electron-electron interactions
DESCRIPTION
Electron-Electron Interactions. Dragica Vasileska Professor Arizona State university. Classification of Scattering Mechanisms. Treatment of the Electron-Electron Interactions. Electron-electron interactions can be treated either in: K-space, in which case one can separate between - PowerPoint PPT PresentationTRANSCRIPT
DRAGICA VASILESKAPROFESSOR
ARIZONA STATE UNIVERSITY
Electron-Electron Interactions
Classification of Scattering Mechanisms
Scattering Mechanisms
Defect Scattering Carrier-Carrier Scattering Lattice Scattering
CrystalDefects Impurity Alloy
Neutral Ionized
Intravalley Intervalley
Acoustic OpticalAcoustic Optical
Nonpolar PolarDeformationpotential
Piezo-electric
Scattering Mechanisms
Defect Scattering Carrier-Carrier Scattering Lattice Scattering
CrystalDefects Impurity Alloy
Neutral Ionized
Intravalley Intervalley
Acoustic OpticalAcoustic OpticalAcoustic Optical
Nonpolar PolarNonpolar PolarDeformationpotential
Piezo-electric
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 long-range 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:
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
2/1
2*
22
eNm
Dpcoll
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:
The equation of motion of a unit volume of the electron gas of concentration n is:
neuEEdneuA
VQ
dA
Cappl
uenneE
dtudnm
22
2
2*
2/122
2
2
*,0
mneu
dtud
pp
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:
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)
em
em
ab
abp
Nqq
Nk
emk min
max0
min
max02
2
ln)1(ln4
*)(
1
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:
Then, one can obtain the scattering rate in the Born approximation as one usually does in Brooks-Herring approach.
DLree e
rerH /
12
2
1212
4)(
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:
k1 k2
k2’k1’
r1 r2
Binary Collisions, Cont’d
The pair transition rate is defined as:
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:
2112'2112
212'2'121
,)(,
2),,,(21'2'1
kkrHkkM
EEEEMkkkkS
ee'
kkkk
2'1212
222
2
12 kk ,/1
1
q
LqVeM
D
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:
Note that a separate sum over k1’ is not needed because of the momentum conservation -function. For non-degenerate semiconductors, we have:
'2 2
)](1)][(1)[(),,,()(
1'2'12'2'121
1 k k
kfkfkfkkkkSk
'2 2
)(),,,()(
12'2'121
1 k k
kfkkkkSk
Binary Collisions Scattering Rate
In summary:
2221
223
24
21 /14
*)(
)(1
2 D
D
k Lkk
kkLnemkf
k
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:
22
02
0
k23
4
0kk
kkkk f
Vemn
ee Screening constant
There are three methods commonly used for the treatment of the electron-electron interaction:
A. Method due to Lugli and FerryB. Rejection algorithmC. Real-space molecular dynamics
• 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:
• The advantages of this method are:1. The scattering rate does not require any assumption on the form of the distribution function2. 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.
N
i Di
iDnee L
Lenm1 2223
24
01kk
kk4
k/
(A) Method Due to Lugli and Ferry
• Within this algorithm, a self-scattering mechanism, internal to the interparticle scattering is introduced by the following substitution:
• 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:
DD LL 21
1kkkk
220
0
/
220
0
1kkkk
DL/
(B) Rejection Algorithm
• If the collision is accepted, then the final state is calculated using:
where:
The azimuthal angle is then taken at random between 0 and 2.
• The final states of the two particles are then calculated using:
)',(,)(
cos gg anglewhereLrg
rr
Dr
22 11
21
'''; 00 kkg kkg
ggkk
ggkk
'
'
'
'
2121
0
00
BULK SYSTEMSSEMICONDUCTOR DEVICE MODELING
Real-Space Treatment of the Electron-Electron
Interactions
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=it, 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:
and:
Here, F(t) is the force arising from the applied field as well as that of the Coulomb interaction:
• The contributions due to the periodic replication of the particles inside V in cells outside is represented with the Ewald sum:
2
21 t
mttttt *)()()( Fvrr
tm
tttt *)()()( Fvv
iitqt )()( rEF
N
iii
i Vet
12
2
321
4ra
rF )(
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 free-flight is added
Earlier Work – k-space treatment of the Coulomb interaction
Good for 2D device simulationsRequires 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.
0%
10%
20%
30%
40%
50%
60%
0 10 20 30 40 50Distance Along the Channel [nm]
Cur
rent
Red
uctio
n
Impurity position varying along the center of the channel
V G = 1.0 VV D = 0.2 V
Source end Drain end
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 togetherMust 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
EXCHANGE -CORRELATION EFFECTSSCREENING OF THE COULOMB
INTERACTION POTENTIAL
More on the Electron-Electron Interactions for
Q2D Systems
Exchange-Correlation Correction to the Ground State
Energy if the System
Space Quantization
• Poisson equation:
d2VH(z)dz2
e2sc
n(z)Na(z)[ ]
• Hohenberg-Kohn-Sham Equation: (Density Functional Formalism)
Finite temperature generalizationof the LDA (Das Sarma and Vinter)
Veff(z)VH(z)Vxc(z)Vim(z)
h2
2mz*¶2
¶z 2 Veff (z) yn (z)nyn (z)[ ]
EF
VG>0
0123
0’
1’ z-axis [100](depth)
[100]-orientation:2-band : mz=ml=0.916m0, mxy=mt=0.196m04-band: mz=mt, mxy= (ml mt)1/2
2-band
4-band
Exchange-Correlation Effects
EEHF EcorrEkinHF Eexchange
HF Ecorr
Total Ground StateEnergy of the System
Hartree-Fock Approximationfor the Ground State Energy
Accounts for the error madewith the Hartree-Fock Approximation
Accounts for the reductionof the Ground State Enerydue to the inclusion of thePauli Exclusion Principle
Ways of Incorporating the Exchange-Correlation Effects:
Density-Functional Formalism (Hohenberg, Kohn and Sham)
Perturbation Method (Vinter)
Subband StructureImportance of Exchange-Correlation Effects
Exchange-Correlation Correction:
Lower subband energies Increase in the subband
separation Increase in the carrier
concentration at which the Fermi level crosses into the second subband
Contracted wavefunctions
Vasileska et al., J. Vac. Sci. Technol. B 13, 1841 (1995)(Na=2.8x1015 cm-3, Ns=4x1012 cm-2, T=0 K)
Thick (thin) lines correspond to thecase when the exchange-correlationcorrections are included (omitted) inthe simulations.
0.02
0.06
0.1
0.14
0.18
0 20 40 60 80 100
second subband
first subband
Veff(z)
Normalized wavefunction
Distance from the interface [Å]
Subband StructureComparison with Experiments
0
10
20
30
40
50
60
0 5x10 11 1x10 12 1.5x10 12 2x10 12 2.5x10 12 3x10 12
Exp. data [Kneschaurek et al.]V
eff(z)=V
H(z)+V
im(z)+V
exc(z)
Veff
(z)=VH(z)
Veff
(z)=VH(z)+V
im(z)
Ene
rgy
E10
[meV
] T = 4.2 K, Ndepl
=1011 cm-2
Ns [cm-2]
Kneschaurek et al., Phys. Rev. B 14, 1610 (1976) Infrared Optical AbsorptionExperiment:
far-ir
radiation
LEDSiO2 Al-Gate
Si-Sample
Vg
Transmission-Line Arrangement
Subband StructureComparison with Experiments
0
10
20
30
40
50
0 5x10 11 10 12 1.5x10 12 2x10 12 2.5x10 12 3x10 12
10 subb. appr. with Vxc
(z)5 subb. appr. with V
xc(z)
Hartree approximation
10 [
meV
]
Ns [cm-2]
Experimental data[Schäffler et al.]
T = 300 KN
depl = 6x1010 [cm-2]
Unprimed ladder Primed ladder
Experimental data from: F. Schäffler and F. Koch (Solid State Communications 37, 365, 1981)
0
10
20
30
40
50
0 5x10 11 10 12 1.5x10 12 2x10 12 2.5x10 12 3x10 12
10 subb. appr. with Vxc
(z)5 subb. appr. with V
xc(z)
Hartree approximationDas Sarma and Vinter
1'0'
[meV
]
Ns [cm-2]
Experimental data[Schäffler et al.]
T = 300 KN
depl = 6x1010 [cm-2]
Screening of the Coulomb Interaction
What is Screening?
+
-
--
-
-
-
- -
-
lD - Debye screening length
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
r
3D: 1r 1
rexp rlD
screeningcloud
Example:
Diagramatic Description of RPAPolarization Diagrams
= + + + . . .
=1 -
Effective interaction (or ‘dressed’ or ‘renormalized’)
Bare interaction
= + + + . . .
bare pair-bubble
Proper (‘irreducible’)polarization parts
Screening:Simulation results are for: Na=1015 cm-3, Ns=1012 cm-
2
0
5x10 5
1x10 6
1.5x106
2x10 6
2.5x106
3x10 6
0 5x10 6 1x10 7 1.5x107 2x10 7
q00
(q)q
11(q)
q0'0'
(q)
s
wavevector [cm-1]
s
s
0
0.2
0.4
0.6
0.8
1
1.2
0 2x10 6 4x10 6 6x10 6 8x10 6 1x10 7
T = 0 KT = 10 KT = 40 KT = 80 KT = 300 K
Wavevector [cm-1]
Relative Polarization Function:P00(0)(q,0)/P00
(0)(0,0)Screening Wavevectors:qnm
s (q,0) e2
2kPnm
(0)(q,0)
T=300 K
2D-Plasma Frequency: pl(q) e2Nsq2kmxy*
Screening:Form-Factors: Na=1015 cm-3, Ns=1012 cm-2
10 -4
10 -3
10 -2
10 -1
0 5x10 6 1x10 7 1.5x107 2x10 7
|F01,00
|
|F01,11
|
|F01,01
|
|F01,0'0'
|
wavevector [cm-1]
10 -3
10 -2
10 -1
10 0
0 5x106 10 7 1.5x107 2x107
|F 00,00 ||F 00,11 ||F 11,11 ||F 00,0'0' ||F 11,0'0' |
wavevector [cm-1]
Diagonal form-factors Off-diagonal form-factors
Fij,nm(q) dz dz'0
ò0
ò y j*(z)yi(z)˜ G (q,z,z')ym
* (z')yn(z') =>i n
mj z'z