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Monte Carlo Simulation of Semiconductors -Chris Darmody Neil Goldsman 2018

Monte Carlo Simulation of Semiconductors

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Page 1: Monte Carlo Simulation of Semiconductors

Monte Carlo Simulation of Semiconductors

-Chris Darmody Neil Goldsman

2018

Page 2: Monte Carlo Simulation of Semiconductors

Background

• What is the Monte Carlo method?

– Use repeated random sampling to build up distributions and averages

• Want to determine electron energy and velocity distributions under applied electric fields in crystal

𝑘, 𝐸 𝑘′, 𝐸 + ħω

𝑞 = 𝑘′ − 𝑘, ħω

𝑘, 𝐸

𝑘′, 𝐸 − ħω

𝑞 = 𝑘 − 𝑘′, ħω

Initial Electron Momentum: 𝑘

Final Electron Momentum: 𝑘′ Phonon Momentum: 𝑞

𝐹

Phys. Rev. Let., 118(10) (2017)

Chris Darmody Neil Goldsman

Page 3: Monte Carlo Simulation of Semiconductors

Jacoboni and Reggiani, Rev. Mod. Phys. 55.3

Slope = μ

𝑣𝑠𝑎𝑡

𝐸𝐶𝑟𝑖𝑡

Silicon Transport Properties

http://www.ioffe.ru/SVA/NSM/Semicond/Si/electric.html

Chris Darmody Neil Goldsman

Page 4: Monte Carlo Simulation of Semiconductors

Simulation Overview

Random flight time: τ

Drift in field for τ

Scatter

t < tmax

Start

Stop

YES

NO

Position Changing in Real Space:

Energy Changing in Momentum

Space:

𝐹

τ

E

𝐸 1 + 𝛼𝐸 =ħ2𝑘2

2𝑚∗

𝑘

F

Electron Drift Motion

Electron Scattering

τ

Chris Darmody Neil Goldsman

Page 5: Monte Carlo Simulation of Semiconductors

Reciprocal Space, Band Structure, and Constant Energy Ellipses

Chris Darmody Neil Goldsman

Page 6: Monte Carlo Simulation of Semiconductors

Schrodinger Eq. in Periodic Potential

−ħ2

2𝑚

𝑑2𝜓 𝑥

𝑑𝑥2 + 𝑉 𝑥 𝜓 𝑥 = 𝐸𝜓 𝑥

• Eigenvalue problem gives allowed eigenvalues (E) for each eigenfunction (𝜓𝑘)

• Only certain E-k pairs allowed 𝑘 = 0 𝜋

𝑎 −

𝜋

𝑎

∆𝑘 =2𝜋

𝐿

𝐸

Allowed k-states (𝜓𝑘)

Allowed energies for each state

𝑉 𝑥 = 𝑉 𝑥 + 𝑛𝑎 , 𝑛 = 1, 2, 3, 4…

Periodic Potential in Crystal

Bloch Solutions:

𝜓𝑘 𝑥 = 𝑢 𝑥 𝑒𝑖𝑘𝑥,

𝑢 𝑥 = 𝑢 𝑥 + 𝑛𝑎 ,

𝑘 =2𝜋𝑛

𝐿=

2𝜋𝑛

𝑁𝑎

Forbidden Gap Eg

Chris Darmody Neil Goldsman

Page 7: Monte Carlo Simulation of Semiconductors

Reciprocal Space

Real (𝑟 ) Space Recip. (𝑘) Space 𝑘𝑧

𝑘𝑥 𝑘𝑦

Λ

Σ

Δ

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

Page 8: Monte Carlo Simulation of Semiconductors

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

Page 9: Monte Carlo Simulation of Semiconductors

Simplified Band Model

𝐸 1 + 𝛼𝐸 =ħ2𝑘2

2𝑚∗≡ 𝛾(𝑘)

𝑘

E

𝐸 =1 + 4𝛼𝛾(𝑘) − 1

2𝛼

ml mt mt

𝑚∗ =1

13

1𝑚𝑙

+2𝑚𝑡

= 𝑚𝑐

Electrons in a crystal move like free particles except with an effective mass

𝑚𝑑 = (𝑚𝑙𝑚𝑡2)1 3

http://math.ucr.edu/home/baez/information/index.html

non-parabolicity factor

Chris Darmody Neil Goldsman

Page 10: Monte Carlo Simulation of Semiconductors

Breakdown of Algorithm Steps

Chris Darmody Neil Goldsman

Page 11: Monte Carlo Simulation of Semiconductors

Monte Carlo Algorithm

Random flight time: τ

Drift in field for τ

Scatter

t < tmax

Start

Stop

YES

NO

Chris Darmody Neil Goldsman

Page 12: Monte Carlo Simulation of Semiconductors

Electron Drift Motion in Electric Field 𝐹

S1 S2

Scattering Mechanisms (Scattering Rates): S1, S2, … S3 S4 S5 ⋯ Virtual

Constant Total Scattering Rate: Γ ~1014 − 1015 1/s

𝑃 𝜏 = Γ𝑒−Γ𝜏dτ Probability of drifting for time 𝜏 then scattering:

𝜏 = −ln(𝑟1)

Γ Choose random flight time:

r1 uniformly random number from 0-1

∆𝑘 = −𝑞𝐹

ħ∆𝑡 Change k while drifting for time ∆𝑡 < 𝜏:

𝑣 =1

ħ𝛻𝑘𝐸 =

ħ𝑘

𝑚∗

1

(1 + 2𝛼𝐸) Instantaneous velocity:

Chris Darmody Neil Goldsman

Page 13: Monte Carlo Simulation of Semiconductors

Monte Carlo Algorithm

Random flight time: τ

Drift in field for τ

Scatter

t < tmax

Start

Stop

YES

NO

Chris Darmody Neil Goldsman

Page 14: Monte Carlo Simulation of Semiconductors

Scattering

S1 S2 S3 S4 S5 ⋯ Virtual

Constant Total Scattering Rate: Γ

Λ1(𝐸) Λ2(𝐸)

Λ3(𝐸) Λ4(𝐸)

Λ5(𝐸) Λ…(𝐸)

Λ𝑛

Γ< 𝑟2 ≤

Λ𝑛+1

Γ Randomly choose scattering mechanism (n+1):

r2, r3, r4 uniformly random numbers from 0-1

𝜑′ = 2𝜋𝑟3, cos 𝜃′ = 1 − 2𝑟4 Randomly choose k’ orientation:

𝑘𝑥′ = 𝑘′ sin(𝜃′) cos(𝜑′)

𝑘𝑦′ = 𝑘′ sin(𝜃′) sin(𝜑′)

𝑘𝑧′ = 𝑘′ cos(𝜃′)

𝑘𝑥 𝑘𝑦

𝑘𝑧

𝑘 𝜃′

ϕ′

𝑘′

After scattering, change energy from E to E’ depending on

mechanism, then calculate 𝑘′ from E’

Chris Darmody Neil Goldsman

Page 15: Monte Carlo Simulation of Semiconductors

Scattering Mechanisms • Acoustic Scattering:

– 𝑆𝑎𝑐 𝐸 =2𝑚𝑑

3 2 𝑘𝐵𝑇𝐷𝑎𝑐

2

𝜋ħ4𝑣𝑠2𝜌

𝐸 + 𝛼𝐸2 1 2 (1 + 2𝛼𝐸)

– 𝐸′ ≈ 𝐸

• Optical Scattering (absorb upper, emit lower):

– 𝑆𝑜𝑝 𝐸 =𝐷𝑡𝐾 𝑜𝑝

2 𝑚𝑑3 2

𝑍

2𝜋𝜌ħ3𝜔𝑜𝑝

𝑁𝑜𝑝

𝑁𝑜𝑝 + 1𝐸′ + 𝛼𝐸′2

1 2 (1 + 2𝛼𝐸′)

– 𝐸′ = 𝐸 ± ħ𝜔𝑜𝑝

– ħ𝜔𝑜𝑝 = 𝑘𝐵𝑇𝑜𝑝 (get temperatures from parameter sheet)

– 𝑁𝑜𝑝 =1

expħ𝜔𝑜𝑝

𝑘𝐵𝑇−1

(# of phonons in mode)

• Virtual Scattering:

– 𝐸′ = 𝐸

– 𝑘′ = 𝑘 – Do nothing: Effectively combines two drift events without scattering Chris Darmody

Neil Goldsman

Page 16: Monte Carlo Simulation of Semiconductors

Intervalley Scattering

𝑓1−3

𝑔1−3 𝑘𝑥

𝑘𝑦

𝑘𝑧

Equivalent Final Valleys in Si 𝒁𝒇 = 𝟒

𝒁𝒈 = 𝟏

Introduce degeneracy factor in optical scattering rate formulas

• 3 different ‘g’ mechanisms with 3 different 𝜔𝑜𝑝

• 3 different ‘f’ mechanisms with 3 additional 𝜔𝑜𝑝

• All 6 mechanisms can absorb or emit a phonon

13 Total Scattering Equations: 12 Intervalley + 1 Acoustic

𝑆𝑜𝑝 𝐸 =𝐷𝑡𝐾 𝑜𝑝

2 𝑚𝑑3 2 𝒁

2𝜋𝜌ħ3𝜔𝑜𝑝

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 ∗ 3 ∗ 4 + 3 ∗ 1 + 1 = 31

Absorb/Emit Acoustic f1, f2, f3 g1, g2, g3

*Intervalley scattering mechanisms treated using optical scattering form

Page 17: Monte Carlo Simulation of Semiconductors

Detailed Monte Carlo Algorithm Start

Calc. Scattering Rates: S(E)

Initialize: 𝐸 =3

2𝑘𝐵𝑇, 𝑘

Random flight time: r1, τ

Randomly Choose Scatter Mechanism: r2, get E’

𝜏 > 0

Drift Flight 𝜏 = 𝜏 − ∆𝑡

𝑘 = 𝑘 −𝑞𝐹

ħ∆𝑡

Sample Data E, 𝑣 ||𝐹

Randomly Choose Scatter Final State: r3, r4, get 𝑘′

Update State: 𝑘 = 𝑘′, E=E’

Max Time? Sample Data

E, 𝑣 ||𝐹

Output Histograms Velocity & Energy Distributions

Stop

Y

N

N

Y

Perform this algorithm for each Field

Chris Darmody Neil Goldsman

Page 18: Monte Carlo Simulation of Semiconductors

Sampling Data Between Scattering Events

𝜏

∆𝑡

Drifting Between Scattering Events • Choose a global sub-flight time step ∆𝑡 • Round 𝜏 to an integer number of sub-flights • Sample E and 𝑣 ||𝐹 at each sub-flight time step

Histograms:

Run simulation for enough real scattering events to obtain smooth histograms

Chris Darmody Neil Goldsman

Page 19: Monte Carlo Simulation of Semiconductors

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 𝑣 ||𝐹 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

Page 20: Monte Carlo Simulation of Semiconductors

Parameter Name Conversion

Remember to convert units!

Powerpoint Parameter Sheet

𝑚𝑙 , 𝑚𝑡 𝑚𝑙∆, 𝑚𝑡∆

𝐷𝑎𝑐 E1∆

𝑇𝑜𝑝 𝜃 𝑓,𝑔 1−3

𝛼 𝛼∆

𝑣𝑠 𝑢𝑙

Chris Darmody Neil Goldsman

Page 21: Monte Carlo Simulation of Semiconductors

Mean Velocity Result Comparison to Lit.

http://www.ioffe.ru/SVA/NSM/Semicond/Si/electric.html Chris Darmody Neil Goldsman

Page 22: Monte Carlo Simulation of Semiconductors

Mean Energy Result Comparison to Lit.

http://www.ioffe.ru/SVA/NSM/Semicond/Si/electric.html Chris Darmody Neil Goldsman