Course Unit Page


This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Quality education

Academic Year 2021/2022

Learning outcomes

Illustration of the functioning principles of the most important categories of solid-state sensors. Sensors realized with process techniques used in Microelectronics and Micro-Electro-Mechanics Systems (MEMS). Application of the quantum theory to few-particle systems. Qbit as the elementary unit of quantum information. Example of solid-state devices where quantum effects play a dominant role. Physical feasibility of a computing device architecture based upon quantum dots and/or quantum wires.

Course contents


Basic concepts of mathematics and physics acquired from earlier courses. Basic concepts about the electron devices. In this course, further mathematical and physical concepts, not necessarily elementary, will be used. They will be explained as necessary during the lessons. The use of such concepts can not be disposed of, they actually constitute the cultural basis of the course itself.

Learning outcomes

Illustrating the functioning principles of the most important categories of solid-state devices and sensors. The students will learn about devices and sensors realized with process techniques used in Microelectronics and Micro-Electro-Mechanical Systems (MEMS). For each device or sensor, the physical effect, the model used for its characterization and the technological aspects used for the overall realization will be discussed. The student will also gain knowledge about the fundamentals of quantum mechanics and band theory of solids, about the physical phenomena underlying the transport of charged carriers in solids. The fundamentals of quantum mechanics will also be used in the last part of the course to provide introductory elements of quantum computing and information.


General: To have critical understanding of technical and scientific tools; communication skills; to be able to work in an international context.

Specific: to understand the methods for investigating advanced solid-state devices, memories and sensors; to determine the important microscopic and macroscopic parameters involved in the functioning of such devices.

Contents (M1)

Crystal lattice. Definition of characteristic vectors. Elementary cell. Direct and reciprocal lattices. First Brillouin zone. Face-centered cubic lattice. Miller indices. Examples: diamond and zinc-blende structures. Particle dynamics in a periodic potential. Form of the conduction and valence band of silicon, germanium, and gallium arsenide. Transitions: intra-valley, inter-valley, direct inter-band, trap-assisted inter-band transitions. Fermi statistics.

Density of states in the k space and in the phase space. Effective mass. Dispersion relation E(k) in a crystal. Density of states in energy. Bands in conductors, insulators, and semiconductors. Dependence of conductivity on temperature. Form of the constant-energy surfaces of Ge, Si, GaAs.

Distribution function. Example of calculation of the density of states. Concept of hole. Calculation of the intrinsic concentration at equilibrium. Estimate of the position of the Fermi level. Dependence of the intrinsic concentration on temperature. Relevant parameters for Ge, Si GaAs. Qualitative explanation of the contribution of electron and holes to the current density of a semiconductor. Calculation of the position of the Fermi level. Average velocity and current density of the electrons and of the holes.

Dynamics of electrons and holes in the parabolic-band approximation. Doped semiconductor of the donor or acceptor type. Compensation effect. Calculation of the equilibrium concentrations. Non-degeneracy and complete-ionization approximations. Effect of temperature and dopant concentration on the position of the Fermi level. Intrinsic, saturation and freeze-out conditions of the dopants. Non-uniform dopant distribution. Expressions of the equilibrium concentrations of electrons and holes. Poisson equation in the semiconductor in the equilibrium and non-equilibrium cases.

Continuity equations. Quasi-static approximation. Mathematical model of semiconductor devices. Definition of the quasi-Fermi potentials. Boundary conditions of the model Trap-assisted transitions. Deduction of the SRH net-recombination rate. Limiting cases of it: depletion and weak-injection conditions. Auger recombination and impact ionization. Direct generations of thermal and optical type. Limiting case of strong ionization. Form of the optical generation inside the material. Macroscopic mobility models: collisions with the lattice, the ionized impurities and the interfaces. Dependence of mobility on temperature, total impurity concentration and field component normal to a boundary. Matthiessen rule. Example of a macroscopic mobility model. Results of mobility calculations obtained from the solution of the Boltzmann transport equation.

Applications: p-n junction: structure and qualitative form of the electric potential. Solution of the Poisson equation at equilibrium: neutral regions and space-charge regions. Drift and diffusion component of the current density. Effect of the contacts. Qualitative functioning of the p-n junction. Full depletion of the space-charge region in the reverse-bias condition. Shockley theory. Weak-injection condition and decoupling of the neutral zones. Current-voltage relation of the p-n junction; dominance of the diffusive current density for the minority carriers in the neutral zones. Shockley boundary conditions. Reverse bias: form of the charge density, electric field and electric potential. Calculation of the width of the space-charge region. Barrier capacitance of the p-n junction. Chynowetz model for the impact-ionization coefficients. Avalanche condition.

Bipolar transistor. Qualitative description. Calculation of the currents. Ebers and Moll model. Equivalent circuit. Optimization of the current gain.

MOS capacitor. Neutral zone and space-charge zone. Relation between charge in the semiconductor and surface potential in the different functioning regimes. Solution of the Poisson equation in the MOS capacitor with a constant substrate doping. Flat-band, depletion and inversion conditions. Approximate relations between charge per unit area and surface potential based on the full-depletion and ACE approximations. Differential capacitance of the MOS capacitor.

MOS transistor. Calculation of the drain current. Gradual-channel approximation and calculation of the steady-state characteristics. Saturation voltage. The linear-parabolic model. Low-frequency differential parameters. Limits of the gradual-channel approximation. Solution of the Poisson equation in the channel in a non-equilibrium case. Surface potential at the source and drain ends of the channel.

Contents (M2)

Introductory part where the basic relations of quantum mechanics are shown; deduction of the energy bands in crystals. Transport theory: Boltzmann transport equation and deduction of the mathematical model of semiconductor devices.

Application of the basic devices as memories. Resistive memories. Phase-change memories.

Interaction of radiation with matter, absorption of light. Photodiode, continuous mode and storage mode. Calculation of the current in the continuous mode. Photocapacitor built in a uniform substrate. Integration interval and sampling interval. Photocapacitor built in an epitaxial layer. MOS phototransistor. Charge-transfer devices. Linear charge-coupled devices. Two-dimensional charge-coupled devices: frame-transfer and line-transfer architectures. Use of a charge-coupled device as a filter. CID sensors: qualitative functioning. Analysis of the floating gate.

Solar cell. Equivalent circuit, fill factor and efficiency. Calculation of the equivalent circuit of the solar cell. The solar constant. Form of the contacts of solar cells.

Life-time measurement using optical generation. Mobility measurement (Haynes and Shockley method). Hall effect. Use of the Hall effect to measure carrier concentration and mobility. Measurement of the dopant profile using the measurement of capacitance. Introduction to scaling rules. Evolution of the integration level of integrated circuits. Constant-field scaling rules. Deviations from the scaling rules. Physical and economic limits to scaling.

Scheme of the top-down and bottom-up design sequence of integrated circuits. Computer-aided design. Numerical methods for simulating solid-state devices: solution of the Poisson equation in the one-dimensional case. Solution of the continuity and transport equations. two- and three-dimensional examples.

Performance metric of devices. Performance analysis of solid-state video cameras.

Introduction to the elasticity theory; determination of the mechanical deformation of cantilever beams and membranes. Lattice vibrations, derivation of the macroscopic elastic properties of solids and applications (e.g., energy-harvesting devices, gyroscopes). Piezo-resistive sensors of acceleration and pressure; piezoelectric accelerometer; vertical and lateral capacitive accelerometers.

Thermal sensors: integrated thermopile sensors; semiconductor-junction temperature sensors; proportional-to-absolute-temperature sensors (PTAT).

Magnetic sensors: Hall plates; differential-amplification magnetic sensors (DAMS); MAGFET and dual-drain MAGFET; vertical and lateral magneto transistors. Energy-efficient gas sensors based on SOI CMOS technology.

Use of MOS transistors as chemical sensors in liquids.

Introduction to quantum computing and information. Bits and Qubits. Binary arithmetic. Linear vector spaces. Example of qubit: polarization of light. An example of quantum circuit. Deutsch’s algorithm. Search algorithms. Quantum Fourier transform. Shor’s algorithm. Grover’s search algorithm. No-cloning theorem. Quantum teleportation. EPR and Bell inequalities. The Di Vincenzo criteria. Quantum error correction.


Refer to the website http://www-micro.deis.unibo.it/cgi-bin/user?rudan

Teaching methods

Lessons + Exercises.

Needless to say, attending all classes is fundamental.

For the sake of the Commission entitled to Programme's certification, the Teacher hereby declares that lessons and exercises conform to the learning's objectives.

Assessment methods

Oral examination. Numerical rating between 0 and 30. The examination is passed if the rating is equal to, or larger than 18.

To timely identify the level of knowledge necessary to fulfill the course's requirements, and remembering that the Italian law prescribes that exams are public, students are firmly invited to be present at other students' exams well before the date of their own.

Teaching tools

Slides. Notes. Books. Material posted in the website, including the audio recording of the lessons of earlier Academic Years.

Office hours

See the website of Massimo Rudan