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Lecture 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. SUMMARY (keywords). Lattice → Wigner-Seitz unit cell.

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Lecture 1

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  1. Lecture 1 Periodicity and Spatial ConfinementCrystal 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 structure A crystal is solid material whose constituent atoms, molecules or ions are arranged in an orderly repeating pattern Crystal = lattice + basis

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

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

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

  7. Body-centered cubic 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.

  8. C ? 4 How many types of lattices exist?

  9. C ? 6 How many types of lattices exist?

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

  11. Bravais Lattices

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

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

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

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

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

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

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

  19. Empty lattice Plane wave

  20. T=0 K Fermi energy

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

  22. Bragg diffraction gap a

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

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

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

  26. 8-band H CB uCBk HH LH uHHk Split-off uLHk k=0 k uSOk 1-band H 1-band H 4-band H 8-band H

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

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

  29. 1. Second order perturbation: H second order in k: Theory of invariants (basic ideas) 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

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

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

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

  33. Four band Hamiltonian:

  34. Exercise:Show that the 2-bands {|1/2,1/2>,|1/2,-1/2>} conduction band k·p Hamiltonian reads H= a k2I, 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

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

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

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

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

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

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

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

  42. 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

  43. Thanks for your attention

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