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Fall 2006. Fundamentals of Semiconductor Physics. 万 歆 Z hejiang I nstitute of M odern P hysics xinwan@zimp.zju.edu.cn http://zimp.zju.edu.cn/~xinwan/. Transistor technology evokes new physics.
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Fall 2006 Fundamentals of Semiconductor Physics 万 歆 Zhejiang Institute of Modern Physics xinwan@zimp.zju.edu.cn http://zimp.zju.edu.cn/~xinwan/
Transistor technology evokes new physics The objective of producing useful devices has strongly influenced the choice of the research projects with which I have been associated. It is frequently said that having a more-or-less specific practical goal in mind will degrade the quality of research.I do not believe that this is necessarily the case and to make my point in this lecture I have chosen my examples of the new physics of semiconductors from research projects which were very definitely motivated by practical considerations. -- William B Shockley Nobel Lecture, December 11, 1956
Chapter 3. Junctions & Contacts 3.1 Metal-semiconductor contacts 3.2 p-n junctions 3.3 Heterojunctions Total 9 hours.
Chapter 3. Junctions & Contacts 3.1 Metal-semiconductor contacts • A review of the principles • Idealized metal-semiconductor junctions • Current-Voltage characteristics • Ohmic contacts 3.2 p-n junctions 3.3 Heterojunctions
Bring M-S into Contact At thermal equilibrium the Fermi levels is constant throughout a system.
Charge Transfer Equilibrium Electrons depleted Assume band structures not changed near the surface. Real situation: Surface states
No Hope to Solve Analytically • Poisson’s equation • the electron density, the hole density and the donor and acceptor densities. To solve the equation, we have to express the electron and hole density, n and p, as a function of the potential, f
Depletion Approximation • For the idealized n-type semiconductor, hole density neglected • Electron density n << Nd, thus depleted, from the interface to x = xd • Beyond xd, n = Nd • Negative charge right at the metal surface
Field, Potential and Charge (Gauss’ law)
Applied Bias • Up to this point, we have been considering thermal equilibrium conditions at the metal-semiconductor junction. • Now, we study the case of an applied voltage; i.e., a nonequilibrium condition. • Electrons transferring from metal to semiconductor see a barrier.
Applied Bias • To the first order, the barrier height is independent of bias because no voltage can be sustained across the metal. • Bias changes the curvature of the semiconductor bands, modifying the potential drop from fi.
Schottky Barrier Lowering • We now explore the statement that the barrier to electron flow from metal to semiconductor is “to first order” unchanged by bias. • Approximations: • Free electron theory • Metal as plane conducting sheet • Semiconductor: effective mass, relative permittivity • Root: Metal plane = image charge of opposite sign
I-V Characteristics w/out Math • At equilibrium, rate at which electrons cross the barrier into the semiconductor is balanced by rate at which electrons cross the barrier into the metal. (flow = counter flow). • When a bias is applied, the potential drop within the semiconductor is changed and we can expect the flux of electrons from the semiconductor toward the metal to be modified. • The flux of electrons from the metalto the semiconductor is not affected. • The difference is the net current.
I-V Characteristics • The current of electrons from the semiconductor to the metal is proportional to the density of electrons at the boundary. • In equilibrium,
I-V Characteristics • When a bias is applied to the junction, • Now, • Therefore, the ideal diode equation reads
Comments • The ideal diode equation arises when a barrier to electron flow affects the thermal flux of carriers asymmetrically. • The essence of the ideal diode equation predicts a saturation current J0 for negative Va and a very large steeply rising current when Va is positive. • J0 in the above ideal case is independent of the applied bias. More careful analysis will modify this slightly.
More Detailed Analysis • Schottky: Integrating the equations for carrier diffusion and drift across the depletion region near the contact. Assumes that the dimensions of the space-charge region are sufficiently large (a few electron mean-free path) so that the use of a diffusion constant and a mobility value are meaningful (small field, no drift velocity saturation). • Bethe: Based on carrier emission from the metal. Valid even when these abovementioned constraints are not met.
Comments • One important implicit assumption is that the system is in quasi-equilibrium, i.e., almost at thermal equilibrium even though currents are flowing. • Electron density at the interface when the bias is applied. • Using Einstein relation to relate drift and diffusion current. • The ultimate test: Agreements between measurements and predictions. • Derivation not valid when Va ~ fi (no barrier). A fraction of the voltage is dropped across the resistance of the semiconductor. The forward voltage across the junction is reduced.
Nonrectifying (Ohmic) Contacts • Definition: The contact itself offers negligible resistance to current flow when compared to the bulk. • The voltage dropped across the ohmic contacts is negligible compared to voltage drops elsewhere in the device. • No power is dissipated in the contacts. • Ohmic contacts can be described as being in equilibrium even in when currents are flowing. • All free-carrier densities at an Ohmic contact are unchanged by current flow. The densities remain at their thermal-equilibrium values.
Tunneling Contacts • By heavily doping the semiconductor, so that the barrier width is very small and tunneling through the barrier can take place.
Pinning of Fermi Energy • To account for the surface effects, the metal-semiconductor contact is treated as if it contained an intermediate region sandwiched between the two crystals. • For a large density of surface states, the Fermi energy is said to be pinned by the high density of states.
Chapter 3. Junctions & Contacts 3.1 Metal-semiconductor contacts 3.2p-n junctions 3.3 Heterojunctions
Metal-Semiconductor Contacts • A system of electrons is characterized by a constant Fermi level at thermal equilibrium. • Systems that are not in thermal equilibrium approach equilibrium as electrons are transferred from regions with a higher Fermi level to regions with a lower Fermi level. • The transferred charge causes the buildup of barriers against further flow, and the potential drop across these barriers increases to a value that just equalizes the Fermi levels. Similar phenomena in a single crystal with non-uniform doping.
Graded Impurity Distributions • We assume initially the majority carrier density equals the dopant density everywhere. We ask how equilibrium is approached. • A gradient in the mobile carrier density diffusion of carriers. • Carrier diffusion leaves dopant ions behind. • Separation of charge field opposing the diffusion flow. • Equilibrium is reached when diffusion is balanced by the field.
The separation of the Fermi level from the conduction-band edge (or intrinsic Fermi level) represents the potential energy of an electron. Potential
Field Using mass-action law,
Poisson’s Equation Cannot be solved in the general case!
(i) Small Gradient Case Quasi-neutrality approx.:
(ii) p-n Junction • Depletion approximation↔ quasi-neutral approximation
Two Idealized Cases • Linearly graded junction: a continuous gradient in dopant between n-type and p-type regions. • Step junction (abrupt junction): a constant n-type dopant density changes abruptly to a constant p-type dopant density – for example, formed by epitaxial deposition
