Lectures on Semiconductor Equations

Overview

References

• P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor Equations, Springer-Verlag, 1990.
• S. M. Sze, Semiconductor Devices Physics and Technology, Wiley 1985.
• M. R. Wehr, J. A. Richards, Jr., and T. W. Adair, III, Physics of the Atom, 4th Ed., Addison-Wesley, 1984.

Chap. 1   Basic Properties of Semiconductors

• Energy: Energy is the ability to do work (W=F*D, force times distance) and it is measured in joules (newton-meter). 1 newton = 1 kg * 1 meter/sec^2.
• Kinetic Energy  = 1/2 (mv^2)
• Heat
• Potential Energy: Diver on a board.
• Electromagnetic Radiation (Light): Electromagnetic Spectrum #1
• Made up of electricity and magnetism. It consists of an electric and a magnetic field at right angles to each other. #2
• All electromagnetic waves travel at the speed of light.
• Mass: Theory of Relativity (E=mc^2)
• A. Einstein (1905): Light is absorbed in the form of packets of energy, now called photons.
• Mass is a form of energy and it can be converted into other form, e.g., heat.
• Conservation of energy (not mass).
• Nuclear Energy: Fission (Split) and Fusion (Join).
• Semiconductor: A semiconductor is a material (silicon, germanium, gallium arsenide) that can behave as a conductor of electricity or an insulator depending on what is done to it. We can control the amount of current that can pass through a semiconductor. The properties of a semiconductor can be changed by adding small impurities. This is called doping. Some impurities increase the number of electrons available for carrying a current to give an n-type semiconductor. Other impurities soak up electrons, creating holes in the semiconductor (p-type) that behave like positive charges and enable a current to flow. Different parts of the same piece of silicon can be doped as p-type and n-type. Many components in electronic circuits rely on what happens at the junctions between p-type and n-type semiconductors.
• Diode: Two pieces of semiconductor material, one n-type and one p-type, may be joined together to make a diode -- a circuit device that allows current to pass through it in one direction only. #3
• Diode allows electricity to flow in one direction only.
• Resistor resists the flow of current.
• Capacitor accumulates and stores electric charge.
• Transistor amplifies electric current passing through it or switches current on or off in response to a controlling signal.
• N-MOSFET: n-type metal-oxide-silicon-field-effect-transistor #4
• Hundreds of semiconductor microprocessors etched (to be made a drawing, design, etc. on metal plate, glass, etc. by the action of an acid) on a single slice of silicon are tested before the slice is cut into individual components. #5
• Circuit is a loop of electrical conductors through which a current of electrons can flow.
• Microprocessor #6 (Intel 1970s) is the control center and electronic calculator of a computer. It was the first time a computer's central processing unit (CPU) could fit on a single chip.
• IC (integrated circuit) is an electronic circuit contained in a paper-thin chip of silicon roughly half an inch square. Microchips are able to perform at almost the speed of light -- at more than a million operations per second.
• Electron: Discovered in 1897. An electron is a subatomic particle that carries a negative electric charge.
• m=0.91095*10^(-30) kg, e=1.6021917 * 10^(-19) coul.
• Charge: Two kinds of electric charge (+ and -). #7
• Charles-Augustin de Coulomb (1736-1806, French physicist) was the first to measure accurately the forces between objects with electric charge. Like charges repel each other, unlike charges attract each other.
• Coulomb's law and Electric field E #8
• F=Eq (Coulomb's Law), q: positive test charge.
• A coul is defined as the amount of that flows through a given cross section of a wire in 1 second if there is a steady current (an assembly of moving charge) of 1 ampere (2*10^(-7) nt/m) in the wire (q=i*t, q: net charge, i: current, t: second). The mks (meter-kilogram-second) unit in which electric charge is measured.
• Andre Ampere (1775-1836, French physicist): Discovered that parallel electric currents attract each other if they move in the same direction, and repel each other if they move in opposite.
• Ampere's Law and Magnetic field B #9
• F=qv x B: If a positive test charge q is moving with velocity v through a point P and if a force acts perpendicularly on the moving charge, a magnetic field B is present at point P. (Magnetic field exerts a side-ways force on a moving charge)
• 1 ampere = The force of attraction per unit length between two parallel wires with 1 meter apart and with the same current is 2*10^(-7) nt/m. The unit for the strength of an electric current.
• Charge is quantized: The electric field is not continuous but is made up of integral multiples of the fundamental charge e.

Chap. 2   Schrodinger's Equation

• Schrodinger's equation: A mathematical way of describing the electrons in an atom or molecule in terms of waves.
• Quantum Mechanics: Explains the structure of the atom and how atoms give off energy in small packets called quanta or photons.
• Max Planck (1900, German): Discovered the idea that atoms emit or absorb energy in separate packets
• Planck's Formula: E=nh\nu, \nu: the frequency of  atomic oscillators, n: quantum number, h=6.626196*10^(-34) joul-sec: Planck's number. #10
• Albert Einstein (1905, German): Suggested that light is absorbed in the form of photons that behave like wave.
• E=mc^2.
• Niels Bohr (1913, Danish): Showed how atoms radiate light.
• The Bohr model #11
• Energy Levels.
• Louis de Broglie (1924): Suggested that if light has a dual, wave-particle nature, perhaps matter has also.
• Erwin Schrodinger (1926, Austrian): Wave mechanics.
• Wave Mechanics #12
• Wave:  wavelength (lambda), frequency (\nu).
• Particle: energy (E), momentum (p).
• Wave-particle relation: h=p*\lambda.
• Matter waves
• Classical Standing Waves
• Quantization of wavelength and thus energy.
• Max Born: Probability wave function.
• Homework 1 (p. 782): What is the wavelength of a beam of electrons whose kinetic energy is 100 eV?
• Homework 2 (p. 786): Consider an electron confined by electrical forces to move between two rigid "walls" separated by 1.0*10^(-9) m, which is about 5 atomic diameters. Find the quantized energy (in eV) values for the three lowest stationary states.
• The Classical Wave Equation
• Traveling Wave
• Schrodinger's equation

• Carrier Transport: The motion of charge carriers (electrons and holes).
• Crystal: A crystal is a small piece of solid that has a regular 3D shape due to the internal arrangement of atoms, ions, or molecules making up the solid. #13
• Silicon atom and crystal
• Ion: An atom or group of atoms that possesses an electric charge through the gain or loss one or more electrons.
• Lattice: The arrangement coupled together by strong spring-like forces between atoms is called the lattice.
• Valency is the combining power of a chemical element. It tells us how many chemical bonds an element can form when it combines with other elements in compounds.
• Energy Bands: Energy bands are formed when atoms are coupled together.
• Conduction band, bandgap, valance band, kinetic energy, crystal momentum, parabolic band structure. #14
• Density of States: The density of allowed energy states per unit volume in crystal momentum space.
• Fields: Action-at-a-distance.
• Gravitational field g in F=mg (Newton's Law of universal gravitation F=(GM/r^2)m)
• Electric field E
• Magnetic field B

• Kinetic transport equations model the flow of charge carriers in semiconductors.
• Classical models: Newton's second law of motion of particle ensembles (total effect) => probabilistic formulation => the classical Liouville equation
• Microscopic models
• Many-body motion: Too many phase (position-momentum) space dimensions (coordinates of all particles).
• Too costly for numerical simulation
• Reduction of the dimension
• Long range forces (e.g., the Coulomb force) => the collisionless Boltzmann (Vlasov) equation
• Long and short (scattering, i.e., collision) range forces => the Boltzmann equation
• Semi-classical models: Modification of  classical Liouville equation by incorporating the quantum effects of the semiconductor crystal lattice via the band-diagram of the material.
• Quantum mechanical models: Schrodinger equation => Wigner transformation => the quantum Liouville equation
• The quantum Liouville equation => the classical Liouville equation in the classical limit of the Planck constant (scaled by 2 pi).
• The quantum Liouville equation => Long and short range forces => the quantum Boltzmann equation

• The quantum Liouville equation => Long and short range forces => the quantum Boltzmann equation

The (Semi-)Classical Liouville Equation

• Particle Trajectories
• Motion of a single electron in a vacuum under the action of an electric field E.
• No magnetic field
• P. 5:  (1.2.1) - (1.2.5)