1. Lecture 1
Periodicity and Spatial Confinement Crystal structure Translational symmetry Energy bands k·p theory and effective mass
Theory of invariants Heterostructures
J. Planelles

2. SUMMARY (keywords) Lattice → Wigner-Seitz unit cell
Periodicity → Translation group → wave-function in Block form
Reciprocal lattice → k-labels within the 1rst Brillouin zone
Schrodinger equation → BCs depending on k; bands E(k); gaps
Gaps → metal, isolators and semiconductors
Machinery: kp Theory → effective mass J character table
Theory of invariants: G GA1; H =  NiG kiG
Heterostructures: EFA
QWell QWire QDot

3. Crystal = lattice + basis
Crystal structure
A crystal is solid material whose constituent atoms, molecules or ions are arranged in an orderly repeating pattern +
Crystal = lattice + basis

4. Bravais lattice: the lattice looks the same when seen from any point
P’ = P + (m,n,p) * (a1,a2,a3)
integers

6. Primitive: smallest unit cells (1 point)
Unit cell: a region of the space that fills the entire crystal by translation
Primitive: smallest unit cells (1 point)

7. Primitive: smallest unit cells (1 point)
Unit cell: a region of the space that fills the entire crystal by translation
Primitive: smallest unit cells (1 point)

8. Wigner-Seitz unit cell: primitive and captures the point symmetry
Centered in one point. It is the region which is closer to that point than to any other.
Body-centered cubic

9. How many types of lattices exist?4

10. How many types of lattices exist?6

11. How many types of lattices exist?
14 (in 3D)
C ? 5

12. Bravais Lattices

13. Translational symmetry
We have a periodic system.
Is there a simple function to approximate its properties? a

14. Eigenfunctions of the linear momentum
basis of translation group irreps

15. Range of k and 3D extension
same character:
3D extension
Bloch functions as basis of irreps:

16. First Brillouin zone: Wigner-Seitz cell of the reciprocal lattice

17. Crystals are infinite... How are we supposed to deal with that?
Solving Schrödinger equation: Von-Karman BCs
Crystals are infinite... How are we supposed to deal with that?
We use periodic boundary conditions
We solve the Schrödinger equation for each k value:
The plot En(k) represents an energy band
Group of translations:
k is a quantum number due to translational symmetry
1rst Brillouin zone

18. How does the wave function look like?
Hamiltonian eigenfunctions are basis of the Tn group irreps
We require
Bloch function
Periodic (unit cell) part
Envelope part
Envelope part
Periodic part

19. How do electrons behave in crystals?
Energy bands
How do electrons behave in crystals?
quasi-free electrons
ions

20. Empty lattice
Plane wave

21. T=0 K
Fermi energy

22. Band folded into the first Brillouin zone
e (k): single parabola
folded parabola

23. Bragg diffraction
gap a

25. Types of crystals Insulator Semiconductor Metal Low conductivity
Switches from conducting to insulating at will
fraction eV
Conduction Band
Fermi level
few eV
gap
Valence Band
Metal
Empty orbitals available at low-energy: high conductivity

27. How do we calculate realistic band diagrams?
k·p Theory
Tight-binding
Pseudopotentials
k·p theory
How do we calculate realistic band diagrams?
The k·p Hamiltonian

28. gap Kane parameter MgO k=0 k k=Γ uCBk uHHk uLHk uSOk CB HH LH
Split-off
uCBk
uHHk
uLHk
uSOk
k=0 k
gap
Kane parameter

29. k=0 k 8-band H 1-band H 1-band H 4-band H 8-band H uCBk uHHk uLHk uSOk
Split-off
uCBk
uHHk
uLHk
uSOk k
1-band H
1-band H
4-band H
8-band H

30. One-band Hamiltonian for the conduction band
This is a crude approximation... Let’s include remote bands perturbationally CB
1/m*
Effective mass
Free electron InAs
m=1 a.u. m*=0.025 a.u.
negative mass? k

31. Theory of invariants
1. Perturbation theory becomes more complex for many-band models
2. Nobody calculate the huge amount of integrals involved
grup them and fit to experiment
Alternative (simpler and deeper) to perturbation theory:
Determine the Hamiltonian H by symmetry considerations

32. Theory of invariants (basic ideas)
1. Second order perturbation: H second order in k:
2. H must be an invariant under point symmetry (Td ZnBl, D6h wurtzite)
A·B is invariant (A1 symmetry) if A and B are of the same symmetry
e.g. (x, y, z) basis of T2 of Td: x·x+y·y+z·z = r2 basis of A1 of Td 32

33. Theory of invariants (machinery)
1. k basis of T2
2. ki kj basis of
3. Character Table:
notation: elements of these basis:
4.Invariant: sum of invariants:
fitting parameter (not determined by symmetry)
irrep
basis element 33

34. Machinery (cont.) How can we determine the matrices?
 we can use symmetry-adapted JiJj products
Example: 4-th band model: 34

35. Machinery (cont.) We form the following invariants
Finally we build the Hamiltonian
Luttinger parameters: determined by fitting 35

36. Four band Hamiltonian:36

37. 2. Angular momentum components in the 1/2 basis: Si=1/2 si, with
Exercise: Show that the 2-bands {|1/2,1/2>,|1/2,-1/2>} conduction band k·p Hamiltonian reads H= a k2 I, where I is the 2x2 unit matrix, k the modulus of the linear momentum and a is a fitting parameter (that we cannot fix by symmetry considerations)
Hints: 1.
2. Angular momentum components in the 1/2 basis: Si=1/2 si, with
3. Character tables and basis of irreps 37

38. Answer:
1. Disregard T1:
2. Disregard E:
3. Disregard T2:
4. A1: 38

39. Brocken translational symmetry
Heterostructures
Brocken translational symmetry
How do we study this? B A z B A z z A
quantum dot
quantum wire
quantum well 39

40. Heterostructures How do we study this? B A z
the same crystal structure
similar lattice constants
no interface defects B A z
If A and B have:
...we use the “envelope function approach”
Project Hkp onto {Ψnk}, considering that:
Envelope part
Periodic part

41. Heterostructures Quantum well In a one-band model we finally obtain: Bz
V(z)
1D potential well: particle-in-the-box problem
Quantum well

42. Most prominent applications:
Laser diodes
LEDs
Infrared photodetectors
Image: CNRS France
Image: C. Humphrey, Cambridge
Quantum well

43. Most prominent applications:B A z
Image: U. Muenchen
Most prominent applications:
Transport
Photovoltaic devices
Quantum wire

44. Most prominent applications:z A
Most prominent applications:
Single electron transistor
LEDs
In-vivo imaging
Cancer therapy
Photovoltaics
Memory devices
Qubits?
Quantum dot

45. SUMMARY (keywords) Lattice → Wigner-Seitz unit cell
Periodicity → Translation group → wave-function in Block form
Reciprocal lattice → k-labels within the 1rst Brillouin zone
Schrodinger equation → BCs depending on k; bands E(k); gaps
Gaps → metal, isolators and semiconductors
Machinery: kp Theory → effective mass J character table
Theory of invariants: G GA1; H =  NiG kiG
Heterostructures: EFA
QWell QWire QDot

46. Thanks for your attention