Kai Sun

University of Michigan, Ann Arbor

Course work and grading

Course grade will be based on • Problem set: 20% • Final Presentation: 20%

You will form teams of up to 3 members. Each team will give a 15 minute presentation on a topic you choose, illustrating an application of solid state physics.

• Midterm: 30% • Final: 30% • Piazza discussions: 5% bonus points

Homework will be assigned on Thursday and will be due the next Thursday Office hour: Tuesday or Wednesday (see the poll on CTools)

How do we know there are atoms or molecules?

Brownian motion: random moving of particles suspended in a fluid

Now we have technique to “see” atom directly (Scanning tunneling microscope, developed in the 80s), but people knows the existences of atoms long before we can see them. Q: How did people know there are atoms before we can really see them? A1: For liquid/gas, from the study on Brownian motion (Einstein 1905) A2: For solids, from the X-ray crystallography (will be discussed next time)

Matter

Matters

gas/liquid:

Atoms/molecules can move around

solids:

Atoms/molecules cannot move

Crystals:

Atoms/molecules form a periodic

structure

Random solids:

Atoms/molecules form a random

structure

Salt

NaCl Na: Sodium (larger spheres) Cl: Chlorine (smaller spheres)

An Ideal Crystal

An ideal crystal: infinite repetition of identical groups of atoms (e.g. NaCl). A group is called the basis (a unit cell) Q: How to describe and classify this periodic structure? A: Using Bravais lattices

Bravais Lattices

Unit cell: each periodicity is called a unit cell. A unit cell may contain 1 or more atoms or molecules Bravais lattice: as easy as 1, 2, 3 1. Choose one point in each unit cell. 2. Make sure that we pick the same point in every unit cell. 3. These points form a lattice, which is known as the Bravais lattice. One can choose the point arbitrarily and different choices give the same Bravais lattice Many different materials can share the same Bravais lattice

1D example

1D crystal 3 atoms/periodicity

Choice I:

Choice II:

Choice III:

2D example

An example of 2D crystal (one atom per unit cell)

Choice I Choice II

Another 2D example

An example of 2D crystal (two atoms per unit cell)

Choice I Choice II

Another 2D example

An example of 2D crystal (three atoms per unit cell)

Choice I Choice II

The translational symmetry of Bravais lattices

Symmetry: invariance under certain operation For Bravais lattices, translational symmetry is one of the most important. Translation: moving every point the same distance in the same direction, without rotation, reflection or change in size. Translational symmetry and translation vectors: If a system goes back to itself when we translate the system by some vector, we say that this system has a translational symmetry and this vector is known as a translation vector. There are an infinite number of translation vectors

If 𝑡 is a translation vector, n ∗ 𝑡 is also a translation vector where 𝑛 is an integer. For Bravais lattices, All Bravais lattices have translational symmetry. Any vector that connects two lattice points is a translational vector. For Bravais lattices, these translational vectors are also known as lattice vectors

Interactive figures

http://www-personal.umich.edu/~sunkai/teaching/Winter_2013/translations.html

Not all lattices are Bravais lattices: examples the honeycomb lattice (graphene)

Not all lattices are Bravais lattices: examples the honeycomb lattice (graphene)

Primitive lattice vectors

Q: How can we describe these lattice vectors (there are an infinite number of them)? A: Using primitive lattice vectors (there are only d of them in a d-dimensional space). For a 3D lattice, we can find three primitive lattice vectors (primitive translation vectors), such that any translation vector can be written as

𝑡 = 𝑛1𝑎 1 + 𝑛2𝑎 2 + 𝑛3𝑎 3 where 𝑛1, 𝑛2 and 𝑛3 are three integers. For a 2D lattice, we can find two primitive lattice vectors (primitive translation vectors), such that any translation vector can be written as

𝑡 = 𝑛1𝑎 1 + 𝑛2𝑎 2 where 𝑛1 and 𝑛2 are two integers. For a 1D lattice, we can find one primitive lattice vector (primitive translation vector), such that any translation vector can be written as

𝑡 = 𝑛1𝑎 1 where 𝑛1 is an integer.

Primitive lattice vectors

Red (shorter) vectors: 𝑎 1 and 𝑎 2

Blue (longer) vectors: 𝑏1 and 𝑏2

𝑎 1 and 𝑎 2 are primitive lattice vectors

𝑏1 and 𝑏2 are NOT primitive lattice vectors

𝑏1 = 2𝑎 1 + 0 𝑎 2 𝑎 1 =1

2𝑏1 + 0𝑏2

Integer coefficients noninteger coefficients

Primitive lattice vectors

The choices of primitive lattice vectors are NOT unique.

Primitive cell in 2D

The Parallelegram defined by the two primitive lattice vectors are called a primitive cell.

the Area of a primitive cell: A = |𝑎 1 × 𝑎 2| Each primitive cell contains 1 site.

Primitive cell in 3D

The parallelepiped defined by the three primitive lattice vectors are called a primitive cell.

the volume of a primitive cell: V = |𝑎 1. (𝑎 2 × 𝑎 3)| each primitive cell contains 1 site.

A special case: a cuboid

Wigner–Seitz cell

the volume of a Wigner-Seitz cell is the same as a primitive cell each Wigner-Seitz cell contains 1 site (same as a primitive cell).

Rotational symmetries

Rotational symmetries: If a system goes back to itself when we rotate it along certain axes by some angle 𝜃, we say that this system has a rotational symmetry. For the smallest 𝜃, 2𝜋/𝜃 is an integer, which we will call 𝑛. We say that the system has a 𝑛-fold rotational symmetry along this axis. For Bravais lattices, It can be proved that 𝑛 can only take the following values: 1, 2, 3, 4 or 6.

Mirror planes

Mirror Planes:

2D Bravais lattices

http://en.wikipedia.org/wiki/Bravais_lattice

3D Bravais lattices

http://en.wikipedia.org/wiki/Bravais_lattice