###### tags: `Solid State Physics`
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Based on
1. **"Solid State Physics"** by **Ashcroft** and **Mermin**
2. **"Solid State Physics** course by **Prof. Chi-Feng Pai**

The concept of the reciprocal space used in solid state physics and semiconductor physics.

Other concepts such as Bloch's theorem will be covered in other notes.
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[TOC]
# Bravais lattice
* Fundamental concept specifying the periodic array of repeating units.
* Units can be single atoms, molecules, ions, etc.
* Bravais lattice only summarize the geometry of the structure.
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Two definitions of the Bravais lattice:

1. Infinite array of discrete points with an arrangement that **appears exactly the same**, from whichever direction the array is viewed.
2. All points with the position vectors $\textbf R = n_1a_1+n_2a_2+n_3a_3$ with $n \in \textbf Z$

we should note that we can have many sets of the **(**$\textbf a_1, \textbf a_2, \textbf a_3$**)** as long as it can represents the structure completely.
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## Primitive unit cell
**==Cell containing only one lattice point.==**
* We can see the picture above
* **左上角$\rightarrow$** vectors is not limited to only one set
* **右上角$\rightarrow$** Three of them can be the primitive unit cell, while the bottom one can't. Because it has two lattice points.
* **左下角$\rightarrow$** The example of primitive unit cell. Note that if we make each lattice point the vertex of the shape, it can still be the primitive unit cell ($1/4 \times 4 = 1$).

## Wigner-Seitz primitive cell
* **上圖右下角$\rightarrow$** One example of such primitive unit cell.
* In **real space** it is the Wigner-Seitz primitive cell.
* In **receprocal space** it will be the the first Brillain Zone.

## (Conventional) Unit cell
**==What we usually see of FCC and the BCC cell==**
* So that the conventional unit cell is what we usually see for the easier representation.But generally it is not the same as the primitive unit cell.

## Basis (for non-Bravais lattice)
**==For some structures we should use basis to fully describe it==**
* Let's see how we deal with the honeycomb cell.
* For example, for the BCC and the FCC, we can use **Simple cubic + Basis** to describe them.
* no. of basis = no. of lattice points in that cell **(BCC has two basis $\rightarrow$ two lattice points in unit cell)**
* We can think like: if we want to merge bravais lattices, we need basis. **Like BCC, is the merge of two simple cubic bravais lattices.**

# X-Ray Diffraction
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1. X-Ray's $\lambda$ should have similar value as the lattice constant to have the diffraction.
2. We can further calculate the crirterion in terms of the energy.
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## Bragg vs. Von Laue formulations
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### Bragg formulation
* k and k' in the Bragg formulation are wave vectors.
* Bragg's one is good and classical, but it doesn't tell you **"Why and how do we choose the plane?"** $\rightarrow$ 我們不知道怎麼選繞射的平面！！！
* Furthermore, the lattice points are virtual points and we didn't consider their volume!!! How to make sure that ray does hit the lattice points?
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### Von Laue formulation
* **More general!!!**
* First of all, we don't assume that there're planes of lattice points waiting to be hit.
* We assume that there're scattering points at which the light hits.
* Distance between lattice points are **described by the Bravais lattice's wave vectors.**
* And then, we only need them to satisfy $\vec{K} \cdot \vec{R} = 2\pi n, \quad n \in \mathbb{Z}$
* 必須注意一下的是說我們的兩個波行進方向的wave vector才會是公式中的K, **which being the reciprocal lattice**.
* See detailed derivation below
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* Then now we can derive for the Bragg's plane and fine the reflection point.
## X-Ray diffraction method

# Reciprocal lattice
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The set of all wave vectors $\vec K$ that yield plane waves with the periodicity of a given Bravais lattice is known as its reciprocal lattice.
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## Derivation
* From the principle of the X-Ray diffraction, and the periodic condition, we can say that $\vec{K} \cdot \vec{R} = 2\pi n, \quad n \in \mathbb{Z}$.
* And actually the $(k_1, k_2, k_3)$ in **$\vec K$** are Miller indices (h, k, l).
* Above has the equation for calculating $\vec b$, and has the example of simple cubic.
* **bigger lattice constant, smaller reciprocal lattice unit vector!!!** $\rightarrow$ 真實空間間隔越大，倒空間間隔越小。
* FCC, BCC 互為reciprocal lattice (Structurewise speaking)
## Lattice planes: Miller indices
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Since we know there are reciprocal lattice vectors normal to any family of lattice planes, it is natural to pick a reciprocal lattice vector to represent the normal. （簡而言之，我們數學上不是都會用法向量來表示面嗎？這邊同理）

To make the choice unique, one uses the shortest such reciprocal lattice vector. In this way one arrives at **the Miller indices of the plane.**
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* The denser the lattice plane, the larger the $\vec K$
* d小一點，$|\vec K| = \frac {2\pi}{d}$ larger
* 你選定了你的lattice plane後，其就對應到了一個 $\vec K$
* For any family of lattice planes separated by a distance d, **there are reciprocal lattice vectors perpendicular to the planes**, the shortest of which have a length of $2\pi/d$. And Vice versa.
* $\vec K = k_1b_1+k_2b_2+k_3b_3$, and $(k_1, k_2, k_3)$ is same as the Miller indices $(h, k, l)$
## Some comments on k-space
* value of $k$ is discrete in $\frac {\hbar^2}{2m}|\vec K|^2$, where $n$ is the band index.
* In ground state, electrons can only be filled up to certain ($k_1, k_2, k_3$).
* 填越外面能量越高，而且會isotropic的填.
* The boundary is called the **fermi surface**, it's radius is called the **fermi wavevector**.
* For the first Brillouin zone, $\frac {-a}{\pi}\leq k \leq \frac {a}{\pi}$
* In VASP, the denser the reciprocal phase is sampled, the more accurate the result will be.
# Two Fourier Conjugate pairs
## Real vs. Reciprocal
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k-domain tells you about electron states in reciprocal space (band structure $E(k)$).
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* Fourier conjugates in space.
* k is the spatial frequency, λ the real-space wavelength.
* $k=2\pi/\lambda$
* Solving the Bloch's wavefuncion, the **$\hbar k$ is called the crystal momentum.**

## Time vs. Frequency
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ν (or ω) domain: tells you about energies/frequencies of excitations (like phonons, photons, electronic transitions).
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* Fourier conjugates in time.
* ν is the temporal frequency, E is directly proportional.
* $E=h\nu=\hbar\omega$