Periodic Boundary Conditions
Why PBC?
● ≤ ≤
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PBC
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Crystal structure
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Different unit cells
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○
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○
How to define a crystal structure?
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○
○
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○ 𝜋 𝜆
○ 𝜆
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○
○ 𝜆 𝜋 𝜆
Wigner-Seitz vs Brillouin zone
Brillouin Zone (Brillouin zone boundary)
K-space
● 𝜆
●
𝜋 𝜆
𝜋
● 𝜆 𝜋 𝜋 𝜆
Bloch theorem for simplified model for a periodic
system
● 𝝍
○ 𝝍→
○ →
○ →
○ →
●
○ 𝝍 −
○ 𝝍
●
How to define energy of a crystal
structure in k-space?
● → 𝜋
● 𝜆→ 𝜋 𝜆
●
● 𝜋
𝜋 𝜋 𝜋𝜋 𝜋 𝜋
𝜋 ≤ ≤𝜋
𝜋 ≤ ≤ 𝜋
𝜋 ≤ ≤ 𝜋
𝜋 ≤ ≤ 𝜋
𝜋 ≤ ≤ 𝜋
…
Band gap
𝜋 𝜋 𝜋𝜋 𝜋 𝜋
Valance
band
Conduction
band
Material classification based on the value of Eg
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○ ≤
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Example of band structure
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How to obtain and visualize
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● https://jp-minerals.org/vesta/en/download.html
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http://www.crystallography.net/cod/
Packages for QM calculations of solids
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○ …
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Task (Band gap calculation using Gaussian)
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https://gaussian.com/iop/
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Goodluck!