Thermal Properties of Materials Thermal conductivity (electrons, phonons) Heat capacity (specific heat) at constant volume Heat capacity (specific heat) at constant pressure Thermal expansion

Thermal Conductivity (Survey) Contribution to thermal conductivity: Phonons (lattice vibrations)  low contribution to thermal conductivity Electrons (connected to electrical conductivity)  high contribution to thermal conductivity

Specific Heat (Heat Capacity) Einstein and Debye models – quantum mechanical description of transport phenomena

Definition of the Heat Assumption, that: 𝑊=0 Δ𝐸 = 𝑄 … change in energy of a thermodynamic system (𝑊 is the work performed by the system, 𝑄 is the heat) Assumption, that: 𝑊=0 Δ𝐸 = 𝑄

Heat Capacity The amount of heat (energy) required to raise the systems temperature by one degree (usually expressed in Kelvin) … heat capacity at a constant volume … heat capacity at a constant pressure (𝐻 is the enthalpy) 𝛼 … coefficient of thermal expansion 𝑇 … (absolute) temperature 𝑉 … volume of material 𝜅 … compressibility

Specific Heat … per unit of mass: … per mole: Temperature dependency

Temperature Dependency of Specific Heat CV = 25 J mol-1 K-1 = 5.98 cal mol-1 K-1 Experimental results: Specific heat of materials with one atom per unit cell is about 25 J mol-1 K-1 at room temperature. At low temperatures the specific heat decreases. Metals: 𝐶 v ≈𝑇 Insulators: 𝐶 v ≈ 𝑇 3 In magnetic materials specific heat increases if ordering of magnetic moments increases Fig. 5.26 a Specific heat capacity 𝐶 v 𝑇 of different materials

Specific Heat at Phase Transition Specific heat capacity of KH2PO4, which has a first-order phase transition at 120 K The material needs additional energy (heat) for the phase transition

Structure Transition in KH2PO4: paraelectric  ferroelectric RG: Fdd2 (orthorhombic) a = 10.467Å, b = 10.467Å, c = 6.967Å … K … P … O … H Paraelectric RG: I -42d (tetragonal) a = 7.444Å, c = 6.967Å

Magnetic Phase Transition of CePtSn Antiferromagnetic with 𝑇N = 7.5 K Change in ordering of magnetic moments

Ideal Gas Na = 6.022 x 1023 mol-1 R = kB Na = 8.314 J mol-1 K-1 = 1.986 cal mol-1 K-1 Kinetic energy of ideal gas 𝑝 … pressure 𝑝 ∗ … impulse 𝐴 … area 𝑁 … number of . atoms 𝑇 … temperature

Classical Theory of Heat Capacity (Ideal Gas) CV = 25 J mol-1 K-1 = 5.98 cal mol-1 K-1 𝐸mol … energy per mole Good compliance with experiments at higher temperatures

Quantum Theory of Heat Capacity 1903: Einstein postulated the quantum behavior of lattice vibrations (lattice oscillations) analogous to the quantum behavior of electrons. The quanta of lattice vibrations are called phonons. … impulse (de Broglie) … energy Longitudinal oscillations Transversal oscillations

Dispersion Relation (Phonon Dispersion) Analogy to the band structure Optical phonons (optical branch) Frequency Acoustic phonons (acoustic branch) Frequency of phonons (THz) Wave vector Optical phonons… higher energy (frequency) Acoustic phonons… lower energy (frequency) 𝐾/𝐾max in [111]-direction

Phonon dispersion as obtained from the neutron diffraction experiments 𝐸′=ℏ𝜔′ 𝑝′ =ℏ 𝑘′ 𝐸=ℏ𝜔 𝑝 =ℏ 𝑘 𝐸 𝑃ℎ =ℏΩ 𝑝 𝑃ℎ =ℏ 𝐾 Fig. 5.17 Acoustic and optical branches determined by neutron diffraction a) for aluminum b) for potassium bromide A – acoustic; O – optical; T – transversal; L – longitudinal

Acoustic and Optical Branches of a Linear Atomic Chain Acoustic branch Optical branch Fig. 5.23 Typical movement pattern of atoms in a linear chain

Energy of a Quantum Mechanical Oscillator … quantum energies … Bose-Einstein distribution … Fermi function (distribution) of electrons

Heat Capacity – The Einstein Model E = 0.01 eV KP QM Number of phonons Temperature [K]

Heat Capacity – The Einstein Model Classical approximation CV = 3R CV  exp(-/kBT) Temperature [K] Extreme case:

Comparison of Theoretical and Experimental Results Specific heat of materials with one atom per unit cell is about 25 J mol-1 K-1 at room temperature. At low temperatures the specific heat decreases. Metals: 𝐶 v ≈𝑇 Insulators: 𝐶 v ≈ 𝑇 3 Theory (Einstein model): Specific heat is about 25 J mol-1 K-1 at high temperatures. At low temperatures the specific heat decreases exponentially. 𝐶 v ≈exp⁡(−𝜔/ 𝑘 B 𝑇) The Einstein model considers only phonons with particular (discrete) frequencies.

Heat Capacity – The Debye Model Phonons with different energies … number of (acoustic) phonons … distribution (density) of oscillation frequency [DOS* of electrons] vs … speed of sound * Density of states

Heat Capacity – The Debye Model

Debye Temperatures Fig. 5.26 a Specific heat capacity 𝐶 v 𝑇 of different materials

Heat Capacity at High and Low Temperatures (Debye Model) 𝐶v  𝑇3: Better compliance with experimental values at low temperatures !!! For insulators !!!

Total Heat Capacity Phonons (Debye model) T < QD Electrons 𝐶V/𝑇 𝛽 … contribution of phonons 𝛾 … contribution of electrons 𝑇2

for the investigation of lattice vibrations Experimental Methods for the investigation of lattice vibrations X-ray diffraction Profile change of electron density (thermal vibrations of electrons) Influence on the intensities of diffraction lines Neutron diffraction Interaction between low-energy (slow) neutrons and the Phonons

Heat Conductivity Thermal conductivity: K Partial differential equation: Solution for definite initial and boundary conditions Temperature dependency – similar to the change in concentration at diffusion processes J = 0 J = 0 𝑇 = const. 𝑇 = const.

Thermal Conductivity 𝑛 … number of electrons 𝑙 … free path between two collisions (electron-phonon) 𝑣 … velocity of electrons

Thermal Conductivity Metals Dielectrics Wiedemann-Franz law: Thermal conductivity, W/cm/K Temperature, K Wiedemann-Franz law: Materials with high electrical conductivity exhibits a high thermal conductivity Material K [W/cm/K] SiO2 0,13 – 0,50 (at 273K or 80K) NaCl 0,07 – 0,27 (at 273K or 80K) Al2O3 200 at 30K Cu 50 at 20K Ga 845 at 1.8 K

Inharmonic vibrations: Thermal Expansion Intramolecular force Harmonic vibrations: Inharmonic vibrations: Thermal expansion

Thermal Expansion Argon (kfz) Change of mean interatomic distance with temperature: Temperature dependency of lattice parameters: Argon (kfz) Lattice parameter increases quadratically with the temperature The thermal expansion at 𝑇 = 0K is zero Lattice parameter [Å] Density [g/cm³] Temperature [K]

Thermal Expansion in GdNiAl