Solids and semiconductors Physics 123 11/18/2018 Lecture XX

Bonding in solids Atoms in solids organize themselves in crystal structures Positions of atoms are determined by a balance of electrostatic attraction and repulsion Minimum of potential energy U0 is called ionic cohesive energy and is equivalent to binding energy in nucleus 11/18/2018 Lecture XX

Metals Metal have 1-2 e on the outer shell, they are loosely bound to the rest of the atom and can be considered “free” to move within the boundaries of metal  electron gas Electrons in potential well – boundaries on metal surface  L is very large Distance between energy levels inversely proportional to L2 Energy levels become energy bands 11/18/2018 Lecture XX

Metals Electrons are fermions, according to Pauli principle not more than one electron can exit for each quantum state How much space does a free electron need to itself? dxdp>h In 3-D Phase space (3 spatial coordinates +3 momentum coordinates) dxdydzdpxdpydpz =dVdP>h3 electrons = balls in phase space each occupying h3 of space Actually two electrons can coexist in h3 – spin up and spin down 11/18/2018 Lecture XX

Density of states Let’s calculate the number of states in unit volume between energy E to E+dE: g(E)dE In momentum space think of a spherical layer of radius p and thickness dp Total phase space volume of this layer V4pp2dp Number of electrons that can live in this volume (number of available apartments) 2(spin)x(total volume)/(volume occupied by one electron) Number of states per unit volume 11/18/2018 Lecture XX

Fermi energy Consider T=0K All electrons must fall into the lowest possible quantum state, but respect each other’s privacy – Pauli principle Suppose you have n electron per unit volume, what is the highest energy that they can have at T=0K - Fermi energy? 11/18/2018 Lecture XX

Fermi-Dirac probability function At T=0 all states below EF are occupied, above EF are free When T increases some electrons get enough energy to get above EF Fermi function – smoothened step 11/18/2018 Lecture XX

Density of occupied states g(E) – density of available states f(E)- probability to find electron with a certain value of E Number of occupied states per unit volume 11/18/2018 Lecture XX

Energy bands insulator semiconductor conductor Conduction band Eg In conductors the highest energy band is partially filled allowing electrons to move freely – conduction band In insulators the highest energy band is completely filled – valence band, there is an energy gap between valence and conduction band – Eg Semiconductors are similar to insulators, but the energy gap is smaller insulator semiconductor conductor Conduction band Eg Conduction band Eg Valence band Valence band 11/18/2018 Lecture XX

Intrinsic semiconductors Since in semiconductors the energy gap is small, thermal energy can be enough for some electrons to jump to conduction band Resistivity of semiconductor decreases (unlike metals) with temperature – more electrons in conduction band Electrons leave vacancies behind – holes, which act as effective positive charge and also carry electric current Conduction band Eg EF Valence band 11/18/2018 Lecture XX

Semiconductors Most commonly used semiconductors Si (Z=14), Ge (Z=32) Si electron structure: 1s22s22p63s23p2 Ge electron structure: 1s22s22p63s23p63d104s24p2 Semiconductors have 4 electrons on outer shell In crystal structure each atom bonds with 4 neighbors to share electrons Si 11/18/2018 Lecture XX

Semiconductors and doping Doping – introduction of impurities with valence 3 (Ga) or 5(As) As incorporates itself into the existing crystal structure sharing 4 of its e with Si- neighbors, one e is free to move around – n-type doping Ga does the same, but instead of extra e it creates a vacancy – hole – p-type doping Resistivity of doped semiconductor is much higher than that of intrinsic material As Si Ga Si 11/18/2018 Lecture XX

P and n-type semiconductors Impurities create extra levels in the band structure Conduction band Conduction band Acceptor level Allows e to jump there Donor level Gives e to conduction band Valence band Valence band p-type n-type 11/18/2018 Lecture XX

p-n junction Suppose that you bring p-type and n-type semiconductor in contact Electrons from n-type will readily fill the vacancies provided in p-type, thus creating the space charge. Mind that before materials were brought together they were electrically neutral Q=-1e Q=+1e As Si Ga Si 11/18/2018 Lecture XX

p-n diode The current flows through p-n junction if electrons have vacancies to jump to, it does not flow in the opposite direction Not entirely true, there still is so called “dark” current, because of thermal excitation to conduction band, this current grows with T P-type vacancies +++ P-type +++ current No current + - - --- Electron flow + --- n-type electrons n-type LED: e+hole=light Reverse bias Forward bias 11/18/2018 Lecture XX

Transistors npn or pnp junction – no current is flowing – logical zero Small current (supply of electrons) on base (p in npn or n in pnp) opens the transistor – larger current is flowing – logical one current P-type +++ --- n-type +++ P-type 11/18/2018 Lecture XX