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

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

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

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

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

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)

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)

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

How many types of lattices exist?
4

How many types of lattices exist?
6

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

Bravais Lattices

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

Eigenfunctions of the linear momentum
basis of translation group irreps

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

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

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

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

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

Empty lattice Plane wave

T=0 K Fermi energy

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

Bragg diffraction gap a

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

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

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

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

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

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

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

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

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

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

Four band Hamiltonian:
36

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

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

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

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

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

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

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

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

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