Journal of Computational Electronics 1: 503–513, 2002
c 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
The Use of Quantum Potentials for Confinement and Tunnelling
in Semiconductor Devices
A. ASENOV, J.R. WATLING AND A.R. BROWN
Device Modelling Group, Department of Electronics and Electrical Engineering,
University of Glasgow, Glasgow G12 8LT, UK
A.Asenov@elec.gla.ac.uk
D.K. FERRY
Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287-5706, USA
Abstract. As MOSFETs are scaled to sub 100 nm dimensions, quantum mechanical confinement in the direction
normal to the silicon dioxide interface and tunnelling (through the gate oxide, band-to-band and from sourceto-drain) start to strongly affect their characteristics. Recently it has been demonstrated that first order quantum
corrections can be successfully introduced in self-consistent drift diffusion-type models using Quantum Potentials.
In this paper we describe the introduction of such quantum corrections within a full 3D drift diffusion simulation
framework. We compare the two most popular quantum potential techniques: density gradient and the effective
potential approaches, in terms of their justification, accuracy and computational efficiency. The usefulness of
their 3D implementation is demonstrated with examples of statistical simulations of intrinsic fluctuation effects in
decanano MOSFETs introduced by discrete random dopants. We also discuss the capability of the density gradient
formalism to handle direct source-to-drain tunnelling in sub 10 nm double-gate MOSFETS, illustrated in comparison
with Non-Equilibrium Green’s Functions simulations.
Keywords: numerical simulations, quantum corrections, quantum potential, density gradient, effective potential,
MOSFETs, intrinsic fluctuations
1.
Introduction
Conventional MOSFETs scaled down to 15 nm
gate lengths have been successfully demonstrated
(Thompson et al. 2001). Scaling below this milestone
involves intolerably thin gate oxides and unacceptably high channel doping and may require a departure from the conventional MOSFET concepts. The
most promising of the new device architectures remains
the double gate MOSFET (Hisamoto 2001) which retains useful field effect action to gate lengths below
5 nm, utilising affordable oxide thicknesses and no
channel doping. The combination of thin gate oxides and heavy doping in conventional MOSFETs, and
the thin silicon body of the double-gate structures,
will result in substantial quantum mechanical (QM)
threshold voltage shifts, gate leakage tunnelling and
transconductance degradation (Jallepalli et al. 1997).
Additionally, below 10 nm gate-lengths direct sourceto-drain tunnelling will rapidly became one of the
major limiting factors for scaling (Ren et al. 2001,
Watling et al. 2002). In conventional MOSFETs gate
assisted band-to-band tunnelling may also increase to
unacceptable levels with the increase of the channel
doping.
Computationally efficient methods for including
QM effects are required for the purpose of practical Computer Aided Design (CAD) of current, and
next generation, devices. However, at present, complete
quantum simulations involving, for example, Wigner
or Green’s functions (Ren et al. 2001, Svizhenko et al.
2002) are computationally expensive and are therefore
504
Asenov
conventional architecture. Finally we discuss the possibility of emulating source-to-drain tunnelling in sub
10 nm double gate MOSFETs using the DG formalism.
2.
Figure 1. Schematic illustration of various possible quantum effects that are likely to be important within a MOSFET. A conventional MOSFET architecture is depicted.
not suitable for inclusion within CAD simulation tools.
First order quantum corrections based on the Density Gradient (DG) formalism (Ancona and Iafrate
1989) have already been successfully introduced in 2D
(Rafferty et al. 1998) and 3D (Asenov et al. 1999)
drift-diffusion simulations. In addition, an Effective
Potential (EP) approach for introducing quantum corrections in classical and semi-classical simulations has
been proposed (Ferry 2000) and demonstrated in Monte
Carlo MOSFET simulations (Ferry, Akis and Vasileska
2000).
The quantum mechanical effects affecting the operation and the performance of conventional MOSFETs
are depicted schematically in Fig. 1. In this paper we
focus mainly on the simulation of QM confinement
effects in conventional MOSFETs using the aforementioned DG and EP approaches. We also discuss
the handling of source-drain tunnelling in double-gate
devices using DG simulations. We discuss and compare the implementation of the above two quantum
correction techniques in our 3D drift diffusion type
simulator, using a self-adjusting technique for updating
the potential (Stern 1970, Watling et al. 2001), similar
to the procedure used in the self-consistent solution
of the Schrödinger-Poisson equations. We start by
discussing the derivation and the interpretation of the
two quantum correction techniques. The calibration of
our quantum corrected simulations in respect of more
comprehensive and accurate quantum simulation techniques then follows. Further, we compare the influence
of the quantum corrections on 3D ‘atomistic’ statistical
simulations of intrinsic fluctuations, introduced by random discrete dopants, in nano-scale MOSFETs with
Quantum Potentials
We are motivated by the need to introduce computationally efficient quantum corrections in our 3D driftdiffusion ‘atomistic’ simulator, presently used to investigate the intrinsic fluctuations in decanano MOSFETs
which are introduced by the discreteness of charge and
atomicity of matter (Asenov et al. 1999, 2001). The
investigation of intrinsic fluctuation effects involves
statistical 3-D simulations of large samples of macroscopically identical but microscopically different devices. To extract averages and standard deviations we
typically simulate samples of 200 devices which are
microscopically different in terms of discrete dopant
distribution, interface pattern and line edge roughness.
Therefore the computational efficiency of the quantum
correction approach is of great importance and the use
of quantum potentials becomes an attractive option.
2.1.
The Density Gradient Formalism
The density gradient formalism may be derived from
the equation of motion for the one particle Wigner
function (Ancona and Iafrate 1989, Carruthers and
Zachariasen 1983):
∂ f (p, r, t)
2
+ v · ∇r f (p, r, t) − V (r)
∂t
h
← → h ∇ r ∇p
× sin
f (p, r, t) = 0
2
(1)
The quantum effects are included through the inherently non-local driving potential in the third term on
the left-hand side, where it is understood that ∇r acts
only on V and ∇p acts only on the distribution function f . The operator within the sum may be written in
terms of a power series, so that the transport equation
for the Wigner distribution function may be written in
the form of a modified Boltzmann Transport Equation
(BTE) as (Iafrate, Grubin and Ferry 1982, Tsuchiya,
Winstead and Ravaioli 2001):
∞
∂f
1
h (2α−1) (−1)α+1
+ v · ∇r f − ∇r V · ∇k f +
∂t
h
4α (2α + 1)!
α=1
∂f
× (∇r V · ∇k f )2α+1 =
(2)
∂t Coll
Quantum Potentials in Semiconductor Devices
where V represents the electrostatic potential. Expanding to first order in h, so that only the first non-local
quantum term is considered, has been shown to be
sufficiently accurate to model non-equilibrium quantum transport and also for the inclusion of tunnelling
phenomena in particle based Monte Carlo simulators
(Tsuchiya and Miyoshi 2000, Tsuchiya, Fischer and
Hess 2000). Although it is tempting to expand the
non-local term to higher powers of Planck’s constant,
this may lead to spurious results since the equation of
motion is not analytic in h (Barker 1992). The additional, non-classical, quantum correction term may be
viewed as a modification to the classical potential and
acts like an additional quantum force term in particle
simulations, similar in spirit to the Bohm interpretation. It should be noted that in the limit of slow spatial
variations, e.g., for gutter-like potentials, the non-local
terms disappear and the equation reduces to the classical BTE. However, in order to obtain a correction
that may be used efficiently in device simulators it is
necessary to make an assumption regarding the carrier
distribution function. The following form of the distribution is usually assumed:
2
h (ki − ki )2
f = exp −β
,
+
V
(r
)
−
E
f
2m ∗
h 2 (ki − ki )2
kB T
≈
∗
2m
2
∂2V
∂ 2 (ln n)
≈
−k
T
B
∂x2
∂x2
(4)
where:
h 2 ∂ 2 (ln n)
∂
V−
=−
∂x
12m ∗ ∂ x 2
Thus using these approximations the current may be
written as:
J = qnµn E + q Dn
∂
ne−βVqc
∂x
n(x) = Nc exp{−β[V (x) + Vqc (x) − E f ]}
(7)
(8)
By expanding the term e−βVqc and taking the lowestorder component we obtain what is termed the density
gradient approximation:
√
∂ 2 n/∂ x 2
√
n
(9)
The additional quantum potential term which is proportional to the second-derivative of the root of the
density may be taken as an additional term in the quasiFermi level so that the electronic equation of state becomes similar to that of an ideal gradient gas1 (Ancona
and Iafrate 1989, Chapman and Cowling 1952) for
typical low-density, high-temperature semiconductors,
which includes an additional term that is dependent
on the gradient of the carrier density (Rafferty et al.
1998):
√
∇2 n
kB T
n
2bn √
= φn − ψ +
ln
q
ni
n
2
(6)
We may in turn write the carrier distribution in equilibrium as:
= n cl (x) exp[−βVqc (x)]
(1) The electron temperature is equal to the lattice temperature and the electron temperature gradient is
zero and
(2) The v · ∇v term is assumed to be small compared
to other terms and can be neglected.
(3)
where β = 1/k B T .
The quantum-corrected BTE may be written in onedimension, without loss of generality, as:
∂f
1
∂f
+ v · ∇r f + F qc · ∇k f =
(5)
∂t
h
∂t coll
Fxqc
h 2 ∂ 2 (ln n) and n represents the
where: Vqc (x) = − 12m
cl
∗
∂x2
classical carrier density without the quantum corrections. The density gradient approximation in a driftdiffusion context may be derived in a manner similar to
that for deriving the drift-diffusion approximation from
the Boltzmann Transport Equation (Snowden 1989)
making the following assumptions:
∂n
qnµn h 2 ∂
J = qnµn E + q Dn
+
∂x
6m ∗ ∂ x
for each independent spatial coordinate, i, along with:
505
(10)
Where bn = 4qhm ∗ r ; all other symbols have their usual
n
meaning and r is a dimensionless parameter. In situations with strong quantum confinement, (when only a
single sub-band is occupied) the parameter r is considered equal to 1. However, as more subbands become
filled, for example due to increase in temperature,
statistical averaging causes r to change, approaching 3 asymptotically (Perrot 1979, Ancona 2000).
506
Asenov
Throughout this paper we have assumed that r is equal
to 3. This results in a quantum potential correction
term in the standard drift-diffusion flux (Rafferty et al.
1998).
√ ∇2 n
Fn = nµn ∇ψ + Dn ∇n + 2nµn ∇ bn √
(11)
n
To avoid the discretisation of fourth order derivatives in multidimensional numerical simulations a generalised electron quasi-Fermi potential φn is usually
introduced:
Fn = nµn ∇φn
(12)
Thus the unipolar drift-diffusion system of equations
with QM corrections, which in most of the cases is
sufficient for MOSFET simulations, becomes:
∇ · (ε∇ψ) = −q( p − n + N D+ − N A− ) (13)
√
n
∇2 n
kB T
(14)
2bn √
ln
= φn − ψ +
q
ni
n
∇ · (nµn ∇φn ) = 0
(15)
The system of Eqs. (13)–(15), where Eq. (13) represents Poisson’s equation and Eq. (15) the current
continuity equation, are solved self-consistently until
convergence.
where
λD =
2πh 2
m∗k B T
(17)
is the thermal de-Broglie wavelength (Fetter and
Walecka 1971). The Fourier transform of Eq. (16), may
be taken to obtain the real space wave packet:
π
πr 2
ψ(r ) = √
(18)
exp − 2
λD
( 2λ D )3/2
Taking into account the spatial extent of the carriers in
the simulations of semiconductor devices leads naturally to the concept of an effective potential. The concept of the effective potential can be understood by
noting that the classical potential, in an inhomogeneous
system, enters the Hamiltonian as:
HV =
drV (r)n(r)
(19)
Using the non-local form for the charge distribution
now leads to:
V = drV (r)
n i (r)
i
=
=
|r − r |2
dr exp −
δ(r − ri )
2
α
i
|r − r |2
drδ(r − ri ) dr V (r ) exp −
α2
drV (r)
i
(20)
2.2.
The Effective Potential Approach
An alternative approach to the DG formalism for including first order quantum corrections is the effective
potential (EP) approach (Ferry 2000, Ferry, Akis and
Vasileska 2000). In this approach a spatially-localised
wave packet is used as a representation of an electron. Moving from the classical particle representation to the quantum particle-like representation, we
go from a point charge representation of the electron to a natural wave packet representation, with size
which is defined roughly by the thermal de-Broglie
wavelength. Considering the momentum space distribution description of the occupied plane wave states,
which contribute to the electron wave packet, the
normalised momentum space distribution is defined
as:
ϕ(k) = 2
λD
2
3/4
2 2
λ k
exp − D
4π
(16)
where the summation over i is a summation over the
carriers themselves. The last form has been achieved
by interchanging the primed and unprimed variables
and rewriting the integrals. The term in the primed integral is now the effective potential and the finite size
of the electron has been resolved by the ‘smoothing’ of
the real classical potential. Here a minimum dispersion
Gaussian form for the electron wave packet is assumed,
although there is no requirement in (20) that this form
be used.
The effective potential, Veff , is thus obtained by convolving the Gaussian wavepacket with the classical
conduction band profile VClassical (obtained from the
solution of Poisson’s equation). Thus Veff is given by:
Veff =
VClassical (x + y)G(y, a0 ) dy
(21)
where G is a Gaussian with standard deviation a0 . A
schematic illustration of the effective potential and the
Quantum Potentials in Semiconductor Devices
Figure 2. Schematic illustration of the effective potential, corresponding to the classical potential of a triangular quantum well. The
important aspects of the effective potential are indicated.
et al. 1997, Hisamoto 2001). The effective mass and
the standard deviation of the wave-packet are considered to be adjustable parameters in the DG and
EP approaches respectively. Although self-consistent
Poisson-Schrödinger simulations are more accurate in
simulating 1D confinement effects there are conceptual
problems associated with the inclusion of the stationary Schrödinger equation in multidimensional transport simulations. Figure 4 shows the quantum mechanical threshold voltage shift for DG and EP techniques as
a function of substrate doping concentration compared
with the results of Jallepalli et al; the inset shows the
sensitivity of the threshold voltage shift to the effective
mass used in the DG formalism (Asenov et al. 2001).
The calibration was carried out for doping concentration N A = 5 × 1017 cm−3 . We find that an effective
mass of 0.15 m0 in DG simulations and a standard deviation of 0.75 nm for the Gaussian used in the effective potential, gave the best results in terms of fitting to
the threshold voltage shift. These single values of the
effective mass and the standard deviation of the wavepacket are used for all other concentrations. Figure 5
shows typical carrier concentration profiles obtained
from the 1-D simulations. All distributions show a peak
in the concentration away from the Si/SiO2 interface,
although the effective potential produces an unrealistically sharper drop-off of the electron concentration
near the Si/SiO2 interface.
3.2.
Figure 3. Effective conduction band potential after convolution
with a gaussian wavepacket calculated for the triangular potential
form by the gate potential at the Si/SiO2 interface. Here a0 , is 0.75 nm.
corresponding classical potential is shown in Fig. 2.
Figure 3 shows the effective potential calculated for the
triangular potential formed by the gate potential at the
Si/SiO2 interface. The effective potential can be related
to the density gradient approach by using a Taylor series
expansion of the effective potential (Ferry, Akis and
Vasileska 2000).
3.
3.1.
Simulation Results
Calibration
We have carefully calibrated both the EP and DG approaches against the results of comprehensive fullband 1D Poisson-Schrödinger simulations (Jallepalli
507
Simulation of MOSFETs
with Continuous Doping
In order to illustrate the importance of the quantum
mechanical confinement effects in the next generation MOSFETs we have simulated a 30 nm × 30 nm
n-MOSFET with simple but robust architecture which
corresponds approximately to the 80 nm technology
node expected to be in production in 2005. The device
has uniform channel doping N A = 5 × 1018 cm−3 and
oxide thickness tox = 1.3 nm. The effect of the quantum mechanical confinement at the Si/SiO2 interface
can clearly be understood by examining and comparing the I D -VG characteristics of this device, simulated
with and without quantum corrections, and depicted in
Fig. 6. The interface confinement results in a threshold voltage shift, approximately 0.22 V, between the
classical and the quantum simulations. The threshold
voltage shift is mainly due to the energy difference between the conduction band edge and the ground electron state in the surface potential well, which increases
508
Asenov
Figure 4. Threshold voltage shift due to quantum effects versus substrate doping. Results for Density Gradient and Effective Potential are
compared to those obtained from Poisson-Schrödinger. Inset shows the sensitivity of the threshold voltage shift to the effective mass.
Electron Concentration [cm-3]
10
-5
10
-6
10
-7
10
-8
ID [A]
with the doping concentration. Both DG and EP capture this effect with a sufficient degree of accuracy using a single parameter for the whole range of doping
concentrations from 1 × 1018 to 1 × 1019 cm−3 relevant to the present and future generations of MOSFETs. In Fig. 6 we have also shifted the I D -VG current curve obtained using DG quantum corrections
by the quantum mechanical threshold voltage shift to
10
-9
10
-10
10
-11
10
-12
10
-13
Density Gradient
shifted by ∆VT
∆ID
∆VT
∆S
Classical
Effective Potential
Density Gradient
-14
10 0.0
Jallepalli (Poisson-Schrödinger)
Density Gradient
Effective Potential
17
10
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
VG [V]
Figure 6. I D -VG characteristic obtained from both classical and
quantum simulators for a 30 nm × 30 nm n-MOSFET, with VD =
0.01 V and a substrate doping of 5×1018 cm−3 . The density gradient
curve has also been shift by the QM threshold voltage shift, VT , in
the dashed line, in order to highlight the decrease in current, I D ,
and the degradation of the sub-threshold slope, S.
16
10
15
10 0
0.1
1
2
3
4
5
6
7
8
Depth from interface [nm]
Figure 5. Electron carrier concentration as a function of distance
from the interface for substrate doping of 5 × 1017 cm−3 . All have
the same net sheet density.
illustrate other important impacts of the quantum confinement on the MOSFET operation. The displacement
of the charge centroid from the interface in the quantum simulations results in an increase in the effective
oxide thickness resulting in a reduction of the gate-tochannel capacitance which sustains the inversion layer
Quantum Potentials in Semiconductor Devices
charge
Q i = Ceff (VG − VT ),
(22)
with
further reduction of the gate oxide complimented with
lo-hi channel doping will be required for proper device
design.
3.3.
Ceff
CSiO2 CSiinv
=
< CSiO2
CSiO2 + CSiinv
(23)
where CSiO2 and CSiinv are the capacitances of the silicon dioxide and silicon inversion layer respectively.
This in turn results in reduction of the drive current
and transconductance (Taur and Ning 1998). The decrease in the effective oxide capacitance also leads to
an increase in short-channel effects and a corresponding increase in the sub-threshold slope (Taur and Ning
1998):
S=
d(log10 I D )
d VG
−1
= 2.3
kB T
Cdm
1+
q
Ceff
(24)
where Cdm represents the depletion layer capacitance,
which does not change significantly moving from
classical to quantum mechanical simulations. These
two effects are depicted in Fig. 6 by I D and S
respectively.
We have also investigated the threshold voltage shift
as a function of channel length in the continuous doping
case as shown in Fig. 7. The threshold voltage rolloff remains acceptable to 30 nm channel length but
the inclusion of quantum effects render the threshold
voltage too high for this generation of devices and a
509
3-D Quantum Atomistic Simulations
We have also compared the impact of the DG and EP
quantum corrections on the threshold voltage fluctuations in decanano MOSFETS, due to discrete random
dopants. Figure 8 shows a typical equi-concentration
contour for a simulated atomistic MOSFET using
the EP method. Simulating 200 atomistic devices,
which gives an uncertainty of approximately 5% in
σ VT (Asenov 1998), we have investigated the average threshold voltage (Fig. 9) and standard deviation
(Fig. 10) at different channel lengths. The EP simulations result in similar average threshold voltages and
standard deviations compared to DG. In this particular
case we have found that the DG approach is computationally more efficient due to the numerically intensive
nature of the 3-D convolution involved in calculating
the effective potential using Eq. (21). However, it is difficult to make a definitive judgement on the efficiency
of the two quantum correction algorithms, since the
relative speed depends on the adopted integration approach and is also dimensionally dependent. A more
efficient algorithm than the one used here, based on
Gaussian Quadratures, may be used in some cases for
0.6
VT [V]
0.5
Classical
Effective Potential
Density Gradient
0.4
0.3
0.2
30
40
50
60
70
80
90
100
Leff [nm]
Figure 7. Dependence of the threshold voltage on the channel
length in MOSFETs with Weff = 50 nm, N A = 5 × 1018 cm−3 and
tox = 1.3 nm, illustrating the quantum mechanical shift in threshold
voltage.
Figure 8. An equi-concentration contour for a individual
30 nm × 50 nm atomistic MOSFET at threshold obtained using our
quantum effective potential simulator. Also shown are the individual
dopant positions throughout the MOSFET structure.
510
Asenov
Figure 9. Average threshold voltage for 50 nm wide atomistic
MOSFETs with different channel lengths, 1.3 nm gate oxide and
channel doping 5 × 1018 cm−3 . Comparison between classical, density gradient and effective potential simulations.
Figure 10. σ VT for atomistic MOSFETs with different channel
lengths (same parameters as in Fig. 9) Comparison between classical,
DG and EP simulations.
evaluating the effective potential (Ramey and Ferry
2002). Additionally, the time required for the solution
of the density gradient equation of state (Eq. (10)) is
bias dependent and increases from depletion to strong
inversion.
3.4.
simulators (Iafrate, Grubin and Ferry 1982, Tsuchiya,
Winstead and Ravaioli 2001, Tsuchiya and Miyoshi
2000, Tsuchiya, Fischer and Hess 2000). In the case of
a tunnelling barrier with voltage applied on the right
hand side, the additional quantum term acts to raise the
classical conduction band potential profile for carriers
flowing from right to left of the barrier and lower the
classical potential barrier for carriers flowing from left
to right.
However, there is controversy over whether the DG
approach includes this effect and therefore is able to
model source-to-drain tunnelling phenomena in extremely short MOSFETs. Although the DG formalism
is unable to cope with cases where tunnelling is dependent on the coherent phase behaviour of electrons
(as in the case of resonant tunnelling (Bowen et al.
1997)) we see no reasons why, in principle, it cannot,
to a first order approximation, account for tunnelling
in what may be termed the scattering-dominated limit
(Ancona 2001). Here, we have performed a series of
numerical experiments to see if the DG formalism can
account for the impact of source-drain tunnelling on
the subthreshold I D -VG characteristics of short double
gate MOSFETs. In a double gate structure the current is
essentially one-dimensional, making theoretical study
and calibration easier than in a conventional MOSFET
device structure.
In search of a qualitative answer to this question we
have simulated a set of double gate MOSFETs with a
simple generic structure illustrated in Fig. 11. The subthreshold slope within the DG simulations (Fig. 12) degrades significantly as the channel length is decreased,
while in classical simulations the subthreshold slope remains nearly constant with channel length. This degradation of the subthreshold slope in our DG simulations
is consistent with quantum mechanical simulations
performed by others (Lundstrom 2001) and provide an
Source-to-Drain Tunnelling
As commented on previously in Section 2.1, expanding the Wigner transport equation to first order in h
results in an additional, non-classical, quantum term
that has been shown to be sufficient to model nonequilibrium quantum transport and the inclusion of
tunnelling phenomena in particle based Monte Carlo
Figure 11. Schematic representation of the double-gate MOSFET
structure considered within this work.
Quantum Potentials in Semiconductor Devices
Figure 12. I D -VG characteristics for a double gate structure, with
gate lengths ranging from 30 nm down to 6 nm, obtained from our
classical and density gradient simulations. VD = 0.01 V and VG is
applied to both top and bottom gate contacts.
indication that source-to-drain tunnelling is included,
at least qualitatively, in the DG simulations.
Further evidence that the DG approach includes
qualitatively the source-drain tunnelling can be gained
by investigating the temperature dependence of the subthreshold slope, given in the classical case by Eq. (24).
The classical subthreshold slope depends linearly on
temperature since the classical subthreshold current is
essentially thermionic in nature, having an exponential dependence on temperature. However, any current
due to tunnelling will have a much weaker temperature dependence (Kawaura, Sakamoto and Baba 2000,
Kawaura et al. 2000). Figures 13 and 14, show the
Figure 13. I D − VG characteristics for a 30 nm channel length
double gate structure from classical and density gradient simulations
for a range of temperatures. VD = 0.01 V and VG is applied to both
top and bottom gate contacts.
511
Figure 14. I D -VG characteristics for an 8 nm channel length double
gate structure from classical and density gradient simulations for a
range of temperatures. VD = 0.01 V and VG is applied to both top
and bottom gate contacts.
temperature dependence of the subthreshold slope in
both classical and DG simulations, for channel lengths
of 30 nm and 8 nm respectively. We observe that the
temperature dependence of the subthreshold slope is
similar for both the classical and DG simulations in the
30 nm gate length device, in agreement with Eq. (24).
The shift in the corresponding I D -VG curve is due to
the QM threshold voltage shift caused by confinement
quantisation in the thin silicon body. However, for the
8 nm gate length device even at room temperature there
is a noticeable degradation of the subthreshold slope in
the DG simulations as compared with the classical simulations. This is consistent with the expected contribution of source-to-drain tunnelling in the subthreshold
region at this channel length in addition to the classical over-barrier (thermionic) current. Such conclusion
is further supported by the observation that in the DG
simulations the subthreshold slope is nearly independent of temperature bearing in mind that the tunnelling
current, as discussed above, is less sensitive to temperature than a thermionic emission current.
All the results presented so far have assumed a single effective mass in both vertical and lateral directions.
However, the lateral effective mass would be used to
calibrated the DG approach in respect of source-drain
tunnelling in the same way as the vertical effective mass
was used to calibrate it in respect of vertical confinement. We have made a first attempt to calibrate the
lateral effective mass in respect of comprehensive NonEquilibrium Green’s Function (NEGF) simulations of
a double gate MOSFET, performed by A. Svizhenko
512
Asenov
Figure 15. I D -VG characteristics obtained from NEGF simulations, with calibrated density gradient for the double gate structure,
with gate lengths of 20 nm and 4 nm. VD = 1 V and VG is applied
to both top and bottom gate contacts. The inset provides a schematic
illustration of the simulated double-gate MOSFET structure.
and M. P. Anantram at NASA Ames Research Center.
In these simulations a more realistic double-gate structure is used (Ren et al. 2001), shown schematically in
the inset of Fig. 15. The results of the calibration are
illustrated in Fig. 15 starting with a 20 nm transistor
hardly affected by the source-to-drain tunnelling and
ending up with a 4 nm one which, although functional,
has a considerable tunnelling current component. We
have found a good agreement between the DG and the
NEGF simulations for a single lateral effective mass
over a wide range of channel lengths and biases with
more details to be reported in a forthcoming paper.
4.
The Density Gradient approach has been used in
the simulation of double gate MOSFETs with channel lengths ranging from 30 nm to 6 nm to investigate whether the density gradient approach can model
source-drain tunnelling in respect of the subthreshold
current characteristics in decanano scale MOSFETs.
We observe that as the channel length is reduced,
there is a corresponding increase in the subthreshold
slope. The temperature dependence of the subthreshold slope has also been studied. It is observed that
the temperature dependence of the 30 nm MOSFET
is in agreement with classical MOSFET theory, however, the subthreshold slope for the 8 nm device is
nearly independent of temperature, presumably due
to the larger source-drain tunnelling in the smaller
device, which is less temperature sensitive than the
classical thermionically dominated subthreshold current. We have been able to calibrate the lateral effective mass in DG simulations in respect of NEGF
results.
Acknowledgments
We gratefully acknowledge the helpful discussions
with Dr. Mario Ancona. Many thanks are also due
to Dr. Alexi Svizhenko and Dr. Anant Anantram of
NASA Ames Research Center for performing the Nonequilibrium Green’s Function simulation of the double
gate MOSFET. JRW would like to acknowledge the
support of EPSRC under grant no GR/L53755. SHEFC
Research Development Grant VIDEOS provided support for ARB. This work was also supported by IBM
through a Shared University Research Grant.
Conclusions
The DG and EP methods provide efficient means
for incorporating quantum corrections in multi dimensional device simulations and in CAD simulation
tools. Both methods agree well with the available data
from Poisson-Schrödinger simulations, although there
is a better agreement between Density Gradient and
Poisson-Schrödinger calculations in respect of carrier
densities. This, however, has little discernable effect
on threshold voltage and I D -VG current characteristics. We have implemented these quantum potential
techniques in large scale 3D statistical ‘atomistic’ simulations where they produce very similar results for
the average threshold voltage, threshold voltage lowering and standard deviation in the threshold voltage for
nano-scale MOSFETs.
Note
1. If the kinetic theory of an ideal gas, is modified to allow for
particle-particle interactions which in turn leading to two-body
correlations, then the resulting equation of state (thermal flux and
pressure tensor) for the classical gas include terms depending on
the concentration gradient of the gas and are known as Burnett
equations (see Chapman and Cowling 1952).
References
Ancona M.G. 2000. Equations of state for silicon inversion layers.
IEEE Transactions on Electron Devices 47: 1449–1456.
Ancona M. 2001. Private Communication.
Ancona M.G. and Iafrate G.J. 1989. Quantum correction to the equation of state of an electron gas in a semiconductor. Physical Review
B 39: 9536–9540.
Quantum Potentials in Semiconductor Devices
Asenov A. 1998. Random dopant induced threshold voltage lowering
and fluctuations in sub 0.1 m MOSFETs: A 3D ‘atomistic’ simulation study. IEEE Transactions on Electron Devices 45: 2505–2513.
Asenov A., Slavcheva G., Brown A.R., Davies J.H., and Saini S.
1999. Quantum mechanical enhancement of the random dopant
induced threshold voltage fluctuations and lowering in sub 0.1 µm
MOSFETs. IEDM Tech. Digest 535–538.
Asenov A., Slavcheva G., Brown A.R., Davies J.H., and Saini S.
2001. Increase in the random dopant induced threshold fluctuations and lowering in sub-100 nm MOSFETs due to quantum effects: A 3-D density gradient simulation study. IEEE Transactions
on Electron Devices 48: 722–729.
Barker J.R. 1992. Fundamental aspects of quantum transport theory. In: Handbook on Semiconductors, Vol 1: Basic Properties of
Semiconductors. North-Holland, Amsterdam.
Bowen R.C., Klimeck G., Lake R., Frensley W.R., and Moise T.
1997. Quantitative resonant tunneling diode simulation. Journal
of Applied Physics 81: 3207–3213.
Carruthers P. and Zachariasen F. 1983. Quantum collision theory
with phase-space distributions. Review of Modern Physics 55:
245–284.
Chapman S. and Cowling T.G. 1952. The mathematical theory
of non-uniform gases. 2nd edn. Cambridge University Press,
Cambridge, UK.
Ferry D.K. 2000. The onset of quantization in ultra-submicron
semiconductor devices. Superlattices and Microstructures 27: 61–
66.
Ferry D.K., Akis R., and Vasileska D. 2000. Quantum effects in
MOSFETs: Use of an effective potential in 3D Monte Carlo simulation of ultra-short channel devices. IEDM Tech. Digest 287–290.
Fetter A.L. and Walecka J.D. 1971. Quantum theory of many-particle
systems. McGraw-Hill, New York.
Hisamoto D. 2001. FD/DG-SOI MOSFETS—A viable approach to
overcoming the device scaling limit. IEDM Tech. Digest 429–432.
Iafrate G.J., Grubin H.L., and Ferry D.K. 1982. The Wigner distribution function. Physics Letters 87A: 145–148.
Jallepalli S., Bude J., Shih W.K., Pinto M.R., Maziar C.M., and Tasch
A.F., Jr. 1997. Electron and hole quantization and their impact on
deep submicron p- and n-MOSFET characteristics. IEEE Trans.
Electron Devices 44: 297–303.
Kawaura H., Sakamoto T., and Baba T. 2000. Observation of sourceto-drain direct tunnelling current in 8 nm gate electrically variable
shallow junction metal-oxide-semiconductor field-effect transistors. Applied Physics Letters 76: 3810–3812.
Kawaura H., Sakamoto T., Baba T., Ochiai Y., Fujita J., and Sone
J. 2000. Transistor characteristics of 14 nm gate-length EJMOSFETs. IEEE Transactions on Electron Devices 47: 856–860.
Lundstrom M. 2001. Private Communication.
Perrot F. 1979. Gradient correction to the statistical electronic free
energy at nonzero termperatures: Application to equation of state
513
of an electron gas in a semiconductor. Physical Review B A20:
586.
Rafferty C.S., Biegel B., Yu Z., Ancona M.G., Bude J., and Dutton
R.W. 1998. Multi-dimensional quantum effects simulation using
a density gradient model and script-level programming technique.
In: SISPAD’98, De Meyer and Biesemans (Eds.), pp. 137–140.
Ramey S.M. and Ferry D.K. 2002. 3D Monte Carlo modeling of
thin SOI MOSFETs including the effective potential and random
dopant distribution. Journal of Computational Electronics 1: 267–
271.
Ren Z., Venugopal R., Datta S., Lundstrom M., Jovanovic D., and
Fossum J. 2001. Examination of design and manufacturing issues
in a 10 nm double gate MOSFET using non-equilibrium Green’s
function simulation. IEDM Tech. Digest 107–110.
Snowden C.M. 1989. Semiconductor device modelling. SpringerVerlag, Wien.
Stern F. 1970. Iteration methods for calculating self-consistent
fields in semiconductor inversion layers. Journal of Computational
Physics 6: 56–67.
Svizhenko A., Anantram M.P., Govindan T.R., Biegel B., and
Venugopal R. 2002. Two-dimensional quantum mechanical modeling of nanotransistors. Journal of Applied Physics 91: 2343–
2353.
Taur Y. and Ning T.H. 1998. Fundamentals of modern VLSI devices.
Cambridge University Press, Cambridge, UK.
Thompson S., Alavi M., Argavani R., Brand A., Bigwood R.,
Brandenburg J., Crew B., Dubin V., Hussein M., Jacob P., Kenyon
C., Lee E., Mcintyre M., Ma Z., Moon P., Nguyen P., Prince M.,
Schweinfurth R., Shvakumar S., Smith P., Stettler M., Tyagi S.,
Wei M., Xu J., Yang S., and Bohr M. 2001. An enhanced 130 nm
generation logic technology featuring 60 nm transistors optimised
for high performance and low power at 0.7–1.4 V. IEDM Tech.
Digest 257–261.
Tsuchiya H., Fischer B., and Hess K. 2000. A full-band Monte Carlo
model for silicon nanoscale devices with a quantum mechanical
correction of the potential. IEDM Tech. Digest 283–286.
Tsuchiya H. and Miyoshi T. 2000. Quantum mechanical Monte Carlo
approach to electron transport at heterointerface. Superlattices and
Microstructures 27: 529–532.
Tsuchiya H., Winstead B., and Ravaioli U. 2001. Quantum potential
approaches for nano-scale device simulation. VLSI Design 13:
335–340.
Watling J.R., Brown A.R., and Asenov A. 2002. Can the density gradient approach describe the source-drain tunnelling in decanano
double-gate MOSFETs? Journal of Computational Electronics 1:
289–293.
Watling J.R., Brown A.R., Asenov A., and Ferry D.K. 2001. Quantum corrections in 3-D drift diffusion simulations of decanano
MOSFETs using an effective potential. In: SISPAD’01, Tsoukalas
D. and Tsamis C. (Eds.), pp. 82–85.