Conductivity

1. Conductivity • Electrical conductivity • Energy bands in solids • Band structure and conductivity • Semiconductors • Intrinsic semiconductors • Doped semiconductors • n-type materials • p-type materials • Diodes and transistors • p-n junction • depletion region • forward biased p-n junction • reverse biased p-n junction • diode • bipolar transistor • operation of bipolar pnp transistor • FET • Superconductivity • Hall effect – lab experiment

2. ELECTRICAL CONDUCTIVITY • in order of conductivity: superconductors, conductors, semiconductors, insulators • conductors: material capable of carrying electric current, i.e. material which has “mobile charge carriers” (e.g. electrons, ions,..) e.g. metals, liquids with ions (water, molten ionic compounds), plasma • insulators: materials with no or very few free charge carriers; e.g. quartz, most covalent and ionic solids, plastics • semiconductors: materials with conductivity between that of conductors and insulators; e.g. germanium Ge, silicon Si, GaAs, GaP, InP • superconductors: certain materials have zero resistivity at very low temperature.

3. resistivities • some representative resistivities (): • R = L/A, R = resistance, L = length, A = cross section area; resistivity at 20o C • resistance(in ) (L=1m, diam =1mm) resistivity in  m • aluminum 2.8x10-8 3.6x10-2 • brass 8x10-8 10.1x10-2 • copper 1.7x10-8 2.2x10-2 • platinum 10x10-8 12.7x10-2 • silver 1.6x10-8 2.1x10-2 • carbon 3.5x10-5 44.5 • germanium 0.45 5.7x105 • silicon  640  6x108 • porcelain 1010 - 1012 1016 - 1018 • teflon 1014 1020 • blood 1.5 1.9x106 • fat 24 3x107

4. ENERGY BANDS IN SOLIDS: • In solid materials, electron energy levels form bands of allowed energies, separated by forbidden bands • valence band = outermost (highest) band filled with electrons (“filled” = all states occupied) • conduction band = next highest band to valence band (empty or partly filled) • “gap” = energy difference between valence and conduction bands, = width of the forbidden band • Note: • electrons in a completely filled band cannot move, since all states occupied (Pauli principle); only way to move would be to “jump” into next higher band - needs energy; • electrons in partly filled band can move, since there are free states to move to. • Classification of solids into three types, according to their band structure: • insulators: gap = forbidden region between highest filled band (valence band) and lowest empty or partly filled band (conduction band) is very wide, about 3 to 6 eV; • semiconductors: gap is small - about 0.1 to 1 eV; • conductors: valence band only partially filled, or (if it is filled), the next allowed empty band overlaps with it

5. Band structure and conductivity

6. INTRINSIC SEMICONDUCTORS • semiconductor = material for which gap between valence band and conduction band is small; (gap width in Si is 1.1 eV, in Ge 0.7 eV). • at T = 0, there are no electrons in the conduction band, and the semiconductor does not conduct (lack of free charge carriers); • at T > 0, some fraction of electrons have sufficient thermal kinetic energy to overcome the gap and jump to the conduction band; fraction rises with temperature; e.g. density of conduction electrons in Si: ≈ 0.9x1010/cm3 at 20o C (293 K); ≈ 7.4x1010/cm3 at 50o C (323 K). • electrons moving to conduction band leave “hole” (covalent bond with missing electron) behind; under influence of applied electric field, neighboring electrons can jump into the hole, thus creating a new hole, etc.  holes can move under the influence of an applied electric field, just like electrons; both contribute to conduction. • in pure Si and Ge: nb. of holes (“p-type charge carriers”) = nb. of conduction electrons (“n-type charge carriers”); • pure semiconductors also called “intrinsic semiconductors”.

7. Intrinsic silicon: • DOPED SEMICONDUCTORS: • “doped semiconductor”: (also “impure”, “extrinsic”) = semiconductor with small admixture of trivalent or pentavalent atoms;

8. n-type material • donor (n-type) impurities: • dopant with 5 valence electrons (e.g. P, As, Sb) • 4 electrons used for covalent bonds with surrounding Si atoms, one electron “left over”; • left over electron is only loosely bound  only small amount of energy needed to lift it into conduction band (0.05 eV in Si) •  “n-type semiconductor” has conduction electrons, very few holes (just the few intrinsic holes) • example: doping fraction of 10-8 Sb in Si yields about 5x1016 conduction electrons per cubic centimeter at room temperature, i.e. gain of 5x106 over intrinsic Si.

9. p-type material • acceptor (p-type) impurities: • dopant with 3 valence electrons (e.g. B, Al, Ga, In)  only 3 of the 4 covalent bonds filled  vacancy in the fourth covalent bond  hole • “p-type semiconductor” has mobile holes, very few mobile electrons (only the intrinsic ones). • advantages of doped semiconductors: • can”tune” conductivity by choice of doping fraction • can choose “majority carrier” (electron or hole) • can vary doping fraction and/or majority carrier within piece of semiconductor • can make “p-n junctions” (diodes) and “transistors”

10. n – type material p– type material

11. Majority and Minority Carriers • n-type material: • majority carrier: electrons • minority carrier: holes • p-type material: • majority carrier: holes • minority carrier: electrons

12. DIODES AND TRANSISTORS • p-n JUNCTION: • p-n junction = semiconductor in which impurity changes abruptly from p-type to n-type ; • “diffusion” = movement due to difference in concentration, from higher to lower concentration; • in absence of electric field across the junction, holes “diffuse” towards and across boundary into n-type and capture electrons; • electrons diffuse across boundary, fall into holes (“recombination of majority carriers”);  formation of a “depletion region” (= region without free charge carriers) around the boundary; • charged ions are left behind (cannot move): • negative ions left on p-side  net negative charge on p-side of the junction • positive ions left on n-side  net positive charge on n-side of the junction •  electric field across junction which prevents further diffusion.

13. p-n junction • Formation of depletion region in p-n junction:

14. DIODE • diode = “biased p-n junction”, i.e. p-n junction with voltage applied across it • “forward biased”: p-side more positive than n-side; • “reverse biased”: n-side more positive than p-side; • forward biased diode: • the direction of the electric field is from p-side towards n-side •  p-type charge carriers (positive holes) in p-side are pushed towards and across the p-n boundary, • n-type carriers (negative electrons) in n-side are pushed towards and across n-p boundary  current flows across p-n boundary

15. Forward biased pn-junction • Depletion region and potential barrier reduced

16. Reverse biased diode • reverse biased diode: applied voltage makes n-side more positive than p-side  electric field direction is from n-side towards p-side  pushes charge carriers away from the p-n boundary  depletion region widens, and no current flows • diode conducts only when positive voltage applied to p-side and negative voltage to n-side • diodes used in “rectifiers”, to convert ac voltage to dc.

17. Reverse biased diode • Depletion region becomes wider, barrier potential higher

18. TRANSISTORS • (bipolar) transistor = combination of two diodes that share middle portion, called “base” of transistor; other two sections: “emitter'' and “collector”; • usually, base is very thin and lightly doped. • two kinds of bipolar transistors: pnp and npn transistors • “pnp” means emitter is p-type, base is n-type, and collector is p-type material; • in “normal operation of pnp transistor, apply positive voltage to emitter, negative voltage to collector;

19. operation of pnp transistor: • if emitter-base junction is forward biased, “holes flow” from battery into emitter, move into base; • some holes annihilate with electrons in n-type base, but base thin and lightly doped  most holes make it through base into collector, • holes move through collector into negative terminal of battery; i.e. “collector current” flows whose size depends on how many holes have been captured by electrons in the base;

20. Transistor operation • Number of holes captured depends on the number of n-type carriers in the base • Number of n-type carriers can be controlled by the size of the current (the “base current”) that is allowed to flow from the base to the emitter; • base current is usually very small; • small changes in the base current can cause a big difference in the collector current; • transistor acts as amplifier of base current, since small changes in base current cause big changes in collector current. • transistor as switch: if voltage applied to base is such that emitter-base junction is reverse-biased, no current flows through transistor -- transistor is “off” • therefore, a transistor can be used as a voltage-controlled switch; computers use transistors in this way.

21. Field-effect transistor (FET) • In FETs, current through “channel” from “source” to “drain” is controlled by voltage (electric field) applied to the “gate” • in a pnp FET, current flowing through a thin channel of n-type material is controlled by the voltage (electric field) applied to two pieces of p-type material (“gate”) on either side of the channel (current depends on electric field). • Advantage of FET over bipolar transistor: very small gate current – small power consumption • Many different kinds of FETs • FETs are the kind of transistor most commonly used in computers.

22. SUPERCONDUCTIVITY • mobile electrons in conductor move through lattice of atoms or ions that vibrate (thermal motion) • cool down conductor  less vibration  “easier” for electrons to get through  resistivity of conductors decreases (i.e. they become better conductors) when they are cooled down • in some materials, resistivity goes to zero below a certain “critical temperature” TC • these materials called superconductors -- critical temperature TC different for different materials; • no electrical resistance  electric current, once started, flows forever! • superconductivity first observed by Heike Kamerlingh Onnes (1911) in Hg (mercury) at temperatures below 4.12 K.

23. Superconductors • many other superconductors with critical temperatures below about 20K found by 1970 -- “high TC superconductors”: (Karl Alex Müller and Johannes Georg Bednorz, 1986) • certain ceramic oxides show superconductivity at much higher temperatures; since then many new superconductors discovered, with TC up to 125K. • advantage of high TC superconductors: • can cool with (common and cheap) liquid nitrogen rather than with (rare and expensive) liquid helium; • much easier to reach and maintain LN temperatures (77 K) than liquid Helium temperatures (few K).

24. Properties of superconductors • electrical resistivity is zero (currents flowing in superconductors without attenuation for more than a year) • there can be no magnetic field inside a superconductor (superconductors ”expel” magnetic field -- “Meissner effect”) • transition to superconductivity is a phase transition (without latent heat). • about 25 elements and many hundreds of alloys and compounds have been found to be superconducting • examples: In, Sn, V, Mo, Nb-Zr, Nb-Ge, Nb-Ti alloys

25. applications of superconductors • superconducting magnets: • magnetic fields stronger, the bigger the current - “conventional” magnets need lots of power and lots of water for cooling of the coils; • s.c. magnets use much less power (no power needed to keep current flowing, power only needed for cooling) • most common coil material is NbTi alloy; liquid He for cooling • e.g. particle accelerator “Tevatron” at Fermi National Accelerator Laboratory (“Fermilab”) uses 990 superconducting magnets in a ring with circumference of 6 km, magnetic field is 4.5 Tesla. • magnetic resonance imaging (MRI): • create images of human body to detect tumors, etc.; • need uniform magnetic field over area big enough to cover person; • can be done with conventional magnets, but s.c. magnets better suited - hundreds in use • magnetic levitation - high speed trains??

26. explanation of superconductivity -- 1 • Cooper pairs: • interaction of the electrons with the lattice (ions) of the material,  small net effective attraction between the electrons; (presence of one electron leads to lattice distortion, second electron attracted by displaced ions) • this leads to formation of “bound pairs” of electrons (called Cooper pairs); (energy of pairing very weak - thermal agitation can throw them apart, but if temperature low enough, they stay paired) • electrons making up Cooper pair have momentum and spin opposite to each other; net spin = 0  behave like ”bosons”.

27. explanation of superconductivity -- 2 • unlike electrons, bosons “like” to be in the same state; when there are many of them in a given state, others also go to the same state • nearly all of the pairs locked down in a new collective ground state; this ground state is separated from excited states by an energy gap; • consequence is that all pairs of electrons move together (collectively) in the same state; electron cannot be scattered out of the regular flow because of the tendency of Bose particles to go in the same state  no resistance • (explanation given by John Bardeen, Leon N. Cooper, J. Robert Schrieffer, 1957)

28. Edwin Hall (1879): magnetic field perpendicular to current  potential difference perpendicular to current and magnetic field allows determination of charge carrier density in metals and semiconductors I Hall Effect

29. magnetic field exerts force onmoving charge carrier of charge q (Lorentz force) in the lateral direction: Lateral displacement of charges  accumulation of charges  electric field (Hall field) perpendicular to current and magnetic field direction force due to Hall field opposite to Lorentz force Equilibrium reached when magnitude of force due to Hall field = mag. of Lorentz force  get drift speed v Current density J, density of charge carriers n, Hall coefficient RH t w I Hall effect explanation

30. In the lab, we measure current I, B-field, Hall voltage VH, size (width w, height t) of sample calculate RH from measurements, and assume |q| = e  find n. sign of VH and thus RH tells us the sign of q t w I Hall effect measurements