Semiconductors n D*n If T>0 Filled valence band but small gap (~1 eV) to an empty (at T=0) conduction band look at density of states D and distribution function n Fermi energy on center of gap for undoped. Always where n(E)=0.5 (problem 13-26) D(E) typically goes as sqrt(E) at top of valence band and at bottom of conduction band n D valence conduction EF D*n If T>0 P461 - Semiconductors

Semiconductors II Distribution function is so probability factor depends on gap energy estimate #electrons in conduction band of semiconductor. Integrate over n*D factors at bottom of conduction band P461 - Semiconductors

Number in conduction band using Fermi Gas model = integrate over the bottom of the conduction band the number in the valence band is about the fraction in the conduction band is then P461 - Semiconductors

Conduction in semiconductors INTRINSIC. Thermally excited electrons move from valence band to conduction band. Grows with T. “PHOTOELECTRIC”. If photon or charged particle interacts with electrons in valence band. Causes them to acquire energy and move to conduction band. Current proportional to number of interactions (solar cells, digital cameras, particle detection….) EXTRINSIC. Dope the material replacing some of the basic atoms (Si, Ge) in the lattice with ones of similar size but a different number (+- 1) of valence electrons P461 - Semiconductors

Superconductivity Res. T Resistance goes to 0 below a critical temperature Tc element Tc resistivity (T=300) Ag --- .16 mOhms/m Cu -- .17 mOhms/m Ga 1.1 K 1.7 mO/m Al 1.2 .28 Sn 3.7 1.2 Pb 7.2 2.2 Nb 9.2 1.3 many compounds (Nb-Ti, Cu-O-Y mixtures) have Tc up to 90 K. Some are ceramics at room temp Res. T P461 - Semiconductors

Superconductors observations Most superconductors are poor conductors at normal temperature. Many good conductors are never superconductors --> superconductivity due to interactions with the lattice practical applications (making a magnet), often interleave S.C. with normal conductor like Cu if S.C. (suddenly) becomes non-superconducting (quenches), normal conductor able to carry current without melting or blowing up quenches occur at/near maximum B or E field and at maximum current for a given material. Magnets can be “trained” to obtain higher values P461 - Semiconductors

Superconductors observations For different isotopes, the critical temperature depends on mass. ISOTOPE EFFECT again shows superconductivity due to interactions with the lattice. If M -> infinity, no vibrations, and Tc--> 0 spike in specific heat at Tc indicates phase transition; energy gap between conducting and superconducting phases. And what the energy difference is plasma -> gas -> liquid -> solid -> superconductor P461 - Semiconductors

What causes superconductivity? Bardeen-Cooper-Schrieffer (BCS) model paired electrons (cooper pairs) coupled via interactions with the lattice gives net attractive potential between two electrons if electrons interact with each other can move from the top of the Fermi sea (where there aren’t interactions between electrons) to a slightly lower energy level Cooper pairs are very far apart (~5,000 atoms) but can move coherently through lattice if electric field ----> resistivity = 0 (unless kT noise overwhelms--> breaks lattice coupling) atoms electron electron P461 - Semiconductors

Conditions for superconductivity Temperature low enough so the number of random thermal phonons is small interactions between electrons and phonons large (-> large resistivity at room T) number of electrons at E = Fermi energy or just below be large. Phonon energy is small (vibrations) and so only electrons near EF participate in making Cooper pairs (all “action” happens at Fermi energy) 2 electrons in Cooper pair have antiparallel spin --> space wave function is symmetric and so electrons are a little closer together. Still 10,000 Angstroms apart and only some wavefunctions overlap (low E --> large wavelength) P461 - Semiconductors

Conditions for superconductivity 2 2 electrons in pair have equal but opposite momentum. Maximises the number of pairs as weak bonds constantly breaking and reforming. All pairs will then be in phase (other momentum are allowed but will be out of phase and also less probability to form) if electric field applied, as wave functions of pairs are in phase - maximizes probability -- allows collective motion unimpeded by lattice (which is much smaller than pair size) P461 - Semiconductors