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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

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**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 GA1; H = NiG kiG Heterostructures: EFA QWell QWire QDot

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**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

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**Bravais lattice: the lattice looks the same when seen from any point**

P’ = P + (m,n,p) * (a1,a2,a3) integers

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**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)

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**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)

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**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

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**How many types of lattices exist?**

4

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**How many types of lattices exist?**

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**How many types of lattices exist?**

14 (in 3D) C ? 5

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Bravais Lattices

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**Translational symmetry**

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

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**Eigenfunctions of the linear momentum**

basis of translation group irreps

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**Range of k and 3D extension**

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

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

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**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

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**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

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**How do electrons behave in crystals?**

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

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Empty lattice Plane wave

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T=0 K Fermi energy

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**Band folded into the first Brillouin zone**

e (k): single parabola folded parabola

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Bragg diffraction gap a

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**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

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**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

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**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

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**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

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**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

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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

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**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

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**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

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**Machinery (cont.) How can we determine the matrices?**

we can use symmetry-adapted JiJj products Example: 4-th band model: 34

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**Machinery (cont.) We form the following invariants**

Finally we build the Hamiltonian Luttinger parameters: determined by fitting 35

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**Four band Hamiltonian:**

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**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

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

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**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

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**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

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**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

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**Most prominent applications:**

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

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**Most prominent applications:**

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

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**Most prominent applications:**

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

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**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 GA1; H = NiG kiG Heterostructures: EFA QWell QWire QDot

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**Thanks for your attention**

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