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Inclusion of Tunneling and SizeQuantization Effects in SemiClassical Simulators
Outline:
 What is Computational Electronics?
 Semi-Classical Transport Theory
 Drift-Diffusion Simulations
 Hydrodynamic Simulations
 Particle-Based Device Simulations
 Inclusion of Tunneling and Size-Quantization Effects in Semi-Classical Simulators
 Tunneling Effect: WKB Approximation and Transfer Matrix Approach
 Quantum-Mechanical Size Quantization Effect
 Drift-Diffusion and Hydrodynamics: Quantum Correction and Quantum
Moment Methods
 Particle-Based Device Simulations: Effective Potential Approach
 Quantum Transport
 Direct Solution of the Schrodinger Equation (Usuki Method) and Theoretical
Basis of the Green’s Functions Approach (NEGF)
 NEGF: Recursive Green’s Function Technique and CBR Approach
 Atomistic Simulations – The Future
 Prologue
Quantum Mechanical Effects
There are three important manifestations
of quantum mechanical effects in nanoscale devices:
- Tunneling
- Size Quantization
- Quantum Interference Effects
Inclusion of Tunneling and SizeQuantization Effects
Tunneling Effect: WKB Approximation and
Transfer Matrix Approach
Quantum-Mechanical Size Quantization Effect
Drift-Diffusion and Hydrodynamics:
• Quantum Correction and
• Quantum Moment Methods
Particle-Based Device Simulations: Effective
Potential Approach
Current (A/ m)
Tunneling Currents vs. Technology Nodes
and Tunneling Mechanisms
10 -4
Io n
10 -6
10 -8
B
I o ff
10 -1 0
10 -1 2
Vox = B
Vox > B
Vox < B
IG
10 -1 4
10 -1 6
0
50
100
150
200
tox
250
Technology Generation (nm)
FN
•
•
•
•
This slide is courtesy of D. K. Schroder.
FN/Direct
Direct
For tox  40 Å, Fowler-Nordheim (FN) tunneling dominates
For tox < 40 Å, direct tunneling becomes important
Idir > IFN at a given Vox when direct tunneling active
For given electric field: - IFN independent of oxide thickness
- Idir depends on oxide thickness
WKB Approximation to Tunneling
Currents Calculation
F
F
B
B
EF
EF
0
- eEx
x-axis
No applied bias
a
0
With applied bias
 The difference between the Fermi level and the top of the barrier is
denoted by FB
 According to WKB approximation, the tunneling coefficient through this
triangular barrier equals to:
 a

T  exp  2   ( x)dx 
 0

where:
 ( x) 
2m *

2
F B  eEx 
WKB Approximation to Tunneling
Currents Calculation
 The final expression for the
Fowler-Nordheim tunneling
coefficient is:
 4 2m *F 3B/ 2 
T  exp 

3
eE



 Important notes:
 The above expression
explains tunneling process
only qualitatively because
the additional attraction of
the electron back to the plate
is not included
 Due to surface
imperfections, the surface
field changes and can make
large difference in the results
Calculated and experimental tunnel
current characteristics for ultra-thin oxide
layers.
(M. Depas et al., Solid State Electronics, Vol.
38, No. 8, pp. 1465-1471, 1995)
Tunneling Current Calculation in ParticleBased Device Simulators
 If the device has a Schottky gate then one must
calculate both the thermionic emission and the
tunneling current through the gate
WKB fails to account for quantum-mechanical
reflections over the barrier
Better approach to use in conjunction with
particle-based device simulations is the
Transfer Matrix Approach
W. R. Frensley, “Heterostructure and Quantum Well Physics,” ch. 1 in
Heterostructure and Quantum Devices, a volume of VLSI Electronics:
Microstructure Science, N. G. Einspruch and W. R. Frensley, eds.,
(Academic Press, San Diego, 1994).
Transfer Matrix Approach
Within the Transfer Matrix approach one
can assume to have either
Piece-wise constant potential barrier
Piecewise-linear potential barrier
D. K. Ferry, Quantum Mechanics for Electrical Engineers, Prentice Hall, 2000.
Piece-Wise Constant Potential Barrier
(PCPBT Tool) installed on the nanoHUB
www.nanoHUB.org
The Approach at a Glance
Slide property of Sozolenko.
The Approach, Continued
Slide property of Sozolenko.
Piece-Wise Linear Potential Barrier
 This algorithm is implemented in ASU’s code for
modeling Schottky junction transistors (SJT)
 It approximates real barrier with piece-wise
linear segments for which the solution of the 1D
Schrodinger equation leads to Airy functions and
modified Airy functions
 Transfer matrix approach is used to calculate the
energy-dependent transmission coefficient
 Based on the value of the energy of the particle
E, T(E) is looked up and a random number is
generated. If r<T(E) than the tunneling process
is allowed, otherwise it is rejected.
Tarik Khan, PhD Thesis, December 2004.
The Approach at a Glance
 1D Schrödinger
equation:

 d 
 V ( x)  E
2
2m dx
2
Vi+1
Vi
2
 Solution for piecewise
linear potential:
 i  Ci(1) Ai ( )  Ci( 2) Bi ( )
 The total transmission
matrix:
MT  M FI M1M 2 ........M N 1M BI
 T(E):
k
T  N 1
K0
1
T 2
m11
E
Vi-1
V(x)
ai-1
ai+1
ai
r1 '
1
[
A
(0)

Ai (0)]
i
2
ik0
M FI  
1
r1 '
[
A
(0)

Ai (0)]
 i
2
ik
0

r
1

[ Bi (0)  1 Bi' (0)]
2
ik0

r1 ' 
1
[ Bi (0)  Bi (0)]
2
ik0

'
'
  rN Bi ( N )  ik N 1Bi ( N ) rN Bi ( N )  ik N 1Bi ( N ) 

M BI  
'
'
rn r A ( )  ik A ( ) r A ( )  ik A ( ) 
N 1 i N
N i N
N 1 i N 
 N i N
'
Bi (i ) 
  ri Bi (i )  Bi (i )   Ai (i )

Mi  

ri r A' ( ) A ( )  ri 1 Ai' (i ) ri 1Bi' (i ) 
i i 
 i i i
Simulation Results for Gate Leakage in
SJT
Current [A/um]
10
-3
10
-4
10
-5
10
-6
10
-7
0.1
Drain current
Gate Current
Tunneling Current
0.2
0.3
0.4
0.5
Gate Voltage [V]
0.6
0.7
T. Khan, D. Vasileska and T. J.
Thornton, “Quantum-mechanical
tunneling phenomena in metalsemiconductor junctions”, NPMS 6SIMD 4, November 30-December 5,
2003, Wailea Marriot Resort, Maui,
Hawaii.
Quantum-Mechanical Size Quantization
Quantum-mechanical size quantization
manifests itself as:
- Effective charge set-back from the
interface
- Band-gap increase
- Modification of the Density of States
function
D. Vasileska, D. K. Schroder and D.K. Ferry, “Scaled silicon MOSFET’s: Part II - Degradation
of the total gate capacitance”, IEEE Trans. Electron Devices 44, 584-7 (1997).
Effective Charge Set-Back From The
Interface
 Schrodinger-Poisson Solvers
 Quantum Correction Models
 Quantum Moment Models
Gate
Cpoly
ε
Cox  ox
t ox
Cdepl
Cinv
Substrate
C tot 
Cox
1
Cox
Cox

Cpoly Cinv  Cdepl
Cox

1
Cox
C
 ox
Cpoly Cinv
D. Vasileska, and D.K. Ferry, "The influence of polysilicon gates on the threshold voltage, inversion layer
and total gate capacitance in scaled Si-MOSFETs,"
Nanotechnology Vol. 10, pp.192-197 (1999).
Schrödinger-Poisson Solvers
At ASU we have developed:
 1D Schrodinger – Poisson Solvers (inversion
layers and heterointerfaces)
 2D Schrodinger – Poisson solvers (Si
nanowires)
 3D Schrodinger – Poisson solvers (Si quantum
dots)
S. N. Miličić, F. Badrieh, D. Vasileska, A. Gunther, and S. M. Goodnick, "3D Modeling of
Silicon Quantum Dots," Superlattices and Microstructures, Vol. 27, No. 5/6, pp. 377-382
(2000).
Space Quantization Literature
Bacarani and Worderman  transconductance degradation
(Proceedings of the IEDM, pp. 278-281, 1982)
Hartstein and Albert  estimate of the inversion layer thickness
(Phys. Rev. B, Vol. 38, pp.1235-1240, 1988)
van Dort et al.  analytical model for Vth which accounts for QM
effects (IEEE TED, Vol. 39, pp. 932-938, 1992)
Takagi and Toriumi  physical origins of Cinv
(IEEE TED, Vol. 42, pp. 2125-2130, 1995)
Vasileska, Schroder and Ferry  influence of many-body effects on
Cinv (IEEE TED, Vol. 44, pp. 584-587, 1997)
Hareland et al.  modeling of the QM effects in the channel
(IEEE TED, Vol. 43, pp. 90-96, 1996)
Krisch et al.  poly-gate capacitance attenuation
(IEEE EDL, Vol. 17, pp. 521-524, 1996)
1D Schrodinger-Poisson Solver for Si
Inversion Layers – SCHRED
• 1D Poisson equation:




  1  
 e N D ( z )  N A ( z )  p ( z )  n( z )


z  ( z ) z 



VG>0
• 1D Schrödinger equation:
EF
 2   1  





V
(
z
)

  ij ( z )  Eij  ij ( z )
 mi ( z ) z 
2

z
 



• Electron density:
z-axis [100]
(depth)
n( z )   N ij  ij2 ( z )
i, j
N ij 
4-band
2-band
m||i k BT
 2

 EF  Eij 

ln 1  exp
 k BT 

2-band
:
2-band :
mm=m
l=0.916m0, m||=mt=0.196m0
=ml=0.916m0, m||=mt=0.196m0
4-band:
4-band:
1/2
mm=m
t=0.196m
0, , m
||==(m
lm
t))1/2
=m
=0.196m
m
(m
m

t
0
||
l
t

Simulation Results Obtained With
SCHRED
2x1020
1.0
10
1x1020
5 x0
5
SC
0 11
10
12
10
QM
19
0.8
C
2
QM
SC
13
10
-2
N [cm ]
s
tot
15
C [F/cm ]
[Å]
av
1.5x10
20
z
-3
n(z) [cm ]
VG= 2.5 V
20
SCNP
25
ox
0.6
SCWP
5
10
15
20
25
30
35
40
Distance from the SiO /Si interface [Å]
2
The classical charge density peaks right
at the SC/oxide interface.
The quantum-mechanically calculated
charge density peaks at a finite distance
from the SC/oxide interface, which leads
to larger average displacement of
electrons from that interface.
QMWP
0.4
1
Ctot
0.2
0
0
QMNP
-0.5
0.0
0.5
V

1
C poly
1.0
G

1
1

Cox Cinv
1.5
2.0
2.5
[V]
Cinv reduces Ctot by about 10%
Cpoly+ Cinv reduce Ctot by about 20%
With poly-depletion Ctot has pronounced gate-voltage dependence
Simulation Results Obtained With
SCHRED
1
0.9
tot
C /C
ox
0.8
T=300 K, N A=1018 cm-3
0.7
classical M-B, metal gates
0.6
classical F-D, metal gates
quantum, metal gates
0.5
19
quantum, poly-gates N =6x10
0.4
D
20
quantum, poly-gates N =10
D
0.3
0.2
1
-3
cm
20
3
-3
quantum, poly-gates N =2x10
cm
4
9
D
2
-3
cm
5
6
7
8
10
Oxide thickness t [nm]
ox
Degradation of the Total Gate Capacitance Ctot
for Different Device Technologies
Simulation Results Obtained With
SCHRED
MOS Capacitor with both Metal and Poly-Silicon Gates
300
300
V
-V
V
-V
th(QMNP)
200
th(QMWP)
100
250
th(SCNP)
th(SCNP)
T=300 K
N = 10
20
D
th
150
Van Dort experimental data
th(SCNP)
-3
cm
V
[mV]
th
V
-V
th(SCWP)
[mV]
V
250
t = 4 nm
50
0
ox
(IEEE TED, Vol.39, pp. 932-938, 1992)
Our simulation results
200
150
100
T=300 K
Metal gates
t = 14 nm
ox
50
16
10
17
18
10
10
-3
N [cm ]
A
0
16
10
17
18
10
10
-3
N [cm ]
A
Vth shows strong substrate doping dependence when poly-gate depletion and QM
effects in the channel are included
There is close agreement between the experimentally derived threshold voltage shift
and our simulation results
Comparison With Experiments
Energy E10 [meV]
60
Kneschaurek et al., Phys. Rev. B 14, 1610 (1976)
11
-2
T = 4.2 K, Ndepl=10 cm
50
Infrared Optical Absorption
Experiment:
Exp. data [Kneschaurek et al.]
Veff(z)=V H(z)+V im (z)+V exc(z)
40
SiO2
LED
Veff(z)=V H(z)
30
Al-Gate
far-ir
Veff(z)=V H(z)+V im (z)
Si-Sample
radiation
20
10
Vg
0
0
11
5x10
12
1x10
12
1.5x10
12
2x10
-2
Ns [cm ]
D. Vasileska, PhD Thesis, 1995.
12
2.5x10
12
3x10
Transmission-Line Arrangement
SCHRED Usage on the nanoHUB
 SCHRED has 92 citations in Scientific Research Papers,
1481 users and 36916 jobs as of July 2009
3D Schrodinger-Poisson Solvers
3D Schrodinger – ARPACK
3D Poisson: BiCGSTAB method
Aluminum
400 nm
Chrome
30 nm
PECVD SiO2
93 nm
20 nm
Built-in gates
Thermal SiO2
p-type bulk silicon
Na = 1016 cm-3
5 nm
3D Schrodinger-Poisson Solvers
1.0
P(s)
0.8
0.6
11
12
13
14
15
16
0.4
0.2
0.0
0
1
2
s
3
4
Left: The energy level spacing distribution as a function of s =E/(E)avg
obtained by combining the results of a number of impurity configurations.
Right: The 11th to 16th eigenstates of the silicon quantum dot.
S. N. Milicic, D. Vasileska, R. Akis, A. Gunther, and S. M Goodnick, "Discrete impurity effects in silicon
quantum dots," Proceedings of the 3rd International Conference on Modeling and Simulation of
Microsystems, San Diego, California, March 27-29, 2000, pp. 520-523 (Computational Publications, 2000).
Quantum Correction Models
- Hansc and Van Dort Approach  These quantum-correction models try to incorporate the
quantum-mechanical description of the carrier density in a
MOSFET device structure via modification of certain device
parameters:
HANSC model - modifies the effective DOS function
N C*  N C 1  exp z / LAMBDA 2
Van Dort model - modifies the intrinsic carrier density via
modification of the energy band-gap. Within this model, the
modification of the surface potential is:
CONV
QM
CONV
 QM





/
q

E

z
,
Δz

z

z
s
s
n
Accounts for the band-gap widening
effect because of the upward shift of
the lowest allowed state
Accounts for the larger displacement
of the carriers from the interface and
extra bend-bending needed for given
population:
4
qEn z  
9
With these modifications, the energy band-gap becomes:
1/ 3
  Si 
QM
CONV 13
Eg  Eg
 ,   

9
 4qk BT 
E2 / 3
B.DORT (MODEL)
This results in modification of the intrinsic carrier density,
which now, anywhere through the depth of the device, takes
the form:


niQM  niCONV exp E gQM  E gCONV / 2k BT

ni  niCONV 1  F ( y )  F ( y )niQM
The function F(y) is introduced to describe smooth transition
between classical and quantum description (pinch-off and
inversion regions)
 


F ( y )  2 exp  a 2 / 1  exp  2a 2 , a  y / yref
N.DORT (MODEL)
The Van Dort model is activated by specifying N.DORT on the
MODEL statement.
Energy
n(z)
Classical density
z
E1
E0

Quantum-mechanical
density
z CONV
z QM
z
distance
Modification of the DOS Function
 The modification of the DOS function affects the
scattering rates and must be accounted for in
the adiabatic approximation via solution of the
1D/2D Schrodinger equation in slices along the
channel of the device
 This is time consuming and for all practical
purposes only charge set-back and modification
of the band-gap are to a very good accuracy
accounted for using either
Bohm potential approach to continuum modeling
Effective potential approach in conjunction with
particle-based device simulators
Quantum Corrected Approaches
Drift Diffusion  Density Gradient
Hydrodynamic  Quantum Hydrodynamic
Particle-based device simulations 
Effective Potential Approaches due to:
- Ferry, and
- Ringhofer and Vasileska
Bohm Theory
 The hydrodynamic formulation is initiated by substituting the wavefunction into the time-dependent SWE:
  Re
iS / 
, R n 
2 2


   V  i
2m
t
 The resultant real and imaginary parts give:
(r, t )
 1

     S   0; (r, t )  R (r, t )2 ; v  S/m
t
 m

S(r, t )
1
S 2  V (r, t )  Q(,r, t ); Hamilton Jacobi eq.
( 2) 

t
2m

(1)
2 1 2
 2 1/ 2 2 1/ 2
Q(, r, t )  
 R

 
2m R
2m
dv
m
 V  Q   fc  fq
dt
Effective Potential Approach Due to Ferry
In principle, the effective role of the potential can be rewritten in terms of the
non-local density as (Ferry et al.1):
Built-in potential
for triangular potential approximation.
V   drV (r ) ni (r )
i
 r  r' 2 
(r ' ri )
~  drV (r )  dr ' exp 
2 

 
i

 r  r' 2 

~   dr(r  ri )  dr ' V (r ' ) exp 
2

 
i

~   dr(r  ri )Veff (r )
Effective potential
approximation
Quantization
energy
i
Classical density
“Set back” of charge -quantum capacitance
effects
Smoothed,
effective potential
1
D. K. Ferry, Superlatt. Microstruc. 27, 59 (2000); VLSI
Design, in press.
Parameter-Free Effective Potential
The basic concept of the thermodynamic approach to effective
quantum potentials is that the resulting semiclassical transport
picture should yield the correct thermalized equilibrium quantum
state. Using quantum potentials, one generally replaces the
quantum Liouville equation
t  
i
H ,   0
for the density matrix (x,y) by the classical Liouville equation
t f  2m* k  x f  1  xV k f  0
for the classical density function f(x,k). Here, the relation between
the density matrix and the density function f is given by the Weyl
quantization
f ( x, k )  W[  ]    ( x  y / 2, x  y / 2)exp(ik y)dy
The thermal equilibrium density matrix in the quantum
mechanical setting is given by eq = e-βH, where =1/kBT is the
inverse energy, and the exponential is understood as a matrix
exponential.
In the semi-classical transport picture, the thermodynamic
equilibrium density function feq is given by the Maxwellian
distribution function.
Consequently, to obtain the quantum mechanically correct
equilibrium states in the semiclassical Liouville equation with the
effective quantum potential VQ, we set:
  2k2

f eq ( x, k )  exp   2m*  V Q   W [  eq ]   e  H  ( x  y / 2, x  y / 2)exp(ik y )dy


D. Vasileska and S. S. Ahmed, “Modeling of Narrow-Width SOI Devices”, Semicond.
Sci. Technol., Vol. 19, pp. S131-S133 (2004).
D. Vasileska and S. S. Ahmed, “Narrow-Width SOI Devices: The Role of Quantum
Mechanical Size Quantization Effect and the Unintentional Doping on the Device
Operation”, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005
Page(s):227 – 236.
Different forms of the effective quantum potential arise from
different approaches to approximate the matrix exponential e-βH.
In the approach presented in this paper, we represent e-βH as the
Green’s function of the semigroup generated by the
exponential.
The logarithmic Bloch equation is now solved asymptotically,
using the Born approximation, i.e. by iteratively inverting the
highest order differential operator (the Laplacian).
This involves successive solution of a heat equation for which
the Green’s function is well known, giving
  2k 
V ( x, k ) 
sinh 
3
2
 2m *
 2   k 

Q
1
 VBQ  VHQ
2m *

  2

 exp  

 8m *
2
 V ( y )e

i  x  y 
dyd 
The Barrier Potential
The total potential is divided into Barrier and Hartree potential,
where Barrier is a Heviside step function and Hartree is the
solution to the Poisson equation.
The barrier field is then calculated using:
  p1  1 
2
m
*sinh



1
B
2m *  i1x1
Q

T

e xVB ( x, p) 
(1, 0, 0)  exp   
e
d1
2
8m * 
 p1  1


2
2
Note: It is evaluated only once at the beginning of the simulation!!!
The Hartree Field
Hartree potential is expanded using the assumption that it is
slowly varying function in space. In that case, one can write:
VHQ ( x,

  2 2  p  2 
x
 exp  
p, t )  1 
2



24m *
2
x 
 VH ( x, t ),
8m * 

2
where:

0
VH ( x, t )  exp  

2
2
x 
 VH ( x, t )
8m 

Then, the Hartree Field is computed using:
 xr VHQ ( x n ,
n
p , t)
  xr VH0 ( x n , t ) 
2
2
2
24m *
2
n n
0 n
 p j pk x j xk xrVH ( x , t ), n  1,
j ,k 1
,N
C. Gardner, C. Ringhofer and D. Vasileska, Int. J. High Speed Electronics and
Systems, Vol. 13, 771 (2003).
Output Characteristics of DG Device
DG SOI Device:
LG
2000
45
Lsd
Drain
Tsi
Back
Gate
BOX
LT
Doping Density [cm -3]
Substrate
1.E+22
Drain Current [uA/um]
Si Channel
1500
1000
W/o quant. (3nm)
QM (3nm)
NEGF (3nm)
W/o quant. (1nm)
QM (1nm)
%Change
500
1.E+19
1.E+16
15
0
1.E+13
0
0
1.E+10
Tox = 1 nm
LG = 9 nm
Lsd = 10 nm
Nb = 0
ΦG = 4.188
30
1 nm
Tsi = 3 nm
LT = 17 nm
Nsd = 2 x 1020 cm-3
g = 1 nm/decade
VG = 0.4 V
0.2
0.4
0.6
Drain Voltage [V]
0.8
1
% Change in Current
V G = 0.4 V
Front
Gate
Source
3 nm
Summary
 Tunneling that utilizes transfer matrix approach can quite
accurately be included in conjunction with particle-based
device simulators
 Quantum-mechanical size-quantization effect can be
accounted in fluid models via quantum potential that is
proportional to the second derivative of the log of the
density
 Effective potential approach has been proven to include
size-quantization effects rather accurately in conjunction
with particle-based device simulators
 Neither the Bohm potential nor the effective potential can
account for the modification of the density of states
function, and, therefore, scattering rates modification
because of the low-dimensionality of the system, and,
therefore, mobility and drift velocity