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Chap. 41: Conduction of electricity in solids Hyun-Woo Lee 41-1 What Is Physics? Q: Why certain materials conduct electricity? Q: Why certain materials do NOT conduct electricity? Solid Many many electrons and atoms Solid material state physics Application of quantum physics to solids 41-2 Electrical Properties of Solids Crystalline solids Lattice structure • Repetition of unit cells Classification criteria Resistivity at room temperature (m) Temperature coefficient of resistivity (K-1) (1 / )( d / dT ) Number density of charge carriers n (m-3) • Can be found from Hall effect measurement Metals, semiconductors, insulators Insulators, semiconductors & metals Insulators Extremely large Ex: Diamond diamond/copper~1024 Semiconductors vs Metals insulator >> semiconductor >> metal • silicon=3103 m, copper=210-8 m semiconductor <0, metal>0 • silicon= -70 10-3 K-1, Copper= +410-3 K-1 nsemiconductor << nmetal • nsilicon=11016 m-3, ncopper=91028 m-3 41-3 Energy Levels in a Crystalline Solids Single 1s2 2s2 2p6 3s2 3p6 3d10 4s1 Two atoms Tunneling between two atoms Three atom (Ex: Cu Z=29) atoms More tunneling Tunneling effects Two wells Level splitting into two levels Tunneling effects in solids N() wells Energy level splitting into N levels Energy bands & energy gaps 41-4 Insulators No partially filled bands For a current to exist, Kinetic energy must increase • Electrons must move to higher-energy levels Pauli exclusion principle • Transition to filled state is prohibited Energy gap (Ex: Eg=5.5 eV in diamond) • Large energy supply needed Current flow strongly suppressed Thermal fluctuation effects Thermal excitations Finite probability to jump Eg Probability P for the jump For Eg=5.5 eV, T=300K P ~ exp( Eg / kT ) e213 3 1093 • cf: # of electron in 1 cm3 ~ 1023 41-5 Metals Partially Easy to induce energy “jump” Fermi level EF Highest occupied level at T=0K Ex: EF=7.0 eV for copper Fermi filled bands speed vF Electron speed at EF Ex: vF=1.6106 m/s for copper No relaxation of vF due to Pauli exclusion principle How Many Conduction Electrons Are There? Number n density n number of conduction electrons in sample sample volume V number of conduction number of atoms number of valence in sample electrons in sample electrons per atom Ex: Magnesium w/ volume 2.0010-6 m3 number of atoms 8.6110 22 in sample Bivalent number of conduction 1.72 1023 electrons in sample Conductivity Above Absolutely Zero Ex: T=1000 K kT=0.086 eV cf: EF=7.0 eV in copper # of charge carriers extremely insensitive to T 41-6 Semiconductors No partially filled bands But small energy gap Ex: Eg=1.1 eV for silicon cf: Eg=5.5 eV for diamond Valence band Highest filled band Conduction band Lowest vacant band Number Density of Charge Carriers Probability P for jump P ~ exp( Eg / kT ) e Charge 42.6 4 10 19 carriers Electrons • Conduction band Holes • Valence band # of charge carriers extremely sensitive to T Motion of charge carriers Electrons in conduction band E Holes in valence band E Efficient description in terms of holes Effective charge of hole: +e Resistivity silicon / copper = 1.51011 Classical estimation m / e 2 n Difference between silicon and copper mainly from carrier density n Temperature Coefficient of Resistivity : 1 d dT Temperature dependence Classical estimation m / e 2 n Semiconductor n increases as T increases < 0 Metal (Ex: silicon) (Ex: copper) decreases as T increases > 0 More about metals How Many Quantum States Are There? Too many states to list all states Density of states N(E) N(E)dE : # of states between E and E+dE per volume Near lower edge of partially filled band 8 2m3 / 2 1/ 2 N (E) E 3 h (m -3 J -1 ) How Many Quantum States Are There ? (continued) Ex: Metal w/ V=210-9 m3 at E=7 eV # of states 4 1019 eV -1 per eV at 7 eV # of states N 11017 in range 0.003 eV at 7 eV The Occupancy Probability P(E) Maxwell distribution Not applicable due to Pauli exclusion principle Fermi-Dirac P( E ) At statistics 1 e ( E E F ) / kT 1 E=EF P(E)=1/2 regardless of T Useful way to define EF at T>0 How Many Occupied States Are There? Density of occupied states N0(E) N0(E)=N(E)P(E) Calculating the Fermi Energy At T=0, Due to Pauli exclusion principle EF n N ( E )dE 0 With N(E) E1/2 8 2m 3 / 2 2 EF3 / 2 n h3 3 3 EF 16 2 2/3 h 2 2 / 3 0.121h 2 2 / 3 n n m m More about semiconductors 41-7 Doped Semiconductors Doping Introducing a small number of replacement atoms (impurities) into semiconductor lattice ~ 1 out of 107 atoms replaced n-Type Semiconductors Pure silicon: Si (Z=14) 1s2 2s2 3p6 3s2 3p2 Valence number: 4 Doping by P (Z=15, valence=5) One extra el. n(egative)-type 5th el. in the “conduction band” Extra electron & proton w/o w/ extra proton extra proton Weakly bound donor levels At room temperature Thermal Ed =0.045 eV for phosphorous doping cf: Eg=1.1 eV Excitations from donor levels to conduction band much easier Majority carriers Electrons in conduction band Minority excitations carriers Holes in valence band Doping level Pure silicon # density of conduction el. at room temp • (n0)no-doping ~ 1016 m-3 Q: Doping for (n0)doping=106 (n0)no-doping (n0)doping= (n0)no-doping + nP nP 1022 m-3 cf: nSi 51028 m-3 nP 1 nSi 5 10 6 p-Type Semiconductors Doping One missing el p(ositive)-type Missing el in “valence band” w/ by Al (Z=13) missing proton Weakly bound acceptor levels At room temperature Thermal Ed =0.067 eV for aluminium doping cf: Eg=1.1 eV Excitations from valence band to acceptor levels much easier Majority carriers Holes in valence band Minority excitations carriers Electrons in conduction band 41-8 The p-n Junction Junction of p-type and n-type semicond. Junction plane Upon contact, …(no bias yet) Motions of the Majority Carriers Diffusion -e Diffusion current Idiff Idiff Space charge +e Depletion zone Contact potential difference V0 Idiff = 0 Motions of the Minority Carriers Minority Drift current Idrift Idrift Space charge somewhat relaxed Majority carriers & minority carriers Balance of Idiff & Idrift 41-9 The Junction Rectifier I vs. V p-n junction as a rectifier AC DC conversion Forward bias Reduce V0 Reduce V0 Narrower depletion zone Backward bias Enhance V0 Enhance V0 Wider depletion zone 41-10 The Light-Emitting Diode (LED) LED Light emission from p-n junction Photon or lattice vibration Forward bias p-n junction as LED Forward biased p-n junction Photon wavelength c c hc f Eg / h Eg Commercial LEDs in visible range Ex: Gallium (valence 3) doped with arsenic (valence 5, 60%) and phosphorous (valence 5, 40%) atoms • Eg=1.8 eV (red color) The Photo-Diode Photo-diode = (LED)-1 Photon Current Photon-induced transition Ex: TV remote control • Remote control : LED Generate a certain sequence of infrared photons • TV : Photo-diode Photon detection Electric signal The Junction Laser Stimulated emission in p-n junction Mirror Mirror Junction laser • Ex: Laser head in compact disc (CD) players 41-11 The Transistor Transistor Intentional control of on-off Application: Amplifier FET (Field Effect Transistor) Integrated circuits Transistors Capacitors Resistors etc. Intel Pentium chip (w/ ~7 million transistors) MOSFET (Metal-Oxide-Semiconductor-FET) MOSFET High speed on-off ~500 nm in length Gate voltage VGS Negatively charge gate Repel el. in n-channel down into substrate Wider depletion zone between p and n n-channel width reduced Larger resistance (off realized) The End