Presentation on theme: "Main Topics from Chapters 3-5 Due to time, not all topics will be on test. Some problems ask to discuss the meaning or implication. Lattice Dynamics (Monatomic,"— Presentation transcript:
Main Topics from Chapters 3-5 Due to time, not all topics will be on test. Some problems ask to discuss the meaning or implication. Lattice Dynamics (Monatomic, Diatomic, Mass Defect, 2D Lattices) Strain (compliance, reduced notation, tensors) Harmonic Oscillator (Destruction/Creation, Hamiltonian & Number Operators, Expectation Values) Energy Density and Heat Capacity (phonons, electrons and photons) Quasiparticle Interactions (e-e, e-phonon, e-photon, defect interations) Electrical and Thermal Conductivity
Lattice Vibrations Longitudinal Waves Transverse Waves When a wave propagates along one direction, 1D problem. Use harmonic oscillator approx., meaning amplitude vibration small. The vibrations take the form of collective modes which propagate. Phonons are quanta of lattice vibrations.
The force on the n th atom; The force to the right; The force to the left; The total force = Force to the right – Force to the left aa U n-1 U n U n+1 Eqn’s of motion of all atoms are of this form, only the value of ‘n’ varies Monatomic Linear Chain Thus, Newton’s equation for the n th atom is
Brillouin Zones of the Reciprocal Lattice 1st Brillouin Zone (BZ=WS) 2nd Brillouin Zone 3rd Brillouin Zone Each BZ contains identical information about the lattice 2 /a Reciprocal Space Lattice: There is no point in saying that 2 adjacent atoms are out of phase by more than (e.g., 1.2 =-0.8 ) Modes outside first Brillouin zone can be mapped to first BZ
Diatomic Chain(2 atoms in primitive basis) 2 different types of atoms of masses m1 and m2 are connected by identical springs U n-2 U n-1 U n U n+1 U n+2 K KK K m1 m2 m a) b) (n-2) (n-1) (n) (n+1) (n+2) a Since a is the repeat distance, the nearest neighbors separations is a/2 Two equations of motion must be written; One for mass m1, and One for mass m2.
As there are two values of ω for each value of k, the dispersion relation is said to have two branches Upper branch is due to the positive sign of the root. Negative sign: k for small k. Dispersion- free propagation of sound waves Optical Branch Acoustical Branch This result remains valid for a chain containing an arbitrary number of atoms per unit cell. 0л/a2л/a–л–л/a k A B C A when the two atoms oscillate in antiphase At C, M oscillates and m is at rest. At B, m oscillates and M is at rest.
Number and Type of Branches Every crystal has 3 acoustic branches, 1 longitudinal and 2 transverse Every additional atom in the primitive basis contributes 3 further optical branches (again 2 transverse and 1 longitudinal)
2D Lattice K U lm U l+1,m U l,m-1 U l,m+1 U l-1,m Write down the equation(s) of motion What if I asked you to include second nearest neighbors with a different spring constant?
2D Lattice C U lm U l+1,m U l,m-1 U l,m+1 U l-1,m Similar to the electronic bands on the test, plot w vs k for the  and  directions. Identify the values of at k=0 and at the edges.
Specific Heat or Heat Capacity The heat energy required to raise the temperature a certain amount The thermal energy is the dominant contribution to the heat capacity in most solids. In non-magnetic insulators, it is the only contribution. Classical Picture of Heat Capacity When the solid is heated, the atoms vibrate around their sites like a set of harmonic oscillators. Therefore, the average energy per atom, regarded as a 3D oscillator, is 3kT, and consequently the energy per mole is = Dulong-Petit law: states that specific heat of any solid is independent of temperature and the same result (3R~6cal/K-mole) for all materials!
Average energy of a harmonic oscillator and hence of a lattice mode at temperature T Energy of oscillator The probability of the oscillator being in this level as given by the Boltzman factor Thermal Energy & Heat Capacity Einstein Model
Mean energy of a harmonic oscillator Low Temperature Limit Zero Point Energy exponential term gets bigger High Temperature Limit is independent of frequency of oscillation. This is a classical limit because the energy steps are now small compared with thermal/vibrational energy <<
Heat Capacity C (Einstein) Heat capacity found by differentiating average phonon energy where T(K) Area = The difference between classical and Einstein models comes from zero point energy. Points:Experiment Curve: Einstein Prediction The Einstein model near T= 0 did not agree with experiment, but was better than classical model. Taking into account the distribution of vibration frequencies in a solid this discrepancy can be accounted for.
1.Approx. dispersion relation of any branch by a linear extrapolation 2.Ensure correct number of modes by imposing a cut-off frequency, above which there are no modes. The cut-off freqency is chosen to make the total number of lattice modes correct. Since there are 3N lattice vibration modes in a crystal having N atoms, we choose so that: Debye approximation to the dispersion Debye approximation has two main steps Einstein approximation to the dispersion
Density of states (DOS) per unit frequency range g( ) The number of modes/states with frequencies and +d will be g( )d . # modes with wavenumber from k to k+dk= for 1D monoatomic lattice
The energy of lattice vibrations will then be found by integrating the energy of single oscillator over the distribution of vibration frequencies. Thus Mean energy of a harmonic oscillator for 1D It would be better to find 3D DOS in order to compare the results with experiment. Debye Model adjusts Einstein Model
3D Example: The number of allowed states per unit energy range for free electron? Each k state represents two possible electron states, one for spin up, the other is spin down.
L L L Octant of the crystal: k x,k y,k z (all have positive values) The number of standing waves;
The Heat Capacity of a Cold Fermi Gas (Metal) Close to E F, we can ignore the variation in the density of states: g( ) g(E F ). By heating up a metal (k B T << E F ), we take a group of electrons at the energy - (with respect to E F ), and “lift them up” to . The number of electrons in this group g(E F )f( )d and each electron increased its energy by 2 : The small heat capacity of metals is a direct consequence of the Pauli principle. Most of the electrons cannot change their energy. kBTkBT
Bam! Random Collisions On average, I go about seconds between collisions with phonons and impurities electron phonon Otherwise metals would have infinite conductivity Electrons colliding with phonons (T > 0) Electrons colliding with impurities imp is independent of T The thermal vibration of the lattice (phonons) will prevent the atoms from ever all being on their correct sites at the same time. The presence of impurity atoms and other point defects will upset the lattice periodicity
Fermi’s Golden Rule Transition rate: Quantum levels of the non-perturbed system Perturbation is applied Transition is induced (E) is the ‘density of states available at energy E’. See Fermi‘s Golden Rule paper in Additional Material on the course homepage
Absorption When the ground state finds itself in the presence of a photon of the appropriate frequency, the perturbing field can induce the necessary oscillations, causing the mix to occur. This leads to the promotion of the system to the upper energy state and the annihilation of the photon. This process is stimulated absorption (or simply absorption). Einstein pointed out that the Fermi Golden Rule correctly describes the absorption process. - degeneracy of state f
Quantum Oscillator Atoms still have energy at T=0. What is for the ground state of the quantum harmonic oscillator? (1D Case) For 3D quantum oscillator, the result is multiplied by 3: ⇒
These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical particles which have no classical analogue. –They behave like particles in momentum space or k space. Phonons are one example of many like this in many different areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids Excitons: Quantized Normal Modes of electron-hole pairs
Phonon spectroscopy = Constraints: Conservation laws of MomentumEnergy Conditions for: elastic scatteringin In all interactions involving phonons, energy must be conserved and crystal momentum must be conserved to within a reciprocal lattice vector.
x=(a-b)/2 or The cubic axes are equivalent, so the diagonal components for normal and shear distortions must be equal. And cubic is not elastically isotropic because a deformation along a cubic axis differs from the stress arising from a deformation along the diagonal. e.g.,  vs.  Zener Anisotropy Ratio: