Temperature & Excitations Lecture 10 3/19/18
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 1 So far we worked on the ground state (T=0)
What happens at finite temperature? When are temperature effects are important for your problem? When are zero-temperature calculations relevant? How to use T=0 calculations to get T-dependent properties?
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 2 Above the ground state
Quantum Mechanics (DFT) gives you the ground state energy When temperature increases, energy increases
Additional energy is contained in excitations Which excitations are relevant for your property?
Average energy above ground state is a measure of temperature
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 3 Changes in Materials with Temperature
Crystal structure, phase changes Surface structure, concentration profile Chemical composition Electrical and thermal conductivity Bulk modulus Volume
Different approaches may be needed for each of these
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 4 Types of excitations
Electronic
Magnetic
Vibrational
Configurational
These can couple and take exotic shapes
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 5 What is temperature?
Temperature is a parameter that is equal for two different systems when they are in thermal equilibrium T is a measure of the mean of the energy of the translational, vibrational and rotational (classical) motions of atoms/molecules
from the CompassMR simulator from Wikipedia
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 6 Microcanonical ensemble
Suppose we have N identical systems with discrete energy levels
quantum system consisting of Microstate is the set of occupancies one particle in a potential well In how many ways can we obtain energy E ?
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 7 Maxwell-Boltzmann statistics
The most probable distribution giving total N and E is
Probability of filling a state decreases exponentially with energy
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 8 Entropy
Entropy is a measure of the number of microscopic configurations Ω that a thermodynamic system can have
S is maximized at equilibrium SAB =SA +SB
A and B are in equilibrium when they have the same parameter
which define temperature
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 9 Laws of thermodynamics
0. A equil. B and B equil. C A equil. C (definition of T)
1. Total energy (heat + work) is conserved
2. Entropy S is maximized at equilibrium, it is the most likely state
3. Only those degrees of freedom which are excited at a given T will contribute to thermodynamic properties. At low temperature, all discrete degrees of freedom “freeze out”. Heat capacity and entropy goes to zero.
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 10 Statistical ensembles
Microcanonical ensemble (NVE) represents an isolated system in which energy (E), volume (V) and the number of particles (N) are all constant. Canonical ensemble (NVT) represents a closed system which can exchange energy with a heat bath at temperature T, but the volume (V) and the number of particles (N) are all constant. Isothermal-isobaric ensemble (NPT) maintains a constant temperature T and pressure P (requires both a thermostat and barostat), corresponds closely to “laboratory” conditions. Grand canonical ensemble(mVT) represents an open system which can exchange energy (E) as well as particles with its surroundings but the volume (V) is kept constant.
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 11 Ensembles
For systems containing many particles (thermodynamic limit), all three ensembles typically give identical behavior NVE: S is maximum at equilibrium NVT: F = E-TS is minimum at equilibrium
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 12 Ensemble averages
For a quantum system in thermal equilibrium with its environment, the weighted average takes the form of a sum over quantum states e.g. weights for a canonical ensemble:
Under the Ergodic Hypothesis, we can assume that the temporal average along a long trajectory is equal to the ensemble-average over the phase space 1 T A A(t) dt T 0
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 13 Bulk modulus of Cu
Narashiman and de Gironcoli, Phys. Rev B, 65, 064302 (2002)
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 14 Why does bulk modulus change with T?
Entropy contribution is small, much more important to take the energy derivative at the correct T-dependent lattice parameter
at T=0
at finite T
Quong, A. and A. Liu, Physical Review B, 56, 7767-7770 (1997) Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 15 Quantum statistics
Bose-Einstein: how many configurations exist for distributing Ni indistinguishable particles to Ai energy levels?
Fermi-Dirac: Only 0 and 1 occupancies... number of possibilities to select Ni out of Ai levels that are populated
Maxwell-Boltzmann: Distinguishable particles, limit of BE and FD distributions at temperatures much higher than excitation energies
Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 16 Excitation entropies
Boson (e.g. phonon) entropy per mode
Fermion (e.g. electron) entropy (0 Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 17 Lattice vibrations Atoms in different lattice cells can vibrate with a certain phase shift with respect to each other: phonon dispersion Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 18 Computing phonon dispersions For a 3D crystal need to compute the harmonic matrix Second derivatives of energy with respect to displacements Equation of motion for atoms assuming ‘plane wave’ solutions Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 19 Phonon spectra Based on total energy differences, vibrational spectra are very accurate Occupancy and entropy per oscillator are Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 20 Quasiharmonic Free Energy F(V ,T) E(V ) EZP (V ) TS(V ,T) Free ground Volume-dependent Energy state phonon frequencies V q, FVTEV, kTB ln 1 exp q, V q,,2 q V(T) is determined as a function of temperature T by minimizing the free energy F(V,T) with respect to V Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 21 Thermal expansion F(V ,T ) E(V ) EZP (V ) TS(V ,T) Volume dependence of the entropy causes thermal expansion At higher volume, force constants become weaker, hence phonon frequencies lower, hence entropy higher Even at T=0 the lattice will expand a bit Li At finite T, the T-dependent term reduces the free energy as the frequencies decrease Quong, A. and A. Liu, Physical Review B, 56, 7767-7770 (1997) Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 22 Cu thermal expansion Only vibrational contribution Narashiman and de Gironcoli, Phys. Rev B, 65, 064302 (2002) Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 23 Heat capacity Recipe: compute phonons at several volumes Find minimum of F with respect to V at each T Half-Heusler alloy TiNiSn Cv — Ab-initio [1] . . . Expt [2] [1] Wee, Kozinsky, et al, JEM, 2012 [2] B. Zhong, M. thesis, Iowa State, 1997 Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 24 Vibration driven phase transition Displacive phase transitions in perovskites Free energies of BaTiO3 based on -point phonons Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 25 Computing lattice thermal conductivity Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 26 Thermal conductivity Temperature dependence from SiGe alloy J. Garg et al, Phys. Rev. Lett. 106 (4), 045901 (2011) Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 27 Thermal conductivity of UO2 What is missing? Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 28 Excitation entropies Boson (e.g. phonon) entropy per mode Fermion (e.g. electron) entropy (0 Spectrum of electrons can be approximated by Kohn-Sham eigenvalues (independent one-electron states) Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 29 Contribution of electronic entropy Enthalpy of Ni as a function of temperature without electronic contribution Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 30 Thermoelectric effect Seebeck coefficient is entropy per unit charge in a semiconductor Density of electronic states (DOS) Fermi-Dirac (FD) distribution Number of electrons = DOS FD Fermi level (n-type) V S Hot Side Cold Side T Hot Side Cold Side metal Energy Energy DOS DOS Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 31 Thermoelectric waste heat recovery Vehicle TE generator TE module Hot side TH Exhaust gas n p Cold side TC Cooling water Recover waste heat from vehicle exhaust Save 5-10% of fuel Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky Seebeck coefficient of CoSb3 Pizzi et al, 2013 Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 33 Electron-driven phase transitions Mott-Hubbard transition in V2O3 Electronic entropy is higher in metal Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 34 Magnetic phase transitions Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky 35