Reciprocal lattice

1D lattices

🧠 Remember: we saw $k$-space is periodic in range $-\pi/a\le k \le \pi/a$ called the Brillouin zone

If we consider the Brillouin zone as a “unit cell” in $k$-space then we can tile it over the whole “crystal”. The Bravais lattice we use to tile the space is called reciprocal lattice.

• 🧠 Remember: Wigner-Seitz unit cell
• Applying the Wigner-Seitz unit cell to the $\rm 1D$ $k$-space we recover the Brillouin zone

💼 Case: consider a periodic function $f(x)$ of period $a$ such that $f(x+a)=f(x)$

• We can express this function as

  $$ f(x)=\sum_n \tilde f_n e^{i2\pi nx/a} \quad \text{where}\quad n\in \Z $$

  • The wave vectors $2\pi n/a$ are the reciprocal lattice vectors
  • The coefficients $\tilde f_n$ are the values of $\tilde f(k)$ at $k=2\pi n/a$

💼 Case: lets consider the opposite, a set of numbers $g_m$ $m\in \Z$ ie $g_m=g(ma)$

• We can define a function reciprocal space

  $$ \tilde g(k)=\sum_m g_m e^{ikma} $$

  • This function is periodic with period $2\pi/a$

💎 Conclusion: you can define the function over an interval to describe the whole function, ie a Brillouin zone

2D lattices: square

$\vec k$ is quantised in both directions $k_x$ is a multiple of $2\pi(N_xa)$ and $k_y$ of $2\pi /(N_y a)$ where $N_x$ and $N_y$ are the numbers of columns and rows in the lattice.

• Again we find a Brillouin region of

  $$ -\pi/a \le k_x, k_y \le \pi/a $$

• We can tile this to get the entire space so the Bravais lattice in $\vec k$-space is a square defined by

  $$ \frac {2\pi}a (m,n) \quad \text{where} \quad m,n\in \Z $$

  • 🗒️ Note: we can use the Wigner-Seitz method again

• For primitive lattice vectors we can choose

  $$ \vec b_1=\frac {2\pi}a (1,0 ) \quad \vec b_2=\frac {2\pi}a (0,1) $$

  which allows us to define the position of each lattice cell as $m \vec b_1+ n \vec b_2$

General definition of primitive vectors of the reciprocal lattice

💼 Case: consider a $\rm 2D$ square lattice with lattice vectors $\vec a_1=a(1,0)$ and $\vec a_2=a(0,1)$