monte carlo simulation of semiconductors
TRANSCRIPT
Monte Carlo Simulation of Semiconductors
-Chris Darmody Neil Goldsman
2018
Background
โข What is the Monte Carlo method?
โ Use repeated random sampling to build up distributions and averages
โข Want to determine electron energy and velocity distributions under applied electric fields in crystal
๐, ๐ธ ๐โฒ, ๐ธ + ฤงฯ
๐ = ๐โฒ โ ๐, ฤงฯ
๐, ๐ธ
๐โฒ, ๐ธ โ ฤงฯ
๐ = ๐ โ ๐โฒ, ฤงฯ
Initial Electron Momentum: ๐
Final Electron Momentum: ๐โฒ Phonon Momentum: ๐
๐น
Phys. Rev. Let., 118(10) (2017)
Chris Darmody Neil Goldsman
Jacoboni and Reggiani, Rev. Mod. Phys. 55.3
Slope = ฮผ
๐ฃ๐ ๐๐ก
๐ธ๐ถ๐๐๐ก
Silicon Transport Properties
http://www.ioffe.ru/SVA/NSM/Semicond/Si/electric.html
Chris Darmody Neil Goldsman
Simulation Overview
Random flight time: ฯ
Drift in field for ฯ
Scatter
t < tmax
Start
Stop
YES
NO
Position Changing in Real Space:
Energy Changing in Momentum
Space:
๐น
ฯ
E
๐ธ 1 + ๐ผ๐ธ =ฤง2๐2
2๐โ
๐
F
Electron Drift Motion
Electron Scattering
ฯ
Chris Darmody Neil Goldsman
Reciprocal Space, Band Structure, and Constant Energy Ellipses
Chris Darmody Neil Goldsman
Schrodinger Eq. in Periodic Potential
โฤง2
2๐
๐2๐ ๐ฅ
๐๐ฅ2 + ๐ ๐ฅ ๐ ๐ฅ = ๐ธ๐ ๐ฅ
โข Eigenvalue problem gives allowed eigenvalues (E) for each eigenfunction (๐๐)
โข Only certain E-k pairs allowed ๐ = 0 ๐
๐ โ
๐
๐
โ๐ =2๐
๐ฟ
๐ธ
Allowed k-states (๐๐)
Allowed energies for each state
๐ ๐ฅ = ๐ ๐ฅ + ๐๐ , ๐ = 1, 2, 3, 4โฆ
Periodic Potential in Crystal
Bloch Solutions:
๐๐ ๐ฅ = ๐ข ๐ฅ ๐๐๐๐ฅ,
๐ข ๐ฅ = ๐ข ๐ฅ + ๐๐ ,
๐ =2๐๐
๐ฟ=
2๐๐
๐๐
Forbidden Gap Eg
Chris Darmody Neil Goldsman
Reciprocal Space
Real (๐ ) Space Recip. (๐) Space ๐๐ง
๐๐ฅ ๐๐ฆ
ฮ
ฮฃ
ฮ
Reciprocal Lattice is the Fourier Transformation of the Real-Space Lattice!
FCC Brillouin Zone
Wessner, IUE Dissertation 2006
Bartolo, Phys. Rev. A 90.3 (2014)
Chris Darmody Neil Goldsman
Plotting Band Structure: E vs k Filled
Valen
ce Ban
ds
Emp
ty CB
s E
G
Irreducible Wedge High Symmetry Points
Constant Energy Ellipsoids Osintsev, IUE Dissertation 1986
Real Silicon Band Structure
(Path through k-space along high symmetry directions) Chris Darmody Neil Goldsman
Simplified Band Model
๐ธ 1 + ๐ผ๐ธ =ฤง2๐2
2๐โโก ๐พ(๐)
๐
E
๐ธ =1 + 4๐ผ๐พ(๐) โ 1
2๐ผ
ml mt mt
๐โ =1
13
1๐๐
+2๐๐ก
= ๐๐
Electrons in a crystal move like free particles except with an effective mass
๐๐ = (๐๐๐๐ก2)1 3
http://math.ucr.edu/home/baez/information/index.html
non-parabolicity factor
Chris Darmody Neil Goldsman
Breakdown of Algorithm Steps
Chris Darmody Neil Goldsman
Monte Carlo Algorithm
Random flight time: ฯ
Drift in field for ฯ
Scatter
t < tmax
Start
Stop
YES
NO
Chris Darmody Neil Goldsman
Electron Drift Motion in Electric Field ๐น
S1 S2
Scattering Mechanisms (Scattering Rates): S1, S2, โฆ S3 S4 S5 โฏ Virtual
Constant Total Scattering Rate: ฮ ~1014 โ 1015 1/s
๐ ๐ = ฮ๐โฮ๐dฯ Probability of drifting for time ๐ then scattering:
๐ = โln(๐1)
ฮ Choose random flight time:
r1 uniformly random number from 0-1
โ๐ = โ๐๐น
ฤงโ๐ก Change k while drifting for time โ๐ก < ๐:
๐ฃ =1
ฤง๐ป๐๐ธ =
ฤง๐
๐โ
1
(1 + 2๐ผ๐ธ) Instantaneous velocity:
Chris Darmody Neil Goldsman
Monte Carlo Algorithm
Random flight time: ฯ
Drift in field for ฯ
Scatter
t < tmax
Start
Stop
YES
NO
Chris Darmody Neil Goldsman
Scattering
S1 S2 S3 S4 S5 โฏ Virtual
Constant Total Scattering Rate: ฮ
ฮ1(๐ธ) ฮ2(๐ธ)
ฮ3(๐ธ) ฮ4(๐ธ)
ฮ5(๐ธ) ฮโฆ(๐ธ)
ฮ๐
ฮ< ๐2 โค
ฮ๐+1
ฮ Randomly choose scattering mechanism (n+1):
r2, r3, r4 uniformly random numbers from 0-1
๐โฒ = 2๐๐3, cos ๐โฒ = 1 โ 2๐4 Randomly choose kโ orientation:
๐๐ฅโฒ = ๐โฒ sin(๐โฒ) cos(๐โฒ)
๐๐ฆโฒ = ๐โฒ sin(๐โฒ) sin(๐โฒ)
๐๐งโฒ = ๐โฒ cos(๐โฒ)
๐๐ฅ ๐๐ฆ
๐๐ง
๐ ๐โฒ
ฯโฒ
๐โฒ
After scattering, change energy from E to Eโ depending on
mechanism, then calculate ๐โฒ from Eโ
Chris Darmody Neil Goldsman
Scattering Mechanisms โข Acoustic Scattering:
โ ๐๐๐ ๐ธ =2๐๐
3 2 ๐๐ต๐๐ท๐๐
2
๐ฤง4๐ฃ๐ 2๐
๐ธ + ๐ผ๐ธ2 1 2 (1 + 2๐ผ๐ธ)
โ ๐ธโฒ โ ๐ธ
โข Optical Scattering (absorb upper, emit lower):
โ ๐๐๐ ๐ธ =๐ท๐ก๐พ ๐๐
2 ๐๐3 2
๐
2๐๐ฤง3๐๐๐
๐๐๐
๐๐๐ + 1๐ธโฒ + ๐ผ๐ธโฒ2
1 2 (1 + 2๐ผ๐ธโฒ)
โ ๐ธโฒ = ๐ธ ยฑ ฤง๐๐๐
โ ฤง๐๐๐ = ๐๐ต๐๐๐ (get temperatures from parameter sheet)
โ ๐๐๐ =1
expฤง๐๐๐
๐๐ต๐โ1
(# of phonons in mode)
โข Virtual Scattering:
โ ๐ธโฒ = ๐ธ
โ ๐โฒ = ๐ โ Do nothing: Effectively combines two drift events without scattering Chris Darmody
Neil Goldsman
Intervalley Scattering
๐1โ3
๐1โ3 ๐๐ฅ
๐๐ฆ
๐๐ง
Equivalent Final Valleys in Si ๐๐ = ๐
๐๐ = ๐
Introduce degeneracy factor in optical scattering rate formulas
โข 3 different โgโ mechanisms with 3 different ๐๐๐
โข 3 different โfโ mechanisms with 3 additional ๐๐๐
โข All 6 mechanisms can absorb or emit a phonon
13 Total Scattering Equations: 12 Intervalley + 1 Acoustic
๐๐๐ ๐ธ =๐ท๐ก๐พ ๐๐
2 ๐๐3 2 ๐
2๐๐ฤง3๐๐๐
โฏ
g mechanisms scatter to ellipses across the zone f mechanisms scatter to neighboring ellipses
Chris Darmody Neil Goldsman
31 ways to scatter from a given valley. 2 โ 3 โ 4 + 3 โ 1 + 1 = 31
Absorb/Emit Acoustic f1, f2, f3 g1, g2, g3
*Intervalley scattering mechanisms treated using optical scattering form
Detailed Monte Carlo Algorithm Start
Calc. Scattering Rates: S(E)
Initialize: ๐ธ =3
2๐๐ต๐, ๐
Random flight time: r1, ฯ
Randomly Choose Scatter Mechanism: r2, get Eโ
๐ > 0
Drift Flight ๐ = ๐ โ โ๐ก
๐ = ๐ โ๐๐น
ฤงโ๐ก
Sample Data E, ๐ฃ ||๐น
Randomly Choose Scatter Final State: r3, r4, get ๐โฒ
Update State: ๐ = ๐โฒ, E=Eโ
Max Time? Sample Data
E, ๐ฃ ||๐น
Output Histograms Velocity & Energy Distributions
Stop
Y
N
N
Y
Perform this algorithm for each Field
Chris Darmody Neil Goldsman
Sampling Data Between Scattering Events
๐
โ๐ก
Drifting Between Scattering Events โข Choose a global sub-flight time step โ๐ก โข Round ๐ to an integer number of sub-flights โข Sample E and ๐ฃ ||๐น at each sub-flight time step
Histograms:
Run simulation for enough real scattering events to obtain smooth histograms
Chris Darmody Neil Goldsman
http://www.ioffe.ru/SVA/NSM/Semicond/Si/electric.html
Extracting Field-Dependent Averages
Average velocity for one input field F
Take time-average of E and ๐ฃ ||๐น for each field to generate final Drift Velocity and Average Energy vs Field plots
Jacoboni and Reggiani, Rev. Mod. Phys. 55.3
Chris Darmody Neil Goldsman
Parameter Name Conversion
Remember to convert units!
Powerpoint Parameter Sheet
๐๐ , ๐๐ก ๐๐โ, ๐๐กโ
๐ท๐๐ E1โ
๐๐๐ ๐ ๐,๐ 1โ3
๐ผ ๐ผโ
๐ฃ๐ ๐ข๐
Chris Darmody Neil Goldsman
Mean Velocity Result Comparison to Lit.
http://www.ioffe.ru/SVA/NSM/Semicond/Si/electric.html Chris Darmody Neil Goldsman
Mean Energy Result Comparison to Lit.
http://www.ioffe.ru/SVA/NSM/Semicond/Si/electric.html Chris Darmody Neil Goldsman