Electron-Electron Interactions
DRAGICA VASILESKA
PROFESSOR
ARIZONA STATE UNIVERSITY
Classification of Scattering Mechanisms
S c a tte rin g M e c h a n is m s
D e fe c t S c a tte rin g
C rys ta l
D e fe c ts
N e u tra l
Im p u rity
C a rrie r-C a rrie r S c a tte rin g
A llo y
Io n ize d
L a ttic e S c a tte rin g
In te rva lle y
In tra va lle y
A c o u s tic
D e fo rm a tio n
p o te n tia l
O p tic a l
P ie zo e le c tric
N o n p o la r
A c o u s tic
P o la r
O p tic a l
Treatment of the Electron-Electron Interactions
 Electron-electron interactions can be treated either
in:

K-space, in which case one can separate between
Collective plasma oscillations
 Binary electron-electron collisions


Real space Molecular dynamics
Bulk systems (Ewald sums)
 Devices (Coulomb force correction, P3M, FMM)

K-space treatment of the
Electron-Electron Interactions
Electron Gas
 As already noted, the electron gas displays both
collective and individual particle aspects.
 The primary manifestations of the collective
behavior are:


Organized oscillations of the system as a whole – plasma
oscillations
Screening of the field of any individual electron within a Debye
length
Collective excitations
 In the collective excitations each electron suffers a small
periodic perturbation of its velocity and position due to the
combined potential of all other electrons in the system.
 The cumulative potential may be quite large since the longrange nature of the Coulomb potential permits a very large
number of electrons to contribute to the potential at a given
point
 The collective behavior of the electron gas is decisive for
phenomena that involve distances that are larger than the
Debye length
 For smaller distances, the electron gas is best considered as a
collection of particles that interact weakly by means of
screened Coulomb force.
Collective behavior, Cont’d
 For the collective description to be valid, it is
necessary that the mean collision time, which tends
to disrupt the collective motion, be large compared
to the period of the collective oscillation. Thus:
 coll 
2
p
 m *
 2 
 N e2
 D




1/ 2
 Examples for GaAs:
 ND=1017 cm-3, p =2×1013, coll >>2/p  1/coll <<3×1012 1/s
 ND=1018 cm-3, p =6.32×1013, coll >>2/p  1/coll <<1013 1/s
 ND=1019 cm-3, p =2×1014, coll >>2/p  1/coll <<3×1013 1/s
Collective Carrier Scattering Explained
 Consider the situation that corresponds to the mode
q=0, when all electrons in the system have been
displaced by the same amount u, as depicted in the
figure below:
u
d
- - - - - - - - - - - - - -
E
+ + + + + + + + + + + + + +
Collective Carrier Scattering Explained
 Because of the positive (negative) surface charge density at the bottom
(top) slab, an electric field is produced inside the slab. The electric field
can be calculated using a simple parallel capacitor model for which:
C 
 A
Q

d
V appl

neuA
 E 
Ed
neu

 The equation of motion of a unit volume of the electron gas of
concentration n is:
2
nm *
d u
dt
2
2
  neE  
2
d u
dt
2
  p u  0,   p
2
2
n e u

 ne 2

 m *






1/ 2
Collective Carrier Scattering Explained
 Comments:
 Plasma oscillation is a collective longitudinal excitation of the
conduction electron gas.
 A PLASMON is a quantum of plasma oscillations. PLASMONS
obey Bose-Einstein statistics.
 An electron couples with the electrostatic field fluctuations due
to plasma oscillations, in a similar manner as the charge of the
electron couples to the electrostatic field fluctuation due to
longitudinal POP.
Collective Carrier Scattering Explained
 The process is identical to the Frӧhlich interaction if
plasmon damping is neglected. Then:
2
ab
 q max
m * e  p 

 N 0 ln  ab
2
 ( k ) 4  k   
 q min

1
em

 q max
  ( N  1) ln 
0

 q em

 min
 


 
 Note on qmax:
 Large qmax refers to short-wavelength oscillations, but one
Debye length is needed to screen the interaction. Therefore,
when qmax exceeds 1/LD, the scattering should be treated as
binary collision.

qc=min(qmax,1/LD)
Collective Carrier Scattering Explained
 Importance of plasmon scattering
 Plasma oscillations and plasmon scattering are important for
high carrier densities
 When the electron density exceeds 1018 cm-3 the plasma
oscillations couple to the LO phonons and one must consider
scattering from the coupled modes
www.engr.uvic.ca/.../Lecture%207%20-%20Inelastic%20Scattering.ppt
Electron-Electron Interactions
(Binary Collisions)
 This scattering mechanism is closely related to
charged impurity scattering and the interaction
between the electrons can be approximated by a
screened Coulomb interaction between point-like
particles, namely:
H e  e ( r12 ) 
e
2
4   r12
e
 r12 / L D
 Then, one can obtain the scattering rate in the Born
approximation as one usually does in BrooksHerring approach.
Binary Collissions
 To write the collision term, one needs to define a pair
transition rate S(k1,k2,k1’,k2’), which represents the
probability per unit time that electrons in states k1
and k2 collide and scatter to states k1’ and k2’, as
shown diagramatically in the figure below:
k2’
k1’
r1
k1
r2
k2
Binary Collisions, Cont’d
 The pair transition rate is defined as:
S ( k 1 , k 2 , k 1' , k 2 ' ) 
2
M 12

2

 E k1 '  E k 2 '  E k1  E k 2

M 12  k 1' , k 2 ' H ee ( r12 ) k 1 , k 2
 Since the interaction potential depends only upon
the distance between the particles, it is easier to
calculate M12 in a center-of-mass coordinate system,
to get:
M 12 
e
V
2

1
2
2
q  1 / LD
,
q
2
 k 12  k 12 '
2
Binary Collisions Scattering Rate
 To evaluate the scattering rate due to binary carrier-carrier
scattering, one weights the pair transition rate that a target
carrier is present and by the probability that the final states
k1’ and k2’ are empty:
1
 ( k1 )

  S (k
k2'
1 , k 2 , k 1' , k 2 ' )
f ( k 2 )[1  f ( k 1 ' )][ 1  f ( k 2 ' )]
k2
 Note that a separate sum over k1’ is not needed because of
the momentum conservation -function. For nondegenerate semiconductors, we have:
1
 ( k1 )

  S (k
k2'
k2
1 , k 2 , k 1' , k 2 ' )
f (k 2 )
Binary Collisions Scattering Rate
 In summary:
1
 ( k1 )


k2
4
f (k 2 )
2
m * ne L D
3 2
4   
k  k2
k1  k 2
2
2
 1 / LD
Incorporation of the electron-electron
interactions in EMC codes
For two-particle interactions, the electron-electron (hole-hole, electron-hole)
scattering rate may be treated as a screened Coulomb interaction (impurity
scattering in a relative coordinate system). The total scattering rate depends
on the instantaneous distribution function, and is of the form:
ee  k 0  
m ne

4
2
V
3
 f k 
k
k  k0

2
k  k
2
0

2

  Screening constant
There are three methods commonly used for the treatment of the electronelectron interaction:
A. Method due to Lugli and Ferry
B. Rejection algorithm
C. Real-space molecular dynamics
(A) Method Due to Lugli and Ferry
• This method starts form the assumption that the sum over the distribution
function is simply an ensemble average of a given quantity.
• In other words, the scattering rate is defined to be of the form:
ee  k 0  
4 2
nm n e L D
3 2
4   
N

k  ki
k  ki
• The advantages of this method are:
i 1
2
 1 / LD
2
1. The scattering rate does not require any assumption on the form of the
distribution function
2. The method is not limited to steady-state situations, but it is also
applicable for transient phenomena, such as femtosecond laser excitations
• The main limitation of the method is the computational cost, since it
involves 3D sums over all carriers and the rate depends on k rather on its
magnitude.
(B) Rejection Algorithm
•
Within this algorithm, a self-scattering mechanism, internal
to the interparticle scattering is introduced by the following
substitution:
k  k0
k  k0
2
 1/
2
LD

1
2 LD
•
When carrier-carrier collision is selected, a counterpart
electron is chosen at random from the ensemble.
•
Internal rejection is performed by comparing the random
number with:
k  k0
k  k0
2
 1 / LD
2
•
If the collision is accepted, then the final state is calculated
using:
cos  r  1 
where:
2r
1  g (1  r
2
2
)L D
,
where
 r  angle ( g, g ' )
g  k  k 0 ; g '  k 'k 0
'
The azimuthal angle is then taken at random between 0
and 2.
•
The final states of the two particles are then calculated
using:
1
'
k0  k0 
k  k0 
'
2
1
2
g '  g 
g '  g 
Real-Space Treatment of the
Electron-Electron Interactions
BULK SYSTEMS
SEMICONDUCTOR DEVICE MODELING
Bulk Systems
(C) Real-space molecular dynamics
•
An alternative to the previously described methods is the
real-space treatment proposed by Jacoboni.
•
According to this method, at the observation time instant
ti=it, the total force on the electron equals the sum of the
interparticle coulomb interaction between a particular
electron and the other (N-1) electrons in the ensemble.
•
When implementing this method, several things need to be
taken into account:
1. The fact that N electrons are used to represent a carrier density n =
N/V means that a simulation volume equals V = N/n.
2. Periodic boundary conditions are imposed on this volume, and
because of that, care must be taken that the simulated volume and the
number of particles are sufficiently large that artificial application from
periodic replication of this volume do not appear in the calculation results.
•
Using Newtonian kinematics, the real-space trajectories of
each particle are represented as:
r (t   t )  r (t )  v t 
and:
1 F(t )
2 m *
F(t )
v(t   t )  v(t ) 
t
m *
t
2
Here, F(t) is the force arising from the applied field as well
as that of the Coulomb interaction:


F ( t )  q E     r ( t ) i 


i
•
The contributions due to the periodic replication of the
particles inside V in cells outside is represented with the
2
Ewald sum:
e N  1
2 
F(t )  
ri 
  2 a i 
4  i 1  ri
3V 
Simulation example of the role of the electron-electron
interaction:
• The effect of the e-e scattering
allows equilibrium distribution
function to approach Fermi-Dirac
or Maxwell Boltzmann distribution.
• Without e-e, there is a phonon
‘kink’ due to the finite energy of the
phonon
Semiconductor Device
Modeling
Ways of accounting for the short-range Coulomb
interactions
 Long-range Coulomb interactions are accounted for
via the solution of the Poisson equation which gives
the so-called Hartree term
 If the mesh is infinitely small, the full Coulomb
interaction is accounted for
 However this is not practical as infinite systems of
algebraic equations need to be solved
 To avoid this difficulty, a mesh size that satisfies the
Debye criterion is used and the proper correction to
the force used to move the carriers during the freeflight is added
Earlier Work – k-space treatment of the Coulomb
interaction
 Good for 2D device simulations
 Requires calculation of the distribution function to
recalculate the scattering rate at each time step and
the screening which is time consuming
 Implemented in the Damocles device simulator
K-space Approach
Present trends – Real-space treatment
 Requires 3D device simulator, otherwise the method
fails
 There are several variants of this method
 Corrected Coulomb approach developed by Vasileska and
Gross
 Particle-particle-particle-mesh (p3m) method by Hockney and
Eastwood
 Fast Multipole method
Real Space Treatment Cont’d
 Corrected Coulomb approach and p3m method are
almost equivalent in philosophy
 FMM is very different
 Treatment of the short-range Coulomb interactions
using any of these three methods accounts for:



Binary collisions + plasma (collective) excitations
Screening of the Coulomb interactions
Scattering from multiple impurities at the same time which is
very important at high substrate doping densities
1. Corrected Coulomb approach
 A resistor is first simulated to calculate the difference




between the mesh force and the true Coulomb force
Cut-off radius is defined to account for the ions
(inner cut-off radius)
Outer cut-off radius is defined where the mesh force
coincides with the Coulomb force
Correction to the force is made if an electron falls
between the inner and the outer radius
The methodology has been tested on the example of
resistor simulations and experimental data are
extracted
Corrected Coulomb Approach Explained
Resistor Simulations
MOSFET: Drift Velocity and Average Energy
2. p3m Approach
Details of the p3m Approach
Impurity located at the very source-end, due to the availability of Increasing
number of electrons screening the impurity ion, has reduced impact on the
overall drain current.
60%
Impurity position varying along the
center of the channel
Current Reduction
50%
V G = 1.0 V
40%
V D = 0.2 V
30%
20%
Source end
Drain end
10%
0%
0
10
20
30
40
Distance Along the Channel [nm]
50
3. Fast Multipole Method
 Different strategy is employed here in a sense that
Laplace equation (Poisson equation without the
charges) is solved. This gives the ‘Hartree’ potential.
 The electron-electron and electron-ion interactions
are treated using FMM
 The two contributions are added together
 Must treat image charges properly. Good news is
that the surfaces are planar and the method of
images is a good choice
Idea
The philosophy of FMM:
Approximate Evaluation
Ideology behind FMM
Simulation Methodology
Method of Images
Resistor Simulations
More on the Electron-Electron
Interactions for Q2D Systems
EXCHANGE-CORRELATION EFFECTS
SCREENING OF THE COULOMB INTERACTION
POTENTIAL
Exchange-Correlation Correction
to the Ground State Energy if the
System
Space Quantization
•
Poisson equation:
3
2
1
0
d 2V (z) e2
H 
[n(z)  Na(z)]
dz2

sc
•
Hohenberg-Kohn-Sham Equation:
(Density Functional Formalism)
VG>0
1’
0’
EF
z-axis [100]
(depth)
4-band
2  2

*
2  Veff (z) y n (z)   n y n (z)
2mz z
[
]
2-band
V (z)  V (z)  V (z) V (z)
eff
H
xc
im
[100]-orientation:
Finite temperature generalization
of the LDA (Das Sarma and Vinter)
2-band :
mz=ml=0.916m0,
mxy=mt=0.196m0
4-band:
mz=mt, mxy= (ml mt)1/2
Exchange-Correlation Effects
HF
E  EHF  E corr  EHF
E

kin
exchange  E corr
Total Ground State
Energy of the System
Accounts for the error made
with the Hartree-Fock Approximation
Hartree-Fock Approximation
for the Ground State Energy
Accounts for the reduction
of the Ground State Enery
due to the inclusion of the
Pauli Exclusion Principle
Ways of Incorporating the Exchange-Correlation
Effects:
 Density-Functional Formalism (Hohenberg, Kohn and Sham)
 Perturbation Method (Vinter)
Subband Structure
Importance of Exchange-Correlation Effects
Vasileska et al., J. Vac. Sci. Technol. B 13, 1841 (1995)
(Na=2.8x1015 cm-3, Ns=4x1012 cm-2, T=0 K)
V
0.18
eff
(z)
 Lower subband energies
 Increase in the subband separation
 Increase in the carrier concentration
0.14
at which the Fermi level crosses into
the second subband
second subband
0.1
 Contracted wavefunctions
Normalized
wavefunction
0.06
first subband
0.02
0
20
40
60
Exchange-Correlation Correction:
80
Distance from the interface [Å]
100
Thick (thin) lines correspond to the
case when the exchange-correlation
corrections are included (omitted) in
the simulations.
Subband Structure
Comparison with Experiments
50
depl
=10
11
cm
-2
Exp. data [Kneschaurek et al.]
V (z)=V (z)+V
(z)+V
(z)
40
H
eff
V (z)=V
10
[m e V ]
T = 4.2 K, N
E n ergy E
Infrared Optical Absorption
Experiment:
Kneschaurek et al., Phys. Rev. B 14, 1610 (1976)
60
eff
30
V (z)=V
eff
H
H
im
SiO2
Al-Gate
LED
exc
(z)
(z)+V
im
far-ir
(z)
Si-Sample
radiation
20
10
Vg
0
0
5x10
11
1x10
12
N
1.5x10
s
12
[cm
2x10
-2
]
12
2.5x10
12
3x10
12
Transmission-Line Arrangement
Subband Structure
Comparison with Experiments
Experimental data from: F. Schäffler and F. Koch
(Solid State Communications 37, 365, 1981)
Unprimed ladder
[m eV ]
Experimental data
[Schäffler et al.]
40
30
T = 300 K
N
20
depl
= 6x10
10
[cm
-2
Experimental data
[Schäffler et al.]
40
T = 300 K
30
N
]
20
10 subb. appr. with V
5 subb. appr. with V
10
Primed ladder
50
1 '0 '
10
[m e V ]
50
xc
xc
= 6x10
depl
10
[cm
10 subb. appr. with V
(z)
5 subb. appr. with V
(z)
-2
10
]
xc
xc
(z)
(z)
Hartree approximation
Das Sarma and Vinter
Hartree approximation
0
0
0
5x10
11
10
12
1.5x10
N [cm
s
12
2x10
-2
]
12
2.5x10
12
3x10
12
0
5x10
11
10
12
1.5x10
12
N [cm
s
2x10
-2
]
12
2.5x10
12
3x10
12
Screening of the Coulomb
Interaction
What is Screening?
-
lD - Debye screening length
r
+
Example:
3D:
1
r
screening
cloud
 r 
1
exp 
r
 lD 
-
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
Diagramatic Description of RPA
Polarization Diagrams
Bare interaction
=
+
+
+...
=
Effective interaction
(or ‘dressed’ or ‘renormalized’)
Proper (‘irreducible’)
polarization parts
1 -
=
+
bare pair-bubble
+
+...
Screening:
Simulation results are for: Na=1015 cm-3, Ns=1012 cm-2
Re lative Polarization Function:
0)
(q,0) /P (0)
P (00
00(0,0)
1.2
3x10
6
2.5x10
6
Screening Wavevectors :
2
s
qnm
(q,0)   e P (0)
nm(q,0)
2k
T=300 K
1
T=0K
q s (q)
00
q s (q)
T = 10 K
0.8
T = 40 K
2x10
11
6
q s (q)
0'0'
T = 80 K
0.6
1.5x10
6
0.4
1x10
6
0.2
5x10
5
T = 300 K
0
0
2x10
6
4x10
6
6x10
Wavevector [cm
6
8x10
-1
]
6
1x10
7
0
0
5x10
6
1x10
7
1.5x10
wavevector [cm
-1
2D-Plasma Frequency:  (q) 
pl
7
2x10
]
e 2N q
s
xy
2km*
7
Screening:
Form-Factors: Na=1015 cm-3, Ns=1012 cm-2


0
0
Fij,nm(q)   dz 
10
10
-1
|F
|F
-2
|F
|F
10
00,00
00,11
11,11
00,0'0'
11,0'0'
10
-1
10
-2
6
z
z'
|
|F
|
10
-3
|F
|
|F
|
10
7
wavevector [cm
1.5x10
-1
]
7
2x10
7
m
n
Off-diagonal form-factors
|F
10
5x10
=>
|
-3
0
j
i
Diagonal form-factors
0
|F
10
dz'y *j (z)y i(z)G˜(q,z,z')y *m(z')y n(z' )
|
01,00
|
01,11
|
01,01
01,0'0'
|
-4
0
5x10
6
1x10
7
wavevector [cm
1.5x10
-1
]
7
2x10
7