Lecture XV Solid state dr hab. Ewa Popko

Measured resistivities range over more than 30 orders of magnitude Material Resistivity (Ωm) (295K) Resistivity (Ωm) (4K) “Pure” Metals Copper Semi- Conductors Ge (pure) 5  InsulatorsDiamond10 14 Polytetrafluoroethylene (P.T.F.E) Potassium 2  Metals and insulators

Metals, insulators & semiconductors? At low temperatures all materials are insulators or metals. Semiconductors: resistivity decreases rapidly with increasing temperature. Semiconductors have resistivities intermediate between metals and insulators at room temperature. Pure metals: resistivity increases rapidly with increasing temperature Resistivity (Ωm) Temperature (K) Diamond Germanium Copper

Bound States in atoms Electrons in isolated atoms occupy discrete allowed energy levels E 0, E 1, E 2 etc.. The potential energy of an electron a distance r from a positively charge nucleus of charge q is V(r) E2E1E0E2E1E0 r 0 Increasing Binding Energy

Bound and “free” states in solids V(r) E2E1E0E2E1E0 The 1D potential energy of an electron due to an array of nuclei of charge q separated by a distance R is Where n = 0, +/-1, +/-2 etc. This is shown as the black line in the figure. r R Nuclear positions V(r) lower in solid (work function). V(r) Solid

Energy Levels and Bands + E position Electron level similar to that of an isolated atom Band of allowed energy states. In solids the electron states of tightly bound (high binding energy) electrons are very similar to those of the isolated atoms. Lower binding electron states become bands of allowed states. We will find that only partial filled band conduct

Solid state N~10 23 atoms/cm 3 2 atoms 6 atoms Energy band theory

Metal – energy band theory

At a temperature T the probability that a state is occupied is given by the Fermi-Dirac function n(E)dE The finite temperature only changes the occupation of available electron states in a range ~k B T about E F. where μ is the chemical potential. For k B T << E F μ is almost exactly equal to E F. Fermi-Dirac function for a Fermi temperature T F =50,000K, about right for Copper. The effects of temperature

Insulator -energy band theory

diamond

semiconductors

Intrinsic conductivity ln(  ) 1/T

ln(  ) Extrinsic conductivity – n – type semiconductor

Extrinsic conductivity – p – type semiconductor

Conductivity vs temperature ln(  ) 1/T

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