Download Quantum Corrections in the Simulation of Decanano MOSFETs

Quantum Corrections in the Simulation of Decanano MOSFETs
A. Asenov*, A. R. Brown and J. R. Watling
Device Modelling Group
Dept. of Electronics and Electrical Engineering
University of Glasgow, G12 8LT, UK
*
[email protected]
ABSTRACT
Quantum mechanical confinement and tunnelling play
an important role in present and future generation decanano
(sub 100 nm) MOSFETs and have to be properly taken into
account in the simulation and design. Here we present a
simple approach of introducing quantum corrections in a
3D drift diffusion simulation framework using the density
gradient (DG) algorithm. We discuss the calibration of the
DG approach in respect of quantum confinement effects in
comparison with more comprehensive but computationally
expensive quantum simulation techniques. We also
speculate about the capability of DG to describe source-todrain tunnelling in sub 10 nm (nano) MOSFETS. The
application of the DG approach is illustrated with examples
of 3D statistical simulations of intrinsic fluctuation effects
in decanano and nano-scale double gate MOSFETs.
1
INTRODUCTION
MOSFETs scaled down to 15nm gate lengths have been
successfully demonstrated [1] and 9nm MOSFETs are
expected in mass production in 2016 according the new
edition of the roadmap. There is a consensus, however, that
scaling around and below 10 nm will require a departure
from the traditional MOSFET architecture. Among the
most promising nanometre scale transistor candidates are
double gate MOSFETs [2]. The combination of thin gate
oxides and heavy doping in the conventional MOSFETs,
and the thin silicon body of the double-gate structures, will
result in substantial quantum mechanical (QM) threshold
voltage shift and transconductance degradation [3]. Below
10 nm gate-lengths direct source-to-drain tunnelling will
rapidly became one of the major limiting factors for scaling
[4]. Computationally efficient methods to include QM
effects are required for the purpose of practical Computer
Aided Design of this generation of devices. First order
quantum corrections based on density gradients (DG) have
already been introduced in 2-D [5] and 3-D [6] driftdiffusion simulations. In this paper we discuss the
implementation of DG quantum corrections in a 3-D drift
diffusion simulation network designed to study intrinsic
parameter fluctuations introduced by the discreteness of
charge and atomicity of matter. We start with the
calibration of the DG approach in respect of quantum
confinement effects in comparison with more sophisticated
quantum simulations [2]. We also investigate to what extent
the DG approach can describe, at least semi-quantitatively,
the expected source-to-drain tunnelling in nano-scale
devices. Finally we illustrate the application of DG
corrected drift-diffusion simulations in the analysis of
various sources of intrinsic fluctuations in decanano double
gate MOSFETs.
2
THE DENSITY GRADIENT APPROACH
We are motivated by the need to include quantum
corrections in the atomistic simulation of intrinsic
fluctuation effects in decanano MOSFETs introduced by
atomicity of charge and matter [6-8]. The investigation of
intrinsic fluctuation effects involves statistical 3-D
simulations of large samples of macroscopically identical
but microscopically different devices and is very
computationally expensive. Therefore the computational
efficiency of the quantum correction approach is of great
importance and the use of the DG approach becomes an
attractive option.
The DG approach may be derived from the one particle
Wigner function [9]:
∂f (k, r, t )
+ v ⋅ ∇ r f (k, r, t ) −
∂t
 h∇ r ∇ k 
 ∂f (k, r, t ) 
2
 f (k, r, t ) = 
V (r )sin

h
 2 
 ∂t
coll
(1)
Quantum effects are included through the inherently
non-local driving potential in the third term on the left-hand
side. Expanding to first order in h, so that only the first nonlocal 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 [10,
11]. 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 the
particle simulations, similar in spirit to the Bohm
interpretation.
(2)
0.1
(3)
Thus the unipolar drift-diffusion system of equations with
QM corrections, which in many cases is sufficient for
MOSFET simulations, becomes:
(
∇ ⋅ (ε∇ψ ) = −q p − n + N D+ − N A−
2bn
)
kT n
∇2 n
ln
= φn − ψ +
q
ni
n
∇ ⋅ (nµn ∇φn ) = 0
(4)
(5)
(6)
We have carefully calibrated the DG approach against
the results of a full-band 1-D Poisson-Schrödinger (PS)
solver [2]. Although Poisson-Schrödinger simulations are
more accurate they are not yet practical for 3-D device
simulations. Fig. 1 shows the quantum mechanical
threshold voltage shift for DG as a function of substrate
doping compared with the results presented by Jallepalli in
[2]. Simulation results obtained using the recently
developed effective potential (EP) quantum correction
approach [12] are also presented for comparison in this and
the following two figures. Utilising a single value of the
electron effective mass of 0.18m0, obtained from matching
the PS results at doping concentration 1018 cm-3 the DG
simulations follow precisely the PS results. Fig. 2 shows
typical carrier concentration profiles obtained from the 1-D
simulations. Both the DG and the EP simulations show a
peak in the concentration away from the Si/SiO2 interface,
although the EP produces a sharper drop-off at the Si/SiO2
interface compared to PS and DG.
1018
1019
Fig. 1. 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.
Jallepalli (Poisson-Schrödinger)
Density Gradient
Effective Potential
17
10
16
10
15
10 0
1
2
3
4
7
6
5
8
Depth from interface [nm]
Fig. 2. Electron carrier concentration as a function of
distance from the interface, for substrate doping of
5×1017cm-3. All have the same net sheet density.
10
-5
10
-6
10
-7
10
-8
ID [A]
CALIBRATION
0 17
10
Substrate Doping [cm-3]
The system of equations (4) – (6) is solved self-consistently
using standard techniques.
3
Effective Potential
Density Gradient
Jallepalli (Poisson-Schrodinger)
0.2
where bn = h /(12qm*n ) , and all other symbols have their
usual meaning. To avoid the discretisation of fourth order
derivatives in (1) in multidimensional numerical
simulations a generalised electron quasi-Fermi potential φ n
is introduced as follows:
Fn = nµn ∇φn
0.3
Electron Concentration [cm-3]
 ∇2 n 

Fn = nµn ∇ψ − Dn ∇n + 2µn ∇bn
n 

0.4
VT (QM) – VT (Classical) [V]
The density gradient approximation may be derived in a
manner similar to that for deriving the drift diffusion
approximation from the Boltzmann Transport Equation and
results in a quantum potential correction term in the
standard drift-diffusion flux [5].
10
-9
10
-10
10
-11
10
-12
10
-13
Classical
Effective Potential
Density Gradient
-14
10 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
VG [V]
Fig. 3. ID-VG characteristic obtained from both classical and
quantum simulators for a 30nm×30nm n-MOSFET, with
VD=0.01V and a substrate doping of 5×1018cm-3.
4
SOURCE-TO-DRAIN TUNNELLING
-5
10
-6
10
-7
10
Density
Gradient
Classical
-8
10
I
D
It still remains unclear to what extent the
approximations involved in deriving the DG approach
remove its ability to model the direct source-to-drain
tunnelling expected in nanometre channel length
MOSFETs. In search of a qualitative answer to this
question we simulate a set of double gate MOSFETs with
generic structure illustrated in Fig 4.
subthreshold slope degrades significantly as the channel
length is decreased, while in the classical simulations the
subthreshold slope remains nearly constant with channel
length. The degradation in the subthreshold slope in the DG
simulations is consistent with the more elaborate quantum
mechanical simulations performed by others [13]. These
observations provide an indication that source-to-drain
tunnelling is included, to some extent, in the DG
simulations.
[A]
Fig. 3, shows an ID-VG characteristic for a 30nm×30nm
n-MOSFET obtained from our 3-D quantum simulator. The
threshold voltage shift between the classical and the
quantum simulations and the overall shape of the current
voltage characteristics obtained using the DG and EP
approaches are very similar.
-9
10
6 nm
8 nm
10 nm
-10
10
Top Gate
12 nm
15 nm
20 nm
-11
Source
Undoped
Channel
n+
Drain
Oxide
n+
30 nm
10
-12
10
-0.4
-0.2
0
G
Oxide
Bottom Gate
Fig. 4. Schematic representation of the double-gate
MOSFET structure considered in this work.
Fig. 5 show the electron concentration distribution normal
to the gate obtained from the DG simulations of a
30×30×1.5nm double-gate MOSFET. As expected, for such
a thin Si body quantum confinement effects produce a peak
in the distribution in the middle of the channel.
20
VG = 0.6 V
VG = 0.4 V
10
10
0.4
0.6
[V]
Fig. 6. ID-VG characteristics for a double gate structure, with
gate lengths ranging from 30nm down to 6nm, obtained
from our classical and density gradient simulations.
VD=0.01V and VG is applied to both top and bottom gate
contacts.
Further evidence can be gained by looking at the
temperature dependence of the subthreshold slope
illustrated in Fig. 7. In the classical drift-diffusion
simulations the subthreshold slope, essentially thermionic
in nature, depends linearly on temperature. However, any
current due to tunnelling will have a much weaker
temperature dependence [14].
-3
10
-3
Electron concentration [cm ]
19
0.2
V
18
10
-4
VG = 0.2 V
17
10
-5
10
10
-6
10
16
[A]
10
15
I
VG = 0.0 V
14
10
-7
10
D
10
Density
Gradient
Classical
-8
10
-9
10
13
10
0
-10
1
0.5
1.5
Depth [nm]
10
300 K
200 K
-11
77 K
10
-12
Fig. 5. Quantum (density gradient) electron concentration
profile through the centre of a 30×30×1.5nm double-gate
MOSFET. The oxide thickness is 1.5nm.
Fig. 6 illustrates the current voltage characteristics for a
set of double gate transistors with channel lengths in the
range from 30 to 6 nm. In the DG simulations the
10
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
V
G
0.2
0.3
0.4
0.5
[V]
Fig. 7. ID-VG characteristics for an 8nm channel length
double gate structure from classical and density gradient
simulations, for a range of temperatures. VD=0.01V and VG
is applied to both top and bottom gate contacts.
5
RANDOM DISCRETE DOPANTS
Theoretically the double-gate MOSFETs do not require
channel doping to operate and therefore are considered to
be inherently resistant to random dopant induced parameter
fluctuations. Here we investigate to what extent the random
dopants in the source and drain region introduce intrinsic
parameter fluctuations in such devices.
for the shorter channel length device. This is associated
with the effective channel length variation along the
channel introduced by the random discrete dopant
distribution in the source and the drain regions.
6
CONCLUSIONS
The density gradient approach provides computationally
efficient means for incorporating quantum corrections in
multi-dimensional device simulations. It agrees well with
the available data from Poisson-Schrödinger simulations. In
the simulation of sub 10 nm double gate MOSFETs the
density gradient approach shows behaviour qualitatively
consistent with source-to-drain tunnelling.
REFERENCES
Fig. 8. Electrostatic Potential in a 30×30×5nm double-gate
atomistic MOSFET at threshold.
Fig. 9. Electron equiconcentration surface in a 30×30×5nm
double-gate atomistic MOSFET with location of dopants
shown.
Figs. 8 and 9 illustrate the impact of the unavoidable
discrete dopants in the source/drain region on the potential
and the electron distribution in a 30×30×5 nm double-gate
MOSFET. The carrier concentration in the source and drain
region is modulated by the potential fluctuations associated
with individual discrete dopants. The maximum in the
carrier concentration is in the middle of the channel.
Channel
Dimensions
(L × W × T)
Threshold
Voltage
Fluctuations
σVT [mV]
10 × 30 × 1.5 nm
1.07
9.56
7.13
30 × 30 × 5 nm
0.66
3.28
1.93
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σID [%]
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the source and drain. The threshold voltage is ~200mV.
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length fluctuations introduce small threshold voltage
fluctuations which increase from 0.66 mV to 1.07mV as the
device is scaled from 30 to 10 nm. The on-state current,
however, does exhibit significant fluctuations, particularly
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