nanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 1Lattice DynamicsTimothy S. FisherPurdue UniversitySchool of Mechanical Engineering, andBirck Nanotechnology [email protected] 595MMay 2007nanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 2Phonon Heat Conduction• Phonons are quantized lattice vibrations• Govern thermal properties in electrical insulators and semiconductors• Can be modeled to first order with spring-mass dynamics• Wave solutions♦ wave vector K=2π/λ♦ phonon energy=ħω♦ dispersion relations gives ω = fn(K)ωKoptical branchacoustic branchsound speed(group velocity)springconstantgatom,massmananoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 3Heat Conduction Through Thin Films• Experimental results for 3-micron silicon films• Non-equilibrium scattering models work fairly well• Crystalline structure often has largerimpact than film thickness3 micronAsheghi et al., 1999nanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 4Heat Conduction Through Multiple Thin Films• Fine-pitch 5 nm superlattices• Cross-thickness conductivity measurement• Measured values are remarkably close to bulk alloy values (nearly within measurement error)• Expected large reduction in conductivity not observed5nmCahill et al., 2003nanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 5Lattice Vibrations• Consider two neighboring atoms that share a chemical bond• The bond is not rigid, but rather like a spring with an energy relationship such as…r0Urr0nanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 6Lattice Vibrations, cont’d• Near the minimum, the energy is well approximated by a parabola♦ u= r –r0and g = spring constant• Now consider a one-dimensional chain of molecules212Ugu=springconstantgatom,massmananoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 7Lattice Energy and Motion• Harmonic potential energy is the sum of potential energies over the lattice• Equation of motion of atom at location u(na)• Simplified notation{}21[] [( 1)]2harmnUguna u n a=−+∑[][]{}22()2( ) ( 1) ( 1)()harmduna UFmgunaunaunadt u na∂==−=−−−−+∂{}21122nnn ndumguuudt−+=− − −nanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 8Lattice Motion, cont’d• Seek solutions of the form• Boundary conditions♦ Born-von Karman: assume that the ends of the chain are connected•uN+1= u1•u0= uN(){}()~expnut iKna t−ω12NnanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 9Lattice Motion, cont’d• Then the boundary conditions become• Let λ be the vibration wavelength, λ = aN/n• Minimum wavelength, λmin= 2a = 2(lattice spacing)(){}[]{}[]11~exp 1~exp1exp 2, where is an integerNuiKNatuiKatiKNa KNa nn++−ω⎡⎤⎣⎦−ω→= → =π22nKaNππ==λK = wave vectornanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 10Solution to the Equations of Motion• Substitute exponential solution into equation of motion• Solve for ω• This is the dispersion relation for acoustic phonons♦ relates phonon frequency (energy) to wave vector (wavelength)()()()()2221cosiKna t iKna tiKa iKaiKna tme g e e egKae−ω−ω−−ω⎡⎤−ω =− − −⎣⎦=− −122(1 cos )() 2 sin( )gKagKKamm−ω= =nanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 11Dispersion Curve• Changing K by 2π/a leaves u unaffected♦ Only N values of K are unique♦ We take them to lie in -π/a < K < π/aω(K)Kπ/a- π/a 2(g/m)1/2nanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 12Wave Velocities• Phase velocity: c = ω/K• Group velocity: vg= ∂ω/∂K = a(g/m)1/2cos(Ka/2)• For small K:• Thus, for small K (large λ), group velocity equals phase velocity (and speed of sound)• We call these acoustic vibration modes00limlimKgKgaKmgva cmK→→ω=ω→===nanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 13Notes on Lattice Vibrations•For K = ±π/a, the group velocity is zero♦ why?♦ neighbors are 180 deg out of phase• The region -π/a < K < π/a is the first Brillouin zone of the 1D lattice• We must extrapolate these results to three dimensions for bulk crystals{} {}1exp exp cos sin 1nnuiKa i iu+==π=π+π=−nanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 14Density of Phonon States• Consider a 1D chain of total length L carrying M+1 particles (atoms) at a separation a♦ Fix the position of atoms 0 and M♦ Each normal vibrational mode of polarization p takes the form of a standing wave♦ Only certain wavelengths (wavevectors) are allowedλmax=2L (Kmin=π/L), λmin=2a (Kmax=π/a=Mπ/L)♦ In general, the allowed values of K are01 MM-1La~sin( )exp( )nKpunKait−ω23 ( 1), , ,...,MKLL L Lπππ −π=Note: K=Mπ/L is not included because it implies no atomic motion, i.e., sin(nMπa/L)=sin(nπ)=0.See Kittel, Ch5, Intro to Solid-StatePhysics, Wiley 1996nanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 15Density of States, cont’d• Thus, we have M-1 allowed, independent values of K♦ This is the same number of particles allowed to move♦ In K-space, we thus have M-1 allowable wavevectors♦ Each wavevector describes a single mode, and one mode exists in each distance π/L of K-space♦ Thus, dK/dN = π/L, where N is the number of modesπ/(M-1)a π/aπ/Lπ/aDiscrete K-space representationnanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 16Density of States, cont’d• The phonon density of states gives the number of modes per unit frequency per unit volume of real space♦ The last denominator is simply the group velocity, derived from the dispersion relation111 11()/dN dN dKDdLdKd ddKLα=ω= = =ωωπω111() cos ()() 2Note singularity for /ggDaKavmKa−⎡⎤⎛⎞ω= =π ω⎢⎥⎜⎟πω⎝⎠⎣⎦=πnanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 17Periodic Boundary Conditions• For more generality, apply periodic boundary conditions to the chain and find♦ Still gives same number of modes (one per particle that is allowed to move) as previous case, but now the allowed wavevectors are separated by ΔK = 2π/L♦ Useful in the study of higher-dimension systems (2D and 3D)240,,,...,MKLL Lπππ=± ±nanoHUB.orgonline simulations and moreT.S. Fisher, Purdue University 182D Density of States• Each allowable wavevector(mode)