Quantum Transport
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
Transport Properties of system/device
using Green’s functions formalism
Low field transport
Linear response theory (ASU)
High field transport
Bulk systems – Airy approach (Rita Bertoncini,
ASU, PhD Thesis)
Devices:
Recursive Green’s Functions Approach (ASU,
Purdue)
CBR Approach (ASU, WSI, Purdue)
Linear Response Theory
 Only the retarded Green’s function is needed as it includes the
collisional broadening of the states
 In the ASU’s simulator for low-field mobility calculation in silicon
inversion layers, strained-Si layers and InGaAs/InAlAs
heterostructures the following features have been implemented:
 Realistic treatmet of scattering within the self-consistent Born
approximation
 Modification of the density of states function is accounted for due to the
collisional broadening of the states and the intersubband scattering
 Random phase approximation in its full implementation is included to
properly treat static screening of Coulomb and Interface-Roughness
scattering
 Bethe-Salpether integral equation is solved in the calculation of the
conductivity
 Excellent agreement is obtained with measured low-field mobility data in
silicon inversion layers and predictions were made for the mobility
behavior in Strained-Si layers and InGaAs/InAlAs heterostructures that
were later confirmed with experimental measurements
Relevant Literature






D. Vasileska, P. Bordone, T. Eldridge and D.K. Ferry, “Calculation of
the average interface field in inversion layers using zero-temperature
Green’s functions formalism”, J. Vac. Sci. Technol. B 13, 1841-7 (1995).
P. Bordone, D. Vasileska and D.K. Ferry, “Collision duration time for
optical phonon emission in semiconductors”, Physical Review B 53,
3846-55 (1996).
D. Vasileska, T. Eldridge and D.K. Ferry, “Quantum transport: Silicon
inversion layers and InAlAs-InGaAs heterostructures”, J. Vac. Sci.
Technol. B 14, 2780-5 (1996).
D. Vasileska, P. Bordone, T. Eldridge and D. K. Ferry, “Quantum
transport calculations for silicon inversion layers in MOS structures”,
Physica B 227, 333-5 (1996).
D. Vasileska and D. K. Ferry, “Scaled silicon MOSFET’s: Part I Universal mobility behavior”, IEEE Trans. Electron Devices 44, 577-83
(1997).
G. Formicone, D. Vasileska and D.K. Ferry, “Transport in the surface
channel of strained Si on a relaxed Si1-xGex substrate”, Solid State
Electronics 41, 879-886 (1997).
Proposed Strained-Si and Strained-SiGe
Devices
Gate
(a)
Gate
(b)
n+poly-Si
Source
n
+
Gate
(c)
n+poly-Si
n+poly-Si
Drain
SiO2
Source
Source
Drain
SiO2
Si1-xGex
Strained-Si
n
+
n+
n+
n
+
Strained-Si
Relaxed Si1-xGex
SiGe Graded Buffer
5% x% of Ge
Strained-Si
Relaxed Si1-xGex
SiGe Graded Buffer
5% x% of Ge
SiGe Graded Buffer
5% x% of Ge
Gate
(e)
Metal
Metal
Source
SiO2
Drain
Source
p+
Strained-Si1-xGex
n-
SiO2
Drain
Si
Si
p+
Strained-Si
Si1-xGex
Relaxed Si1-xGex
Gate
(d)
SiO2
p+
Strained-Si1-xGex
p+
p+ modulation doping
Si Substrate
n- Si Substrate
Drain
n+
Is Strain Beneficial in Nanoscale MOSFETs
With High Channel Doping Densities?
’
’
’
4 -band
’
’



00’

’
Biaxial tension
Strained Silicon
x=0.1
x=0.2
x=0.4
x=0.1
x=0.2
x=0.4
1.8
strained-Si /Si
2
Mobility [cm /V-s]
1500
2
Exp. data
Silicon
1.6
1.4
1.2
1 16
10
1000
10
17
10
18
-3
Substrate doping NA [cm ]
500
0
16
10
-3
17
-3
17
-3
17
-3
18
-3
A
Regular Silicon
2500
17
N =1x10 cm




2000
2.2
00’+
Mobility enhancement ratio
2 -band
2
10
Substrate doping N
10
18
A
1.8
N =7x10 cm
A
N =1x10 cm
A
1.6
1.4
1.2
12
10
-3
A
A
N =5x10 cm
1
17
N =2x10 cm
[cm ]
13
10
-2
Inversion charge density N [cm ]
s
High Field Transport in Devices:
Recursive Green’s Functions Approach
 The most complete 1D transport in resonant tunneling diodes
(RTDs) that operate on purely quantum mechanical principles was
accomplished with the NEMO1D Code
 The NEMO 1D Code was developed by Roger Lake, Gerhard
Klimeck, Chris Bowen and Dejan Jovanovich while working at Texas
Instruments/Raytion
 It solves the retarded Green’s function (spectral function) in
conjuction with less-than Green’s function (occupation function) selfconsistently
 References for NEMO1D:
 Roger. K. Lake, Gerhard Klimeck, R. Chris Bowen, Dejan Jovanovic, Paul
Sotirelis and William R. Frensley,
"A Generalized Tunneling Formula for Quantum Device Modeling",VLSI Design,
Vol. 6, pg 9 (1998).
 Roger Lake, Gerhard Klimeck, R. Chris Bowen and Dejan Jovanovic,
"Single and multiband modeling of quantum electron transport through layered
semiconductor devices", J. of Appl. Phys. 81, 7845 (1997).
The Philosophy Behind the Recursive
Green’s Function Approach
K. B. Kahen, Recursive-Green’s-function analysis of wave propagation in two-dimensional
nonhomogeneous media, .Phys. Rev. E 47, 2927 - 2933 (1993).
S. Datta, From Atom to Transistor, 2008.
Representative Simulation Results
High Field Transport in Devices:
Contact Block Reduction Method
The retarded Green’s function of an open system:
- 1
GR (E ) = [IE - H] = [IE - H0 - S ]- 1
where H 0  closed system Hamiltonian , S  self-energy matrix
The Dyson equation,
- 1
GR (E ) = éëI - G0 (E )S (E )ùû G0 (E ),
- 1
where G0 (E ) º éëIE - H0 + i h ù
û =
å
a
a a
,
E - ea + i h h= 0+
H0 a = E a a
0
G (E ) describes closed system (decoupled device)
To determine Green’s function of an open system
we need to invert a huge matrix
D. Mamaluy, D. Vasileska, M. Sabathil, T. Zibold, and P. Vogl, “Contact block reduction

method for ballistic transport and carrier densities of open nanostructures”, Phys. Rev. B
71, 245321 (2005).
Retarded Green’s Function
GR of an open system in CBR formalism:
éGCR
GR = êê R
êëGDC
R
ù é
GCD
AC- 1GC0
ú= ê
ú ê
0
GRD ú ê- A DC AC- 1GC0 + GDC
û ë
0
AC- 1GCD
0
- A DC AC- 1GCD
ù
ú,
ú
+ GD0 ú
û
C
0
AC = 1C - GC0 S C , ADC = - GDC
SC
GC0 is the contact portion of the G0
where,
index D denotes the interior device region
index C denotes the contact ( boundary ) region
D
All elements of GR can be determined from inversion of small matrix AC
The left upper block GCR fully determine the transmission function
The left lower block GRDC determines density of states, charge density etc.
Transmission Function and Local Density
of States Calculation
 Transmission Function
' R†
T l l ' (E ) = T r ( Gl G R Gl G
)
CBR Formalism
'
T l l ' (E ) = T r ( GCl G CR GCl G CR † ) , where G CR = [1 - G C0 S C ]- 1G C0
GCl = i[S C - S C† ]
 Local Density of States Function
r (r, E ) = r | GR GGR † | r 2p
CBR Formalism
r (r, E ) =
1
2p
å
2
G rRm Gmm ,
m
0
G rRm = r | GDC
BC- 1 | m = å
m ', a
r|a a |m '
E - ea + i h
m ' | BC- 1 | m
h = 0+
Properties of Widely Acceptable 2D
Simulators
 Exactness
- Accomplished with comparison
with experiments
 Speed (Optimization and Process Variation)
Experimental FinFET*
Gate length Lg = 10 nm
Fin width tSi = 12 nm,
Gate oxide thickness tox= 1.7 nm
(110) channel orientation
*Bin Yu et. al., “FinFET Scaling to 10
nm Gate Length”, IEDM Tech. Digest,
2002
Y
B′
Z
X
A
B
Source
Gate
A′
Drain
Buried Oxide
3D view
H. R. Khan, D. Mamaluy and D. Vasileska, “Approaching Optimal Characteristics of 10 nm High
Performance Devices” a Quantum Transport Simulation Study of Si FinFET, IEEE Trans. Electron Devices,
Vol. 55(1), pp. 743-753 (2008).
H. R. Khan, D. Mamaluy and D. Vasileska, “Simulation of the Impact of Process Variation on the Optimized
10-nm FinFET”, IEEE Transactions Electron Dev. Vol. 55(8), pp. 2134 – 2141, August 2008.
NeedNeed
for 3D
Device
for 3D
DeviceSimulations
Simulations

2-D simulator does not give us the
opportunity to analyze:
the effect of fin height on carrier transport
 device characteristics of tri-gate structure
 the effect of an unintentional dopant on
device characteristics

Gate
h
tSi
Drain
Side Gate Side Gate
Z
Buried Oxide
Top Gate
oxide
Top Gate
Y
Y
Y
tox
tox
Simulated geometry
Side Gate
 Gate length =10 nm
 Fin width = 4 nm
 Gate oxide thickness = 1.2 nm
 Gate dielectric – SiO2
 Fin height = 4 nm ~ 8 nm
tSi
Z
h
VGS = 0.1V
tox
h
Side Gate
Lg
Top Gate
Y
tox
tSi
B
Source
Y
Y
Z X
4
Top Gate
View along B-B'
Y
Drain current [ A]
B'
Side Gate Side Gate
h
tSi
3
h
tSi
Side Gate
Z
Z
2
1
Side Gate
Z
Z
 Source/drain doping = n type-2×1019 cm-3
 Body doping = 2×1015 cm-3
 Doping gradient = 1.25 nm/dec
 Gate doping = uniform, n type-2×1019 cm-3
0
DG FinFET
TG FinFET
0.0
0.1
0.2
Drain voltage [V]
0.3
0.4
DG vs. TG FinFET
1
10
0.08
10
0.04
-2
10
6
-3
10
4
-4
10
Drain current, I
Drain current, I
DS
DS
8
2
Net gate leakage [nA]
-1
10
[µA]
0
10
[µA]
12
VDS = 0.4V
VDS = 0.4V
0.00
-0.04
-0.08
-0.12
-5
10
DG FinFET
TG FinFET
-6
DG FinFET
TG FinFET
0
10
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
-0.16
-0.4
-0.3
-0.2
-0.1
0.0
Gate voltage [V]
Gate voltage [V]
Parameter
DG
TG
ION = IDS @ VDS = 0.4V, VGS = 0.3V [μA]
7.57
10.18
ISD,LEAK = IDS @ VDS = 0.4V, VGS = -0.4V [nA]
0.0319
0.0048
Subthreshold swing [mV/dec]
73
70
|IG| @ VDS = 0.4V, VGS = -0.4V [nA]
0.027
0.122
0.1
0.2
0.3
ON-State Electron density
along the dotted line
Y [nm ]
4
2
tox
0
<2.
2.
8
6.
1.
1.
Y [nm ]
6
2.
tox
4
3.
3.
> 4.
2
0
0
2
4
6
8
10
12
12
3.6E19
3.0E19
2.4E19
1.8E19
1.2E19
6.0E18
> 4.2E19
0
Electron density (TG) > electron density (DG)
2.0E18
VGS=0.2, VDS=0.4V
<2.0E18
Z [nm]
Atomistic Simulations – The Future of
Nano-Devices
Examples of devices for which atomistic
simulations will be necessary include:
 Devices in which local Strain exists
 Alloy Disorder has to be properly described
• Group of Gerhard Klimeck, Purdue University, West Lafayette, IN, USA
• Group of Aldo di Carlo, Tor Vergata, Rome, Italy.
Why Tight-Binding ?
 Allows us to describe the band structure over the entire Brillouin zone
 Relaxes all the approximations of Envelope Function approaches
 Allows us to describe thin layer perturbation (few Å)
 Describes correctly band mixing
 Gives atomic details
 The computational cost is low
 It is a real space approach
 Molecular dynamics
 Scalability (from empirical to ab-initio)
Scalability of TB approaches
Empirical Tight-Binding
Hamiltonian matrix elements are obtained by comparison of calculated
quantities with experiments or ab-initio results. Very efficient, Poor
transferability.
Semi-Empirical Hartree-Fock
Methods used in the chemistry context (INDO, PM3 etc.). Medium
transferability.
Density Functional based Tight-Binding (DFTB, FIREBALL, SIESTA)
DFT local basis approaches provide transferable and accurate
interaction potentials. The numerical efficiency of the method
allows for molecular dynamics simulations in large super cells,
containing several hundreds of atoms.
The sp3s* Hamiltonian
[Vogl et al. J. Phys. Chem Sol. 44, 365 (1983)]
In order to reproduce both valence and conduction band of
covalently bounded semiconductors a s* orbital is introduced
to account for high energy orbitals (d, f etc.)
The sp3d5s* Hamiltonian
[Jancu et al. PRB 57 (1998)]
Many parameters, but works quite well !
Tight-Binding sp3d5s* model for nitrides
Ab-Inito Plane Wave DFT-LDA
Band Structure for GaN Wurtzite
TB Wurtzite GaN Band Structure
Nearest-neighbours sp3d5s* tightbinding parametrization for wurtzite
GaN, AlN and InN compare well with
Ab-Initio results.
Boundary conditions
Periodic
After P planes the structure
repeats itself. Suitable for
superlattices
H=
P
Finite chain
After P planes the structure end.
Suitable for quantum wells
H=
Open boundary conditions
After P planes there is a semiinfinite crystal
Suitable for current calculations
∞
P
BULK P
BULK
∞
Where do we put the atoms ?
To describe the electronic and optical properties of a
nanostructure we need to know where the atoms are.
1) We know “a priori” the atom positions (for example X-ray
information)
2) We need to calculate the atomic positions
Simple analytic espressions
Full calculation
Classical calculations Continuum theory
Atomistic (Valence Force Field)
Quantum calculation
Example: Strain and Pseudomorphic
growth
An epitaxial layer is grown, on a substrate with different lattice constant.
The epilayer deforms (strain)
as
a0
as
a
a0
as
as
as
as
C12
a  a0  2 as  a0 
C11
as
as
R'  (1   )R
Strain tensor
Strain in a AlGaN/GaN Nanocolumn
Calleja’s pillars
z, [0001]
20nm
Al0.28Ga0.72 N
GaN
y, [1210]
x, [1010]
AlGaN/GaN Nanocolumns
Potential
piezo-electric
polarization
pyro-electric
polarization
(  4 (P pz  P py ))  4
The Poisson equation
piezo-electric
moduli tensor
Pi pz  dijk jk
How do we describe alloys ?
Usually, tight-binding parameterizations are made for single elments and binary
compounds (Si, Ge, GaAs, InAs etc.). However, nanostructure are usually build by
using also ternary (AlGaAs etc.) and quatrnary (InGaAsP etc.) alloys.
1) Supercell calculations
A0.5B0.5C
Average over an ensamble
of configurations
2) Virtual crystal approximation
A new crystal is defined with averaged properties (P)
P(AxB1-xC)=x P(AC) + (1-x) P(BC)
3) Other methods (Modified VCA, CPA, T-matrix etc.)
Self-Consistent Tight-Binding
Schrodinger
Poisson
Charge transfer is important in semiconductor nanostructures.
Self-consistent solution of Schredinger and Poisson equations are common in
envelope function approaches
Tight-binding allows for a full (with all the electrons) self-consistent solution
of the nanostructure problem
Full self consistent approach only suitable for small systems like molecules
Self-consistent approach for only the free charge
With the aim of Self-Consistent treatment of external electrostatic
potential, Tight-Binding can be applied to semiconductor device
simulations.
Self-Consistent Tight-Binding
[A. Di Carlo et. al., Solid State Comm. 98, 803 (1996); APL 74, 2002 (1999)]
The electron and hole densities in each 2D layer are given by:
n( z ) 
1
2  
2
d 2k //  z Ec k //
x
y
1
2  
2
f Ec  Fn 
2
1  f Fh  Ev 
c
BZ //
p( z ) 
2
d 2k //  z Ev k //
BZ //
v
z
The influence of free carrier charge redistribution and macroscopic
polarization fields are included by solving the Poisson equation:
d
d
d

D( z )     VH  P   e p  n  N D  N A 
dz
dz  dz

H  H C  VH
+
boundary
conditions
Ei k //
Summary
 Linear response and solution of the Beth-Salpether
equation in conjunction with the Dyson equation for
the retarded Green’s function is useful when
modeling low-field mobility of inversion layers
 When modeling high field transport both Dyson
equation for the retarded Green’s function and the
kinetic equation for the less-than Green’s function
have to be solved self-consistently
 CBR approach and recursive Green’s function
method have both their advantages and their
disadvantages
 When local strains and stresses have to be
accounted for in ultra-nano-scale devices then
atomistic approaches become crucial
Prologue
What are the lessons that we have
learned?
 Semi-classical simulation is still a very important part of Today’s
semiconductor device modeling as power devices and solar cells
(traditional ones) operate on semi-classical principles
 Quantum corrections can quite accurately account for the
quantum-mechanical size quantization effect which gives about
10% correction to the gate capacitance
 For modeling ultra-nano scale devices one can successfully
utilize both Poisson-Monte Carlo-Schrodinger solvers and fully
quantum-mechanical approaches based on NEGF (tunelling +
size quantization)
 Full NEGF is a MUST when quantum interference effects need to
be captured and play crucial role in the overall device behavior
 For a subset of ultra-nano scale devices that are in the focus of
the scientific community now, in which band-structure, local strain
and stresses, play significant role, atomistic simulations are
necessary.
Simulation Strategy for Ultra-Nano-Scale
Devices
Calibrate semi-empirical
approaches with ab-initio band
structure simulations
Perform
BANDSTRUCTURE/TRANSPORT
calculations on systems containing
millions of atoms
1000 atoms
Millions of atoms
Atomistic Simulations Selected Literature
 Mathieu Luisier and Gerhard Klimeck,
"A multi-level parallel simulation approach to electron transport in nano-scale
transistors", Supercomputing 2008, Austin TX, Nov. 15-21 2008. Regular paper - 59
accepted papers, 277.
 Mathieu Luisier, Neophytos Neophytou, Neerav Kharche, and Gerhard Klimeck,
"Full-Band and Atomistic Simulation of Realistic 40 nm InAs HEMT", IEEE IEDM, San
Francisco, USA, Dec. 15-17, 2008, DOI : 10.1109/IEDM.2008.4796842,
 Mathieu Luisier, and Gerhard Klimeck,
"Performance analysis of statistical samples of graphene nanoribbon tunneling
transistors with line edge roughness",
Applied Physics Letters, Vol. 94, 223505 (2009), DOI:10.1063/1.3140505,
 Mathieu Luisier, and Gerhard Klimeck,
"Atomistic, Full-Band Design Study of InAs Band-to-Band Tunneling Field-Effect
Transistors ", IEEE Electron Device Letters, Vol. 30, pp. 602-604 (2009),
DOI:10.1109/LED.2009.2020442.
Questions or Comments?