1.753108 eV/cm and 11.0
3 108 eV/cm. 9,11 To simplify the fit procedure we have tried,
and succeeded, to use only one zeroth-order deformation potential D 0 , for all zeroth-order processes, and one first-order
deformation potential D 1 , for all first-order processes. The
values D 0 53.43108 eV/cm and D 1 53.0 eV lead to computed velocity-field characteristics in very good agreement
with experimental ones8 in undoped material, at 300K and 77
K, along both ^100& and ^111& directions ~Fig. 10!. The computed drift mobility, plotted in Fig. 11 as a function of temperature, is also in good agreement with experimental values
on a wide range of temperatures ~20–300 K!.
TABLE I. Characteristics of electron intervalley D–D transitions in Si used
in the model presented in this work.
Transition
Symbol
Value
Units
g 1 ~g-TA!
\v
D1
\v
D1
\v
D0
\v
D1
\v
D0
\v
D0
11.4
3.0
18.8
3.0
63.2
3.4
21.9
3.0
46.3
3.4
59.1
3.4
meV
eV
meV
eV
meV
108 eV/cm
meV
eV
meV
108 eV/cm
meV
108 eV/cm
g 2 ~g-LA!
g 3 ~g-LO!
f 1 ~f -TA!
f 2 ~f -LA!
f 3 ~f -TO!
J. Appl. Phys., Vol. 82, No. 8, 15 October 1997
FIG. 11. Drift mobility in undoped bulk Si as a function of temperature. The
crosses represent simulation results ~this work!, and circles are experimental
results ~from Ref. 8!.
Philippe Dollfus
3915
To model the electron transport in bulk Si12x Gex ~for
x<0.3!, we assume the conduction-band structure to remain
Si-like with six D valleys undistorted in the presence of Ge.
The only effect of alloying is, thus, the occurrence of alloy
scattering. The model of Harrison and Hauser19 is used for
the calculation of alloy scattering rates. This model is based
on the concept of the alloy potential U all that characterizes
the potential fluctuations due to alloy disorder. The value
U all50.8 eV proposed by Pejčinović et al.20 and confirmed
by Ershov and Ryzhii21 leads to a satisfying agreement between the calculated and experimental mobilities in relaxed
SiGe alloys. This value is close to the result obtained by
Fischetti and Laux22 (U all50.7 eV) using a full-band Monte
Carlo model.
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Philippe Dollfus