# ece lecture power EE Lecture D MOS Electrostatics Fundamental Analysis by mikeholy

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```									          EE-656:
Lecture 1
Introduction:
Why Take Another Course?
Electrical and Computer Engineering
Purdue University
West Lafayette, IN USA
Fall 2007
NCN
www.nanohub.org
Alam / Lundstrom ECE-656 F07   1
outline

1) The reason to take EE656
2) Relation to other courses you have taken
3) Introduction to Boltzmann Transport Equation
4) A few official things

Alam / Lundstrom ECE-656 F07        2
EE656 and other courses at Purdue
Device-specific         EE 695F:         EE xx:             ??
system design           RF Design        Opto-system        Bio-system Design

Application specific    EE 654:                  EE 612:       ??
device operation        Opto/MW/Magnetics        CMOS          Bio-Systems

Physical Principle of   EE 606:              EE 604
device Operation        Basic Semi Dev.      EM, Magnetics

EE 656:             EE 659:                ??
Semi-Transport      Quantum Transport      Bio-physics

Foundation
Quantum            Statistical
Mechanics          Mechanics

Alam / Lundstrom ECE-656 F07                       3
Drift-diffusion and Poisson equations
gN

n 1                              J(x)               J(x+dx)
   J N  rN  g N          The equations are pretty good
n
t q                            and have served us well for the last
J N  qn N E  qDN n           50 years…
rN
p 1                             We can analyze complex devices
   J P  rP  g P
t q                             like MOSFETs, bipolar transistors,
J P  qp P E  qDP p           lasers, etc. with very good
agreement with experiments ….
  E  q  p  n  ND  N A 
     

Then why do we need EE656 ?

Alam / Lundstrom ECE-656 F07           4
…. validity of the eqs. no longer guaranteed!
Consider a n-type semiconductor in SS without R-G ….

n 1
   J N  rN  g N
t q
J N  qn N E  qDN n              1
0    JN
q
p 1
   J P  rP  g P
t q
J N  qn N E
J P  qp P E  qDP p

  E  q  p  n  ND  N A 
     

Alam / Lundstrom ECE-656 F07              5

I
J N  q n N E

V                                  Carrier   velocity
Density
Depends on chemical composition,
crystal structure, temperature, doping, etc.

Quantum Mechanics + Equilibrium Statistical Mechanics
Encapsulated into concepts of effective masses
and occupation factors (Ch. 1-4)

Transport with scattering, non-equilibrium Statistical Mechanics
 Encapsulated into drift-diffusion equation with
recombination-generation (Ch. 5 & 6)
Alam / Lundstrom ECE-656 F07                        6
…. n and f computed without E-field!

J N  qn N E
Fn

 
dk g 2 D (k )  f ( x, k )                             Ec

1                    E(k)
m*          f ( x, k ) 
g 2 D (k )                                        
  EC 
2 2
k       
   2
1 e    
        2 m*
 FN 


Fn

… no distinction between +ve &                                                     k0
-ve k; is velocity zero !?
Alam / Lundstrom ECE-656 F07                     7
… velocity computed without reference to k!

J N  qn N E                                            x   x x
x

d ( m N * )       m *
 qE  N
dt             N

q N
      *
E  N E
m N

Which k are we talking about ?
Average v perhaps, but how should we do the averaging?
Alam / Lundstrom ECE-656 F07               8
inconsistent definition of f and v !

J N  qn N E

E(k)                                   E(k)

Fn                                     Fn

k0                                     k0

To resolve this puzzle, need to track of k-specific population …
Alam / Lundstrom ECE-656 F07                    9
…. even DOS g(E) has problems!
J N  qn N E



dk g 2 D (k )  f ( x, k )

f ( x, k ) 
1                        q N
g 2 D (k ) 
m*                                   
  EC 
2 2
k       
 FN 
          E  N E

*
2
1 e    
        2 m*      
        m N

Alam / Lundstrom ECE-656 F07                           10
…. even DOS g(E) has problems!
2
          2 (r)  U C  r   r   E  r 
2m0
U C  r   UC  r  a 

 n,k r   un,k r exp ik  r 

2
         2 (r)  U C  r   V  r   r   E  r 
                   
2m0
?    ?                   UC r   V r   UC r  a   V r  a 

 n,k r   un,k r  exp ik  r 

g ( x, k )?
Alam / Lundstrom ECE-656 F07                               11
Position variable x ….
J N  qn N E




dk g 2 D ( x, k )  f ( x, k )

1                                    q N
m*                   f ( x, k )                                  ?    ( x, k )         E  N E ?
g 2 D ( x, k )                  ?                                  
  EC 
2 2
k       
 FN 

*
2                                                                            mN
1 e           2 m*      

Homogenous space
All possible directions are
available for electron scattering

Alam / Lundstrom ECE-656 F07                                     12
Random Dopants and Inhomogenous Space

Random Dopants
Effective             Percolation
Media

TCAD people do this by Monte Carlo all the time,
but a more fundamental analysis is possible …
Alam / Lundstrom ECE-656 F07                 13
Scattering, effective-mass in nanodevices?

What to do when scattering is small?

V q N V
  N  *
L mN L             What happens when L is undefined?

What happens when m* is defined?

Alam / Lundstrom ECE-656 F07       14
mobility at high fields?
q N                       0
     *
E  N E                      1
E
m N                   E n         n

1    
  EC  
        

What causes velocity
                             saturation at high fields?

Where does all the mobility
formula in Medici come from?
E

Alam / Lundstrom ECE-656 F07               15
Power-dissipation, energy-flux, etc …
Vacuum
Bipolar                 MOSFET                 Future ??
Tubes
Spintronics

1906-1950s        1947-1980s             1960-until now

?
Temp

Tubes               bipolar           MOS

1900      1920     1940        1960        1980       2000   2020
Alam / Lundstrom ECE-656 F07                    16
Power-dissipation, energy-flux, etc …

Power dissipation ….
n 1
P  (J N  J P )  E    ???                            J N  rN  g N
t q

Power dissipation in  J N  qn N E  qDN n
forward-biased diode?
p 1
   J P  rP  g P
t q
J P  qp P E  qDP p
Heat flux …
  E  q  p  n  ND  N A 
     
Q  (J N  J P )  E  
… what velocity should I use?
Alam / Lundstrom ECE-656 F07                     17
Electric field, Magnetic Field, Temperature

Many semiconductor devices
n 1
   J N  rN  g N           work under Temperature
t q
difference between contacts
J N  qn N E  qDN n            (Peltier cooler), or with strong
magnetic field (MRAM memories).
p 1
   J P  rP  g P
t q
J P  qp P E  qDP p

  E  q  p  n  ND  N A 
     
How to modify the equations to

Alam / Lundstrom ECE-656 F07                  18
… so we do need a course!
Part 1:
q N V                      High field ballistic transport with N
 *                          High field transport with undefined m*
mN L
Part 2:
Scattering dominated transport at low E, T,
0                 and B Fields; Balance Equations
                  1
E
  E n        n

1                    Part 3:
  EC                  When L is undefined – Percolation theory
            
g(B,T )  ....
Part 4:
Nonequilibrium high field transport with
rigorously computed scattering events and
  g(E,t, E(k))              bandstructures

Alam / Lundstrom ECE-656 F07                 19
outline

1) The reason to take EE656
2) Relation to other courses you have taken
3) Introduction to Boltzmann Transport Equation
4) A few official things

Alam / Lundstrom ECE-656 F07        20
How is the course related to others at Purdue
Device-specific         EE 695F:         EE xx:             ??
system design           RF Design        Opto-system        Bio-system Design

Application specific    EE 654:                  EE 612:       ??
device operation        Opto/MW/Magnetics        CMOS          Bio-Systems

Physical Principle of   EE 606:              EE 604
device Operation        Basic Semi Dev.      EM, Magnetics

EE 656:             EE 659:                ??
Semi-Transport      Quantum Transport      Bio-physics

Foundation
Quantum            Statistical
Mechanics          Mechanics

Alam / Lundstrom ECE-656 F07                       21
outline

1) The reason to take EE656
2) Relation to other courses you have taken
3) Introduction to Boltzmann Transport Equation
4) A few official things

Alam / Lundstrom ECE-656 F07    22
energy/position resolved transport by f(x,k,t)

E(k)
E(k0 )       particle                           E(k0 )

E
k
k0                                  E(k)
EC (x)

ETOT  EC (x)  E(k)                                                  k
k0
x
Alam / Lundstrom ECE-656 F07                   23
f(x,k,t) are defined by the Boltzmann Equation

n 1
   J N  rN  g N                  f N
t q                                                                        ö
    r f N  Fe   p f N  Cf N
t
J N  qn N E  qDN n

p 1
   J P  rP  g P                  f P
t q                                                                        ö
    r f P  Fe   p f P  Cf P
t
J P  qp P E  qDP p

  E  q  p  n  ND  N A 
     
  E  q  p  n  ND  N A 
     

Alam / Lundstrom ECE-656 F07                               24
n vs. f(x,k,t): Why BTE is difficult

N(x,y,z;t) …………….. 4 independent variables
F(x,y,z;kx,ky,kz;t) ………7 independent variables

100x100x100x10 …. 107 million unknowns
(100x100x100)x(100x100x100)x10 = 1013 unknowns!

A pretty difficult problem ... We will need to
make approximations (very large or very small tau,
low vs. High-field transport, etc. ……

Alam / Lundstrom ECE-656 F07         25
Assumptions of the BTE

Independent particle transport -- originally developed
for dilute, uncharged gas molecules like H, He, Ar, etc.

Application to transport of charge carriers should be
always be done carefully and results frequently checked
against experiments.

Assumes that at each x, an E(k) diagram exists and is
governed by local material composition. Modification
of DOS due to remote interference requires generalization
of BTE to NEGF formulations.

Alam / Lundstrom ECE-656 F07           26
consistent n, v, P, Q should follow from f(x,k,t)


n( x, t )   dk gC (k )  f ( x, k )


J ( x, t )   dk gC (k )  f ( x, k )  (k )


P( x, t )   dk gC (k )  f ( x, k )  m * (k )



Q( x, t )   dk gC (k )  f ( x, k )  m * (k )



............
Calculation of f(x,k,t) addresses aforementioned problems …
but what equations do we use to compute f(x,k,t)
Alam / Lundstrom ECE-656 F07   27
Summary

1) Semiclassical transport equations involving drift-diffusion and
Poisson equations have had tremendous success in last 50
years in guiding design of electronic devices.

2) Some of the fundamental assumptions of the theory are falling
apart for nanodevices. Also the range of new devices require
broadening of the new equations. Moreover, physical origin of
mobility models are seldom explored in elementary course.

3) Energy-resolved BTE will help us address many of these
challenges.

Alam / Lundstrom ECE-656 F07                  28
Summary

EE656
EE659

Analyze the                                      Simple models
assumptions                                      for new devices

EE606

The goal is to understand why simple models have worked
so well so far and to see if we can develop similar elegant
models for modern ultra-scaled semiconductor devices.
Alam / Lundstrom ECE-656 F07               29
outline

1) The reason to take EE656
2) Relation to other courses you have taken
3) Introduction to Boltzmann Transport Equation
4) A few official things

Alam / Lundstrom ECE-656 F07         30
Course Information
Instructors   Prof. Lundstrom and Prof. Alam

Pre-requisite EE 606 or equivalent elementary Solid State Devices Course

Books
 Advanced Semiconductor Fundamentals (QM, SM, Transport)
 Fundamentals of Carrier Transport (2nd Ed) by M. Lundstrom.

Website http://cobweb.ecn.purdue.edu/~ee656/

-Th
Office hours 5:00-6:00 T @ (EE 313D or EE-310)

HW/Exams
5-6 HW (25%), Exam 1 (25%), Exam 2 (25%), Final (25%)
Discussing HW is okay, copying is not. Will result in F grade

Alam / Lundstrom ECE-656 F07                 31

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