monte carlo simulation of semiconductors
TRANSCRIPT
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Monte Carlo Simulation of Semiconductors
-Chris Darmody Neil Goldsman
2018
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Background
β’ What is the Monte Carlo method?
β Use repeated random sampling to build up distributions and averages
β’ Want to determine electron energy and velocity distributions under applied electric fields in crystal
π, πΈ πβ², πΈ + Δ§Ο
π = πβ² β π, Δ§Ο
π, πΈ
πβ², πΈ β Δ§Ο
π = π β πβ², Δ§Ο
Initial Electron Momentum: π
Final Electron Momentum: πβ² Phonon Momentum: π
πΉ
Phys. Rev. Let., 118(10) (2017)
Chris Darmody Neil Goldsman
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Jacoboni and Reggiani, Rev. Mod. Phys. 55.3
Slope = ΞΌ
π£π ππ‘
πΈπΆπππ‘
Silicon Transport Properties
http://www.ioffe.ru/SVA/NSM/Semicond/Si/electric.html
Chris Darmody Neil Goldsman
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Simulation Overview
Random flight time: Ο
Drift in field for Ο
Scatter
t < tmax
Start
Stop
YES
NO
Position Changing in Real Space:
Energy Changing in Momentum
Space:
πΉ
Ο
E
πΈ 1 + πΌπΈ =Δ§2π2
2πβ
π
F
Electron Drift Motion
Electron Scattering
Ο
Chris Darmody Neil Goldsman
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Reciprocal Space, Band Structure, and Constant Energy Ellipses
Chris Darmody Neil Goldsman
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Schrodinger Eq. in Periodic Potential
βΔ§2
2π
π2π π₯
ππ₯2 + π π₯ π π₯ = πΈπ π₯
β’ Eigenvalue problem gives allowed eigenvalues (E) for each eigenfunction (ππ)
β’ Only certain E-k pairs allowed π = 0 π
π β
π
π
βπ =2π
πΏ
πΈ
Allowed k-states (ππ)
Allowed energies for each state
π π₯ = π π₯ + ππ , π = 1, 2, 3, 4β¦
Periodic Potential in Crystal
Bloch Solutions:
ππ π₯ = π’ π₯ ππππ₯,
π’ π₯ = π’ π₯ + ππ ,
π =2ππ
πΏ=
2ππ
ππ
Forbidden Gap Eg
Chris Darmody Neil Goldsman
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Reciprocal Space
Real (π ) Space Recip. (π) Space ππ§
ππ₯ ππ¦
Ξ
Ξ£
Ξ
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
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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
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Simplified Band Model
πΈ 1 + πΌπΈ =Δ§2π2
2πββ‘ πΎ(π)
π
E
πΈ =1 + 4πΌπΎ(π) β 1
2πΌ
ml mt mt
πβ =1
13
1ππ
+2ππ‘
= ππ
Electrons in a crystal move like free particles except with an effective mass
ππ = (ππππ‘2)1 3
http://math.ucr.edu/home/baez/information/index.html
non-parabolicity factor
Chris Darmody Neil Goldsman
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Breakdown of Algorithm Steps
Chris Darmody Neil Goldsman
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Monte Carlo Algorithm
Random flight time: Ο
Drift in field for Ο
Scatter
t < tmax
Start
Stop
YES
NO
Chris Darmody Neil Goldsman
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Electron Drift Motion in Electric Field πΉ
S1 S2
Scattering Mechanisms (Scattering Rates): S1, S2, β¦ S3 S4 S5 β― Virtual
Constant Total Scattering Rate: Ξ ~1014 β 1015 1/s
π π = ΞπβΞπdΟ Probability of drifting for time π then scattering:
π = βln(π1)
Ξ Choose random flight time:
r1 uniformly random number from 0-1
βπ = βππΉ
Δ§βπ‘ Change k while drifting for time βπ‘ < π:
π£ =1
Δ§π»ππΈ =
Δ§π
πβ
1
(1 + 2πΌπΈ) Instantaneous velocity:
Chris Darmody Neil Goldsman
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Monte Carlo Algorithm
Random flight time: Ο
Drift in field for Ο
Scatter
t < tmax
Start
Stop
YES
NO
Chris Darmody Neil Goldsman
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Scattering
S1 S2 S3 S4 S5 β― Virtual
Constant Total Scattering Rate: Ξ
Ξ1(πΈ) Ξ2(πΈ)
Ξ3(πΈ) Ξ4(πΈ)
Ξ5(πΈ) Ξβ¦(πΈ)
Ξπ
Ξ< π2 β€
Ξπ+1
Ξ Randomly choose scattering mechanism (n+1):
r2, r3, r4 uniformly random numbers from 0-1
πβ² = 2ππ3, cos πβ² = 1 β 2π4 Randomly choose kβ orientation:
ππ₯β² = πβ² sin(πβ²) cos(πβ²)
ππ¦β² = πβ² sin(πβ²) sin(πβ²)
ππ§β² = πβ² cos(πβ²)
ππ₯ ππ¦
ππ§
π πβ²
Οβ²
πβ²
After scattering, change energy from E to Eβ depending on
mechanism, then calculate πβ² from Eβ
Chris Darmody Neil Goldsman
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Scattering Mechanisms β’ Acoustic Scattering:
β πππ πΈ =2ππ
3 2 ππ΅ππ·ππ
2
πΔ§4π£π 2π
πΈ + πΌπΈ2 1 2 (1 + 2πΌπΈ)
β πΈβ² β πΈ
β’ Optical Scattering (absorb upper, emit lower):
β πππ πΈ =π·π‘πΎ ππ
2 ππ3 2
π
2ππΔ§3πππ
πππ
πππ + 1πΈβ² + πΌπΈβ²2
1 2 (1 + 2πΌπΈβ²)
β πΈβ² = πΈ Β± Δ§πππ
β Δ§πππ = ππ΅πππ (get temperatures from parameter sheet)
β πππ =1
expΔ§πππ
ππ΅πβ1
(# of phonons in mode)
β’ Virtual Scattering:
β πΈβ² = πΈ
β πβ² = π β Do nothing: Effectively combines two drift events without scattering Chris Darmody
Neil Goldsman
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Intervalley Scattering
π1β3
π1β3 ππ₯
ππ¦
ππ§
Equivalent Final Valleys in Si ππ = π
ππ = π
Introduce degeneracy factor in optical scattering rate formulas
β’ 3 different βgβ mechanisms with 3 different πππ
β’ 3 different βfβ mechanisms with 3 additional πππ
β’ All 6 mechanisms can absorb or emit a phonon
13 Total Scattering Equations: 12 Intervalley + 1 Acoustic
πππ πΈ =π·π‘πΎ ππ
2 ππ3 2 π
2ππΔ§3πππ
β―
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 β 3 β 4 + 3 β 1 + 1 = 31
Absorb/Emit Acoustic f1, f2, f3 g1, g2, g3
*Intervalley scattering mechanisms treated using optical scattering form
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Detailed Monte Carlo Algorithm Start
Calc. Scattering Rates: S(E)
Initialize: πΈ =3
2ππ΅π, π
Random flight time: r1, Ο
Randomly Choose Scatter Mechanism: r2, get Eβ
π > 0
Drift Flight π = π β βπ‘
π = π βππΉ
Δ§βπ‘
Sample Data E, π£ ||πΉ
Randomly Choose Scatter Final State: r3, r4, get πβ²
Update State: π = πβ², E=Eβ
Max Time? Sample Data
E, π£ ||πΉ
Output Histograms Velocity & Energy Distributions
Stop
Y
N
N
Y
Perform this algorithm for each Field
Chris Darmody Neil Goldsman
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Sampling Data Between Scattering Events
π
βπ‘
Drifting Between Scattering Events β’ Choose a global sub-flight time step βπ‘ β’ Round π to an integer number of sub-flights β’ Sample E and π£ ||πΉ at each sub-flight time step
Histograms:
Run simulation for enough real scattering events to obtain smooth histograms
Chris Darmody Neil Goldsman
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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 π£ ||πΉ 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
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Parameter Name Conversion
Remember to convert units!
Powerpoint Parameter Sheet
ππ , ππ‘ ππβ, ππ‘β
π·ππ E1β
πππ π π,π 1β3
πΌ πΌβ
π£π π’π
Chris Darmody Neil Goldsman
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Mean Velocity Result Comparison to Lit.
http://www.ioffe.ru/SVA/NSM/Semicond/Si/electric.html Chris Darmody Neil Goldsman
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Mean Energy Result Comparison to Lit.
http://www.ioffe.ru/SVA/NSM/Semicond/Si/electric.html Chris Darmody Neil Goldsman