8/12/2019 D Quantum Corrections
1/41
Inclusion of Tunneling and Size-Quantization Effects in Semi-
Classical Simulators
8/12/2019 D Quantum Corrections
2/41
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 QuantumMoment Methods
Particle-Based Device Simulations: Effective Potential Approach
Quantum Transport
Direct Solution of the Schrodinger Equation (Usuki Method) and TheoreticalBasis of the Greens Functions Approach (NEGF)
NEGF: Recursive Greens Function Technique and CBR Approach
Atomistic SimulationsThe Future
Prologue
8/12/2019 D Quantum Corrections
3/41
Quantum Mechanical Effects
There are three important manifestations
of quantum mechanical effects in nano-
scale devices:
- Tunneling
- Size Quantization
- Quantum Interference Effects
8/12/2019 D Quantum Corrections
4/41
Inclusion of Tunneling and Size-
Quantization Effects
Tunneling Effect: WKB Approximation and
Transfer Matrix Approach
Quantum-Mechanical Size Quantization EffectDrift-Diffusion and Hydrodynamics:
Quantum Correction and
Quantum Moment Methods
Particle-Based Device Simulations: EffectivePotential Approach
8/12/2019 D Quantum Corrections
5/41
Tunneling Currents vs. Technology
Nodes and Tunneling Mechanisms
10-16
10-14
10-12
10-10
10-8
10-6
10-4
0 50 100 150 200 250
Cu
rrent(A/
m)
Technology Generation (nm)
Ion
IG
Iof f
For tox 40 , Fowler-Nordheim (FN) tunneling dominates For t
ox< 40 , 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
B Vox>
B
Vox=
B
Vox<
B
FN FN/Direct Direct
tox
B Vox>
B
Vox=
B
Vox<
B
FN FN/Direct Direct
tox
This slide is courtesy of D. K. Schroder.
8/12/2019 D Quantum Corrections
6/41
WKB Approximation to Tunneling
Currents Calculation
0
EF
B
0
EF
B
a
No applied bias With applied bias
- eEx
x-axis
The difference between the Fermi level and the top of the barrier isdenoted by B
According to WKB approximation, the tunneling coefficient through thistriangular barrier equals to:
a
dxxT 0 )(2exp where: eExm
x B 2*2
)(
8/12/2019 D Quantum Corrections
7/41
WKB Approximation to Tunneling
Currents Calculation
eE
mT B
3
*24exp
2/3
Calculated and experimental tunnelcurrent characteristics for ultra-thin oxidelayers.
(M. Depas et al., Solid State Electronics, Vol.38, No. 8, pp. 1465-1471, 1995)
The final expression for theFowler-Nordheim tunnelingcoefficient is:
Important notes:
The above expressionexplains tunneling processonly qualitatively becausethe additional attraction ofthe electron back to the plateis not included
Due to surfaceimperfections, the surface
field changes and can makelarge difference in the results
8/12/2019 D Quantum Corrections
8/41
Tunneling Current Calculation in Particle-
Based 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-mechanicalreflections 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).
8/12/2019 D Quantum Corrections
9/41
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.
8/12/2019 D Quantum Corrections
10/41
Piece-Wise Constant Potential Barrier
(PCPBT Tool) installed on the nanoHUB
www.nanoHUB.org
8/12/2019 D Quantum Corrections
11/41
The Approach at a Glance
Slide property of Sozolenko.
8/12/2019 D Quantum Corrections
12/41
The Approach, Continued
Slide property of Sozolenko.
8/12/2019 D Quantum Corrections
13/41
Piece-Wise Linear Potential Barrier
This algorithm is implemented in ASUs code formodeling Schottky junction transistors (SJT)
It approximates real barrier with piece-wiselinear segments for which the solution of the 1D
Schrodinger equation leads to Airy functions andmodified Airy functions
Transfer matrix approach is used to calculate theenergy-dependent transmission coefficient
Based on the value of the energy of the particleE, T(E) is looked up and a random number isgenerated. If r
8/12/2019 D Quantum Corrections
14/41
The Approach at a Glance
E
ai-1 aiai+1
Vi
Vi+1
Vi-1 V(x)
1D Schrdingerequation:
Solution for piecewiselinear potential:
The total transmissionmatrix:
T(E):
ExVdx
d
m
)(
2 2
22
)()( )2()1( iiiii BCAC
1 2 1........T FI N BI M M M M M M
1
2011
1 NT
kT
K m
' '1 1
0 0
' '1 1
0 0
' '1 1
' '1 1
1 1[ (0) (0)] [ (0) (0)]
2 2
1 1[ (0) (0)] [ (0) (0)]
2 2
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
i i i i
FI
i i i i
N i N N i N N i N N i N
BI
n N i N N i N N i N N i N
r rA A B B
ik ik M
r rA A B B
ik ik
r B ik B r B ik BM
r r A ik A r A ik A
'
' ''1 1
( ) ( )( ) ( )
( ) ( )( ) ( )
i i i ii i i i i
i
i i i i i i ii i i i i
A Br B B
Mr r A r Br A A
8/12/2019 D Quantum Corrections
15/41
Simulation Results for Gate Leakage in
SJT
10-7
10-6
10-5
10-4
10-3
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Drain currentGate CurrentTunneling Current
Current[A/u
m]
Gate Voltage [V]
T. Khan, D. Vasileska and T. J.
Thornton, Quantum-mechanical
tunneling phenomena in metal-
semiconductor junctions, NPMS 6-
SIMD 4, November 30-December 5,
2003, Wailea Marriot Resort, Maui,
Hawaii.
8/12/2019 D Quantum Corrections
16/41
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 Statesfunction
D. Vasileska, D. K. Schroder and D.K. Ferry, Scaled silicon MOSFETs: Part II - Degradationof the total gate capacitance, IEEE Trans. Electron Devices44, 584-7 (1997).
8/12/2019 D Quantum Corrections
17/41
Effective Charge Set-Back From The
Interface
Schrodinger-Poisson Solvers
Quantum Correction Models
Quantum Moment Models
Substrate
Gate
ox
oxox
t
C
polyC
invC deplC inv
ox
poly
ox
ox
deplinv
ox
poly
ox
oxtot
C
C
C
C1
C
CC
C
C
C1
C
C
D. Vasileska, and D.K. Ferry, "The influence of poly-
silicon gates on the threshold voltage, inversion layer
and total gate capacitance in scaled Si-MOSFETs,"NanotechnologyVol. 10, pp.192-197 (1999).
8/12/2019 D Quantum Corrections
18/41
Schrdinger-Poisson Solvers
At ASU we have developed:
1D SchrodingerPoisson Solvers (inversion
layers and heterointerfaces)
2D SchrodingerPoisson solvers (Sinanowires)
3D SchrodingerPoisson solvers (Si quantum
dots)
S. N. Milii, 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).
8/12/2019 D Quantum Corrections
19/41
8/12/2019 D Quantum Corrections
20/41
8/12/2019 D Quantum Corrections
21/41
Simulation Results Obtained With
SCHRED
0
5 x019
1x1020
1.5x1020
2x1020
0 5 10 15 20 25 30 35 40
n(z)[cm-3]
Distance from the SiO2
/Si interface []
QM
VG= 2.5 V
SC0
5
10
15
20
25
1011 1012 1013
QMSC
zav
[]
Ns [cm-2]
Cinvreduces Ctotby about 10%
Cpoly+ Cinvreduce Ctotby about 20%
With poly-depletion Ctothas pronoun-
ced gate-voltage dependence
0.2
0.4
0.6
0.8
1.0
-0.5 0.0 0.5 1.0 1.5 2.0 2.5
Ctot
[F/cm
2]
VG [V]
Cox
SCNP QMNP
SCWPQMWP
invoxpolytot CCCC
1111
The classical charge density peaks rightat the SC/oxide interface.
The quantum-mechanically calculatedcharge density peaks at a finite distancefrom the SC/oxide interface, which leadsto larger average displacement of
electrons from that interface.
8/12/2019 D Quantum Corrections
22/41
Simulation Results Obtained With
SCHRED
0.2
0.3
0.4
0.50.6
0.7
0.8
0.91
1 2 3 4 5 6 7 8 9 10
classical M-B, metal gates
classical F-D, metal gatesquantum, metal gates
quantum, poly-gates ND=6x1019cm-3
quantum, poly-gates ND=1020cm-3
quantum, poly-gates ND=2x1020cm-3
Ctot/
Cox
Oxide thickness tox
[nm]
T=300 K, NA
=1018cm-3
Degradation of the Total Gate Capacitance Ctot
for Different Device Technologies
8/12/2019 D Quantum Corrections
23/41
Simulation Results Obtained With
SCHRED
8/12/2019 D Quantum Corrections
24/41
Comparison With Experiments
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0 5x1011
1x1012
1.5x1012
2x1012
2.5x1012
3x1012
Exp. data [Kneschaurek et al.]V
eff(z)=V
H(z)+V
im(z)+V
exc(z)
Veff
(z)=VH(z)
Veff
(z)=VH(z)+V
im(z)
Exp. data [Kneschaurek et al.]V
eff(z)=V
H(z)+V
im(z)+V
exc(z)
Veff
(z)=VH(z)
Veff
(z)=VH(z)+V
im(z)
EnergyE10[meV] T = 4.2 K, Ndepl=10
11 cm-2
Ns [cm-2 ]
Kneschaurek et al., Phys. Rev. B 14, 1610 (1976) Infrared Optical Absorption
Experiment:
far-ir
radiation
LED
SiO2 Al-Gate
Si-Sample
Vg
Transmission-Line Arrangement
Infrared Optical Absorption
Experiment:
far-ir
radiation
LED
SiO2 Al-Gate
Si-Sample
Vg
far-ir
radiation
LEDLED
SiO2 Al-Gate
Si-Sample
Vg
Transmission-Line Arrangement
D. Vasileska, PhD Thesis, 1995.
8/12/2019 D Quantum Corrections
25/41
SCHRED Usage on the nanoHUB
SCHRED has 92 citations in Scientific Research Papers,
1481 users and 36916 jobs as of July 2009
8/12/2019 D Quantum Corrections
26/41
3D Schrodinger-Poisson Solvers
3D SchrodingerARPACK
3D Poisson: BiCGSTAB method
Aluminum
Chrome
PECVD SiO2
Thermal SiO2
p-type bulk silicon
Na = 1016
cm-3
400 nm
30 nm
93 nm20 nm
5 nm
Built-in gates
Aluminum
Chrome
PECVD SiO2
Thermal SiO2
p-type bulk silicon
Na = 1016
cm-3
400 nm
30 nm
93 nm20 nm
5 nm
Built-in gates
8/12/2019 D Quantum Corrections
27/41
3D Schrodinger-Poisson Solvers
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.
0 1 3 42
s
0.0
0.8
0.6
0.4
0.2
1.0
P(s) 11 12 13
14 15 16
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 ofMicrosystems, San Diego, California, March 27-29, 2000, pp. 520-523 (Computational Publications, 2000).
8/12/2019 D Quantum Corrections
28/41
Quantum Correction Models
- Hansc and Van Dort Approach -
These quantum-correction models try to incorporate thequantum-mechanical description of the carrier density in aMOSFET device structure via modification of certain deviceparameters:
HANSC model - modifies the effective DOS function
Van Dort model - modifies the intrinsic carrier density viamodification of the energy band-gap. Within this model, themodification of the surface potential is:
2* /exp1 LAMBDAzNN CC
CONVQMn
CONVs
QMs zzzzEq ,/
Accounts for the band-gap wideningeffect because of the upward shift ofthe lowest allowed state
Accounts for the larger displacementof the carriers from the interface andextra bend-bending needed for givenpopulation:
94zqEn
8/12/2019 D Quantum Corrections
29/41
With these modifications, the energy band-gap becomes:
This results in modification of the intrinsic carrier density,which now, anywhere through the depth of the device, takes
the form:
The function F(y) is introduced to describe smooth transitionbetween classical and quantum description (pinch-off andinversion regions)
3/23/1
4
,
9
13
E
Tqk
EE
B
SiCONVg
QMg
B.DORT (MODEL)
QMi
CONVii
BCONV
gQM
gCONVi
QMi
nyFyFnn
TkEEnn
)()(1
2/exp
refyyaaayF /,2exp1/exp2)( 22
N.DORT (MODEL)
8/12/2019 D Quantum Corrections
30/41
The Van Dort model is activated by specifying N.DORT on theMODEL statement.
E0
E1
distance
Energy n(z)
z
z
CONVz
QMz
Classical density
Quantum-mechanicaldensity
8/12/2019 D Quantum Corrections
31/41
Modification of the DOS Function
The modification of the DOS function affects thescattering rates and must be accounted for inthe adiabatic approximation via solution of the1D/2D Schrodinger equation in slices along the
channel of the device
This is time consuming and for all practicalpurposes only charge set-back and modificationof the band-gap are to a very good accuracyaccounted for using eitherBohm potential approach to continuum modeling
Effective potential approach in conjunction withparticle-based device simulators
8/12/2019 D Quantum Corrections
32/41
Quantum Corrected Approaches
Drift DiffusionDensity Gradient
HydrodynamicQuantum Hydrodynamic
Particle-based device simulations
Effective Potential Approaches due to:
- Ferry, and
- Ringhofer and Vasileska
8/12/2019 D Quantum Corrections
33/41
Bohm Theory
The hydrodynamic formulation is initiated by substituting the wavefunc-
tion into the time-dependent SWE:
The resultant real and imaginary parts give:
nRReiS ,/ t
iVm
22
2
qc ffQVdt
dm
mR
RmtQ
tQtVSmt
tS
S/mtRtSmt
t
v
r
eq.JacobiHamiltonrrr
vrrr
2/122/12
22
2
2
2
1
2),,(
);,,(),(2
1),()2(
;),(),(;01),(
)1(
8/12/2019 D Quantum Corrections
34/41
Effective Potential Approach Due to Ferry
ieffi
ii
ii
ii
Vd
Vdd
dVd
ndrVV
)r()rr(r~
'rrexp)'r('r)rr(r~
)r'r(
'rr
exp'r)r(r~
)r()r(
2
2
2
2
In principle, the effective role of the potential can be rewritten in terms of thenon-local density as (Ferry et al.1):
Classical densitySmoothed,
effective potential
Buil t-in potential
for tr iangular po-
tent ial approx ima-
t ion.
Effective potential
approximation
Quantization
energy
Set backof ch arge --
quantum capacitance
effects
Buil t-in potential
for tr iangular po-
tent ial approx ima-
t ion.
Effective potential
approximation
Quantization
energy
Set backof ch arge --
quantum capacitance
effects
1 D. K. Ferry, Superlatt. Microstruc.27, 59 (2000); VLSI
Design, in press.
8/12/2019 D Quantum Corrections
35/41
Parameter-Free Effective Potential
The basic concept of the thermodynamic approach to effectivequantum 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
for the density matrix (x,y) by the classical Liouville equation
for the classical density function f(x,k). Here, the relation betweenthe density matrix and the density function f is given by the Weyl
quantization
, 0it H
12 *
0t x x k mf k f V f
( , ) [ ] ( / 2, / 2)exp( )f x k W x y x y ik y dy
8/12/2019 D Quantum Corrections
36/41
The thermal equilibrium density matrix in the quantum
mechanical setting is given by eq = e-H, 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 correctequilibrium states in the semiclassical Liouville equation with the
effective quantum potential VQ, we set:
22
2 *
( , ) exp [ ] ( / 2, / 2)exp( )k Q eq H
eq m
f x k V W e x y x y 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. 2005Page(s):227236.
8/12/2019 D Quantum Corrections
37/41
Different forms of the effective quantum potential arise from
different approaches to approximate the matrix exponential e-H.
In the approach presented in this paper, we represent e-Has theGreens 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 thehighest order differential operator (the Laplacian).
This involves successive solution of a heat equation for which
the Greens function is well known, giving
2 2
23 2
1 2 *( , ) sinh exp ( )2 * 8 *2
i x yQ
Q QB H
m kV x k V y e dyd m mk
V V
8/12/2019 D Quantum Corrections
38/41
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:
Note: It is evaluated only once at the beginning of the simulation!!!
1 1
1 1
22 11
1 1
2 *sinh 2 *( , ) (1, 0, 0) exp2 8 *
Q i xTx B
p
mB me V x p e d m p
8/12/2019 D Quantum Corrections
39/41
The Hartree Field
8/12/2019 D Quantum Corrections
40/41
Output Characteristics of DG Device
Tox = 1 nm Tsi = 3 nmLG = 9 nm LT = 17 nmLsd = 10 nm Nsd = 2 x 10
20 cm-3
Nb = 0 g= 1 nm/decade
G = 4.188 VG = 0.4 V
1.E+10
1.E+13
1.E+16
1.E+19
1.E+22
DopingDensity[cm-3]
Source DrainSi Channel
Substrate
BOXBackGate
FrontGate
LG
LT
Tsi
Lsd
Tox = 1 nm Tsi = 3 nmLG = 9 nm LT = 17 nmLsd = 10 nm Nsd = 2 x 10
20 cm-3
Nb = 0 g= 1 nm/decade
G = 4.188 VG = 0.4 V
1.E+10
1.E+13
1.E+16
1.E+19
1.E+22
DopingDensity[cm-3]
Source DrainSi Channel
Substrate
BOXBackGate
FrontGate
LG
LT
Tsi
Lsd
Source DrainSi Channel
Substrate
BOXBackGate
FrontGate
LG
LT
Tsi
Lsd
Source DrainSi Channel
Substrate
BOXBackGate
FrontGate
LG
LT
Tsi
Lsd
0
500
1000
1500
2000
0 0.2 0.4 0.6 0.8 1Drain Voltage [V]
DrainCurrent[uA/um]
0
15
30
45
%C
hangeinC
urrent
W/o quant. (3nm)QM (3nm)NEGF (3nm)W/o quant. (1nm)
QM (1nm)%Change
3 nmVG= 0.4 V
1 nm
8/12/2019 D Quantum Corrections
41/41
Summary
Tunneling that utilizes transfer matrix approach can quiteaccurately be included in conjunction with particle-baseddevice simulators
Quantum-mechanical size-quantization effect can beaccounted in fluid models via quantum potential that is
proportional to the second derivative of the log of thedensity
Effective potential approach has been proven to includesize-quantization effects rather accurately in conjunctionwith particle-based device simulators
Neither the Bohm potential nor the effective potential canaccount for the modification of the density of statesfunction, and, therefore, scattering rates modificationbecause of the low-dimensionality of the system, and,therefore, mobility and drift velocity
Top Related