Lecture contents. A few concepts from Quantum Mechanics. Tight-binding model Solid state physics review

Transcription

1 Lecture contents A few concepts from Quantum Mechanics Particle in a well Two wells: QM perturbation theory Many wells (atoms) BAND formation Tight-binding model Solid state physics review Approximations

2 Few concepts from Quantum Mechanics lectrons are waves: Psi-function (, ) Schrödinger equation Hamiltonian General (time dependent)schrödinger equation: ( r, t) i t Hˆ ( r, t) If Ĥ does not depend on time t i ( r, t) ( r) e ; plane _ wave Time-independent Schrödinger equation: H ( r) ( r) Hˆ ( r) m V ( r) ( r) ( r) Uncertainty principle (Fourier theorem+ w.-p. dualism) p x h Compare to wave packet k : ˆ kx

3 Quantum Mechanics: particle in a single-dimensional well 3 V(x) Delocalized states (continuous spectrum ) V L x Localized states (discrete spectrum ) No states d ( x) m dx V ( x) ( x) ( x) V(x)=V for x < V(x)= for x L V(x)=V for x > L For infinite barrier, V k n n n L k m n ml n n L nx sin L

4 Two wells 4 V(x) V (x) L V (x) x First approximation of perturbation theory: What will happen if the wells are resonant? * V x) dx V ( Hˆ ( r) ( T V V ) Negative Solution in the form: C C

5 Two wells: energy level splitting 5 V(x) x L Negative anti-bonding bonding V V esonant levels split Splitting increases at small distances

6 When two atoms brought together, wave functions interact each other and energy levels split. Bonding and antibonding 6 Wave function Probability density function Antibonding Bonding

7 Tight-binding model: periodic array of atoms 7 V(x) a Hcryst V r m x ( ) What do we know? Formation of energy band Width depends mainly on interaction of the closest neighbors Wavefunctions are renormalized Start with valence state: r) ( ) H at ( r Consider crystal Hamiltonian as atomic + perturbation : Solution in the form : H cryst H at V C ( r ); ma; m,...

8 C ( r ) Tight-binding model: solution, - translation vector 8 dr C N e ik Applying (periodic) boundary conditions k quasi wavevector is a quantum number to innumerate the electron states

9 Tight-binding model: solution 9 Solution for band constructed from a single atomic orbital (s-band): k ( r) V ( r ) e ( r) ( r ) e ik ik When overlap is negligible: r) ( r ) ( ) ( Transfer integrals : ( r) V ( r ) Diagonal element: ( r r) V ( ) S-band: k ( ) e Width of a D S-band = ik Over nearest neibors

10 Tight-binding model: s-band in FCC crystal FCC st Brillouin zone k a a k

11 Tight-binding model: effective mass Compare: Parabolic band dispersion (e.g. free electron) V k m

12 Orbital structure and energy levels of Si atom nergy levels of electrons in silicon 3p 3s Orbital structure Silicon: s s p 6 3s 3p nergy levels are filled according to Pauli principle

13 nergy levels of Si as a function of interatomic spacing 3 Low epulsion Attraction When N Si atoms brought together Potential energy 3p 3s High Interatomic spacing Close Far

14 Few concepts from Solid State Physics. Adiabatic approximation When valence and core electrons are separated, general Schrödinger equation for a condensed medium without spin 4 = H L + H e H(, r) (, r) Mass of ions > times greater than mass of electrons Ion velocities > times slower lectrons adjust instantaneously to the positions of atoms Separate ion and electron motion (accuracy ~m/m) (, r) ( r, ) ( ) H L ( ) ( ) H ( r, ) ( r, ) e L e

15 5 Few concepts from Solid State Physics. Phonons anhar m l m l m l m l m l l l l L U u u C U M p H,,, exp ) ( kt n t i ikr k e u u, Hamiltonian for lattice motion (harmonic oscillations) : Displacements show up as plane waves with weak interaction via anharmonicity: Phonon dispersion relation in GaAs, k, n k nergy in a mode: quilibrium distribution (Bose instein):

16 Quantum harmonic oscillator: Hamiltonian Quantum harmonic oscillator H p M Cx M C M 6 Solution gives resonance frequency (as in classical mechanics) C M And quantum oscillation spectrum: (n may be considered as number of quasiparticles ) n n x

17 Quantization of lattice vibrations: phonons 7 For a single oscillator the frequency is fixed, but when many oscillators interact we have a number of modes (normal modes) k ach mode is occupied by n k phonons k n k k For a D chain states are determined as: Occupancy of modes is given by Bose-statistics: k n( ) n ; for n,,... N Na exp kt Bose-instein distribution function

18 Few concepts from Solid State Physics 3. One-electron (mean-field) approximation 8 Schrödinger equation for the electrons (no spin): Introduce one-electron states, i : H p m ei pi m V l V r ( r) ( r) Ve ( ri ) i V i ee ( r, ) ( r, ) r i l Ve ph Vee One-electron Schrödinger equation (each state can accommodate up to electrons): i i Small perturbation V(r) has periodicity of the crystal