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Solids and semiconductors

Solids and semiconductors. Physics 123. Bonding in solids. Atoms in solids organize themselves in crystal structures Positions of atoms are determined by a balance of electrostatic attraction and repulsion

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Solids and semiconductors

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  1. Solids and semiconductors Physics 123 Lecture XX

  2. 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 U0is called ionic cohesive energy and is equivalent to binding energy in nucleus Lecture XX

  3. 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 Lecture XX

  4. 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 Lecture XX

  5. 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 Lecture XX

  6. 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? Lecture XX

  7. Fermi-Dirac probability function • At T=0 all states below EFare occupied, above EF are free • When T increases some electrons get enough energy to get above EF • Fermi function – smoothened step Lecture XX

  8. 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 Lecture XX

  9. Energy bands • 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 Conduction band Eg Eg Valence band Valence band Lecture XX

  10. 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 Lecture XX

  11. Semiconductors Si Si Si Si Si • 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 Lecture XX

  12. Semiconductors and doping Si Si Si Si As Ga Si Si Si Si • 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 Lecture XX

  13. 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 Lecture XX

  14. p-n junction Si Si Si Si As Ga Si Si Si Si • 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 Lecture XX

  15. 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 Lecture XX

  16. 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 Lecture XX

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