Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
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