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OUTLINE Review Ionic / Covalent Molecules Types of Solids (ER 13.2)
Band Theory (ER ) basic ideas description based upon free electrons descriptions based upon nearly-free electrons ‘Free’ Electron Models (ER ) Temperature Dependence of Resistivity (ER 14.1) 3
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Ionic Bonds RNave, GSU at 4
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Ionic Bonds 5
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Ionic Bonding RNave, Georgia State Univ at hyperphysics.phy-astr.gsu.edu/hbase/molecule 6
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Covalent Bonds RNave, GSU at 7
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Covalent Bonding space-symmetric tend to be closer SYM ASYM
spatial spin ASYM SYM spatial spin space-symmetric tend to be closer 8
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Covalent Bonding space-symmetric tend to be closer
not really parallel, but spin-symmetric Stot = 1 Stot = 0 not really anti, but spin-asym space-symmetric tend to be closer 9
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TYPES OF SOLIDS (ER 13.2) CRYSTALINE BINDING
molecular ionic covalent metallic 11
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most organics inert gases O2 N2 H2 Molecular Solids orderly collection of molecules held together by v. d. Waals gases solidify only at low Temps easy to deform & compress poor conductors 12
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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 13
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Ge Si diamond Covalent Solids 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 14
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Metallic Solids (weaker version of covalent bonding)
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 15
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BAND STRUCTURE 17
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Isolated Atoms 18
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Diatomic Molecule 19
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Four Closely Spaced Atoms
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Six Closely Spaced Atoms as fn(R)
the level of interest has the same nrg in each separated atom 21
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ref: A.Baski, VCU 01SolidState041.ppt
Two atoms Six atoms Solid of N atoms ref: A.Baski, VCU 01SolidState041.ppt 22
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Four Closely Spaced Atoms
conduction band valence band 23
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Solid composed of ~NA Na Atoms as fn(R)
1s22s22p63s1 24
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Sodium Bands vs Separation
There’s only one band for sodium. An isolated s-orbital can hold 2 electrons. But since each Na only has one electron, the ‘3s’ band would be half full. Suppose we wanted to sketch the function N(e) Rohlf Fig 14-4 and Slater Phys Rev 45, 794 (1934) 25
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Copper Bands vs Separation
An individual copper atom is 3d10 4s1. At the separation in bulk copper, all the band overlap and there is no band gap. Rohlf Fig 14-6 and Kutter Phys Rev 48, 664 (1935) 26
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Differences down a column in the Periodic Table: IV-A Elements
same valence config Sandin 27
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The 4A Elements 28
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Band Spacings in Insulators & Conductors
electrons free to roam electrons confined to small region RNave: 29
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How to choose F and Behavior of the Fermi function at band gaps
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Fermi Distribution for a selected F
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How does one choose/know F
If in unfilled band, F is energy of highest energy electrons at T=0. If in filled band with gap to next band, F is at the middle of gap. 32
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Fermions T=0 RNave: 33
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Fermions T > 0 34
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Number of Electrons at an Energy
In QStat, we were doing distrib fn Number of ways to have a particular energy Number of electrons at energy 35
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# states probability of this nrg occurring # electrons at a given nrg
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Semiconductors ER13-9, -10 39
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Semiconductors ~1/40 eV ~1 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 40
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Intrinsic Semiconductors
Silicon Germanium RNave: 41
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Doped Semiconductors lattice p-type dopants n-type dopants 42
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5A doping in a 4A lattice 5A in 4A lattice 3A in 4A lattice 43
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5A in 4A lattice 3A in 4A lattice 44
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‘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 . 46
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Free-Electron Model Spatial Wavefunctions Energy of the Electrons Fermi Energy Density of States dN/dE E&R 13.5 Number of States as fn NRG 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 47
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Free-Electron Model (ER13-5)
classical description 48
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Quantum Mechanical Viewpoint
In a 3D slab of metal, e’s are free to move but must remain on the inside Solutions are of the form: With nrg’s: 49
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At T = 0, all states are filled up to the Fermi nrg
A useful way to keep track of the states that are filled is: 1. In a partially filled band, eF is the max energy electron at T=0. 50
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total number of states up to an energy fermi:
Suppose we wanted to count all the dots. We can use this expression to write eF in terms of quantities that we can actually know easily. # states/volume ~ # free e’s / volume 51
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Sample Numerical Values for Copper slab
= gm/cm3 1/63.6 amu 6e23 = 8.5e22 #/cm3 = 8.5e28 #/m3 fermi = 7 eV nmax = 4.3 e 7 so we can easily pretend that there’s a smooth distrib of nxnynz-states 52
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Density of States How many combinations of are there
within an energy interval to + d ? 53
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At T ≠ 0 the electrons will be spread out among the allowed states
How many electrons are contained in a particular energy range? 54
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this assumes there are no other issues
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Distribution of States: Simple Free-Electron Model vs Reality
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Problems with Free Electron Model (ER13-6, -7)
* * * * * * * * * * * * * * * * * * * * * * * * * * * * Bragg reflection . 58
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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 . 59
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Nearly-Free Electron Model version 1 – SP221
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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 61
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~~~~~~~~~~ (x) ~ u e i(kx-t) (x) ~ u(x) e i(kx-t) Bloch’s Theorem
amplitude In reality, lower energy waves are sensitive to the lattice: (x) ~ u(x) e i(kx-t) Bloch’s Theorem Amplitude varies with location u(x) = u(x+a) = u(x+2a) = …. 62
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What it is ? (x) ~ u(x) e i(kx-t) u(x+a) = u(x)
(x+a) e -i(kx+ka-t) (x) e -i(kx-t) (x+a) e ika (x) Something special happens with the phase when e ika = 1 ka = +/ m m = not a surprise m = 1, 2, 3, … 1. Of course this is simply restating our Bloch assumption. What it is ? 63
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Consider a set of waves with +/ k-pairs, e.g.
k = + /a moves 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 64
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+ ~ 2 cos kx ~ 2i sin kx +|2 ~ 4 cos2 kx |2 ~ 4 sin2 kx
So there are two types of solutions for the interwoved wavefunctions at one of these “special k-values”. In this figure one sees that in the cos-type solution has electrons concentrated at the lattice sites and the sin-type solution, the electrons are concentrated on the barriers. These two solutions will have different average KE, and taking into account the electrons interactions with the lattice, diff total energy. Kittel 65
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Free-electron Nearly Free-electron Kittel
This figure shows the split in energy that occurs at these “special k-values” This gives rise to a forbidden set of energies. There are certain energies, i.e. wavelengths where the electrons cannot propagate down the lattice. Waves are strongly reflected at the lattice sites and free movement is prohibited. Kittel Discontinuities occur because the lattice is impacting the movement of electrons. 66
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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 1. We are going to discuss several (4) ways to interpret the discontinuity. ER Ch 13 p461 starting w/ eqn (13-19b) 67
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Effective Mass m* -- describing the balance between applied ext-E and lattice site reflections
m* a = Fext q Eext 68
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2) 1) 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, net effect of ext-E and lattice interaction de-accelerates electrons At inflection pt No distinction between m & m*, m = m*, “free electron”, lattice structure does not apply additional restrictions on motion. 1) 69
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Another way to look at the discontinuities
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 70
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Even when above barrier, reflection and transmission coefficients can
From earlier: Even when above barrier, reflection and transmission coefficients can increase and decrease depending upon the energy. 1. These discontinuities are related to reflections of the electron waves off the lattice structure. The lattice presents potential barriers. We know from our study earlier that interesting things can happen to the transmission and reflection coefficients when the energy goes above the barrier height. 71
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enhanced by change in reflection coefficients
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 72
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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. 73
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A reminder: 74
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Waves from the left behave like:
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Waves from the right behave like:
1. Big K is related to the electron kinetic energy. 76
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Bloch’s Theorem defines periodicity of the wavefunctions:
unknown weights Bloch’s Theorem defines periodicity of the wavefunctions: The sum of waves in a region are a linear combination of left & right Small k is related to the lattice spacing. Big K is related to the electron kinetic energy. Related to Lattice spacing 77
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Applying the matching conditions at x a/2
A + B left right A + B left right A + B left right A + B left right And eliminating the unknown constants A & B leaves: 1. Note the complicated details of the ‘barriers’ are hidden in r & t. 78
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For convenience (or tradition) set:
1. Note the complicated details of the ‘barriers’ are hidden in r & t. 79
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allowed solution regions
Related to Energy Related to possible Lattice spacings 1. K and k are different which makes the cosines oscillate with different … fast and slow. Here we have k big and K small. 2. The problem can be solved graphically. The is drawn using a trivial model for t that assumes t increases linearly. allowed solution regions 80
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allowed solution regions
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Superconductivity ER 14-1, 13-4
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Temperature Dependence of Resistivity
R Nave: Joe Eck: superconductors.org 83
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Temperature Dependence of Resistivity
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Semiconductors & Insulators
Resistivity increases with increasing Temp Temp but same # conduction e-’s Semiconductors & Insulators Resistivity decreases with increasing Temp Temp but more conduction e-’s 85
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First observed Kamerlingh Onnes 1911
Groenigen, Netherlands 1908 Liquified 4He 1911 reached 0.9 K Nobel Prize 1913 William Thomson (developed Kelvin scale) thought electron motion would cease at 0K, therefore no conduction. Onnes and others thought resistance would gradually decrease to zero. First observed Kamerlingh Onnes 1911 86
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Note: The best conductors & magnetic materials tend not to be superconductors (so far)
Superconductors.org Only in nanotubes 87
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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 88
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Superconductor Classifications
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Type I Bardeen, Cooper, Schrieffer 1957, 1972 “Cooper Pairs” e e
1. Bardeen. 2 Nobels. Transitor & bcs e e Symmetry energy ~ 0.01 eV Q: Stot=0 or 1? L? J? 90
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Popular Bad Visualizations:
correlation lengths Sn nm Al Pb Nb Pairs are related by momentum ±p, NOT position. 1. Here are some popular visualizations of Cooper pairs, but the are NOT correct. Best conductors best ‘free-electrons’ no e – lattice interaction not superconducting 91
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More realistic 1-D billiard ball picture:
Cooper Pairs are ±k sets Furthermore: Q: Are 2 e-’s in correlated in position ? A: NO! Wave-particle duality From a particle viewpt pairs are constantly forming & breaking From a wave viewpt, the ‘correlation length’ describes the region of space over which the superconducting effects occur about a distortion. “Pairs should not be thought of as independent particles” -- Ashcroft & Mermin Ch 34 92
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Experimental Support of BCS Theory
Isotope Effects Measured Band Gaps corresponding to Tcrit predictions Energy Gap decreases as Temp Tcrit Heat Capacity Behavior Isotope effects – inertia of lattice ions. Superconductivity difficult to achieve when lattice harder to deform. HC shows strong growth from a zero value. Suggests that there’s some sort of threshold that we’ve gone over, such as a band - like in a semiconductor. Band Gaps – If do scatter plot of measured band gap energies vs critical temp, discover there is a correlation. Band gaps small and consistent with BCS estimates of KE. The band gap energy is directly related to the “kinetic energy” at the transition temp. Egap as a function of temp. Note that there is a common scaling law indicating some type of universal behaviour. – as approach Tc, thermal energy makes pairing difficult. Thermal agitation hides the correlation structures in the lattice. 93
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Semiconductor or Superconductor Normal Conductor
In a normal conductor, there is no band gap between the filled and unfilled levels. Normal conductors to not exhibit macroscopic quantum features. In a superconducting material, there is a very tiny band gap ~10-4 eV, which is very small compared to the 1-2 eV occurring in semiconductors. This tiny band gap would only be apparent at very low temperatures. 94
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Another fact about Type I: -- Interrelationship of Bcrit and Tcrit
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Type II Tc Yr Composition 150 138 30 93 mixed normal/super
May 2006 InSnBa4Tm4Cu6O18+ 150 2004 Hg0.8Tl0.2Ba2Ca2Cu3O8.33 138 1986 (La1.85Ba.15)CuO4 30 YBa2Cu3O7 93 1. Type-IIs have mixed sc/normal phases in the sample. mixed normal/super Q: does BCS apply ? 96
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actual ~ 8 m Sandin 97
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Type II – mixed phases fluxon Q: does BCS apply ? 98
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La2-x Bax Cu O2 solid solution
Y Ba2 Cu3 O crystalline may control the electronic config of the conducting layer La2-x Bax Cu O solid solution 99
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Another fact about Type II: -- Interrelationship of Bcrit and Tcrit
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Applications OR Other Features of Superconductors
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Meissner Effect 102
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Magnetic Levitation – Meissner Effect
1. One could envision a surface process as diagrammed to explain the Meissner Effect. However the sketched idea for the induced currents is not quite right. If one stacks two type-II disks, the magnet will be raised higher. There’s some sort of bulk process going on. Kittel states this explusion effect is not clearly directly connected to the = 0 effects Q: Why ? 103
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Magnetic Levitation – Meissner Effect
MLX01 Test Vehicle 581 km/h mph ,000+ riders 2005 tested passing trains at relative 1026 km/h 104
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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. 105
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Josephson Junction ~ 2 nm
Josephson Junction is created by placing a non-superconductor between two different superconductors. If the gap is thin, then electrons can tunnel through the barrier. If the gap is really thin, on the order of “the Cooper-pair correlation length”, then the lattice distortions responsible for the “Cooper pairs” can be transmitted. This current is observable. This is the “DC Josephson Effect”. If a voltage difference is applied btw the two superconductors, the tunneling is modified, the phase difference across the gap increases, and also the effective current. Since our wavefunctions are periodic, the constantly increasing phase difference will result in an oscillating current. This is the “AC Josephson Effect”. NIST in Gaithersburg has a development program to construction more accurate voltage standards using this effect. 106
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Recall: Aharonov-Bohm Effect -- from last semester
affects the phase of a wavefunction A Source B 107
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SQUID superconducting quantum interference device
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Add up change in flux as go around loop
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Typical B fields (Tesla) (# flux quanta) 110
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Finding 'objects of interest' at sea with MAGSAFE
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.
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Phillip Schmidt etal. Exploration Geophysics 35, 297 (2004).
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Arian Lalezari
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SQUID 2 nm 1014 T SQUID threshold Heart signals 10 10 T
Superconducting Quantum Interference Device One starts with a bias current in the device. If a B-field threads the center, then a vector potential will exist in the ring region, modifying the phases near each junction. Because wavefunctions have to be single valued at any location in the ring, if one were to make a loop around the ring measuring the phase change, one would have to come up with a multiple of 2pi. This constraint and the fact that we’re observing wavefunction phases makes the device incredible sensitive to changes in the central B-field. 1014 T SQUID threshold Heart signals 10 10 T Brain signals 10 13 T 114
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Fundamentals of superconductors:
Basic Introduction to SQUIDs: Detection of Submarines Fancy cross-referenced site for Josephson Junctions/Josephson: SQUID sensitivity and other ramifications of Josephson’s work: Understanding a SQUID magnetometer: Some exciting applications of SQUIDs: 115
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Relative strengths of pertinent magnetic fields
The 1973 Nobel Prize in physics Critical overview of SQUIDs Research Applications Technical overview of SQUIDs: 116
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Sn nm Al 1600 Pb Nb 38 Redraw LHS Best conductors best ‘free-electrons’ no e – lattice interaction not superconducting 117
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