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Physics of solids 1

A.Y. 2019/2020

Learning objectives

This course targets a general understanding of a broad range of fundamental phenomena and properties of solid-state matter. The course covers electronic,

vibrational, and spectroscopic properties of crystals, plus heat and electricity transport

vibrational, and spectroscopic properties of crystals, plus heat and electricity transport

Expected learning outcomes

The student is expected to learn in detail:

1. Periodic crystals and crystal lattice. Direct and reciprocal lattice.

X-ray diffraction. Miller indices. Form factor and structure factor.

Packing fraction. Quasicrystals (brief).

2. Total cohesive energy of the crystals. Simple examples of crystals of

noble gases and ionic solids. The density functional theory (DFT) for

calculating total adiabatic energy of a solid. Similarities and differences

with Hartree-Fock. Approximate density functionals: the Thomas-Fermi and

Kohn-Sham methods; the local density approximation (LDA).

3. Linear elastic response and the elastic constants of a solid.

4. Atomic motions in a crystalline solid: the harmonic approximation,

lattice vibrations and phonons. Phonon dispersions in special directions

of cubic crystals: longitudinal modes and transverse modes. Phonons in

3-dimensional solids in arbitrary crystal directions. The LO-TO splitting

in cubic crystals.

5. Methods for measuring the dispersion curves of phononic frequencies.

Brief review of the thermal properties of phonons, the Debye model.

6. Anharmonic effects in crystals: the Gruneisen theory of the thermal

expansion of solids, collisions between phonons, thermal conductivity.

7. A simplified model for the motion of electrons in metallic solids: the

jellium model, and its DFT solutions. Charge and heat conduction in the

relaxation-time approximation. Hall coefficient.

8. Band theory of solids: electrons in a periodic potential, models and

methods for the calculation of the electronic bands (brief). General

features of the structural and electronic band calculations of crystals

within the DFT approach. Semiclassical motion of the electrons in

crystals, effective mass. Holes and their motion.

9. Semiconductors: the valence and conduction bands, intrinsic and doped

semiconductors, carriers, mobility, conductivity, Hall effect, cyclotron

resonance. A few transport and out-of-equilibrium effects. The p-n

junction. A panorama of inhomogeneous semiconductor applications.

10. Metals: AC response and conductivity. The Boltzmann equation. Optical

response of the electrons and spectroscopies. Out-of-equilibrium and

thermoelectric effects.

11. Elements of electron-electron correlation effects and screening effects. Excitonic

effects: optical gap and quasiparticle gap. Optical excitations: plasmons,

polarons.

1. Periodic crystals and crystal lattice. Direct and reciprocal lattice.

X-ray diffraction. Miller indices. Form factor and structure factor.

Packing fraction. Quasicrystals (brief).

2. Total cohesive energy of the crystals. Simple examples of crystals of

noble gases and ionic solids. The density functional theory (DFT) for

calculating total adiabatic energy of a solid. Similarities and differences

with Hartree-Fock. Approximate density functionals: the Thomas-Fermi and

Kohn-Sham methods; the local density approximation (LDA).

3. Linear elastic response and the elastic constants of a solid.

4. Atomic motions in a crystalline solid: the harmonic approximation,

lattice vibrations and phonons. Phonon dispersions in special directions

of cubic crystals: longitudinal modes and transverse modes. Phonons in

3-dimensional solids in arbitrary crystal directions. The LO-TO splitting

in cubic crystals.

5. Methods for measuring the dispersion curves of phononic frequencies.

Brief review of the thermal properties of phonons, the Debye model.

6. Anharmonic effects in crystals: the Gruneisen theory of the thermal

expansion of solids, collisions between phonons, thermal conductivity.

7. A simplified model for the motion of electrons in metallic solids: the

jellium model, and its DFT solutions. Charge and heat conduction in the

relaxation-time approximation. Hall coefficient.

8. Band theory of solids: electrons in a periodic potential, models and

methods for the calculation of the electronic bands (brief). General

features of the structural and electronic band calculations of crystals

within the DFT approach. Semiclassical motion of the electrons in

crystals, effective mass. Holes and their motion.

9. Semiconductors: the valence and conduction bands, intrinsic and doped

semiconductors, carriers, mobility, conductivity, Hall effect, cyclotron

resonance. A few transport and out-of-equilibrium effects. The p-n

junction. A panorama of inhomogeneous semiconductor applications.

10. Metals: AC response and conductivity. The Boltzmann equation. Optical

response of the electrons and spectroscopies. Out-of-equilibrium and

thermoelectric effects.

11. Elements of electron-electron correlation effects and screening effects. Excitonic

effects: optical gap and quasiparticle gap. Optical excitations: plasmons,

polarons.

**Lesson period:** Second semester
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Unique edition

Responsible

Lesson period

Second semester

**Course syllabus**

1. Periodic crystals and crystal lattice. Direct and reciprocal lattice.

X-ray diffraction. Miller indices. Form factor and structure factor.

Packing fraction. Quasicrystals (brief).

2. Total cohesive energy of the crystals. Simple examples of crystals of noble gases and ionic solids. The density functional theory (DFT) for calculating total adiabatic energy of a solid. Similarities and differences with Hartree-Fock. Approximate density functionals: the Thomas-Fermi and Kohn-Sham methods; the local density approximation (LDA).

3. Linear elastic response and the elastic constants of a solid.

4. Atomic motions in a crystalline solid: the harmonic approximation, lattice vibrations and phonons. Phonon dispersions in special directions of cubic crystals: longitudinal modes and transverse modes. Phonons in

3-dimensional solids in arbitrary crystal directions. The LO-TO splitting in cubic crystals.

5. Methods for measuring the dispersion curves of phononic frequencies.

Brief review of the thermal properties of phonons, the Debye model.

6. Anharmonic effects in crystals: the Gruneisen theory of the thermal expansion of solids, collisions between phonons, thermal conductivity.

7. A simplified model for the motion of electrons in metallic solids: the jellium model, and its DFT solutions. Charge and heat conduction in the relaxation-time approximation. Hall coefficient.

8. Band theory of solids: electrons in a periodic potential, models and methods for the calculation of the electronic bands (brief). General features of the structural and electronic band calculations of crystals within the DFT approach. Semiclassical motion of the electrons in crystals, effective mass. Holes and their motion.

9. Semiconductors: the valence and conduction bands, intrinsic and doped semiconductors, carriers, mobility, conductivity, Hall effect, cyclotron resonance. A few transport and out-of-equilibrium effects. The p-n junction.

A panorama of inhomogeneous semiconductor applications.

10. Metals: AC response and conductivity. The Boltzmann equation. Optical response of the electrons and spectroscopies. Out-of-equilibrium and thermoelectric effects.

11. Electron-electron correlation effects and electron-phonon effects. Excitons, plasmons, polaritons, polarons, superconductivity (brief).

X-ray diffraction. Miller indices. Form factor and structure factor.

Packing fraction. Quasicrystals (brief).

2. Total cohesive energy of the crystals. Simple examples of crystals of noble gases and ionic solids. The density functional theory (DFT) for calculating total adiabatic energy of a solid. Similarities and differences with Hartree-Fock. Approximate density functionals: the Thomas-Fermi and Kohn-Sham methods; the local density approximation (LDA).

3. Linear elastic response and the elastic constants of a solid.

4. Atomic motions in a crystalline solid: the harmonic approximation, lattice vibrations and phonons. Phonon dispersions in special directions of cubic crystals: longitudinal modes and transverse modes. Phonons in

3-dimensional solids in arbitrary crystal directions. The LO-TO splitting in cubic crystals.

5. Methods for measuring the dispersion curves of phononic frequencies.

Brief review of the thermal properties of phonons, the Debye model.

6. Anharmonic effects in crystals: the Gruneisen theory of the thermal expansion of solids, collisions between phonons, thermal conductivity.

7. A simplified model for the motion of electrons in metallic solids: the jellium model, and its DFT solutions. Charge and heat conduction in the relaxation-time approximation. Hall coefficient.

8. Band theory of solids: electrons in a periodic potential, models and methods for the calculation of the electronic bands (brief). General features of the structural and electronic band calculations of crystals within the DFT approach. Semiclassical motion of the electrons in crystals, effective mass. Holes and their motion.

9. Semiconductors: the valence and conduction bands, intrinsic and doped semiconductors, carriers, mobility, conductivity, Hall effect, cyclotron resonance. A few transport and out-of-equilibrium effects. The p-n junction.

A panorama of inhomogeneous semiconductor applications.

10. Metals: AC response and conductivity. The Boltzmann equation. Optical response of the electrons and spectroscopies. Out-of-equilibrium and thermoelectric effects.

11. Electron-electron correlation effects and electron-phonon effects. Excitons, plasmons, polaritons, polarons, superconductivity (brief).

**Prerequisites for admission**

Basic mechanics, thermoyinamics, statistics, electromagnetism, quantum mechanics, and structure of matter.

**Teaching methods**

Lectures. The topics are discussed verbally and through illustrations and equations at the blackboard.

Specific contents such as tables and illustrations are provided in paper and electronic format.

For selected topics, flipped classes are implemented: students learn the subjects on textbooks, and then the lecture consists of a discussion guided by the students themselves.

Specific contents such as tables and illustrations are provided in paper and electronic format.

For selected topics, flipped classes are implemented: students learn the subjects on textbooks, and then the lecture consists of a discussion guided by the students themselves.

**Teaching Resources**

- C.Kittel, Introduction to Solid State Physics (Wiley New York 1996)

- J.R. Hook and H.E. Hall, Solid State Physics (Wiley Chichester 1991)

- G. Grosso and G. Pastori Parravicini, Solid State Physics (Academic, San Diego, 2000)

- N.W. Ashcroft and N.D. Mermin, Solid State Physics (Holt Sanders NY 1976)

- J.M. Ziman, Principes of the theory of solids (Cambridge University Press 1972)

- J.Callaway, Quantum Theory of the Solid State (Academic 1991)

- J.R. Hook and H.E. Hall, Solid State Physics (Wiley Chichester 1991)

- G. Grosso and G. Pastori Parravicini, Solid State Physics (Academic, San Diego, 2000)

- N.W. Ashcroft and N.D. Mermin, Solid State Physics (Holt Sanders NY 1976)

- J.M. Ziman, Principes of the theory of solids (Cambridge University Press 1972)

- J.Callaway, Quantum Theory of the Solid State (Academic 1991)

**Assessment methods and Criteria**

Ora exam: a few course topics are discussed for 45-60 minutes.

The exam will gauge the student's acquired competences and critical skills, through the discussion of problems in solid-state physics.

The exam will gauge the student's acquired competences and critical skills, through the discussion of problems in solid-state physics.

Educational website(s)

Professor(s)