Monte Carlo Simulation of Semiconductors

Monte Carlo Simulation of Semiconductors

-Chris Darmody Neil Goldsman

2018

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

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

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

Reciprocal Space, Band Structure, and Constant Energy Ellipses

Chris Darmody Neil Goldsman

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

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

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

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

Breakdown of Algorithm Steps

Chris Darmody Neil Goldsman

Monte Carlo Algorithm

Random flight time: τ

Drift in field for τ

Scatter

t < tmax

Start

Stop

YES

NO

Chris Darmody Neil Goldsman

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

Monte Carlo Algorithm

Random flight time: τ

Drift in field for τ

Scatter

t < tmax

Start

Stop

YES

NO

Chris Darmody Neil Goldsman

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

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

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

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

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

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

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

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