Monte Carlo Simulation of Semiconductors
-Chris Darmody Neil Goldsman
2018
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
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
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
Reciprocal Space, Band Structure, and Constant Energy Ellipses
Chris Darmody Neil Goldsman
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
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
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
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
Breakdown of Algorithm Steps
Chris Darmody Neil Goldsman
Monte Carlo Algorithm
Random flight time: Ο
Drift in field for Ο
Scatter
t < tmax
Start
Stop
YES
NO
Chris Darmody Neil Goldsman
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
Monte Carlo Algorithm
Random flight time: Ο
Drift in field for Ο
Scatter
t < tmax
Start
Stop
YES
NO
Chris Darmody Neil Goldsman
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
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
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
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
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
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
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
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
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