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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: A 1 ; H = N i k i Heterostructures: EFA QWell QWire QDot

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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) P a1 a2 integers P a1 a2

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

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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? C ? 4

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How many types of lattices exist? C ? 6

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

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Eigenfunctions of the linear momentum basis of translation group irreps

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

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

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

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How does the wave function look like? Hamiltonian eigenfunctions are basis of the T n group irreps We require Bloch function Periodic (unit cell) partEnvelope part Periodic part

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

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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 (k): single parabola folded parabola

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a Bragg diffraction gap

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

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

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MgO k=Γ k=0 CB HH LH Split-off u CB k u HH k u LH k u SO k k gap Kane parameter

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k=0 CB HH LH Split-off u CB k u HH k u LH k u SO k k 8-band H 4-band H 1-band H 8-band H 1-band H

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

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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) 2. H must be an invariant under point symmetry (T d ZnBl, D 6h wurtzite) A·B is invariant (A 1 symmetry) if A and B are of the same symmetry e.g. (x, y, z) basis of T 2 of T d : x·x+y·y+z·z = r 2 basis of A 1 of T d 1. Second order perturbation: H second order in k:

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

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Machinery (cont.) How can we determine the matrices? we can use symmetry-adapted J i J j products Example: 4-th band model:

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

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Four band Hamiltonian:

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Exercise: Show that the 2-bands {|1/2,1/2>,|1/2,-1/2>} conduction band k·p Hamiltonian reads H= a k 2 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: Angular momentum components in the 1/2 basis: S i =1/2 i, with 3. Character tables and basis of irreps

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Answer: 1. Disregard T 1 : 2. Disregard E: 3. Disregard T 2 : 4. A 1 :

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

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

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Heterostructures BAB z In a one-band model we finally obtain: 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

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

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

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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: A 1 ; H = N i k i Heterostructures: EFA QWell QWire QDot

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Thanks for your attention

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