Lattice vibrations and using diffraction
- 1. Lattice vibrations Elastic vibrations of a 1-D homogeneous line Extends from x to x+Δx, length Δx Displacement u(x) depends on position Strain Hooke’s law limit : Tension at x : F(x) = cs(x) Tension at x+Δx : F(x+Δx) = cs(x+Δx) Mass of the string element m = ρΔx
- 2. Lattice vibrations where Elastic vibrations of a 1-D homogeneous line Wave equation with phase velocity Solution is a wave represented as Dispersion relation is thus or
- 3. Lattice vibrations Dispersion relation for a 1D continuous line obeying Hooke’s law is linear Elastic vibrations of a 1-D homogeneous line Phase velocity : Group velocity : Both phase velocity and group velocity (speed at which information travels) are independent of ω.
- 4. Lattice vibrations The frequency spectrum is now discrete, each value corresponding to a normal mode. Elastic vibrations of a 1-D homogeneous line fixed at both ends u(x = 0) = u(x = L) = 0 ⇒ ⇒ Solutions are now standing waves with
- 5. Lattice vibrations Thus positions a, 2a, 3a…..(n-1)a, na, (n+1)a Elastic vibrations of a 1-D chain of atoms Force on the nth atom Fn = β(un+1 –un) + β(un-1 –un)
- 6. Lattice vibrations Elastic vibrations of a 1-D chain of atoms trial solution of the form which automatically implies
- 7. Lattice vibrations Elastic vibrations of a 1-D chain of atoms trial solution of the form which automatically implies
- 8. Lattice vibrations Elastic vibrations of a 1-D chain of atoms
- 10. Lattice vibrations Phase velocity : Group velocity : At the first Brillouin zone boundary ( ), group velocity = 0. It changes from +ve to –ve across the zone boundary. A wave packet will travel backwards (opposite to the phase propagation direction) if k crosses the B.Z. boundary (Bragg reflection). This treatment is general and is applicable to any wave, elastic or electromagnetic
- 11. Lattice vibrations The dotted line represents a wave with λ < 2a (< a actually). Its equivalent has λ ≈ 12a (k ≈ π 6𝑎 which lies within the 1st B.Z.) A wave of wavelength < 2a (ie. k > π 𝑎 ) is physically identical to another wave in with wavelength > 2a (ie. k < π 𝑎 ). Thus all allowed waves inside the lattice can be mapped onto some lattice point (k) inside the first Brillouin zone. Small k (large λ) near zone centre Large k (small λ) near zone boundary
- 12. Lattice vibrations In 2 or 3 dimensions, the first Brillouin zone is the Wigner-Seitz primitive unit cell of the reciprocal lattice. First and other Brillouin zones of a 2D square reciprocal lattice (corresponding to a square real-space lattice). If ‘a’ is the lattice constant of the real-space lattice, the lattice constant of the reciprocal lattice will be 2π 𝑎 . The four corners of the first Brillouin zone are thus ( 2π 𝑎 , 2π 𝑎 ), ( 2π 𝑎 , - 2π 𝑎 ), (- 2π 𝑎 , - 2π 𝑎 ) and (- 2π 𝑎 , 2π 𝑎 )
- 13. Lattice vibrations The first Brillouin zone of (a) simple cubic (SC) lattice (W-S cell of its SC reciprocal lattice) (b) body-centered cubic (BCC) lattice (W-S cell of its FCC reciprocal lattice) (c) face-centered cubic (FCC) lattice (W-S cell of its FCC reciprocal lattice) (d) hexagonal close packed (HCP) lattice equivalent to a simple hexagonal lattice with a 2-atom basis. Its first B.Z. is another simple hexagonal lattice roated 30O w.r.t. the real-space lattice. (e) rhombohedral lattice and (f) base-centered orthorhombic lattice.
- 14. Lattice vibrations Finite lattice : Density of states If there are N+1 atoms in the 1D lattice with lattice constant a, L=Na. The first and last ones are fixed, all others are free to move (fixed boundary conditions). The solutions become standing waves, which are a superposition of propagating waves with wave vectors +k and -k k values greater than k = π 𝑎 may be excluded. becomes Thus the minimum interval between any two allowed k-values is π 𝑁𝑎 = π 𝐿 With fixed boundary conditions, the 1D ‘volume’ in k-space occupied by an allowed k- point is π 𝐿 . The 1D ‘density of states’ in k-space is 𝐿 π
- 15. Lattice vibrations The 1D ‘density of states’ in k-space is 𝐿 π In 2D ‘density of states’ in k-space is In 3D ‘density of states’ in k-space is Fixed boundary conditions
- 16. Lattice vibrations An equivalent way of describing a finite system is by applying periodic boundary conditions : The 1st atom is always in an identical state as the last atom. un+N =un. So the N+1 atoms may be thought of as being arranged in a ‘ring’ with N atoms, the 1st having merged with the Nth As opposed to the fixed boundary conditions where the solutions to the wave equation were standing waves, for periodic boundary conditions, we consider propagating waves with both +ve and –ve wavevectors. Using the above 2 conditions, the allowed values of k are given by With periodic boundary conditions, the 1D ‘volume’ in k-space occupied by an allowed k-point is 2π 𝐿 . The 1D density of states in k-space is 𝐿 2π
- 17. Lattice vibrations The 1D ‘density of states’ in k-space is 𝐿 2π In 2D ‘density of states’ in k-space is In 3D ‘density of states’ in k-space is Periodic boundary conditions (Born-von Karman boundary conditions)
- 18. Lattice vibrations Dispersion in a 1D periodic lattice with identical atoms The largest value of k is decided by the smallest length scale of the system, the lattice constant a. The smallest interval of k is decided by the largest length scale of the system, the sample size L.
- 19. Lattice vibrations Trial solutions are of the form 1D diatomic lattice
- 20. Lattice vibrations This has 2 solutions for ω2 Substituting in the equations of motion a pair of simultaneous homogenous linear equations in u0 and v0. To have a solution, determinant of coefficients must be zero. or
- 21. Lattice vibrations In the limit k -> 0 (near Brillouin zone centre) Optical branch of dispersion relation Acoustic branch of dispersion relation In the limit k -> π 𝑎 (near Brillouin zone edge) Optical branch Acoustic branch
- 22. Lattice vibrations ω->0 as k->0 for acoustic branch, for optical branch ω is a maximum at k=0. Transverse and longitudinal oscillations lead to different pairs of optical (TO and LO) and acoustic (TA and LA) branches of the dispersion curve. Further branches of the dispersion relation arise due to other vibrational modes such as torsional modes (ZO and ZA)
- 23. Lattice vibrations Γ Center of the Brillouin zone Simple cube M Center of an edge R Corner point X Center of a face Face-centered cubic K Middle of an edge joining two hexagonal faces L Center of a hexagonal face U Middle of an edge joining a hexagonal and a square face W Corner point X Center of a square face Body-centered cubic H Corner point joining four edges N Center of a face P Corner point joining three edges Hexagonal A Center of a hexagonal face H Corner point K Middle of an edge joining two rectangular faces L Middle of an edge joining a hexagonal and a rectangular face M Center of a rectangular face
- 24. Lattice vibrations Experimental data for Al : FCC Different theoretical models are compared with experiment Muraleedharan et al. AIP Advances 7, 125022 (2017)
- 25. Lattice vibrations Experimental data for diamond : FCC with 2-atom basis
- 26. Lattice vibrations Experimental data for graphite : simple hexagonal. Coloured dots are experimental data points, black lines are theoretical estimates. L. Wirtz, A. Rubio / Solid State Communications 131 (2004) 141–152
- 27. Lattice vibrations Derivation of the above : Thermal conductivity jU : flux of thermal energy, K : thermal conductivity From kinetic theory of an ideal gas, cv : volume heat capacity, v : average particle velocity, l : mean free path between collisions Temperature gradient in the z direction
- 28. Lattice vibrations Thermal conductivity Let each particle (atom/molecule) in the gas have a mean thermal energy ϵ. Thermal energy per unit area, per second, carried by particles whose speeds lie between v and v+dv, and whose directions of motion subtend an angle lying between θ and θ +dθ with the z-axis, that cross the x-y plane, is F(v) is the distribution of molecular speeds, ϵ’ = ϵ(T’) where T’ is the temperature where the molecules last made a collision On average, the molecules move a distance l (i.e., the mean free path) between collisions. Hence, dz =-l cos θ Therefore C : heat capacity per particle
- 29. Lattice vibrations Thermal conductivity with all these substitutions: Can be integrated to find the total heat flux in z-direction: or, where is the thermal conductivity
- 30. Lattice vibrations Mean free path of phonons : Thermal conductivity σ : cross sectional area of each ‘particle’ Since average energy per DOF is kBT, no. of phonons with frequency ω is
- 31. Show that the density of states D(ω) for phonons in 3D is given by Using Debye’s simplifying assumption
- 33. Show that the average energy of phonons of a normal mode of frequency ω0 is given by
- 35. Find the total energy contained in lattice vibrations at a temperature T