Quantum Dots in Photonic Structures

1. Lecture 6: Electronic band structure of solids Quantum Dots in PhotonicStructures Wednesdays, 17.00, SDT Jan Suffczyński Projekt Fizyka Plus nr POKL.04.01.02-00-034/11 współfinansowany przez Unię Europejską ze środków Europejskiego Funduszu Społecznego w ramach Programu Operacyjnego Kapitał Ludzki

2. Plan for today Reminder 2. Band structure of solids- Nearlyfreeelectron model 3. Band structure of solids – tightbindningapproximation

3. Reminder: Fundamental postulates of the quantum mechanics Postulate 1: All information about a system is provided by the system’s wavefunction Postulate 2:The motion of a nonrelativistic particle is governed by the Schrodinger equation Postulate 3: Measurement of a system is associated with a linear, Hermitian operator Time-independent S.E.:

4. De Broglie’s Hypothesis • ALLmaterial particles possess wave-like properties, characterized by the wavelengthλB,related to the momentum p of the particlein the same way as for light Planck’s Constant Momentum of the particle de Broglie wavelength of the particle Frequency:

5. Wave Picture of Particle • A(k)is spiked at a given k0, and zero elsewhere Onlyone wave with k = k0 (λ = λ0) contributes; thus one knows momentum exactly, and the wavefunction is a traveling wave – particle is delocalized • A(k)is shaped as a bell-curve Gives a wave packet – “partially” localized particle • A(k)is the same for all k No distinctions for momentums, so particle’s position is well defined - the wavefunction is a “spike”, representing a “very localized” particle

6. + + Crystallattice We define lattice points; these are points with identical environments Crystal = lattice + basis

7. Wigner-Seitz cell The smallest (“primitive”) cell which displays the full symmetry of the lattice is the Wigner-Seitz cell. Construction method: surfaces passing through the middle pointsto the nearest lattice points

8. Wigner-Seitz Cellconstruction Form connection to all neighbors and span a plane normal to the connecting line at half distance

9. Wigner-Seitz cellconstruction The smallest (“primitive”) cell which displays the full symmetry of the lattice is the Wigner-Seitz cell.

10. Body-centered cubic Wigner-Seitz cellconstruction The smallest (“primitive”) cell which displays the full symmetry of the lattice is the Wigner-Seitz cell. First Brilluoinzone

11. Realistic Potential in Solids • niare integers • Example: 2D Lattice

12. Bloch theorem Felix Bloch 1905, Zürich - 1983, Zürich

13. Envelope part Periodic (unit cell) part Bloch waves Bloch’stheorem: Solutions of the Schrodingerequation Felix Bloch 1905, Zürich - 1983, Zürich for the wave in periodicpotential U(r) = U(r+R) are: Bloch function: modified slide from Rob Engelen

14. Bloch waves Twoequivalentviews on the Bloch wave (1D example): modified slide from Rob Engelen

15. Bloch’s Theorem • What is probability density of finding particle at coordinate x? • But |uk(x)|2is periodic, so P(x) is as well Probabilityof finding matter at position x scales with ||2 or ×* Compare: probability of detecting light scales with |E(x,t)|2 or E×E*

16. Bloch’s Theorem The probability of finding an electron at anyatom in the solid is the same!  Each electron in a crystalline solid “belongs” to each and every atom forming the solid

17. Periodicity in reciprocal space Reciprocal lattice vector Bloch theorem Remark: (in 1D case) • IfV(x) has lattice periodicity [“translational invariance”, V(x)=V(x+a)]: • the electron densityr(x) has also lattice periodicity, however, • the wave function does NOT:

18. Electron in a crystalperiodicpotential ions The case of the „empty”lattice If:

19. Nearlyfreeelectron model Freeelectronenergy

20. Nearlyfreeelectron model Remark: E Energy of an electron with 1 A wavelength  150 eVEnergy of a photon with 1 A wavelength  12 keV

21. Brilluoinzones e (k): single parabola folded parabola

22. Nearlyfreeelectron model Consider a set of waves with +/- k-pairs, e.g. wave moving right wave moving left Superposition ofthesewaves also a solution of Schr. equation:

23. Nearlyfreeelectron model Origin of a band gap! Kittel

24. Electronicenergybands allowed energybands

25. Effective Mass:m* A method thatthe free electron model to work in the situations where there arelatticecrystalperturbations

26. Effective Mass m* -- describing the balance between applied ext-E and lattice site reflections m* a = S Fext q Eext

27. 2) greater curvature, 1/m* > 1/m > 0,  m* < m  net effect of ext-E and lattice interaction provides additional acceleration of electrons m = m* greater |curvature| but negative, net effect of ext-E and lattice interaction de-accelerates electrons At inflection pt No distinction between m & m*, m = m*, “free electron”, lattice structure does not apply additional restrictions on motion. 1)

28. Isolated Atoms

29. Diatomic Molecule

30. Four Closely Spaced Atoms

31. Six Closely Spaced Atomsas fn(R) the level of interest has the same Energyin each separated atom

32. Two atoms Six atoms Solid of N atoms ref: A.Baski, VCU 01SolidState041.ppt www.courses.vcu.edu/PHYS661/pdf/01SolidState041.ppt

33. Four Closely Spaced Atoms conduction band valence band

34. Solid composed of ~NA Na Atomsas fn(R) 1s22s22p63s1

35. Schrödinger Equation Revisited • If a wavefunctions ψ1(x) and ψ2(x) are solutions for the Schrödinger equation for energy E, then functions • -ψ1(x), -ψ2(x), and ψ1(x)±ψ2(x) are also solutions of this equations • the probability density of -ψ1(x) is the same as for ψ1(x)

36. Hydrogenmolecule • Consider an atom with only one electron in s-state outside of a closed shell • Both of the wavefunctions below are valid and the choice of each is equivalent • If the atoms are far apart, as before, the wavefunctions are the same as for the isolated atoms

37. The sum of them is shown in the figure These two possible combinations represent two possible states of two atoms system with different energies Once the atoms are brought together the wavefunctions begin to overlap There are two possibilities Overlapping wavefunctions are the same (e.g., ψs+(r)) Overlapping wavefunctions are different Hydrogenmolecule

38. Covalent Bonding Revisited • When atoms are covalently bonded electrons supplied by atoms are shared by these atoms since pull of each atom is the same or nearly so • H2, F2, CO, • Example: the ground state of the hydrogen atoms forming a molecule • If the atoms are far apart there is very little overlap between their wavefunctions • If atoms are brought together the wavefunctions overlap and form the compound wavefunction, ψ1(r)+ψ2(r), increasing the probability for electrons to exist between the atoms

39. Interatomic Binding • All of the mechanisms which cause bonding between the atoms derive from electrostatic interaction between nuclei and electrons. • The differing strengths and differing types of bond are determined by the particular electronic structures of the atoms involved. • The existence of a stable bonding arrangement implies that the spatial configuration of positive ion cores and outer electrons has less total energy than any other configuration (including infinite separation of the respective atoms). • The energy deficience of the configuration compared with isolated atoms is known as cohesive energy, and ranges in value from 0.1 eV/atom for solids which can muster only the weak van der Waals to 7ev/atom or more in some covalent and ionic compounds and some metals.

40. This typical curve has a minimum at equilibriumdistance R0 R > R0 ; the potential increases gradually, approaching 0 as R∞ the force is attractive R < R0; the potential increases very rapidly, approaching ∞ at small separation. the force is repulsive V(R) Repulsive 0 R0 Attractive R r R • Force between the atoms is the negative of the slope of this curve. At equlibrium, repulsive force becomes equals to the attractive part.

41. Electron vs photon