Lattice Dynamics related to movement of atoms
about their equilibrium positions determined by electronic structure Physical properties of solids Sound velocity Thermal properties: -specific heat -thermal expansion -thermal conductivity (for semiconductors) Hardness of perfect single crystals (without defects) Uniform Solid Material
There is energy associated with the vibrations of atoms. They are tied together with bonds, so they can't vibrate independently. The vibrations take the form of collective modes which propagate through the material. Last time we learned that there is energy associated with the vibrations of atoms. But they are tied together, so they dont vibrate independently. Sound velocity, thermal expansion, thermal conductivity, specific heat need lattice dynamics to get correct (X-1) (X) (X+1) Wave-Particle Duality
They dont like to be seen together Phonons are another quasi-particle. What did it mean for something to be a quasi-particle? It acted like a free particle, but could be due to collective interactions including collisions. We are much more familiar with treating light as a particle, what does a phonon particle look like? Unlike static lattice model , which deals with average positions of atoms in a crystal, lattice dynamics extends the concept of crystal lattice to an array of atoms with finite masses that are capable of motion. Just as light is a wave motion that is considered as composed of particles called photons, we can think of the normal modes of vibration in a solid as being particle-like. Quantum of lattice vibration is called the phonon. Phonon: A Lump of Vibrational Energy
Propagating lattice vibrations can be considered to be sound waves, and their propagation speed is the speed of sound in the material. Roughly how big is ? Phonon: Sound Wavepackets So how big is this?(animation and picture from Wikipedia) If you wanted to probe this oscillation, what wavelength would you want your probe to have? We will come back to this. Reminder to the physics of oscillations and waves
Harmonic oscillator in classical mechanics Equation of motion: Example: vertical springs or Hookes law where Solution with where X=A sin t X Kx x Traveling plane waves:
Displacement as a function of time and k Y or in particular X (Phonon wave vector also often given as q instead of k) Consider a particular state of oscillation Y=const traveling along You sometimes see q used instead of k for electrons too, but I have seen it more with phonons. In order for Y to be constant, the argument inside of the cosine must be constant. If its constant, its derivative is equal to zero; lets see what that implies. Also get the same solution if you solve for the wave equation. solves wave equation Transverse wave Longitudinal wave Standing wave
What if you subtracted the waves? Crystal can be viewed as a continuous medium: good for
Large wavelength 10-10m Crystal can be viewed as a continuous medium: good for Speed of longitudinal wave: where Bs: bulk modulus with (ignoring anisotropy of the crystal) Bs determines elastic deformation energy density compressibility (click for details in thermodynamic context) dilation E.g.: Steel Bs= N/m2 =7860kg/m3 > interatomic spacing continuum approach fails
One success of this approach is that we can finally get an accurate calculation for specific heat In addition: vibrational modes quantized phonons Vibrational Modes of a Monatomic Lattice
Linear chain: Remember: two coupled harmonic oscillators Symmetric mode Anti-symmetric mode Superposition of normal modes ? generalization Infinite linear chain
How to derive the equation of motion in the harmonic approximation n n-2 n-1 n+1 n+2 a K un-2 un-1 un+1 un+2 un un un+1 un+2 un-1 un-2 fixed ? Total force driving atom n back to equilibrium n n
equation of motion Old solution for continuous wave equation was.Use similar? ? approach for linear chain Let us try! , , Continuum limit of acoustic waves:
k Note: here pictures of transversal waves although calculation for the longitudinal case Continuum limit of acoustic waves: Technically, only have longitudinal modes in 1D
(but transverse easier to see whats happening) x = na un un Will need to trim out some of this but wait until see how much we get through in prior lecture (a) Chain of atoms in the absence of vibrations. (b) Coupled atomic vibrations generate a traveling longitudinal (L) wave along x. Atomic displacements (un) are parallel to x. (c) A transverse (T) wave traveling along x. Atomic displacements (un) are perpendicular to the x axis. (b) and (c) are snapshots at one instant. k Region is called first Brillouin zone , here h=1 1-dim. reciprocal
lattice vector Gh Region is called first Brillouin zone Vibrational Spectrum for structures with 2 or more atoms/primitive basis
Linear diatomic chain: 2n 2n-2 2n-1 2n+1 2n+2 K a 2a u2n u2n+1 u2n+2 u2n-1 u2n-2 Equation of motion for atoms on even positions: You will see shortly why I choose the axis this way. By doing this, Im able to directly compare my result to the monatomic lattice as I make the masses more similar Equation of motion for atoms on odd positions: Solution with: and , Click on the picture to start the animation M->m
note wrong axis in the movie Only got here in 75 minute class (slide 17) 2 2 , Transverse optical mode for diatomic chain
Amplitudes of different atoms A/B=-m/M Transverse acoustic mode fordiatomic chain A/B=1 Amplitude of vibration is strongly exaggerated!
Analogy with classical mechanical pendulums attached by spring Amplitude of vibration is strongly exaggerated! Longitudinal Eigenmodes in 1D
What if the atoms were opposite charged? Optical Mode: These atoms, if oppositely charged, would form an oscillating dipole which would couple to optical fields with ~a For example, an ionic crystal, where we know an electron transfers from one atom to the other. What would be different between these two? Summary: What is phonon?
Consider the regular lattice of atoms in a uniform solid material. There should be energy associated with the vibrations of these atoms. But they are tied together with bonds, so they can't vibrate independently. The vibrations take the form of collective modes which propagate through the material. Such propagating lattice vibrations can be considered to be sound waves. And their propagation speed is the speed of sound in the material.