Transcript Document

Computational Electronics
Computational Electronics
A.1 Derivation of the Boltzmann Transport Equation
Kinetic theory: We need to derive an equation for the single
particle distribution function f(v,r,t) (classical) which gives the
probability of finding a particle with velocity between v and
v+dv and in the region r to r+dr
• We assume that v and r are given simultaneously which neglects
quantum mechanical nature of particles.
• f(v,r,t) allows us to calculate ensemble averages over velocity and
space (particle density, current density, energy density, etc.):
At    dr  dvAv, r, t f v, r, t 
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• Consider a hypervolume in phase space
ds
jv,r, t 
j(r,v,t) is the flux density
V
j(r,v,t)ds is flux through
hypersurface ds
S
• Consider the particle balance through the hyper-volume V

n

 drdvnv, r, t     jv, r, t   ds   drdv
t V
t Coll
S
V
Time rate of change
of # particles in V
Leakage
through S
Time rate of change
due to collisions
  drdvGr, v, t   R r, v, t 
V
Time rate of change due to G-R mechanisms
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• The flux density is written in terms of the time derivatives of
the ‘position’ variables in 6D:
j x, y , z,v x ,v y ,v z   v x n v, r, t aˆ x  v y naˆy  v z naˆz 
Fy
Fx ˆ
Fz ˆ
F
ˆ
nbv  nbv  nbv with v 
m
m
m
m
x
y
z
• Applying the divergence theorem in 6D
 jv, r, t   ds   drdv  jv, r, t 
S
V
where the divergence of j is
n
n
n Fx n Fy n Fz n
  j  vx
 vy
 vz



x
y
z m v x m v y m v z
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which is written more compactly as:
F
  j  v  r n  v n
m
• Particle balance is therefore:
 n

F
n
n

0
 drdv  v  r n   v n 
m
t Coll t G R 
V
 t
Normalizing, we get the classical form of the Boltzmann
transport equation:
f r, v, t 
F
f
f
  v  r f   v f 

t
m
t Coll t



First two terms on the rhs
are the streaming terms
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GR
• For Bloch electrons in a semiconductor, we could have
considered a 6D space x,y,z,kx,ky,kz where k is the
wavevector and
1
v   k E k 

• The semi-classical BTE for transport of Bloch electrons is
therefore
f r, k, t 
1
F
f
f
  k E k   r f   k f 

t


t Coll t G R
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A.2 Collisional Integral
Assume instantaneous, single collisions which are
independent of the driving force and take particles from k to k
(out scattering) or from k to k (in scattering).
Out scattering
kz
k
ky
k
kx
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In scattering
(A) Out Scattering
nr, k, t   nr, k, t kkt
where kk is the transition rate per particle from k to k
Distribution function is: f r, k, t  
nr, k, t 
N
Take limit as t0
f r, k, t 
 f r, k, t kk 1  f r, k, t 
t OUT
where the last term in the brackets accounts for the Pauli
exclusions principle (degeneracy of the final state after
scattering).
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(B) In Scattering
By an analogous argument, the rate of change of the
distribution function due to in scattering is:
f r, k, t 
 f r, k, t kk 1  f r, k, t 
t
IN
Total rate of change of f (r,k,t) around k is a sum over all
possible initial and final states k:
f r, k, t 
  f r, k, t 1  f r, k, t kk 
t Coll k
In scattering
f r, k, t 1  f r, k, t kk 
Out scattering
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(C) Boltzmann Equation with Collision Integral
The sum over final states k may be converted to an integral
due to the small volume of k-space associated with each state:
V
  3  dk 
8
k
The BTE becomes:
fk 1
F
  k E   r fk   k fk 
t 

V
dkfk  1  fk kk fk 1  fk  kk 
3
8
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A.3 Scattering Theory
What contributes to kk ? How do we calculate kk’?
Scattering Mechanisms
Defect Scattering
Crystal
Defects
Impurity
Neutral
Carrier-Carrier Scattering
Alloy
Lattice Scattering
Intervalley
Intravalley
Acoustic
Ionized
Deformation
potential
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Optical
Piezoelectric
Nonpolar
Acoustic
Polar
Optical
B.1 Time Evolution of Quantum States
When the Hamiltonian is time dependent, the state or the wavefunction of
the system will be also time dependent. In other words, an electron will
have a probability to transfer from one state (molecular orbital) to another.
The transition probability can be obtained from the time-dependent
Schrödinger Equation
 (t ) 
i
 H ( t )
t
One the initial wavefunction, (0), is known, the wavefunction at a given later
time can be determined. If H is time independent, we can easily find that
(t )   an e iEnt /  n
n
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Suppose that a system has initial (t=0) Hamiltonian, H0 (time independent), and is
at an initial eigenstate, k. Under an external influence, described by H’ (time
dependent), the system will change state. For example, a molecule moves close to
an electrode surface to feel an increasing interaction with the electrode. The
combined Hamiltonian is
Hˆ (t )  Hˆ 0 (0)  Hˆ ' (t )
the combined Hamiltonian should be a linear combination of the initial eigenstates,
The wavefunction of the system corresponding to
(t )   Cnk (t ) n
n
From mathematical point of view, this is always possible since the initial eigenstates,
n, form a complete set of basis. The physical picture is that the system under the
influence of the external perturbation will end up in a different state with a probability
given by |Cnk|2. The indices nk mean a transition from kth eigenstate to nth
eigenstate. How fast the transition is or the transition rate is given by
wnk 
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d
|C nk (t ) |2
dt
B.2 Time-Dependent Perturbation Theory
Now we determine the transition rate according to the above definition. We assume
that the initial state of the system is
(0)   k
the external perturbation, H’, is switched on at t=0. The time dependent Schrödinger
Eq. is


(t )
i
 ( H 0  H ' )  (t )
t
For simplicity, we can rewrite this equation as
(t )   Cnk '(t )e iEnt /  n
n
Note that Cnk’(t) is different from Cnk(t), but |Cnk’(t)|2=| Cnk(t)|2 and we can omit the
prime.

dCnk iEnt / 
iEn t / 
i
e
 n   Cnk e
H
' n.
dt
n
n
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Multiplying by k’ and integrate, we obtain

dCk 'k
 i ( Ek '  En ) t / 
i
 e
 k ' | H ' | n Cnk
dt
n
After considering that n are normalized orthogonal functions. Note that the initial
condition becomes
Cnk (0)   nk
In general, solving above equation set is not easy, but we can obtain approximate
solution using perturbation theory when H’ is small comparing to H0. Let us denote
the solution in the absence of H’ as Cnk(0), we have
dC
i k 'k
dt
( 0)
0
So Ck’k(0) is independent of time and the initial condition is
Ckk ' (0)   kk '
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Cnk (0)   nk
We replace Cnk on the right hand side with Cnk(0) and obtain the first order correction
dC
i k 'k
dt
(1)
e
i ( Ek '  Ek ) t / 

 k'| H '| k 

 k ' | H ' | k  in the above equation is often denoted as H’k’k and it measured the
coupling strength between the k’ and k states. Solving we have
t
C k 'k
1
  H k' 'k e i ( Ek '  Ek ) t /  dt,
i 0
One important case is that H’ is fixed once switched on. In this case,
1 ' e  i ( E k '  Ek ) t /   1
C k 'k ( t )  H k 'k
i
i( Ek '  Ek ) / 
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B.3 Fermi-Golden Rule
Thus, we can obtain
2
1
2t
t 
'
2 sin (( E k '  E k )t /  )
'
2
| C k 'k ( t ) |  2 | H k 'k |




|
H
|
 ( Ek '  Ek )
k 'k

[( Ek '  Ek ) / ]2
2
2
So the transition rate is
wk 'k 
2
'
2
|
H
|
 ( Ek '  Ek )
k 'k
2

We can conclude that (1) the transition rate is independent of time, (2) the
transition can occur only if the final state has the same energy as the initial state.
The later one reflects energy conservation. In the case when the energy levels are
continuous band, the number of states near Ek’ for an interval of dEk’ is In the case
when the energy levels are continuous band, the number of states near Ek’ for an
interval of dEk’ is rEk’)dEk’ , where r is the density of states. The transition rate
from k state to the states near Ek’ is then
w   r ( E k ' ) wk 'k dE k ' 
This is Fermi Golden rule,
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2
'
2
|
H
|
r ( Ek )
k
'
k
2

(23.18)
Assumptions made:
(1) Long time between scattering (no multiple scattering events)
(2) Neglect contribution of other c’s (Collision broadening ignored)
kk 
Pkk  2 kk  2


Vs
Ek   Ek  
t

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B.4 Total Scattering Rate Calculation
• For the case when we have general matrix element (with qdependence), the procedure for calculating the scattering rate
out of state k is the following
(k )    kk ' 
k'

where
2
1
 2 
3
1
 2 
3
2
k
'
 dk ' d (cos  )d kk '

1
2  k ' dk ' d (cos  ) M (q)  ( Ek '  Ek
2
2
1
0
k'  k  q

1
2

k'
2
1
dk ' d (cos  ) M (q)  ( Ek '  Ek
0
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1
2
0 )
0 )
• The integration over k’ can be converted into integration over qwavevector and the integration over cos() together with the function that denotes conservation of energy will put limits on
the q-values: qmin and qmax for absorption and emission. The
final expression that needs to be evaluated is:
 ab ,em (k ) 
qmax
m
2 k
3
 q  M (q) 
2
dq
qmin
• where
ab
min
q
em
min
q
 k  k 1 
 k  k 1

Ek

Ek
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,
ab
max
,
q
q
ab
max
 k  k 1
 k  k 1

Ek

Ek
q-vector
qmax(ab)
qmax(em)
qmin(ab)
qmin(em)
-1
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+1
cos()
3.5
x 10
5
3.5
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0.5
1
1.5
2
2.5
3
polar angle
5
Absorption Process
3
Emission Process
0
x 10
3.5
0
0
0.5
1
1.5
2
2.5
3
3.5
polar angle
Histogram of the polar angle for polar optical phonon emission and
absorption. In accordance to the graph shown on the previous figure for
emission forward scattering is more preferred for emission than for absorption.
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Special Case: Constant Matrix
Element
For the special case of constant matrix element, the expression
for the scattering rate out of state k reduces to:
( k ) 
mM 02 k

3
1

Ek
The top sign refers to absorption and the bottom sign refers
To emission. For Elastic scattering we can further simplify to get:
( k ) 
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mM 02 k

3
C.1 Elastic Scattering Mechanisms
(A) Ionized Impurities scattering
(Ionized donors/acceptors, substitutional impurities, charged
surface states, etc.)
• The potential due to a single ionized impurity with net charge
Ze is:
2
Ze
0
Vi r   
4r
mks units
• In the one electron picture, the actual potential seen by
electrons is screened by the other electrons in the system.
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What is Screening?
lD - Debye screening length
-
r
Example:
+
3D:
1
screening
cloud
r
 r 
exp 

r
 l D 
1
-
Ways of treating screening:
• Thomas-Fermi Method
static potentials + slowly varying in space
• Mean-Field Approximation (Random Phase Approximation)
time-dependent and not slowly varying in space
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• For the scattering rate due to impurities, we need for Fermi’s
rule the matrix element between initial and final Bloch states
n, k Vi r  n, k  V 1  drun* ,k eikrVi r un,k eikr
Since the u’s have periodicity of lattice, expand in reciprical
space
 V 1  dre ikrVi r e ikre iGrUnnkk  G 
G
1
 V  dre
Vi r e e
ikr
G
ik r iGr
*
iGr



 dr un,k  r un,k r e

• For impurity scattering, the matrix element has a 1/q type
dependence which usually means G0 terms are small
1
 V  dre
Vi r eikr  drun* ,k  run,k r  Vi q Ikk 
ikr
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
nn 
• The usual argument is that since the u’s are normalized within
a unit cell (i.e. equal to 1), the Bloch overlap integral I, is
approximately 1 for n=n [interband(valley)]. Therefore, for
impurity scattering, the matrix element for scattering is
approximately
k Vi r  k
2
 Vi q 
2
2 4
Z e
 2 2
; V  volume
2 2
V q  l sc


where the scattered wavevector is: q  k  k
• This is the scattering rate for a single impurity. If we assume
that there are Ni impurities in the whole crystal, and that
scattering is completely uncorrelated between impurities:
Vi kk 
N i Z 2e 4
ni Z 2e 4
 2 2

2 2
2
V q  l sc V q 2  l2 sc




where ni is the impurity density (per unit volume).
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• The total scattering rate from k to k is given from Fermi’s
golden rule as:
ki k 
2ni Z 2e 4

Ek  Ek 
2
2 2
V q  l sc


If  is the angle between k and k, then:
q  k  k  k 2  k 2  2kk  cos   2k 2 1  cos 
• Comments on the behavior of this scattering mechanism:
- Increases linearly with impurity concentration
- Decreases with increasing energy (k2), favors lower T
- Favors small angle scattering
- Ionized Impurity-Dominates at low temperature, or room
temperature in impure samples (highly doped regions)
• Integration over all k gives the total scattering rate k :
2

ni Z e m *
4k
i
k 
2 3 3  2
8 sc
 k qD 4k 2  qD2
2 4
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

; qD  1 / l


(A1) Neutral Impurities scattering
• This scattering mechanism is due to unionized donors, neutral
defects; short range, point-like potential.
• May be modeled as bound hydrogenic potential.
• Usually not strong unless very high concentrations
(>1x1019/cm3).
(B) Alloy Disorder Scattering
• This is short-range type of interaction as well.
• It is calculated in the virtual crystal approximation or coherent
potential approximation.
• Limits mobility of ternary and quaternay compounds,
particularly at low temperature.
• The total scattering rate out of state k for this scattering
mechanism is of the form:
3/ 2
kalloy
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2

E   2m * 

 2 
2   
E1/ 2
(C) Surface Roughness Scattering
• This is a short range interaction due to fluctuations of
heterojunction or oxide-semiconductor interface.
• Limits mobility in MOS devices at high effective surface fields.
High-resolution transmission electron micrograph of the
interface between Si and SiO2
(Goodnick et al., Phys. Rev. B 32, pp. 8171, 1985)
2.71 Å
3.84 Å
Modeling surface-roughness
scattering potential:
H' (r,z)  Voz  (r)  Vo z
 Vo(z)(r)
random function that describes the
deviation from an atomically flat interface
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• Extensive experimental studies have led to two commonly
used forms for the autocovariance function.
• The power spectrum of the autocovariance function is found to
be either Gaussian or exponentially correlated.
Comparison of the fourth-order AR spectrum with the
fits arising from the Exponential and Gaussian models
(Goodnick et al., Phys. Rev. B 32, pp. 8171, 1985)
SPECTRUM OF HRTEM ROUGHNESS
=0.24 nm
AR model
Commonly assumed power spectrums
for the autocovariance function :
Gaussian model
x=0.74 nm
Exponential model
x=0.94 nm
• Gaussian:
• Exponential:
 q2 2 
SG (q)    exp
 4 


2 2
SE (q) 
2 2


1 q2 2 2
32
Wave vector (Å-1)
• Note that  is the r.m.s of the roughness and  is the roughness correlation length.
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• The total scattering rate out of state k for surface-roughness
scattering is of the form:
ksr


m*  e
1
k


Ndepl  0.5Ns 

E
3 2
 sc
1  k 2 2  1  k 2 2 
2 2 4
where E is a complete elliptic integral, Ndepl is the depletion
charge density and Ns is the sheet electron density.
• It is interesting to note that this scattering mechanism leads to
what is known as the universal mobility behavior, used in
mobility models described earlier.
m
Increasing substrate doping
Eeff 
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e
0.5Ns  Ndepl 
sc
The Role of Interface Roughness:
D. Vasileska and D. K. Ferry, "Scaled silicon MOSFET's: Part I - Universal
mobility behavior," IEEE Trans. Electron Devices 44, 577-83 (1997).
Phonon
2
Mobility [cm /V-s]
Coulomb
-1
(aN + bN
400
s
)
depl
N
300
200
-1/3
s
experimental data
uniform
step-like (low-high)
retrograde (Gaussian)
100
12
10
13
10
-2
Inversion charge density N [cm ]
s
Interface-roughness
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C.2 Inelastic Scattering Mechanisms
C.2.1 Some general considerations
• The Electron Lattice Hamiltonian is of the following form:
H  He  Hl  Hep
He  Electron Hamiltonia n; Hl  lattice Hamiltonia n
Hep  Electron  Phonon coupling
where He n,k  En,k  n,k
n,k  eikrun,k Bloch states
• For the lattice Hamiltonian we have:
H l l  E l l
El 
x
 q
x,q
l  nq1nq 2nq 3...
n
x
q

1
2

Second quantized representation, where nq is the
number of phonons with wave-vector q, mode x.
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• Phonons:
The Fourier expansion in reciprocal space of the coupled
vibrational motion of the lattice decouples into normal modes
which look like an independent set of Harmonic oscillators with
frequency xq
x labels the mode index, acoustic (longitudinal, 2 transverse modes) or
optical (1 longitudinal, 2 transverse)
q labels the wavevector corresponding to traveling wave solutions for
individual components,
• The velocity and the occupancy of a given mode are given by:
v xq 
nqx 
xq
e
q
1
xq / kBTl
1
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; Bose  Einstein distribution
(1) For acoustic modes, lim
q 0
vxq

xq
q
 u x , acoustic velocity.
(2) For optical modes, velocity approaches zero as q goes to zero.
Room temperature dispersion curves for the acoustic and the optical
branches. Note that phonon energies range between 0 and 60-70 meV.
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• The Electron-Phonon Interaction is categorized as to mode
(acoustic or optical), polarization (transverse or longitudinal),
and mechanism (deformation potential, polar, piezoelectric).
During scattering processes between electrons and phonon,
both wavevector and energy are conserved to lowest order in
the perturbation theory. This is shown diagramatically in the
figures below.
Absorption:
k  k  q
Ek   Ek   q
k, Ek
q ,  q
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Emission:
k  k  q
Ek   Ek  q
k, Ek
q ,  q
• For emission, E k  q must hold, otherwise it is prohibited
by conservation of energy. Therefore, there is an emission
threshold in energy
• Emission: nq  nq  1
Absorption: nq  nq  1
C.2.2 Deformation Potential Scattering
Replace Hep with the shift of the band edge energy produced by
a homogeneous strain equal to the local strain at position r
resulting from a lattice mode of wavevector q
(A) Acoustic deformation potential scattering
• Expand E(k) in terms of the strain. For spherical constant
energy surface
 
E k   E 0 k   E1   e2
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where:
    u r   dilation of volume of unit cell
E1  Deformatio n potential const.
E1  Deformatio n potential
and u is the displacement operator of the lattice
1/ 2
   
 eq,x aq,xe iqr  aq*,xe iqr
u r    
x
q  2NMq 



eq,x  polarization vector
x

• Taking the divergence gives factor of e·q of the form:

eq,x  q  q for longitudin al modes

eq,x  q  0 for transverse modes
Therefore, only longitudinal modes contribute.
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
• For ellipsoidal valleys (i.e. Si, Ge), shear strains may contribute
to the scattering potential
E k   E 0 k   Ed   Eu ezz
u
ezz 
 zˆ ; ezz is component of the strain tensor
z
Scattering Matrix Element:
Assuming q  ul q, then:
Vac
2
E12q nq  1  1

2Vrul
 upper absorption 


 lower emission 
• At sufficient high temperature, (equipartition approximation):
kBTl
nq  nq  1 
q
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• Substituting and assuming linear dispersion relation, Fermi’s
rule becomes
kack
2
2
2

E
2
1 kBTl
 Vac Ek  Ek  q  
Ek  Ek  q 
2

Vrul
• The total scattering rate due to acoustic modes is found by
integrating over all possible final states k’
ac
k

2E12kBTl V
2


4  dk k Ek  Ek  q 

2
3
Vrul 8
0
where the integral over the polar and azimuthal angles just
gives 4.
• For acoustic modes, the phonon energies are relatively small
since
q  0 as q  0
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• Integrating gives (assuming a parabolic band model)
kac
m *kE12 kBT l
2

;
c

r
u
l
l
3
 cl
where cl is the longitudinal elastic constant. Replacing k, using
parabolic band approximation, finally leads to:
kac 
2m
E12 kBT l 1 / 2
E
4
 cl
*3 / 2
• Assumptions made in these derivations:
a) spherical parabolic bands
b) equipartition (not valid at low temperatires)
c) quasi-elastic process (non-dissipative)
d) deformation potential Ansatz
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(B) Optical deformation potential scattering
(Due to symmetry of CB states, forbidden for -minimas)
• Assume no dispersion:
q  0 as q  0
Out of phase motion of basis atoms creates a strain called the
optical strain.
• This takes the form (D0 is optical deformation potential field)
Vdo

 
 D0  ur  ; D0  D0  eq
zeroth order
The matrix element for spherical bands is given by
ac 2
Vkk 
 D0 2 
n k  k  q   n  1 k  k  q 
 

2
r
V

0

0
which is independent of q .
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
0

• The total scattering rate is obtained by integrating over all k’ for
both absorption and emission
kdo
1/ 2


1
m D0 n E  0  




3
1
/
2
do
2r 0  n  1 E  0  E  0 


*3 / 2
2

0
0

where the first term in brackets is the contribution due to
absorption and the second term is that due to emission
• For non-spherical valleys, replace m
*3 / 2
 mt m1l / 2
• The non-polar scattering rate is basically proportional to density
of states
kdo  rE  0 
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(C) Intervalley scattering
• May occur between equivalent or nonequivalent sets of valleys
- Intervalley scattering is important in explaining room temperature mobility in multi-valley semiconductors, and the NDR observed (Gunn effect) in III-V compounds
- Crystal momentum conservation requires that qk where k is
the vector joining the two valley minima
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• Since k is large compared to k, assume q  q and treat
the scattering the same as non-polar optical scattering replacing
D0 with Dij the intervalley deformation potential field, and
the phonon coupling valleys i and j
q  ij
• Conservation of energy also requires that the difference in initial
and final valley energy be accounted for, giving
n E  Eij  ij 1 / 2 

kiv  


1/ 2
2r ij  n  1 E  Eij  ij  E  Eij  0 
j


3/ 2 2
md j Dij
3

ij
ij

where the sum is over all the final valleys, j and
Eij  Emin j  Emin i
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C.2.3 Phonon Scattering in Polar Semiconductors
• Zinc-blend crystals: one atom has Z>4, other has Z<4.
• The small charge transfer leads to an effective dipole which, in
turn, leads to lattice contribution to the dielectric function.
• Deformation of the lattice by phonons perturbs the dipole
moment between the atoms, which results in electric field that
scatters carriers.
• Polar scattering may be due to:
optical phonons
=> polar optical phonon scattering
(very strong scattering mechanism
for compound semiconductors such
as GaAs)
acoustic phonons => piezoelectric scattering
(important at low temperatures in
very pure semiconductors)
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(A) Polar Optical Phonon Scattering (POP)
Scattering Potential:
Microscopic model is difficult. A simpler approach is to consider
the contribution of this dipole to the polarization of the crystal and
its effect on the high- and low-frequency dielectric constants.
• Consider a diatomic lattice in the long-wavelength limit (k0),
for which identical atoms are displaced by a same amount.
• For optical modes, the oppositely charged ions in each
primitive cell undergo oppositely directed displacements, which
gives rise to nonvanishing polarization density P.
Transverse mode:
Longitudinal mode:
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k
k
• Associated with this polarization are macroscopic electric field E
and electric displacement D, related by:
D = E + P
Here, we have taken into account the contribution
to the dielectric function due to valence electrons
• Assume D, E, P  eik.r. Then, in the absence of free charge:
·D = ik ·D = 0 and E = ik E = 0
kD or D=0
k||E or E=0
• Longitudinal modes: P||k => D=0, (LO)=0
• Transverse modes:
Pk => E=0, (TO)=
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• The equations of motion of the two modes (in the k0 limit),
for the relative displacement of the two atoms in the unit cell
w=u1-u2 are:
Transverse mode:
u2
u2
u1
u1
u2
u1
2
d w
2
 TOw  0
2
dt
2
TO
 1
1  2C
 2C 


 M1 M 2  M
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Longitudinal mode:
u2
u2
k
u1
u2
u1
u1
d 2w
1 *
2
 TOw  e E
2
M
dt
E ( t )  Ee
*
jt
e E /M
w 2
2
TO  
• The longitudinal displacement of the two atoms in the unit cell
leads to a polarization dipole:
*2
N *
Ne / 2V M
P
ew 2
E
2
2V
TO  
• The existence of a finite polarization dipole modifies the dielectric function:
*2
Ne / 2V M
D  E  P  E  2
E  ()E
2
TO  
2
2 


 LO
S


TO 
    1 
()   1  2
2
2 
   2 




TO
TO




*2
1
Ne
1 
2
Polarization
S
 LO   

2V M
constant
  (0) 
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• The electric field associated with the perturbed dipole moment is
obtained from the condition that, in the absence of macroscopic
free charge:
  D  ik  D    Eind  P   0  D  0 and Eind
• Consider only one Fourier component:

P



Eind  ind  e1 V (q )  e1  Vqe iqr  i e1 qV (q )
e
e
V (q )  i
q  P  V (q )  i
P
2
 q
 q

e N *

1
iqr
  iqr
V (r )  i
e
 aqe  aq e
 2V
2M (N / 2)LO q q
1/ 2
 e 

 i 
 2VLO 
2



1
1
1 
iqr
  iqr
2  1
,  LO
 aqe  aq e



q q



(0
)
 

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Scattering Rate Calculation:
• Matrix element squared for this interaction:
Vk ,k '
2
e 2 1
1  1 k'k  q


N

0
2
2
2
2VLO q
• Transition rate per unit time from state k to state k’:
k ,k '
2
2
 Vk ,k ' εk'  εk  ωL 0 

2
e 1
1  1 k'k  q     


N

0
k'
k
L0
2
2
2
VLO q
• Total scattering rate per unit time out of state k:
k   k ,k '  k ,q
k'
q

V 2 1
2

d

d
(cos

)
q

 k ,q dq
3 
( 2) 0 1
0
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• Momentum and energy conservation delta-functions limit the
values of q in the range [qmin,qmax]:
absorption: qmin   k  k 1  LO E ( k )
qmax  k  k 1  LO E ( k )
emission:
qmin  k  k 1  LO E ( k )
qmax  k  k 1  LO E ( k )
E ( k )  LO , emission threshold
• Final expression for k
m * e2
k 
2 2kLO


1  E ( k ) 
1  E ( k )
N0 sinh     N0  1 sinh    1
LO 
LO




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Discussion:
1. The 1/q2 dependence of k,k’ implies that polar optical phonon
scattering is anisotropic, i.e. favors small angle scattering
2. It is inelastic scattering process
3. k is nearly constant at high energies
4. Important for GaAs at room-temperature and II-VI compounds
(dominates over non-polar)
Scattering rate
Momentum relaxation
rate
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The larger momentum relaxation
time is a consequence of the fact
that POP scattering favors small
angle scattering events that have
smaller influence on the momentum
relaxation.
(B) Piezoelectric scattering
• Since the polarization is proportional to the acoustic strain, we
have
P  epz   u
• Following the same arguments as for the polar optical phonon
scattering, one finds that the matrix element squared for this
mechanism is:
2
Vkk '
2
  eepz 


 Nq  12  12 k  k'q
2rVq   
• The scattering rate, in the elastic and the equipartition
approximation, is then of the form;
2
m * kBT  eepz  
k2 
k 

 ln1  4 2 
3
4 kr  v s  
qD 
where qD is the screening wavevector.
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Total Electron-Phonon Scattering Rate Versus Energy:
Intrinsic Si
GaAs
In both cases the electron scattering rates were calculated
by assuming non-parabolic energy bands.
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1. Acoustic Phonon Scattering
2. Intervalley Phonon Scattering
3. Ionized Impurity Scattering
4. Polar Optical Phonon Scattering
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5. Piezoelectric Scattering
5. Piezoelectric Scattering
6. Dislocation Scattering (e.g. GaN)
where n’ is the effective screening concentration
Ndis is the Line dislocation density
7. Alloy Desorder Scattering (Al xGa1-xAs)
Where: d is the lattice disorder (0≤d≤1)
Dalloy is the alloy disorder scattering potential
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