Solids Eisberg & Resnick Ch 13 & 14 RNave: http://hyperphysics.phy-astr.gsu.edu/hbase/solcon.html#solcon Alison Baski: http://www.courses.vcu.edu/PHYS661/pdf/01SolidState041.ppt Carl Hepburn, “Britney Spear’s Guide to Semiconductor Physics”. http://britneyspears.ac/lasers.htm http://hyperphysics.phy-astr.gsu.edu/hbase/solcon.html#solcon OUTLINE • Review Ionic / Covalent Molecules • Types of Solids (ER 13.2) • Band Theory (ER 13.3-.4) – basic ideas – description based upon free electrons – descriptions based upon nearly-free electrons • ‘Free’ Electron Models (ER 13.5-.7) • Temperature Dependence of Resistivity (ER 14.1) Ionic Bonds RNave, GSU at http://hyperphysics.phy-astr.gsu.edu/hbase/chemical/bond.html#c4 Ionic Bonds Ionic Bonding RNave, Georgia State Univ at hyperphysics.phy-astr.gsu.edu/hbase/molecule Covalent Bonds RNave, GSU at http://hyperphysics.phy-astr.gsu.edu/hbase/chemical/bond.html#c4 Covalent Bonding SYM spatial ASYM spin ASYM spatial SYM spin space-symmetric tend to be closer Covalent Bonding not really parallel, but spin-symmetric Stot = 1 Stot = 0 not really anti, but spin-asym space-symmetric tend to be closer TYPES OF SOLIDS (ER 13.2) CRYSTALINE BINDING • • • • molecular ionic covalent metallic Molecular Solids • • • • most organics inert gases O2 N2 H2 orderly collection of molecules held together by v. d. Waals gases solidify only at low Temps easy to deform & compress poor conductors Ionic Solids NaCl NaI KCl • individ atoms act like closed-shell, spherical, therefore binding not so directional • arrangement so that minimize nrg for size of atoms • • • • • tight packed arrangement poor thermal conductors no free electrons poor electrical conductors strong forces hard & high melting points lattice vibrations absorb in far IR to excite electrons requires UV, so ~transparent visible Ge Covalent Solids Si diamond • 3D collection of atoms bound by shared valence electrons • difficult to deform because bonds are directional • high melting points (b/c diff to deform) • no free electrons poor electrical conductors • most solids adsorb photons in visible opaque Fe Ni Co Metallic Solids config dhalf full • (weaker version of covalent bonding) • constructed of atoms which have very weakly bound outer electron • large number of vacancies in orbital (not enough nrg available to form covalent bonds) • electrons roam around (electron gas ) • excellent conductors of heat & electricity • absorb IR, Vis, UV opaque BAND STRUCTURE Isolated Atoms Diatomic Molecule Four Closely Spaced Atoms Six Closely Spaced Atoms as fn(R) the level of interest has the same nrg in each separated atom Two atoms Six atoms Solid of N atoms ref: A.Baski, VCU 01SolidState041.ppt www.courses.vcu.edu/PHYS661/pdf/01SolidState041.ppt Four Closely Spaced Atoms conduction band valence band Solid composed of ~NA Na Atoms as fn(R) 1s22s22p63s1 Sodium Bands vs Separation Rohlf Fig 14-4 and Slater Phys Rev 45, 794 (1934) Copper Bands vs Separation Rohlf Fig 14-6 and Kutter Phys Rev 48, 664 (1935) Differences down a column in the Periodic Table: IVA Elements same valence config Sandin The 4A Elements Band Spacings in Insulators & Conductors electrons free to roam electrons confined to small region RNave: http://hyperphysics.phy-astr.gsu.edu/hbase/solcon.html#solcon How to choose eF and Behavior of the Fermi function at band gaps Fermi Distribution for a selected eF Probability of an energy occuring (not normalized) 1.5 1 T=0 1000 5000 0.5 0 0 1 2 3 4 Energy n(e ) 1 e (e e F ) / kT 1 How does one choose/know eF If in unfilled band, eF is energy of highest energy electrons at T=0. If in filled band with gap to next band, eF is at the middle of gap. Fermions T=0 RNave: http://hyperphysics.phy-astr.gsu.edu/hbase/solcon.html#solcon Fermions T > 0 Number of Electrons at an Energy e In QStat, we were doing Tot KE e ne N e de 0 distrib fn Number of ways to have a particular energy Number of electrons at energy e # states probability of this nrg occurring # electrons at a given nrg Semiconductors ER13-9, -10 Semiconductors ~1/40 eV • • Types – Intrinsic – by thermal excitation or high nrg photon – Photoconductive – excitation by VIS-red or IR – Extrinsic – by doping • n-type • p-type ~1 eV Intrinsic Semiconductors Silicon Germanium RNave: http://hyperphysics.phy-astr.gsu.edu/hbase/solcon.html#solcon Doped Semiconductors lattice p-type dopants n-type dopants 5A doping in a 4A lattice 5A in 4A lattice 3A in 4A lattice 5A in 4A lattice 3A in 4A lattice ‘Free-Electron’ Models • Free Electron Model (ER 13-5) • Nearly-Free Electron Model (ER13-6,-7) – Version 1 – SP221 – Version 2 – SP324 – Version 3 – SP425 • . ********************************************************* • Free-Electron Model – – – – – • • Spatial Wavefunctions Energy of the Electrons Fermi Energy Density of States dN/dE Number of States as fn NRG E&R 13.5 E&R 13.5 Nearly-Free Electron Model (Periodic Lattice Effects) – v2 E&R 13.6 Nearly-Free Electron Model (Periodic Lattice Effects) – v3 E&R 13.6 Free-Electron Model (ER13-5) classical description e p2 2m 2 K 2 2m Quantum Mechanical Viewpoint In a 3D slab of metal, e’s are free to move but must remain on the inside 2 2 0 E 2m Solutions are of the form: xyz 8 L3 sin k x x sin k y y sin k z z nz L With nrg’s: e h2 2 2 2 n n n x y z 8mL2 At T = 0, all states are filled up to the Fermi nrg e fermi h2 2 2 2 n n n x y z 8mL2 max A useful way to keep track of the states that are filled is: nx2 ny2 nz2 n2 max e fermi h2 8mL2 n 2 max total number of states up to an energy efermi: N 2 e fermi h 3N 8m V 2 1 8 volume 21 of 8 sphere 3 4 nmax 3 2/3 # states/volume ~ # free e’s / volume Sample Numerical Values for Copper slab N V e fermi = 8.96 gm/cm3 1/63.6 amu h 3N 8m V 2 6e23 = 8.5e22 #/cm3 = 8.5e28 #/m3 2/3 efermi = 7 eV nmax = 4.3 e 7 so we can easily pretend that there’s a smooth distrib of nxnynz-states Density of States Tot KE 0 How many combinations of are there within an energy interval e to e + de ? e fermi N dN dN dE h 3N 8m V 2 V 8mE 2 3 h V 3 2/3 3/ 2 1/ 2 3 8mE 2 2 h 8 V 3 2 m 3 h 1/ 2 8m h2 E 1/ 2 dE e ne N e de At T ≠ 0 the electrons will be spread out among the allowed states How many electrons are contained in a particular energy range? number of ways to have a particular energy 8 V 3 2m 3 h 1/ 2 E probabilit y of this energy occuring 1 1/ 2 e ( E e f ) / kT 1 this assumes there are no other issues Distribution of States: Simple Free-Electron Model vs Reality Problems with Free Electron Model (ER13-6, -7) **************************** 1) 2) 3) Bragg reflection . . Other Problems with the Free Electron Model • • • • • • • graphite is conductor, diamond is insulator variation in colors of x-A elements temperature dependance of resistivity resistivity can depend on orientation of crystal & current I direction frequency dependance of conductivity variations in Hall effect parameters resistance of wires effected by applied B-fields • . • . • . Nearly-Free Electron Model version 1 – SP221 k /2 a /2 2 /2 k 2 a /2 /2 k Nearly-Free Electron Model version 2 – SP324 This treatment assumes that when a reflection occurs, it is 100%. • • • Bloch Theorem Special Phase Conditions, k = +/- m /a the Special Phase Condition k = +/- /a ~~~~~~~~~~ (x) ~ u e i(kx-wt) amplitude In reality, lower energy waves are sensitive to the lattice: Bloch’s Theorem (x) ~ u(x) e i(kx-wt) Amplitude varies with location u(x) = u(x+a) = u(x+2a) = …. (x) ~ u(x) e i(kx-wt) u(x+a) = u(x) (x+a) e -i(kx+ka-wt) (x) e -i(kx-wt) (x+a) e ika (x) Something special happens with the phase when e ika = 1 ka = +/ m k a , 2 What it is ? m = 0 not a surprise m = 1, 2, 3, … a , ... Consider a set of waves with +/ k-pairs, e.g. k = + /a moves k k = /a moves This defines a pair of waves moving right & left Two trivial ways to superpose these waves are: + ~ e ikx + e ikx ~ e ikx e ikx + ~ 2 cos kx ~ 2i sin kx a Kittel + ~ 2 cos kx ~ 2i sin kx |+|2 ~ 4 cos2 kx ||2 ~ 4 sin2 kx Free-electron Nearly Free-electron Kittel Discontinuities occur because the lattice is impacting the movement of electrons. Effective Mass m* A method to force the free electron model to work in the situations where there are complications free electron KE functional form e 2 k 2 2 m* ER Ch 13 p461 starting w/ eqn (13-19b) Effective Mass m* -- describing the balance between applied ext-E and lattice site reflections 1 m* 1 2e 2 k 2 m* a = S Fext q Eext 2) greater curvature, 1/m* > 1/m > 0, m* < m net effect of ext-E and lattice interaction provides additional acceleration of electrons m = m* greater |curvature| but negative, At inflection pt net effect of ext-E and lattice interaction de-accelerates electrons 1) No distinction between m & m*, m = m*, “free electron”, lattice structure does not apply additional restrictions on motion. Another way to look at the discontinuities e 2 k 2 2m apply perturbation from lattice 2 k 2 2 m* Shift up implies effective mass has decreased, m* < m, allowing electrons to increase their speed and join faster electrons in the band. The enhanced e-lattice interaction speeds up the electron. Shift down implies effective mass has increased, m* > m, prohibiting electrons from increasing their speed and making them become similar to other electrons in the band. The enhanced e-lattice interaction slows down the electron From earlier: Even when above barrier, reflection and transmission coefficients can increase and decrease depending upon the energy. change in motion due to applied field enhanced by change in reflection coefficients change in motion due to reflections is more significant than change in motion due to applied field Nearly-Free Electron Model version 3 à la Ashcroft & Mermin, Solid State Physics This treatment recognizes that the reflections of electron waves off lattice sites can be more complicated. A reminder: Waves from the left behave like: from eiKx r e iKx the left from t eiKx the left e 2 K 2 2m Waves from the right behave like: from the right t e iKx from e iKx r eiKx the right e 2 K 2 2m sum A left B right unknown weights Bloch’s Theorem defines periodicity of the wavefunctions: sum x a eika sum x x a eika sum x sum Related to Lattice spacing Applying the matching conditions at x a/2 x a eika sum x sum sum x a eika sum x A + B left right A + B left right A + B left A + B right left right And eliminating the unknown constants A & B leaves: t 2 r 2 iKa 1 iKa cos ka e e 2t 2t e 2 K 2 2m For convenience (or tradition) set: t t ei 1 t r 2 2 r i r ei cosKa cos ka t Related to Energy e cosKa cos ka t Related to possible Lattice spacings 2 K 2 2m allowed solution regions allowed solution regions Superconductivity ER 14-1, 13-4 Temperature Dependence of Resistivity R Nave: http://hyperphysics.phy-astr.gsu.edu/hbase/solids/supcon.html#c1 Joe Eck: superconductors.org Temperature Dependence of Resistivity R L A • Conductors – Resistivity increases with increasing Temp – Temp t but same # conduction e-’s • Semiconductors & Insulators – Resistivity decreases with increasing Temp – Temp t but more conduction e-’s First observed Kamerlingh Onnes 1911 Note: The best conductors & magnetic materials tend not to be superconductors (so far) Superconductors.org Only in nanotubes Superconductor Classifications • Type I – tend to be pure elements or simple alloys – = 0 at T < Tcrit – Internal B = 0 (Meissner Effect) – At jinternal > jcrit, no superconductivity – At Bext > Bcrit, no superconductivity – Well explained by BCS theory • Type II – tend to be ceramic compounds – Can carry higher current densities ~ 1010 A/m2 – Mechanically harder compounds – Higher Bcrit critical fields – Above Bext > Bcrit-1, some superconductivity Superconductor Classifications Type I Bardeen, Cooper, Schrieffer 1957, 1972 “Cooper Pairs” e Q: Stot=0 or 1? L? J? e Symmetry energy ~ 0.01 eV Popular Bad Visualizations: correlation lengths Pairs are related by momentum ±p, NOT position. Sn 230 nm Al 1600 Pb 83 Nb 38 Best conductors best ‘free-electrons’ no e – lattice interaction not superconducting More realistic 1-D billiard ball picture: Cooper Pairs are ±k sets Furthermore: “Pairs should not be thought of as independent particles” -- Ashcroft & Mermin Ch 34 • Experimental Support of BCS Theory – Isotope Effects – Measured Band Gaps corresponding to Tcrit predictions – Energy Gap decreases as Temp Tcrit – Heat Capacity Behavior Normal Conductor Semiconductor or Superconductor Another fact about Type I: -- Interrelationship of Bcrit and Tcrit Type II Yr mixed normal/super Q: does BCS apply ? Composition Tc May 2006 InSnBa4Tm4Cu6O18+ 150 2004 Hg0.8Tl0.2Ba2Ca2Cu3O8.33 138 1986 (La1.85Ba.15)CuO4 30 YBa2Cu3O7 93 actual ~ 8 mm Sandin Type II – mixed phases fluxon Q: does BCS apply ? Y Ba2 Cu3 O7 crystalline may control the electronic config of the conducting layer La2-x Bax Cu O2 solid solution Another fact about Type II: -- Interrelationship of Bcrit and Tcrit Applications OR Other Features of Superconductors http://superconductors.org/Uses.htm Meissner Effect Magnetic Levitation – Meissner Effect Kittel states this explusion effect is not clearly directly connected to the = 0 effects Q: Why ? Magnetic Levitation – Meissner Effect MLX01 Test Vehicle 2003 581 km/h 361 mph 2005 80,000+ riders 2005 tested passing trains at relative 1026 km/h http://www.rtri.or.jp/rd/maglev/html/english/maglev_frame_E.html Maglev in Germany (sc? idi) 32 km track 550,000 km since 1984 Design speed 550 km/h NOTE(061204): I’m not so sure this track is superconducting. The MagLev planned for the Munich area will be. France is also thinking about a sc maglev. Josephson Junction ~ 2 nm Recall: Aharonov-Bohm Effect -- from last semester affects the phase of a wavefunction ~ e i ( p eA ) r1 / ~ eikx ~ eipx / A Source B ~ e i ( p eA ) r2 / SQUID superconducting quantum interference device o ~ oe ileft ~ o e i right ~ o ei fn (location ) Add up change in flux as go around loop dl n 2 loop qB Aharonov Bohm 2 B n q 2 (2e) 2.07 10 15 Telsa m 2 Typical B fields (Tesla) (# flux quanta) http://www.csiro.au/science/magsafe.html Finding 'objects of interest' at sea with MAGSAFE MAGSAFE is a new system for locating and identifying submarines. Operators of MAGSAFE should be able to tell the range, depth and bearing of a target, as well as where it’s heading, how fast it’s going and if it’s diving. Building on our extensive experience using highly sensitive magnetic sensors known as Superconducting QUantum Interference Devices (SQUIDs) for minerals exploration, MAGSAFE harnesses the power of three SQUIDs to measure slight variations in the local magnetic field. MAGSAFE will be able to locate targets without flying close to the surface. Image courtesy Department of Defence. MAGSAFE has higher sensitivity and greater immunity to external noise than conventional Magnetic Anomaly Detector (MAD) systems. This is especially relevant to operation over shallow seawater where the background noise may 100 times greater than the noise floor of a MAD instrument. http://www.csiro.au/science/magsafe.html Phillip Schmidt etal. Exploration Geophysics 35, 297 (2004). Arian Lalezari SQUID 2 nm 1014 T SQUID threshold Heart signals 10 10 T Brain signals 10 13 T • • • • • • • Fundamentals of superconductors: – http://www.physnet.uni-hamburg.de/home/vms/reimer/htc/pt3.html Basic Introduction to SQUIDs: – http://www.abdn.ac.uk/physics/case/squids.html Detection of Submarines – http://www.csiro.au/science/magsafe.html Fancy cross-referenced site for Josephson Junctions/Josephson: – http://en.wikipedia.org/wiki/Josephson_junction – http://en.wikipedia.org/wiki/B._D._Josephson SQUID sensitivity and other ramifications of Josephson’s work: – http://hyperphysics.phy-astr.gsu.edu/hbase/solids/squid2.html Understanding a SQUID magnetometer: – http://hyperphysics.phy-astr.gsu.edu/hbase/solids/squid.html#c1 Some exciting applications of SQUIDs: – http://www.lanl.gov/quarterly/q_spring03/squid_text.shtml • • • • • Relative strengths of pertinent magnetic fields – http://www.physics.union.edu/newmanj/2000/SQUIDs.htm The 1973 Nobel Prize in physics – http://nobelprize.org/physics/laureates/1973/ Critical overview of SQUIDs – http://homepages.nildram.co.uk/~phekda/richdawe/squid/popular/ Research Applications – http://boojum.hut.fi/triennial/neuromagnetic.html Technical overview of SQUIDs: – http://www.finoag.com/fitm/squid.html – http://www.cmp.liv.ac.uk/frink/thesis/thesis/node47.html Redraw LHS Sn 230 nm Al 1600 Pb 83 Nb 38 Best conductors best ‘free-electrons’ no e – lattice interaction not superconducting

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