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Vijay K. Arora
Wilkes University
E-mail: [email protected]
Emerging Technologies
Our Motivation and Economics
Adam Smith, “An Enquiry into Nature and Causes of the Wealth
of Nations” (1776)
The wealth is created by laisse-faire economy and
free trade
John Maynard Keynes, “The General Theory of Employment,
Interest, and Money” (1936)
The wealth is created by careful government
planning and government stimulation of economy
1990’s and Beyond
The wealth is created by innovations and inventions
20th Century Paradigm

Formulate a hypothesis or theory

Accumulate data

Do extensive experimentation and Check

Publish if newsworthy

Respect others’ work helping them to grow in the
profession

Demonstrate character ethics that puts community
interests above personal aggrandizement
21st Century Paradigm

Formulate a hypothesis or theory or design

Make a prototype structure

Patent it

Raise 17 million dollars and start an IPO

Sue your competitor for stealing your idea

Demonstrate personality ethics that lubricates the
process of human interaction for personal
aggrandizement
Gross world product and sales
volumes
Exponential Growth
SIA roadmap
Historical Trends

New Technology generation every three years

For each generation, memory density
increase by 4 times and logic density
increases by 2.5 times

Rule of Two: In every two generations (6
years), the feature size decreased by 2,
transistor current density, circuit speed, chip
area, chip current and maximum I/O pins
increased by 2
Research Scenario

A comprehensive transport theory for
quantum processes at nanosacle

High-field distribution in quantum wells

Optimization of the shape and size of
quantum wells for high frequencies

Quantum Computing: Multi-state logic by
using quantum states

Failure of Ohm’s Law: Re-assessment of the
circuit theory principles
Goals for High Speed Performance
 Large transistor current
• Time constants
• Interconnects
• Cross talk
 Reduced transit time
• Increased Mobility
• High Saturation Velocity
• Reduced Size
RC and Transit Time Delays
Source:Cadence
Cadence
Source:
Interconnect Problems
RC Time Delays
RC time delay is increasing rapidly
 Wire resistance is rising
 Wires have larger cross-section … introduce
coupling
 Electromigration imposes current limits
 System performance, area and reliability
are determined by interconnect
quality, not devices!!!

Interconnect Performance
0.25 
Decreasing Coupling Effect
0.5 
Increasing Performance
1
Increased cross-section improves performance but
also increases noise and capacitive and inductive
coupling
RC Delay Considerations
R3
R2
R1
Cc
layer m
Cf
Cf
Co
layer n
Cf
Cf
R4
Cf Cs
Cf
layer m
Cf
Cs
Cf
substrate
Cint = Cf + Cs + Co + Cload
 = Rint * ( Cint + Cc/(Cint+ Cc) )
 = Rint * (Cint2 + Cint.Cc +Cc)/(Cint + Cc)
• Cc depends on dimensional shrink due to increased in cross-section
• In VLSI, make Cc becomes insignificant as possible, then
 = Rint * Cint
Physical Effects
 Quantum Effects
L  D , a few nm
High-Field Effects
V 5V
kV
E 
 50
L 1m
cm
Field Broadening qED 


or qE  k BT
Nano-Scale
Quantum Engineering
D
h

p

h
3m * k BT
Bulk Semiconductors
Lx , y , z  D
All 3 cartesian directions analog-type
Density of States:
3
* 2
1

2me 
1 dN
gc (E) 
 4  2  E  Eco 2
V dE
 h 
Quasi-Two-Dimensional QW
Lz  D
Lx , y  D
z-direction digital-type
x,y-directions analog-type
 ( k k )
2
Enk Eco 
2
x
2
y
2me *
n  1, 2, 3,......
Density of States:
 n  oz
2
2 2
 oz  * 2
2 me Lz
 E  Eco 
1 dN m

gc2 ( E) 

Int 
A dE 
  oz 
*
e
2
AlGaAs/GaAs/AlGaAs
Prototype Quantum Well
Quasi-One-Dimensional QW
L y , z  D
Lx  D
y, z-direction digital-type
x-directions analog-type (QWW)
2 2
 kx
2
2
Enk Eco 
 m  oy  n  oz
*
2 me
m, n  1, 2, 3,......
Density of States:
 
 o( y, z)  * 2
2me Ly , z
  E  ( E
* 1/ 2
e
1 dN 2m
g c1 ( E ) 

Lx dE

2 2
2
2

m


n
 oz )
co
oy


1
2
Quasi-Zero-Dimensional
Quantum Well
All 3 cartesian directions digital-type
Quantum box (dot)
Lx , y , z  D
Enk Ec    ox  m  oy  n  oz
2
2
m, n,   1, 2, 3,......
2
 
 o( x, y , z )  * 2
2 me Lx , y , z
2 2
Quantum Well Wire
Quantum Box (Dot)
AlGaAs
Quantum box
Quantum
wire
GaAs inside
Quantum Well Arrays
Density of States
DENSITY OF STATES ( 1026 eV-1m-3 )
1
N ( E) 
Lx Ly Lz
 E  E 


s
1.2
1.0
0.8
0.6
0.4
3D
2D
1D
0.2
0.0
0.0
0.2
0.4
0.6
E - Ec (eV)
0.8
1.0
Quantum Well with Finite
Boundaries
 1 
L z  1  a
P
1
2
2m * E  a
P  
2
 2
2 
nz 


Z n z 
sin
L z  L z 
Triangular Quantum Well
Approximate:
Zn z 
nz
2

sin
Ln
 Ln 
2
Ln  2 zo
an
0.53556
an 
Ai'  n 
Exact:
Z n ( z) 
1
Ai'   n  z1o/ 2
 z

Ai
 z  n 

 o

Quantum-Confined
Mobility Degradation
 Changes in the Density of States
 QW
 bulk
 Lz


D
isotropic
Lz  D
 Changes in the relative strength
each scattering interaction
of
Mobility Degradation Versus
Quantum Confinement
0.7
0.6
 2D /  b
0.5
0.4
T=4K
T = 77 K
T = 300 K
0.3
0.2
0.1
0
0.1
-1
1 / Lz (nm )
1
Gate-Field Confinement
Mobility Degradation in a TQW
MOBILITY (m 2 / V.s)
0.08
0.075
Theory
Experiment
0.07
0.065
0.06
0.055
0.05
0.045
10
15
20
25
30
35
40
45
ELECTRIC FIELD (V / m)
50
Electron and Hole Mobility in
Submicron CMOS
Courtesy: Y. Taur and E. Novak, IBM Microelectronics, IEDM97 Invited Talk.
Random Thermal Motion
e
h
h
e
e
h
Ions
e Electrons
h Holes
h

vth  0
3k BT
vth 
 105 m / s
m*
Quantum Emission
Q Q
e
h
o
h
h
h
Atoms
h
e Electrons
h Holes
e
Q
h
h o
qE Q   o

E
 o
Q 
qE
Randomness to Streamlining
Velocity Vectors in Equilibrium
Randomness:


vd  vth  0
Velocity Vectors in a Very High Field
Streamlined:


vd  vth   2 vthˆ
Saturation Velocity-Bulk
v
F1  

vth
F1/ 2  

2
3D

2 k BT
vth 
m*
j
1
x
Fj   
x  Fermi Integral

0
( j  1) 1  e
  Ec Normalized Fermi Energy

k BT
Saturation Velocity Limits
vsat
8k BT

vth 
*
m

vsat
2
3 h

*
4m
 3n 


 8 
1
3
Non-degenerate
limit
Degenerate
limit
Saturation Velocity-Q2D
v
F1/ 2  

vth
2
F0  

2D

F0    ln 1  e

  Ec
k BT


Ec  Eco   oz
Saturation Velocity-Q1D
v

F0  

vth
F1/ 2  

1
1D
  Ec
k BT
Ec  Eco   oy   oz
Modeling Transport
dv
qE v  vth

dt
m*
c
=0
 
q c 
c 
E 1 e
Transient Response: v  

m * 

t
q c
Steady State t   c : vd  
E  oE
m*
Quantum Emission
qE Q   o
 o
Q 
qE
Effective Collision time:  eff
 o
Q 
qEvth
Q


c

  c 1 e






Q


Effective collision length:    o 1  e o





1-D Random Walk in a Bandgap
semiconductor
Modeling the Distribution

f ( , E ) =
1
e
x
  
k BT

    qE  
k BT



1
1
e
x 
1
Q



o
o 1  e


q E o E V
o 
 
k B T Eco Vco





o
Q 
k BT
Left-Right Asymmetry
Itinerant Electron Population
n x 
e
e
   
n( x ) e  e
2 cosh ( )


Streamlining the Randomness
1
0.8
0.6
n +/n
n-/n
0.4
0.2
0
0
0.5
1

1.5
2
2.5
Drift-Diffusion
J ( x)  n( x) q vth tanh  
dn
 q vth 
dx
vd vth tanh  

Dn  vth   no Vt
o
q o
q cn
 no 
mn* Vth

mn*
Drift
Diffusion
Drift Velocity
Diffusion Coefficient
k BT
Vt 
q
Single-Valley v-E Characteristics
Velocity-Field Characterisitcs
2.0 10
5
Drift Velocity, vd (in m/sec)
3D
1.6 10
1.2 10
8.0 10
4.0 10
2D
5
5
1D
4
4
0
0
2
4
6
Normalized Electric Field, 
8
10
Normalized Drift Velocity (vd /( 1/2 vth /2 ))
Effect of Degeneracy (2-D)
N  ns  D
2
1.6
1.4
1.2
1
0.8
D 
N=.01
0.6
N=.1
0.4
N=1
0.2
Non-Degen
0
0
4
8

12
16
20
h
2 m * k BT
Mobility Degradation
Diffusion Coefficient
Degradation
I-V Characteristics
Microresistors
Normalized I-V Characteristics
1.00
0.75
L=1 µm
L=10 µm
0.50
L=100 µm
0.25
V/Vc
10.00
7.50
5.00
2.50
0.00
0.00
I/Isat
Resistance Blow-Up
10
R/Ro (Experiment)
R/Ro(Theory)
r/Ro(Experiment)
r/Ro(Theory)
R/Ro
8
6
4
2
0
0
0.2
0.4
0.6
I/Isat
0.8
1
Multi-Valley Transport in GaAs
Intervalley Electron Transfer
Multi-Valley Transport in GaAs
Velocity-Field Characteristics
High-Frequency Transport
j t
E  Edc   o e
 E  Edc   dc Conductivity Degradation

o
o
E
 hf ( E, ) 
ac Conductivity Degradation
2 2
1    eff
Conclusions
Quantum Confinement

Transport properties function of
confinement length in QW’s because of the
change in the Density of States

Relative strength of each scattering
changes.

Electrons tend to stay away from the
interface as wave function vanishes near
the interface
Conclusions
High-Field Driven Transport

Electric field puts an order into otherwise
completely random motion

Higher mobility may not necessary lead to higher
saturation velocity

Saturation velocity is limited by Fermi /thermal
velocity depending on degeneracy

Saturation velocity is lowered by the quantum
remission process
RC time constants will dominate over transit time
delay because of enhanced resistance

Conclusions
Failure of Ohm’s Law
 Effective resistance may rise
dramatically as current approaches
saturation level
 Familiar voltage divider and current
divider rule may not be valid on
submicron scales
Golden Rule

No matter what the size, make it smaller

No matter what the speed, make it faster

No matter what the function, make it larger

No matter what the cost, make it cheaper

No matter how little it heats up, make it cooler