Computational Solid State Physics 計算物性学特論　第２回

Computational Solid State Physics 計算物性学特論　第２回 2.Interaction between atoms and the lattice properties of crystals

Atomic interaction • Lennard-Jones potential: for inert gas atoms: He, Ne, Ar, Kr, Xe • Stillinger- Weber potential: for covalent bondingatoms: C, Si, Ge

repulsive force attractive force Lennard-Jones potential (1) r: inter-atomic distance VLJ/ε VLJ(r) minimum at r / σ

r p1 p2 (r) temporal dipole moment induced dipole moment Lennard-Jones potential (2) 1st term: repulsive interaction caused by Pauli’s principle 2nd term:Van der Waals interaction (attractive) E1: electric field generated by a temporal dipole moment p1

x a 1-dimensional crystal a: lattice constant Energy per atom: E minimum at a=1.12σ cohesive energyεc=1.04ε

Bulk modulus B : Bulk modulus N : the number of atoms in a crystal a : lattice constant

a Lattice vibration xn displacement Na: length of a crystal assume: neglect the 2nd neighbor interaction x • The first derivative of the inter-atomic potential vanishes because atoms are located at the equilibrium positions. • The second derivative of the inter-atomic potential gives thespring constant κ between atoms.

xn-1 xn xn+1 a Equation of motion for atoms m: mass of an atom Force on the n-th atom: Equation of motion for atoms:

Assume: Solution for equation of motion k: wave vector Periodic boundary condition: 1st Brillouin zone N modes

Dispersion relation of lattice vibration acoustic mode sound velocity: phase velocity at k=0 ω(k)/ω0 v becomes larger for largerκ and smaller m. ka

for Phonon Energy quantization of lattice vibration l=0,1,2,3 :zero point oscillation Bose distribution function for phonon number:

Role of the acoustic phonon in semiconductors at a room temperature • Main electron scattering mechanism in crystals • Determine the lattice heat capacity • Determine the thermal conductivity

Lattice heat capacity: Debye model (1) Density of states of acoustic phonos for 1 polarization phonon dispersion relation Debye temperatureθ Nk: Allowed number of k points in a sphere with a radius k N: number of unit cell

Thermal energy U and lattice heat capacity CV : Debye model (2) 3 polarizations for acoustic modes

Debye model (3) ・Low temperature T<<θ ・High temperature T>>θ Equipartition law: energy per 1 freedom is kBT/2

Heat capacity CV of the Debye approximation: Debye model (4) kB=1.38x10-23JK-1 kBmol=7.70JK-1 3kBmol=23.1JK-1

Heat capacity of Si, Ge and solid Ar: Debye model (5) Solid Ar Si and Ge cal/mol K=4.185J/mol K 3kB mol=5.52cal K-1

3kBT(x) Energy vxτ Energy emission c vxτdT/dx x Thermal conductivity (1) Diffusive energy flux T: temperature c: heat capacity per particle n: average number of phonons v: group velocity of phonon τ: scattering time

C: heat capacity per unit volume, l=vτ: phonon mean free path v: sound velocity of acoustic phonon Thermal conductivity (2) Thermal conductivity coefficient K is largest for diamond because of the high sound velocity!

r: position of an atom v: velocity a: acceleration F: force t: time m: mass of an atom Molecular dynamics simulation for atoms Equation of motion for atoms:

(1) velocity Verlet’s method Time evolution for small time interval :

Proof of (1)

(2) Verlet method Time evolution for small time interval

Temperature Equipartition theorem Temperature is determined from the average kinetic energy.

Periodic boundary condition 2-dimensional system

Trajectories of 20 atoms interacting via Lennard-Jones potential

Setting of energy and temperature melting triangular crystal

Time-lapse snapshots of interacting particles (1) formation of triangular crystal

Time-lapse snapshotswith increasing Temperatures (2) melting

Problems 2-1 • Calculate two branches of the dispersion relation of the lattice vibration for a diatomic linear lattice using a simple spring model, and describe the characteristics of each branch. • Calculate the dispersion relation for a graphen sheet using a simple spring model between nearest neighbor atoms. • Study the role of the optical phonon in semiconductor physics.

Problems 2-2 • Find the most stable 2-dimensional crystal structure, using the Lennard Jones potential. • Find the most stable 3-dimensional crystal structure, using the Lennard Jones potential. • Write a computer simulation program to study the motion of 3 atoms interacting with Lennard-Jones potential. Assume the space of motion to be within a 2-dimensional square region.

Problems 2-3 • Study experimental methods to observe the dispersion relation of phonons. • Study the phonon dispersion relations for Si and Ge crystals and discuss about the similarity and the difference between them. • Study the phonon dispersion relations for Ge and GaAs crystals and discuss about the similarity and the difference between them.

Computational Solid State Physics 計算物性学特論　第２回