Hong-Hao Tu (ITP, TU Dresden)

Solid State Theory (SS2020)

SFB 1143

Lecture 3: Lattice dynamics

April 15th, 2020

Email: hong-hao.tu@tu-dresden.deZoom: tuhonghao@gmail.com

§1.2 Quantum monoatomic chain

• Last lecture: Quantum theory of a monoatomic chain

successive canonical transformations

➢ Collective excitations: phonons

§1.2 Quantum monoatomic chain

• Next step: lattice dynamics in higher dimensions (d=2,3 => more realistic).

𝐻𝑖𝑜𝑛 =

𝑖

Ԧ𝑝𝑖2

2𝑀𝑖+1

2

𝑖≠𝑗

𝑉(Ԧ𝑟𝑖 − Ԧ𝑟𝑗)

Figure from Wikimedia

• Before that, we need a better understanding of the lattice structure…

➢ Find out common and distinct features!

§1.3 Lattice structure

Equilibrium positions of atoms!

• Bravais lattice: defined by a set of basis vectors (primitive vectors)

• Full classification available (see Wikipedia)Figure from Wolfram Demonstrations project

➢ 3d: 7 lattice systems, 14 Bravais lattices

§1.3 Lattice structure

• Unit cell (primitive cell): smallest repeating volume in the crystal

Different choices possible, usually choose the most convenient one.

§1.3 Lattice structure

• Unit cell (primitive cell): smallest repeating volume in the crystal

• Wigner-Seitz cell: special choice of unit cell (with atom at the center)

Different choices possible, usually choose the most convenient one.

Draw perpendicular lines (planes in 3d) through the center of all lines connecting neighboring sites of the Bravais lattice

§1.3 Lattice structure

• Unit cell (primitive cell): smallest repeating volume in the crystal

• Wigner-Seitz cell: special choice of unit cell (with atom at the center)

Different choices possible, usually choose the most convenient one.

Draw perpendicular lines (planes in 3d) through the center of all lines connecting neighboring sites of the Bravais lattice

§1.3 Lattice structure

• Reciprocal lattice

Bravais lattice:

➢ Ԧ𝐺𝑚 is called reciprocal vectors and defines the reciprocal lattice.

§1.3 Lattice structure

• The reciprocal lattice corresponds to the Fourier transform of the Bravais lattices.

Example: periodic potential

𝑅𝑛 = 𝑛1 Ԧ𝑎1 + 𝑛2 Ԧ𝑎2

𝑉 Ԧ𝑟 =

Ԧ𝐺

𝑉 Ԧ𝐺 𝑒𝑖 Ԧ𝐺⋅ Ԧ𝑟

𝑉 Ԧ𝑟 + 𝑅𝑛 =

Ԧ𝐺

𝑉 Ԧ𝐺 𝑒𝑖 Ԧ𝐺⋅( Ԧ𝑟+𝑅𝑛)

𝑒𝑖 Ԧ𝐺⋅𝑅𝑛 = 1

𝑉 Ԧ𝑟 = 𝑉(Ԧ𝑟 + 𝑅𝑛)

Solutions of Ԧ𝐺 are the discrete reciprocal vectors Ԧ𝐺𝑚.

§1.3 Lattice structure

• Reciprocal lattice

§1.3 Lattice structure

• Reciprocal lattice

§1.3 Lattice structure

• Reciprocal lattice

§1.3 Lattice structure

• Reciprocal lattice

§1.3 Lattice structure

§1.3 Lattice structure

§1.3 Lattice structure

Reciprocal vectors for the triangular lattice:

§1.3 Lattice structure

• First Brillouin zone (FBZ):

§1.3 Lattice structure

• First Brillouin zone (FBZ):

§1.3 Lattice structure

• First Brillouin zone (FBZ):

§1.3 Lattice structure

• First Brillouin zone (FBZ):