[{"data":1,"prerenderedAt":1262},["ShallowReactive",2],{"subject:condensed-matter":3,"course-wordcounts":75,"nav:condensed-matter":987},{"id":4,"title":5,"blurb":6,"body":7,"brief":18,"category":56,"description":57,"draft":58,"extension":59,"meta":60,"module":15,"navigation":25,"path":61,"practice":62,"rawbody":63,"readingTime":64,"seo":69,"sources":70,"status":71,"stem":72,"summary":15,"topics":73,"__hash__":74},"course\u002F07.condensed-matter\u002Findex.md","Condensed Matter Physics","How atoms assemble into solids and where their electrons go — bonding and\ncrystal structure, phonons, the free-electron gas and band theory,\nsemiconductors, magnetism, and superconductivity.\n",{"type":8,"value":9,"toc":14},"minimark",[10],[11,12,13],"p",{},"Condensed matter physics is the study of the solid and liquid states as the\ncollective behavior of vast numbers of atoms and electrons. The sequence begins\nwith bonding and molecular spectra, then builds crystal structure and the\nreciprocal lattice — the periodic scaffolding underneath everything that follows.\nFrom there it develops lattice dynamics and phonons, the free-electron Fermi gas,\nand the band theory of periodic solids, which together explain why some materials\nconduct and others do not. The later chapters apply that machinery: semiconductors\nand their devices, dielectrics and ferroelectrics, the microscopic origins of\nmagnetism, and superconductivity, before closing on the low-dimensional physics of\nnanostructures and graphene. It follows Kittel, Ashcroft & Mermin, and Simon, with\nTipler & Llewellyn for the foundational material. Each topic rests on the\nperiodicity and counting arguments established before it.",{"title":15,"searchDepth":16,"depth":16,"links":17},"",2,[],[19,21,26,30,32,34,38,40,42,46,48,52,54],{"p":20},"Condensed matter physics asks what happens when you bring \u003Cstrong>10²³\natoms\u003C\u002Fstrong> together and let them settle. The answer is almost never the\nsum of the parts — order, rigidity, conduction, and magnetism are\n\u003Cem>collective\u003C\u002Fem> properties that no single atom possesses.\n",{"fig":22,"n":23,"caption":24,"large":25},"cm-lattice","001","A crystal is a lattice: identical atoms repeat with perfect periodicity,\nand one unit cell tiles the whole solid.\n",true,{"fig":27,"n":28,"caption":29},"cm-bands","002","Band theory: a filled valence band and an empty conduction band, split by a\nforbidden gap E\u003Csub>g\u003C\u002Fsub>.\n",{"p":31},"The starting point is structure. Most solids are \u003Cstrong>crystals\u003C\u002Fstrong> —\natoms arranged on a periodic lattice — and that periodicity is the single\nfact from which nearly everything else follows. Choose a unit cell and a\nbasis, and the reciprocal lattice hands you the natural language for waves\ninside the solid.\n",{"p":33},"Periodicity reshapes the electrons. In a periodic potential the allowed\nenergies collect into \u003Cstrong>bands\u003C\u002Fstrong> separated by \u003Cem>gaps\u003C\u002Fem>, and\nwhere the last electrons land — mid-band or at a band edge — decides whether\na material is a metal, an insulator, or a semiconductor.\n",{"fig":35,"n":36,"caption":37},"cm-phonon","003","A phonon: a quantized lattice vibration travelling through the atoms, each\nbobbing about its rest position.\n",{"p":39},"The lattice itself is never still. Its atoms vibrate in collective normal\nmodes, and quantizing those modes gives \u003Cstrong>phonons\u003C\u002Fstrong> — the\nparticle-like carriers of sound and heat that also scatter electrons and set\na metal's resistance.\n",{"p":41},"Before bands, the crudest model already carries a metal a long way: treat the\nconduction electrons as a \u003Cem>free gas\u003C\u002Fem>. Pauli exclusion stacks them into\nmomentum space up to the Fermi energy, and the heat capacity, magnetism, and\ntransport of a metal all trace back to what happens at that surface.\n",{"fig":43,"n":44,"caption":45},"cm-fermi","004","The free-electron Fermi sphere: states fill momentum space up to k\u003Csub>F\u003C\u002Fsub>,\nand the surface governs the metal.\n",{"p":47},"From this foundation the subject fans out: \u003Cstrong>semiconductors\u003C\u002Fstrong>\nand the doped junctions that make devices, dielectrics and ferroelectrics,\nand the several distinct origins of \u003Cstrong>magnetism\u003C\u002Fstrong> in solids.\n",{"fig":49,"n":50,"caption":51},"cm-meissner","005","Superconductivity: a superconductor expels magnetic flux (B = 0 inside),\nlevitating a magnet by the Meissner effect.\n",{"p":53},"The dramatic finale is \u003Cstrong>superconductivity\u003C\u002Fstrong> — below a critical\ntemperature, electrons bind into Cooper pairs, resistance vanishes, and the\nmaterial expels magnetic flux entirely, a purely quantum effect made visible\nat human scale.\n",{"p":55},"Throughout, one move recurs: \u003Cem>find the symmetry, count the states, and let\nthe collective behavior emerge\u003C\u002Fem>. The course follows Kittel, Ashcroft &\nMermin, and Simon, ending in the low-dimensional physics of nanostructures\nand graphene.\n","physics","Condensed matter physics explains the solid and liquid states as the collective\nbehavior of enormous numbers of atoms and electrons. This course builds from\nmolecular bonding and spectra through crystal structure and the reciprocal\nlattice, lattice dynamics and phonons, the free-electron Fermi gas, the band\ntheory of periodic solids, semiconductors and their devices, dielectrics and\nferroelectrics, magnetism in solids, superconductivity, and the low-dimensional\nphysics of nanostructures and graphene. It follows Kittel, Ashcroft & Mermin,\nand Simon, with Tipler & Llewellyn for the foundational material.\n",false,"md",{},"\u002Fcondensed-matter",[],"---\ntitle: Condensed Matter Physics\nstatus: available\ncategory: physics\nblurb: |\n  How atoms assemble into solids and where their electrons go — bonding and\n  crystal structure, phonons, the free-electron gas and band theory,\n  semiconductors, magnetism, and superconductivity.\ndescription: |\n  Condensed matter physics explains the solid and liquid states as the collective\n  behavior of enormous numbers of atoms and electrons. This course builds from\n  molecular bonding and spectra through crystal structure and the reciprocal\n  lattice, lattice dynamics and phonons, the free-electron Fermi gas, the band\n  theory of periodic solids, semiconductors and their devices, dielectrics and\n  ferroelectrics, magnetism in solids, superconductivity, and the low-dimensional\n  physics of nanostructures and graphene. It follows Kittel, Ashcroft & Mermin,\n  and Simon, with Tipler & Llewellyn for the foundational material.\nbrief:\n  - p: |\n      Condensed matter physics asks what happens when you bring \u003Cstrong>10²³\n      atoms\u003C\u002Fstrong> together and let them settle. The answer is almost never the\n      sum of the parts — order, rigidity, conduction, and magnetism are\n      \u003Cem>collective\u003C\u002Fem> properties that no single atom possesses.\n  - fig: cm-lattice\n    n: \"001\"\n    caption: |\n      A crystal is a lattice: identical atoms repeat with perfect periodicity,\n      and one unit cell tiles the whole solid.\n    large: true\n  - fig: cm-bands\n    n: \"002\"\n    caption: |\n      Band theory: a filled valence band and an empty conduction band, split by a\n      forbidden gap E\u003Csub>g\u003C\u002Fsub>.\n  - p: |\n      The starting point is structure. Most solids are \u003Cstrong>crystals\u003C\u002Fstrong> —\n      atoms arranged on a periodic lattice — and that periodicity is the single\n      fact from which nearly everything else follows. Choose a unit cell and a\n      basis, and the reciprocal lattice hands you the natural language for waves\n      inside the solid.\n  - p: |\n      Periodicity reshapes the electrons. In a periodic potential the allowed\n      energies collect into \u003Cstrong>bands\u003C\u002Fstrong> separated by \u003Cem>gaps\u003C\u002Fem>, and\n      where the last electrons land — mid-band or at a band edge — decides whether\n      a material is a metal, an insulator, or a semiconductor.\n  - fig: cm-phonon\n    n: \"003\"\n    caption: |\n      A phonon: a quantized lattice vibration travelling through the atoms, each\n      bobbing about its rest position.\n  - p: |\n      The lattice itself is never still. Its atoms vibrate in collective normal\n      modes, and quantizing those modes gives \u003Cstrong>phonons\u003C\u002Fstrong> — the\n      particle-like carriers of sound and heat that also scatter electrons and set\n      a metal's resistance.\n  - p: |\n      Before bands, the crudest model already carries a metal a long way: treat the\n      conduction electrons as a \u003Cem>free gas\u003C\u002Fem>. Pauli exclusion stacks them into\n      momentum space up to the Fermi energy, and the heat capacity, magnetism, and\n      transport of a metal all trace back to what happens at that surface.\n  - fig: cm-fermi\n    n: \"004\"\n    caption: |\n      The free-electron Fermi sphere: states fill momentum space up to k\u003Csub>F\u003C\u002Fsub>,\n      and the surface governs the metal.\n  - p: |\n      From this foundation the subject fans out: \u003Cstrong>semiconductors\u003C\u002Fstrong>\n      and the doped junctions that make devices, dielectrics and ferroelectrics,\n      and the several distinct origins of \u003Cstrong>magnetism\u003C\u002Fstrong> in solids.\n  - fig: cm-meissner\n    n: \"005\"\n    caption: |\n      Superconductivity: a superconductor expels magnetic flux (B = 0 inside),\n      levitating a magnet by the Meissner effect.\n  - p: |\n      The dramatic finale is \u003Cstrong>superconductivity\u003C\u002Fstrong> — below a critical\n      temperature, electrons bind into Cooper pairs, resistance vanishes, and the\n      material expels magnetic flux entirely, a purely quantum effect made visible\n      at human scale.\n  - p: |\n      Throughout, one move recurs: \u003Cem>find the symmetry, count the states, and let\n      the collective behavior emerge\u003C\u002Fem>. The course follows Kittel, Ashcroft &\n      Mermin, and Simon, ending in the low-dimensional physics of nanostructures\n      and graphene.\n---\n\nCondensed matter physics is the study of the solid and liquid states as the\ncollective behavior of vast numbers of atoms and electrons. The sequence begins\nwith bonding and molecular spectra, then builds crystal structure and the\nreciprocal lattice — the periodic scaffolding underneath everything that follows.\nFrom there it develops lattice dynamics and phonons, the free-electron Fermi gas,\nand the band theory of periodic solids, which together explain why some materials\nconduct and others do not. The later chapters apply that machinery: semiconductors\nand their devices, dielectrics and ferroelectrics, the microscopic origins of\nmagnetism, and superconductivity, before closing on the low-dimensional physics of\nnanostructures and graphene. It follows Kittel, Ashcroft & Mermin, and Simon, with\nTipler & Llewellyn for the foundational material. Each topic rests on the\nperiodicity and counting arguments established before it.\n\n",{"text":65,"minutes":66,"time":67,"words":68},"1 min 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and Chemical Bonding",1,"molecules-and-bonding",[993,998,1003,1009],{"title":994,"path":995,"lessonNumber":990,"topics":996,"summary":997},"Bonding Mechanisms","\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms",[989],"A molecule forms when the total energy of two atoms drops below the energy of the separated pair. This lesson works through the four mechanisms that produce that minimum: the ionic bond from charge transfer, the covalent bond from shared electron wave functions, the metallic bond, and the weak dipole-dipole and hydrogen bonds, computing bond lengths and dissociation energies for NaCl, H₂, and H₂⁺.\n",{"title":999,"path":1000,"lessonNumber":16,"topics":1001,"summary":1002},"The Molecular-Orbital Method and H₂⁺","\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus",[989],"The hydrogen molecule ion is the two-center problem that fixes the language of chemical bonding. This lesson builds the molecular orbital as a linear combination of atomic orbitals, minimizes the energy through the variational secular equation, and reduces the result to three two-center integrals: the overlap, the Coulomb term, and the exchange (resonance) integral. The bonding and antibonding levels, their potential-energy curves, and the charge piled between the nuclei follow from those integrals.\n",{"title":1004,"path":1005,"lessonNumber":1006,"topics":1007,"summary":1008},"The Hydrogen Molecule, Exchange, and Hybridization","\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange",3,[989],"Adding the second electron turns the one-electron ion into the two-electron hydrogen molecule, where electron-electron repulsion and the Pauli principle govern the bond. This lesson contrasts the Heitler-London valence-bond and molecular-orbital wave functions, derives the singlet-triplet splitting as an exchange energy, shows why naive molecular orbitals fail at dissociation, and builds the sp, sp², and sp³ hybrids that fix the directed geometry of covalent bonds.\n",{"title":1010,"path":1011,"lessonNumber":1012,"topics":1013,"summary":1014},"Van der Waals Forces","\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces",4,[989],"The bond of last resort acts between all atoms, even closed-shell noble gases with no permanent moment. This lesson separates the three van der Waals contributions — Keesom orientation, Debye induction, and London dispersion — derives the London 1\u002Fr⁶ attraction from the coupled-oscillator and second-order perturbation pictures, and assembles the Lennard-Jones potential to compute the equilibrium spacing and cohesive energy of the noble-gas crystals, argon in particular.\n",{"module":1016,"moduleNumber":16,"slug":1017,"lessons":1018},"Molecular Spectra","molecular-spectra",[1019,1024,1029,1034],{"title":1020,"path":1021,"lessonNumber":990,"topics":1022,"summary":1023},"Rotational and Vibrational Spectra of Molecules","\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra",[1016],"A diatomic molecule stores energy in three well-separated ledgers: electronic, vibrational, and rotational. Quantizing the rigid rotor gives levels spaced as ℓ(ℓ+1); quantizing the bond as a harmonic oscillator gives equally spaced vibrational levels. Their combination produces the P and R branches of an infrared absorption band, from which the bond length and force constant are read directly.\n",{"title":1025,"path":1026,"lessonNumber":16,"topics":1027,"summary":1028},"Anharmonicity and Rovibrational Structure","\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure",[1016],"The rigid rotor and harmonic oscillator are first approximations. A real bond follows the Morse potential, whose levels converge toward dissociation; a real rotor stretches centrifugally; and vibration couples to rotation, so the rotational constant depends on the vibrational level. This lesson works out the anharmonic and centrifugal corrections, the Birge-Sponer route to the dissociation energy, the isotope shift, and the thermal band envelope.\n",{"title":1030,"path":1031,"lessonNumber":1006,"topics":1032,"summary":1033},"Raman Scattering and Electronic Bands","\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands",[1016],"Not every vibration absorbs in the infrared. Raman scattering reaches modes that modulate the polarizability, giving Stokes and anti-Stokes lines whose intensity ratio measures temperature, and the mutual-exclusion rule pairs it with infrared absorption. Electronic transitions add the vibronic structure of band spectra, governed by the Franck-Condon principle, and the radiative fates of an excited state are sorted by the Jablonski diagram into fluorescence and phosphorescence.\n",{"title":1035,"path":1036,"lessonNumber":1012,"topics":1037,"summary":1038},"Lasers, Masers, and Stimulated Emission","\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers",[1016],"Einstein's three radiative processes — absorption, spontaneous emission, and stimulated emission — and the coefficients that relate them. Stimulated emission produces coherent photons, and inverting the level populations turns it into net amplification. We build the ruby three-level laser and the helium-neon four-level laser, and show why the fourth level makes inversion easy.\n",{"module":1040,"moduleNumber":1006,"slug":1041,"lessons":1042},"Crystal Structure","crystal-structure",[1043,1048,1053,1058],{"title":1044,"path":1045,"lessonNumber":990,"topics":1046,"summary":1047},"The Structure of Solids","\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids",[1040],"A crystal is a unit cell repeated in three dimensions. We classify the common cubic lattices, compute the Coulomb energy of an ionic crystal through the Madelung constant, and show how the divergent naive lattice sum is tamed by cubic shells. The cohesive energy that results predicts melting points and connects the diatomic bond of an earlier lesson to the bulk solid.\n",{"title":1049,"path":1050,"lessonNumber":16,"topics":1051,"summary":1052},"Bravais Lattices, Bases, and Crystal Structures","\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems",[1040],"A crystal is a Bravais lattice decorated by a basis. This lesson separates the two, builds primitive and Wigner-Seitz cells, enumerates the seven crystal systems and fourteen Bravais lattices, and fixes the language of point and space groups. Miller indices label planes and directions, and the packing fractions of the close-packed, cubic, and diamond structures follow from the geometry.\n",{"title":1054,"path":1055,"lessonNumber":1006,"topics":1056,"summary":1057},"The Reciprocal Lattice and Brillouin Zones","\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones",[1040],"Every periodic crystal has a dual lattice in wavevector space. This lesson defines the reciprocal lattice through the condition b_i dot a_j equals two pi delta, derives its properties, shows the reciprocal of fcc is bcc, links reciprocal vectors to families of lattice planes, and builds the first Brillouin zone as the Wigner-Seitz cell of the reciprocal lattice, including the higher zones.\n",{"title":1059,"path":1060,"lessonNumber":1012,"topics":1061,"summary":1062},"X-ray and Neutron Diffraction","\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors",[1040],"A crystal diffracts radiation whose wavelength matches its atomic spacing. This lesson derives the Bragg condition, the equivalent Laue condition 2k dot G equals G squared, and the Ewald-sphere construction, then computes the geometric structure factor that produces systematic absences for bcc and fcc, the atomic form factor, and the powder method. It closes on why neutrons and electrons complement X-rays.\n",{"module":1064,"moduleNumber":1012,"slug":1065,"lessons":1066},"Lattice Dynamics","lattice-dynamics",[1067,1072,1077,1082],{"title":1068,"path":1069,"lessonNumber":990,"topics":1070,"summary":1071},"The Harmonic Crystal and Phonon Dispersion","\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion",[1064],"Atoms in a crystal vibrate about their equilibrium sites, and expanding the potential to second order turns the whole lattice into a set of coupled harmonic oscillators. This lesson sets up the harmonic approximation and the dynamical matrix, solves the monatomic linear chain for its dispersion omega(k) = 2 sqrt(K\u002FM) times the absolute sine of ka over two, explains why wavevectors outside the first Brillouin zone are redundant, and extends the chain to two atoms per cell to produce acoustic and optical branches with a frequency gap.\n",{"title":1073,"path":1074,"lessonNumber":16,"topics":1075,"summary":1076},"Phonons, Density of States, and Crystal Momentum","\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos",[1064],"Quantizing the normal modes of a harmonic crystal turns each vibrational mode into a quantum oscillator whose excitations are phonons. This lesson counts phonons with Bose-Einstein statistics, defines crystal momentum and the normal versus Umklapp distinction in momentum conservation, builds the density of states with its van Hove singularities, and shows how inelastic neutron scattering measures a dispersion curve point by point.\n",{"title":1078,"path":1079,"lessonNumber":1006,"topics":1080,"summary":1081},"Thermal Properties — Einstein and Debye Models","\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity",[1064],"The lattice heat capacity follows the classical Dulong-Petit value at high temperature but collapses toward zero as T approaches zero, a purely quantum effect. This lesson derives that behavior from the Einstein model of a single frequency, then the Debye model of a linear phonon spectrum with a cutoff, obtaining the Debye T-cubed law at low temperature and the Debye interpolation across all temperatures, and closes with thermal expansion and the Gruneisen parameter.\n",{"title":1083,"path":1084,"lessonNumber":1012,"topics":1085,"summary":1086},"Anharmonicity, Thermal Expansion, and Heat Conduction","\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport",[1064],"A perfectly harmonic crystal neither expands when heated nor resists heat flow. Both effects come from the cubic and higher terms the harmonic approximation discards. This lesson derives thermal expansion from an asymmetric interatomic potential, treats phonon-phonon scattering as the decay channel these terms open, shows why Umklapp processes are what make lattice thermal conductivity finite, and traces the temperature dependence of the conductivity and the phonon mean free path.\n",{"module":1088,"moduleNumber":1089,"slug":1090,"lessons":1091},"Free-Electron Fermi Gas",5,"free-electron-fermi-gas",[1092,1097,1102,1107],{"title":1093,"path":1094,"lessonNumber":990,"topics":1095,"summary":1096},"Conduction and the Free-Electron Gas","\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction",[1088],"Drude's classical free-electron model gets Ohm's law right but the resistivity, its temperature dependence, and the heat capacity wrong. Replacing the Maxwell-Boltzmann distribution with the Fermi-Dirac distribution and treating electron-lattice collisions as wave scattering repairs all three: the Fermi energy, Fermi speed, and a mean free path set by thermal lattice vibrations.\n",{"title":1098,"path":1099,"lessonNumber":16,"topics":1100,"summary":1101},"The Sommerfeld Model: Ground State and Heat Capacity","\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity",[1088],"Quantizing the free-electron gas in a box fills a Fermi sphere in k-space. The density of states grows as the square root of energy in three dimensions, and the Fermi energy, temperature, and wavevector follow for real metals. The Sommerfeld expansion shows only a thermal shell of width k_BT near E_F is excited, giving an electronic heat capacity linear in T that sits beneath the phonon T-cubed term.\n",{"title":1103,"path":1104,"lessonNumber":1006,"topics":1105,"summary":1106},"Transport, Wiedemann–Franz, and the Hall Effect","\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect",[1088],"The relaxation-time picture displaces the Fermi sphere under an applied field and gives the electrical conductivity ne-squared-tau over m. The same electrons carry heat, and their ratio yields the Wiedemann–Franz law with the universal Lorenz number. A magnetic field bends the carriers into cyclotron orbits and produces the Hall voltage, whose sign reveals the charge of the carriers.\n",{"title":1108,"path":1109,"lessonNumber":1012,"topics":1110,"summary":1111},"Screening, Plasmons, and the Limits of Free Electrons","\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons",[1088],"A mobile electron gas rearranges to screen any foreign charge, turning the bare Coulomb potential into a short-ranged Yukawa form over the Thomas–Fermi length. Displaced collectively, the gas rings at the plasma frequency, whose quantum is the plasmon and whose value sets the reflectivity edge of metals. A ledger of free-electron successes and failures then motivates band theory.\n",{"module":1113,"moduleNumber":1114,"slug":1115,"lessons":1116},"Band Theory",6,"band-theory",[1117,1122,1127,1132],{"title":1118,"path":1119,"lessonNumber":990,"topics":1120,"summary":1121},"Bloch's Theorem and Energy Bands","\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands",[1113],"An electron in a periodic potential has stationary states that are plane waves modulated by a lattice-periodic envelope. This lesson proves Bloch's theorem two ways, defines crystal momentum and the band index, counts the allowed wavevectors from Born–von Kármán boundary conditions, and sets up the extended, reduced, and repeated-zone descriptions of a band.\n",{"title":1123,"path":1124,"lessonNumber":16,"topics":1125,"summary":1126},"The Nearly-Free-Electron Model","\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model",[1113],"A weak periodic potential leaves the free-electron parabola almost intact except near Brillouin-zone boundaries, where two nearly degenerate plane waves mix. This lesson solves the resulting two-by-two secular problem, shows the gap of size twice the potential component opening at each boundary, identifies the two standing waves that pile charge on and between the ions, and works the exactly solvable Kronig–Penney model.\n",{"title":1128,"path":1129,"lessonNumber":1006,"topics":1130,"summary":1131},"The Tight-Binding Method","\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method",[1113],"The opposite limit to nearly-free electrons builds bands from atomic orbitals. A Bloch sum of one orbital per site gives a dispersion set by the hopping integral between neighbours; the band widens from a sharp atomic level as the atoms approach. This lesson derives the s-band cosine dispersion, extends it to p-bands, and introduces Wannier functions as the localized dual of Bloch states.\n",{"title":1133,"path":1134,"lessonNumber":1012,"topics":1135,"summary":1136},"Fermi Surfaces, Effective Mass, and Metals vs Insulators","\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics",[1113],"Filling the bands settles which crystals conduct. A filled band carries no current, so a crystal with filled bands and a gap is an insulator, while a partly filled band makes a metal. This lesson derives the no-current theorem for a filled band, defines the Fermi surface and Harrison's construction, introduces holes and the effective mass from band curvature, and states the semiclassical equations of motion that lead to Bloch oscillations.\n",{"module":1138,"moduleNumber":1139,"slug":1140,"lessons":1141},"Semiconductors",7,"semiconductors",[1142,1147,1152,1157,1162],{"title":1143,"path":1144,"lessonNumber":990,"topics":1145,"summary":1146},"Band Theory and Semiconductors","\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions",[1138],"The periodic lattice splits atomic levels into allowed energy bands separated by forbidden gaps. Whether the highest occupied band is full or partly full, and how wide the gap above it is, sorts every solid into conductor, insulator, or semiconductor. Doping adds donor or acceptor levels inside the gap, and a p-n junction built from doped regions gives the diode, the solar cell, the LED, and the transistor.\n",{"title":1148,"path":1149,"lessonNumber":16,"topics":1150,"summary":1151},"Carrier Statistics: Intrinsic and Extrinsic Semiconductors","\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors",[1138],"The number of mobile electrons and holes in a semiconductor follows from the density of states near each band edge and the Fermi-Dirac tail that reaches into it. This lesson derives the effective densities of states, the intrinsic concentration and its exponential gap dependence, the law of mass action, the temperature march of the Fermi level, and the freeze-out, saturation, and intrinsic regimes of a doped crystal.\n",{"title":1153,"path":1154,"lessonNumber":1006,"topics":1155,"summary":1156},"Carrier Transport and Recombination","\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination",[1138],"Carriers move by drift in a field and by diffusion down a concentration gradient, the two tied together by the Einstein relation. This lesson derives mobility and its scattering-limited temperature dependence, the drift and diffusion currents, the continuity equations, band-to-band and trap-assisted recombination, and the minority-carrier lifetime and diffusion length that set the length scale of every junction device.\n",{"title":1158,"path":1159,"lessonNumber":1012,"topics":1160,"summary":1161},"The p-n Junction in Depth","\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction",[1138],"Joining p-type and n-type silicon aligns their Fermi levels and leaves a depletion region of fixed charge with a built-in potential. This lesson derives the space-charge field and potential from Poisson's equation in the depletion approximation, the built-in voltage from Fermi-level alignment, the Shockley diode equation from minority-carrier diffusion, junction and diffusion capacitance, and the avalanche and Zener breakdown mechanisms.\n",{"title":1163,"path":1164,"lessonNumber":1089,"topics":1165,"summary":1166},"Transistors and Optoelectronic Devices","\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics",[1138],"Two junctions in series make a bipolar transistor whose thin base gives current gain; a gate over an oxide makes a MOSFET whose inversion channel switches digital logic. Run in reverse, a junction converts photons to current. This lesson derives the transistor current gain and the MOSFET channel current, then treats the LED, the diode laser, and the illuminated solar-cell characteristic.\n",{"module":1168,"moduleNumber":1169,"slug":1170,"lessons":1171},"Dielectrics and Ferroelectrics",8,"dielectrics-and-ferroelectrics",[1172,1177],{"title":1173,"path":1174,"lessonNumber":990,"topics":1175,"summary":1176},"Dielectrics, Polarization, and the Local Field","\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization",[1168],"An insulator responds to an electric field by polarizing. This lesson builds the macroscopic polarization and the dielectric constant, sorts the microscopic response into electronic, ionic, and orientational polarizability, and corrects the field an atom actually feels to the Lorentz local field E + P\u002F3 epsilon-0. The Clausius-Mossotti relation links the measured permittivity to the atomic polarizability, and the frequency dependence of each mechanism explains why the static and optical dielectric constants differ.\n",{"title":1178,"path":1179,"lessonNumber":16,"topics":1180,"summary":1181},"Ferroelectrics, Piezoelectrics, and Structural Transitions","\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics",[1168],"Some crystals carry a polarization with no applied field and switch it under a reversing field, tracing a hysteresis loop. This lesson develops the ferroelectric transition through the perovskite BaTiO3 displacive instability and its soft transverse-optical mode, builds the Landau free-energy theory of first- and second-order polar transitions, derives the Curie-Weiss divergence of the dielectric constant, and closes with piezoelectricity and pyroelectricity and their devices.\n",{"module":1183,"moduleNumber":1184,"slug":1185,"lessons":1186},"Magnetism in Solids",9,"magnetism",[1187,1192,1197,1202],{"title":1188,"path":1189,"lessonNumber":990,"topics":1190,"summary":1191},"Diamagnetism and Paramagnetism","\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism",[1183],"Every solid responds to a magnetic field. Filled shells give a small negative diamagnetic susceptibility from induced Larmor currents; localized moments give a positive Curie paramagnetism described by the Brillouin function, with the ground-state moment fixed by Hund's rules. The conduction electrons add a temperature-independent Pauli paramagnetism from the thermal shell near the Fermi surface, partly cancelled by Landau diamagnetism of their orbital motion.\n",{"title":1193,"path":1194,"lessonNumber":16,"topics":1195,"summary":1196},"Exchange and Ferromagnetism","\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism",[1183],"Magnetic ordering at hundreds of kelvin cannot be dipolar; it is an exchange effect, the Coulomb repulsion sorted by the Pauli principle into a spin-dependent energy captured by the Heisenberg Hamiltonian. Weiss molecular-field theory replaces the exchange field by an average proportional to the magnetization, giving a self-consistent equation whose solution is spontaneous magnetization below a Curie temperature and a Curie–Weiss susceptibility above it. Itinerant ferromagnetism follows from the Stoner criterion on the band density of states.\n",{"title":1198,"path":1199,"lessonNumber":1006,"topics":1200,"summary":1201},"Antiferromagnetism, Ferrimagnetism, and Domains","\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains",[1183],"A negative exchange coupling orders neighboring spins antiparallel. Two-sublattice molecular-field theory gives a Néel temperature marked by a cusp in the susceptibility, and unequal sublattices leave a net moment — ferrimagnetism, the magnetism of the ferrites. A ferromagnet breaks into domains to reduce its magnetostatic energy, separated by Bloch walls whose width is set by the competition between exchange and magnetocrystalline anisotropy, and the irreversible motion of those walls produces the hysteresis loop.\n",{"title":1203,"path":1204,"lessonNumber":1012,"topics":1205,"summary":1206},"Spin Waves and Magnons","\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons",[1183],"The lowest excitations of a ferromagnet are not single flipped spins but collective precessions in which every moment tips slightly and its phase advances along the crystal. These spin waves have a quadratic dispersion at long wavelength, quantize into magnons obeying Bose statistics, and their thermal population removes magnetization as the Bloch T-to-the-three-halves law. Antiferromagnetic magnons disperse linearly, and inelastic neutron scattering measures both.\n",{"module":1208,"moduleNumber":1209,"slug":1210,"lessons":1211},"Superconductivity",10,"superconductivity",[1212,1217,1222,1227,1232],{"title":1213,"path":1214,"lessonNumber":990,"topics":1215,"summary":1216},"Superconductivity: Phenomenology and BCS","\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology",[1208],"Below a critical temperature some materials lose all resistance and expel magnetic flux — the Meissner effect that defines the state. The isotope effect points to lattice vibrations, and BCS theory binds electrons into Cooper pairs through phonon exchange. The paired condensate opens an energy gap, quantizes magnetic flux, and drives the Josephson effects.\n",{"title":1218,"path":1219,"lessonNumber":16,"topics":1220,"summary":1221},"London Theory and the Meissner Effect","\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect",[1208],"A perfect conductor freezes the field it was cooled in; a superconductor expels it. The distinction needs a constitutive law beyond zero resistance — the two London equations — whose solution is exponential flux decay over the penetration depth. The same rigidity follows from a macroscopic condensate wave function, and the thermodynamics of the critical field fixes the condensation energy, the latent heat, and the specific-heat jump.\n",{"title":1223,"path":1224,"lessonNumber":1006,"topics":1225,"summary":1226},"Ginzburg–Landau Theory, Vortices, and Type-II","\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory",[1208],"A complex order parameter and a free-energy expansion turn the superconducting transition into a Landau theory. Two lengths emerge — the coherence length and the penetration depth — whose ratio kappa sorts superconductors into type I and type II. Type-II materials admit flux as an Abrikosov lattice of vortices, each threading exactly one quantum h\u002F2e, between a lower and an upper critical field.\n",{"title":1228,"path":1229,"lessonNumber":1012,"topics":1230,"summary":1231},"Microscopic BCS Theory","\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory",[1208],"A phonon-mediated attraction, however weak, binds two electrons above the Fermi sea — the Cooper problem shows the sea is unstable. The BCS variational ground state pairs all electrons near the Fermi surface and, through a self-consistent gap equation, opens an energy gap. Weak-coupling solution gives the exponential T_c and the universal ratios 2 Delta(0) = 3.53 k_B T_c and Delta C \u002F C_n = 1.43.\n",{"title":1233,"path":1234,"lessonNumber":1089,"topics":1235,"summary":1236},"Josephson Effects and Unconventional Superconductors","\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc",[1208],"Two superconductors joined by a thin barrier carry a supercurrent set by their phase difference — the dc Josephson effect — and oscillate at 2eV\u002Fh under a voltage. A two-junction loop turns flux quantization into a magnetometer of single-quantum sensitivity. The cuprates superconduct in CuO2 planes with a doping-dependent dome, d-wave pairing, and a pseudogap that lie outside the phonon picture.\n",{"module":1238,"moduleNumber":1239,"slug":1240,"lessons":1241},"Nanostructures",11,"nanostructures",[1242,1247,1252,1257],{"title":1243,"path":1244,"lessonNumber":990,"topics":1245,"summary":1246},"Quantum Wells, Wires, and Dots","\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots",[1238],"When a crystal is shrunk until one or more of its dimensions approaches the electron wavelength, the continuous bands of the bulk break into discrete subbands. Confining in one direction gives a quantum well with a step-like density of states, in two directions a quantum wire with inverse-square-root singularities, and in all three a quantum dot whose levels are sharp like an atom's. This lesson derives the density of states in each case and applies it to size-tunable dot emission and the Coulomb blockade of a single-electron transistor.\n",{"title":1248,"path":1249,"lessonNumber":16,"topics":1250,"summary":1251},"The 2D Electron Gas and the Integer Quantum Hall Effect","\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect",[1238],"A two-dimensional electron gas in a strong perpendicular magnetic field has its continuous density of states collapse into macroscopically degenerate Landau levels. As the field is swept, the Hall resistance locks onto exact plateaus at h over an integer times e squared, while the longitudinal resistance drops to zero. This lesson derives the Landau levels and their degeneracy, explains the plateaus through disorder-localized states and current-carrying edge channels, and states why the von Klitzing constant is now a resistance standard.\n",{"title":1253,"path":1254,"lessonNumber":1006,"topics":1255,"summary":1256},"The Fractional Quantum Hall Effect and Topological Order","\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology",[1238],"When the lowest Landau level is only partly filled, the non-interacting theory predicts no gap, yet a plateau appears at filling one-third. It is a many-body effect: Coulomb repulsion selects a correlated ground state, the Laughlin wavefunction, whose excitations carry a fraction of the electron charge. This lesson builds the Laughlin state, introduces composite fermions that map the fractional effect onto an integer one, and explains how the quantum Hall effect brought the Chern number and topology into condensed-matter physics.\n",{"title":1258,"path":1259,"lessonNumber":1012,"topics":1260,"summary":1261},"Graphene and Dirac Materials","\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials",[1238],"Graphene is one atomic layer of carbon on a honeycomb lattice. A tight-binding calculation on its two-atom basis gives valence and conduction bands that touch at the corners of the Brillouin zone, where the dispersion is linear and the electrons behave as massless two-dimensional Dirac particles with a fixed speed. This lesson derives the Dirac cones, the Berry phase of pi and the sublattice chirality, the anomalous half-integer quantum Hall effect that follows, and how opening a gap in a Dirac cone points toward topological insulators.\n",1786059479597]