Hall Effect, Fermi Level Constancy, and Excess Carriers in Semiconductors

In this lecture we build on drift current and mobility, complete our discussion of Hall measurements, and introduce one of the most important ideas in semiconductor devices: excess carriers. We begin with a quick recap of key concepts from earlier classes: Carrier statistics and the Fermi level How the Fermi level moves with doping and how to go between EF and doping concentration Electrons and holes as charge carriers in semiconductors Law of mass action (np = ni squared) and its validity only at thermal equilibrium We then revisit drift current and mobility: Electric field in a homogeneous bar, E equals V divided by L, with units of volts per centimeter Drift current under low electric fields and definition of drift velocity Mobility as the proportionality constant between drift velocity and electric field Current density J as the sum of electron and hole contributions Ohm's law in semiconductors, resistivity, resistance, conductivity and conductance Majority and minority carriers in n-type and p-type materials Next, we discuss the limitations of pure electrical measurements: Why resistance measurements give only n multiplied by mu, not n and mu independently Dependence of mobility on field, temperature and doping Velocity saturation and why current becomes independent of voltage at high fields We then develop the Hall effect in detail: Geometry of the Hall bar with dimensions L, W and T Applying a magnetic field in the z direction and understanding the Lorentz force Deflection of electrons and holes and the resulting drop in measured current Applying a transverse electric field Ey to counteract the magnetic force The condition for maximum current when electric and magnetic forces balance Derivation of the Hall coefficient and extraction of majority carrier concentration Using the sign of the Hall voltage to identify n-type versus p-type material Practical limitations: only majority carriers are measured, intrinsic and minority carrier measurements require different techniques We then introduce a powerful equilibrium concept: constancy of the Fermi level. When two materials are brought together and reach equilibrium, their Fermi levels must align Using density of states and Fermi–Dirac probabilities to show that carrier exchange rates balance Interpretation: a system with a single uniform Fermi level is at equilibrium, even though microscopic motion continues Finally, we begin Chapter 4 and introduce excess carriers: What excess carriers really represent: departure from thermal equilibrium, not simply doping Distinction between majority carriers, which are an equilibrium property, and excess carriers, which arise under non-equilibrium conditions Ways to create excess carriers: current injection, optical generation and other forms of excitation Photon absorption in semiconductors: If photon energy h times nu is greater than or equal to the band gap, one photon generates one electron–hole pair If photon energy is below the band gap, the semiconductor is transparent at that wavelength Why excess carriers form the foundation of semiconductor devices: p-n diodes rely on injected excess carriers MOSFETs and BJTs operate by controlling excess carriers to control terminal currents This lecture is ideal for: Undergraduate and early graduate students in electronic devices, solid state devices, VLSI or microelectronics GATE and ESE aspirants seeking a strong conceptual foundation Practicing engineers refreshing Hall effect physics, equilibrium concepts and non-equilibrium carrier dynamics If you want a deep, intuitive and mathematically grounded understanding of Hall measurements, Fermi level alignment and excess carriers, this lecture will help you build a strong foundation for all semiconductor devices.