Commit 43309deb authored by Anton Akhmerov's avatar Anton Akhmerov
Browse files

now for real

parent 40c3ece7
......@@ -6,9 +6,7 @@ In this lecture we will:
- discuss specific heat of a solid based on atomic vibrations (_phonons_)
- disregard periodic lattice $\rightarrow$ consider homogeneous medium
- _(chapter 9: discuss phonons in terms of atomic masses and springs)_
- discuss the Einstein model
- discuss the Debye model
- introduce reciprocal space, periodic boundary conditions and _density of states_
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......@@ -92,8 +92,8 @@ $$
Comparable to phonons, but: electrons are _fermions_.
- Only 2 (due to spin) allowed per $k$-value
- Fill up from the lowest energy until you run out of electrons
- Only 2 (due to spin) allowed per $k$-value
- Fill up from the lowest energy until you run out of electrons
$\rightarrow$ Calculate when you are out of electrons $\rightarrow$ _Fermi energy_.
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......@@ -13,32 +13,28 @@ In this lecture we will:
### Crystal classification
- **_Lattice_**
+ periodic pattern of *lattice points*, which all have an identical view
+ lattice points are not necessarily the same as atom positions
+ there can be multiple atoms per lattice point
+ freedom of translation
+ multiple lattices with different point densities possible
+ periodic pattern of *lattice points*, which all have an identical view
+ lattice points are not necessarily the same as atom positions
+ there can be multiple atoms per lattice point
+ freedom of translation
+ multiple lattices with different point densities possible
- **_Lattice vectors_**
+ from lattice point to lattice point
+ $N$ vectors for $N$ dimensions
+ multiple combinations possible
+ not all combinations provide full coverage
+ from lattice point to lattice point
+ $N$ vectors for $N$ dimensions
+ multiple combinations possible
+ not all combinations provide full coverage
- **_Unit cell_**
+ spanned by lattice vectors
+ has 4 corners in 2D, 8 corners in 3D
+ check if copying unit cell along lattice vectors gives full lattice
+ spanned by lattice vectors
+ has 4 corners in 2D, 8 corners in 3D
+ check if copying unit cell along lattice vectors gives full lattice
- **_Primitive unit cell_**
+ smallest possible $\rightarrow$ no identical points skipped
+ not always most practical choice
+ smallest possible $\rightarrow$ no identical points skipped
+ not always most practical choice
- **_Basis_**
+ only now we care about the contents (i.e. atoms)
+ gives element and position of atoms
+ properly count partial atoms $\rightarrow$ choose which belongs to unit cell
+ positions in terms of lattice vectors, *not* Cartesian coordinates!
+ only now we care about the contents (i.e. atoms)
+ gives element and position of atoms
+ properly count partial atoms $\rightarrow$ choose which belongs to unit cell
+ positions in terms of lattice vectors, *not* Cartesian coordinates!
### Example: graphite
......@@ -47,12 +43,11 @@ In this lecture we will:
1. Choose origin (can be atom, not necessary)
2. Find other lattice points that are identical
3. Choose lattice vectors, either primitive (red) or not primitive (blue)
- lengths of lattice vectors and angle(s) between them fully define the crystal lattice
- for graphite: $|{\bf a}_1|=|{\bf a}_2|$ = 0.246 nm = 2.46 Å, $\gamma$ = 60$^{\circ}$
- lengths of lattice vectors and angle(s) between them fully define the crystal lattice
- for graphite: $|{\bf a}_1|=|{\bf a}_2|$ = 0.246 nm = 2.46 Å, $\gamma$ = 60$^{\circ}$
4. Specify basis
- using ${\bf a}_1$ and ${\bf a}_2$: C$(0,0)$, C$\left(\frac{2}{3},\frac{2}{3}\right)$
- using ${\bf a}_1$ and ${\bf a}_{2}'$: C$(0,0)$, C$\left(0,\frac{1}{3}\right)$, C$\left(\frac{1}{2},\frac{1}{2}\right)$, C$\left(\frac{1}{2},\frac{5}{6}\right)$
- using ${\bf a}_1$ and ${\bf a}_2$: C$(0,0)$, C$\left(\frac{2}{3},\frac{2}{3}\right)$
- using ${\bf a}_1$ and ${\bf a}_{2}'$: C$(0,0)$, C$\left(0,\frac{1}{3}\right)$, C$\left(\frac{1}{2},\frac{1}{2}\right)$, C$\left(\frac{1}{2},\frac{5}{6}\right)$
An alternative type of unit cell is the _Wigner-Seitz cell_: the collection of all points that are closer to one specific lattice point than to any other lattice point. You form this cell by taking all the perpendicular bisectrices or lines connecting a lattice point to its neighboring lattice points.
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......@@ -10,9 +10,8 @@ In this lecture we:
* Formulate a general way of computing the electron band structure, the **Bloch theorem**.
* Derive the electron band structure when the interaction with the lattice is weak using the **Nearly free electron model**. This model:
- Helps to understand the relation between tight-binding and free electron models
- Describes the properties of metals.
- Helps to understand the relation between tight-binding and free electron models
- Describes the properties of metals.
All the different limits can be put onto a single scale as a function of the strength of the lattice potential $V(x)$:
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