Notes on Introduction to

Xupeng Yang

Contents

1 Basic Introduction 1

2 Conventional Metal Physics: and Phonons 1

3 Transport 3

4 Metal Insulator Transition (MIT) 5

5 Low Dimensional Systems 7

6 Quantum Hall Effect (QHE) 9

7 Magnetism 10

8 Superconductor 11

9 Others 11

1 Basic Introduction

1. Condensed Matter: 1023 /cm3 ∼ 2. Major Study: Electrons, Phonons, The interactions between

3. Drives: New materials & New technologies

2 Conventional Metal Physics: Electrons and Phonons

1. Basic Properties of Normal Metals: Ductile Excellent electrical conductor Excellent thermal con- • • • ductor Most are weak paramagnet, some ferromagnet Opaque • • 1 At low T: ρ increases with T χ Const. c T • ∼ V ∝ 2. Drude Free Electron Model: Assumptions: Free electrons (Ignore interaction with lattice) Independent electrons (Ignore interac- • • •

tions between electrons • Electrons were treated as independent classical particles

• Maxwell-Boltzmann distribution Successes: Electrical conductivity and thermal conductivity Wiedemann-Franz law (by luck!) • • • •

The Hall effect and magnetoresistance • AC conductivity and optical properties of metals Problems: Heat capacity puzzle: c = 3 nk = Const. The susceptibility puzzle: χ does not change • • V 2 B • with temperature, non-Curie like χ 1/T ∼ 3. The Sommerfeld Model: Free Electron Gas + Schrodinger Equation + Fermi Statistics • 2/3 8 1 The Fermi Surface: In 3D E n Typical Values: E 7 eV, k 1 10 cm− , T • F ∼ • F ∼ F ∼ × F ∼ 8 104 K, v 2 108 cm/s DOS In 3D: D(ε) ε1/2 Linear T Heat Capacity at Low T × F ∼ × • ∼ • Pauli Paramagnetism (at low T): χ = µ2 D(E ) = const. • B F Successes: Realized the importance of Fermi distribution Established the k-space language for • • •

electrons • Introduced Fermiology • Resolved the Pauli susceptability puzzle • Resolved the heat

capacity puzzle • Resolved the thermopower puzzle • Explained the Wiedemann-Franz Law

4. Landau's Fermi Liquid Theory: Quasi-particle: Same charge, spin, momentum as non-interacting electron • Adiabatic Continuity, Only valid at low T and low energy Qualitatively explain the susceptibility, heat • • capacity Only require a Fermi sea Entropy, distribution function unchanged Energy modified by the • • 2 2 effective mass & the Fermi interaction function Low energy excitation like single particle ε = ! k • • 2m∗ π2k2 1 m∗kF 2 χ m∗kF 1 µ2 Landau parameter: , a Wilson ration: B χ For cV = 3 !2 kBT = !2π2 1+Fa B m∗ F0 RW = 3µ2 γ • • 0 • • B • non-interacting electron gas R = 1 T 2 Law (Experimental Signature): qp-qp scattering Scattering W • • rate 1 k T k T T 2 But electron seems to be a wrong place to start for many novel phenomena τ ∼ B · B ∝ • • Spinon, holon, fractional charge: Collective mode looks like a fraction of an electron

ik r 5. Bloch Theory: Bloch Theorem: For periodic potential, ψ (r) = e · u (r), where u (r) = u (R+r) • nk nk nk nk Momentum is no longer a good quantum number Band index n • • NFE model: Perturbation to the free electron plane wave states (maximum mixing) (highly delocalized) • V 2 1 2 | g| First order: εk = V Second order (non-degenerate): εk = g!0 0 0 Second order (degenerate): • • εK εk g • − − ε2 = V Energy gap: Level repulsion, Explains metal or insulator k ±| n| • !

• +2 Metals: Band overlap Tight Binding Model: (nearly localized) such as transition metal and rare earth metal with partially filled • ik R d and f orbitals. ψ (r) = e · ψ (r R) Overlap integral: t = ψ∗(r + R)(∆V)ψ (r)dr • k R a − • R a a → ! " 2 Nearest neighbor approximation Band width: W 2zt, where z is coordination number, t is overlap • ∼

integral • Useful starting point

6. Lattice Vibrations: Harmonic Approximation: V(a + δx) = V + 1 β(δx)2 ε = (n + 1 )!ω • 0 2 • k k 2 • Phonons: The quantum of the lattice vibration, 1 Mono-atomic 1D Chain: ω β sin aq nk = e!ω/kT 1 != 2 m 2 − • # $ $ Di-atomic 1D Chain: Acoustic & Optical phonon Phonon specific heat: Debye model: • Assume$ linear% &$ • • $ $ dispersion • Define a cutoff in the integral: Debye frequency T 0 c T 3 Blackbody radiation • → V ∼ ⇐ 7. Specific Heat: Directly related to internal energy To extract important microscopic parameters To • • • study phase transition Calorimetry: What, How, Better resolution and accuracy, Flecibility Adiabatic • • Nernst Calorimeter: Slow, Heat leak problem, Need big sample

t/τ • Relaxation time calorimeter: ∆T = ∆T0e− , where τ = cV l/κS (with addenda) Advantage: Accurate, Fast, Microgram crystals, Small, Work in extreme conditions Disadvantage: The addenda Membrane calorimeter: Nano-gram crystals, Measure in-situ evaporated thin films, Extreme conditions • Heater: Resistance stable with T Thermometer: Resistance has Linear relationship with T • • 8. Anharmonic Potential: Universal V = 0 when r Phonon-phonon interaction: No longer • ⇐ → ∞ • independent excitations Phonon heat conduction (lowT, highT) Thermal Expansion: α = ⇒ • • 1 ∂2l 1 ∂2V l ∂T∂p = 3V ∂T∂p • Provide similar information as specific heat • Bad in engineering XRD Measure l or V, High resolution, Hard to use in extreme physical conditions • →

• Capacitive dilatometer: High resolution (capacitance bridge) (0.01 Å), ultra low T and large B (compact design) Negative thermal expansion: ZrW O Rigid Unit Modes • 2 8 → 9. Main Frame: Landau Fermi Liquid Theory + Band Theory

3 Transport

1. Basic Notions: Movement of Particles or Quantities Non-equilibrium steady state J = L F • • • · Very informative and instructive, esp. on Novel materials and in Extreme conditions • Normally the first to be carried out Close relations to device applications • • 2. Fractional Quantum Hall Effect: Ultra low T, Super strong B, Very clean Strong electron correlations • • Most precise method to measure h •

3 3. Cryogenic Technology: Dilution fridge method: He-3 rich & He-3 poor phase at T < 0.87 K • He-3 diffuse, absorb heat Down to 10 mK • • ∼ Superconducting magnet: up to 20 tesla critical field • ⇐ Super high megnetic field: Florida-Bitter resistive magnet, Hybrid magnet •

∂ fk ∂ fk ∂ fk ∂ fk 4. The Boltzmann Transport Equation: + + = 0 = r˙ r fk • ∂t diffusion ∂t field ∂t scattering • ∂t diffusion − · ∇ 0 ∂ f ∂ f fk f k ˙ k $− k $ $ $ = k k fk = $ $ $ $ • ∂t field − · ∇ • ∂t scattering − $τ $ $ $ $ $ 5. Electrical$ Transport: J = σ$ E Measurements: Four-probe, Low frequency ac lock-in method • e • 2 Drude model: σ = ne τ Semi classical: δk = eτE Only the surface of the FS changed ! • m • ! •

• Ignore the diffusion effect, Complexity of the FS 2 1 e τ vkvkdS F The Boltzmann transport equation: ←→σ = π3 ! • 4 vk 2 Cubic symmetry: σ = e v lD(ε ) " • x,y,z 3 F F Matthiessen’s rule: Different scattering mechanisms don’t interfere each other 1 = 1 + 1 + • ⇒ τ τimp τph · · · Electron-electron scattering: 1 T 2 • τ ∼ Electron-lattice scattering: ρ T , at high T ρ T 5 , at low T • ∼ ∼ Electron-impurity scattering: Roughly, Temperature-independent Residual resistivity: ρ(T = 0) • • • • Residual resistivity ratio (RRR): ρ300K Higher the better ρ0

6. Thermal Transport: J = κ( T) Measurement: One-heater, Two-thermometer • Q −∇ • 2 2 1 1 π kB Drude: JQe,x = nvx[ε(Tx vτ Tx+vτ)] = cV vl JQ = 2 fk(εk µ)vkdk κe = 2 Tσ • 2 − − 3 • − • 3 e Phonon Thermal Conductivity: Good metals 1% " • ∼

E cV 7. Thermoelectric Power: Seebeck Coefficient: S = T = 3ne • The piece of heat carried by each charge e • ∇ 2 π kB ∂ ln σ(ε) Inversely proportional to εF S = kBT Reveal abrupt change of electronic stucture • • 3 e ∂ε ε=µ • Study novel electronic phases and phase transitions,% But&$ poorly understood • $ Thermal couple: V = (S S )(T T ) Thermoelectric power generation: Π-junction consisting of • B − A x − 0 • N type & P type material V = ( S + S )(T T ) No moving part, reliable Environmental • | N| | P| h − c • • friendly Arbitrary Shape & Size Radioisotope thermoelectric generator: For unmanned situations, • • Low power, Long durations Thermoelectric refrigeration: Π-junction consisting of N type & P type • material J = J ( Π + Π ) • Q e | N| | P| 8. Peltier Effect: J = ΠJ • Q e

Je σ σS T E 9. Onsager reciprocal relations: = Π = S T • rT • JQ σΠ κT   ∇     −   T             

4 2 10. The Thermoelectric Figure of Merit: ZT = σS T ZT 3, for applilcation, Now ZT 1 • κ • ∼ ∼ Now focusing on heavily doped narrow-band , n 1019 1020/cm3, Not promising • ∼ − because of W-F law Minimize phonon thermal conductivity Low Dimension, Amorphous, Nano- • ⇒ materials Bi S e & Bi Te : Quasi-2D system (Quintuple-layer) Future focus: Considering Spin, • 2 3 2 3 • Strong-correlation system

11. Magnetic Field: Free electron gas: No magnetoresistance Hall coefficeint: R = 1 • • • H ne Ey Beτ Hall angle: tan θ = = = ωcτ Quantum oscillations: ωcτ 1, Shubnikov-de Haas oscillations • Ex m • ≫ 12. Thermo-magnetic Transport: Thermal Hall effect: Heat current (x) produces T (y) • ∇ Ey Nernst effect: ν = T B , Powerful technique for novel metals & superconducting vortices in type-II • ∇ x z superconductors

4 Metal Insulator Transition (MIT)

1. I-M Transition within Band Theory: Doping: Donors & Acceptors impurity bands • ⇒ Pressure: Structure change Overlap Tight binding model Wilson transition • ⇒ ⇐ • 2. : Mott’s Gedanken Experiment: Increase the distance between atoms: Smaller hopping • integral (t) and carrier density (n) Thomas-Fermi Theory: A negative charge added to the Fermi Sea δV δn = D(ε )δV • ⇒ ⇒ − F ⇒ 2 2 2 2 e kr 1 4me kF 1/6 (δV) = k δV Yukawa Potential: δV = e− Screening length: λ = , k = n for ∇ • r • k π!2 ∝ 3D FEG Good metals Non-interacting FEG # • ⇒ Mott Insulator: Coulomb energy cost will exceed the kinetic energy gain Low dimensional materials • •

with large lattice constant • IMT: Tune the U/W ratio; Change band filling by doping. • Pressure induced

IMT: Lattice contraction at IMT ( 0.2%) Metallic bonds Increase of m∗ due to strong electron ∼ ⇐ •

interaction in doped Mott insulator • Many transition metal oxides (TMO): Separated by O, small density; Inner d electrons, weak overlap

The Hubbard Model: H = t i, j ,σ ci†σc jσ + U i ni ni , where t = ϕ∗j [V(r) vi(r)] ϕidr • − 〈 〉 ↑ ↓ − 2 e2 2 st nd U = ϕ (r1) ϕ (r2) dr 1 tern: Hopping tern, 2 tern: on-site"Coulomb repulsion tern i r12 i ! !

Band" $ split: $U > $W (W is$ the width of the original band) Energy gap: Eg (U W) • $ $ $ $ • ∼ − Perovskite Structure: ABO A only donates electronic charge and stabilizes the structure For electronic • 3 • •

properties, the BO6 octahedral is most relevant • RNiO3 system: Charge transfer insulators: O's p orbits and Ni's d orbits strongly hybridized. gap 10-30 meV Bonding angle: W is the largest for straight ∼ • 3+ 2 3+ bond (Ni O − Ni 180◦) and smaller in distorted case. Becomes better insulator with increasing − − •

5 R atomic number (smaller radius, more twisted bond of B O B) Different transition metal: Different − − • d-electron configuration, Different p d hybridization, Different U Different A ions: Different ion − •

size, Different bonding angle, Different W • Substitution of A: Different carrier density • Extra O or O

deficiency: Different hole concentration • Substitution of B: Different on site configuration • Different dimensionality Magnetic Structure: Most have Antiferromagnetically ordered ground state • Typical Strongly Correlated Materials: Incompletely filled d or f electron shells with narrow bands • Wigner Crystal: Crystal of electrons Potential e2 Kinetic !2 when is large • r mr2 • r0 • ∼ 0 ∼ 0 3. Anderson Localization Dilute, Nonmagnetic Impurity, T = 0 , The distribution of the interaction • The Spin Diffusion Puzzle: The relaxation time of donor electron spin is way longer at low concentrations • Electron Spin Resonance (ESR): Unpaired electrons, Resonance frequency microwave (9GHz for • →

0.3T), challenging, Study electron spin dynamics. • Phase sensitive (lock-in) detector, The first derivative of absorption line.

The Anderson Hamiltonian: H = ε n + t c†c V = 0, as ε is const, Tight-binding model • i i i i, j i j i j • i t = 0, Atomic orbitals at each site V 1 Impurity scattering of Bloch waves V > 1 • ! • W ≪! ⇒ • W ⇒ Localization Mobility Edge: ε : 0 < V < 1, Separating localized and non-localized states Tuned by changing the • ± c W • level of disorder. MIT: ε tuned by doping level or pressure Trait: No energy gap in DOS near ε • F • • F

(Psesdogap) • The electron number need not be integer • Coulomb repulsion in unnecessary ◮ Near the localization transition Still Controversy − 2 Mott's Minimum Conductivity: Localization transition is discontinuous. lk > 1, l a σ3D 1 e 1 • • F ≥ • min ≈ 3π2 ! a Scaling Theory: ! G(L) = g(L) ∆E β(g) = d ln g The transition is continuous β(g ) = 0, g • e2 ≈ ∆V • d ln L • • c c ⇒ Unstable fixed point g > g Conductor; g < g Insulator No extended states for any degree • 0 c ⇒ 0 c ⇒ •

of disorder in 1D and 2D. • Doesn't consider other effects can destroy the localization (Magnetic field, S-O coupling, E-E interaction ...) Weak Localization: l l < L < ξ , where l is for elastic scattering, l is for inelastic scattering, L is the • ≪ i i sample's scale, ξ is the localization length , k l 1 Self-crossing loops: P = A + A 2 = 2 2 A 2, F ≫ • | 1 2| × | |

decreases the conductivity, Localization • Dephasing length: τφ , Loss of phase coherence will destroy weak localization 1/2 τφ η/2 τ = T − , d = 1 δσ τφ η! ! = γd  %! ln& = ln , d = 2 σ −  τ 2 kBT  1/2  2 τφ − 2 η/2  ! %τ & = ! T , d = 3  Wave Property: Reported for light waves, microwaves,% & sound waves, matter waves (BEC) •  Non-interacting theory, hard to ideally realize in • 6 4. Charge Density Wave (CDW): (Lattice distortion induced) Peierls' Theorem: 1D materials are insulating The distortion of lattice opens an energy gap around the • • π λ original EF The lattice distortion induced a CDW with λ = , k = 2kF, can be irrational Mechanism: • kF a •

e-ph coupling an electron-hole pair is created by a phonon • Kohn Anomaly: Electron-phonon interaction has a strong influence on the phonon spectrum near q = 2k Phonon softening 1D, Phonon Frequency F ⇒ •

can drop to 0 Hamiltonian: H = k,σ εkck†,σck,σ + q !ωqbq†bq + k,σ,q g(k)ck†+q,σck,σ bq + b† q • − 1/g ∆(0) CDW gap: ∆ = 2We− CDW transition T: TP = , From BCS theory % & • • ! ! 1.76kB ! TTF (donor)-TCNQ (acceptor): MIT: σ drops at 55K and 38K (superstructure with 3.4a), has a thermal • • activation behavior Energy gap Kohn anomaly: Neutron inelastic scattering Fermi Surface Nesting: ⇒ • • two pieces of parallel FS Quasi-1D: FS as parallel planes Warped, Still has FS nesting • ⇒ Pressure: Pressure T Imperfect nesting • ↑→ CDW ↑ ⇐ NbS e : Two sharp increases of ρ at 144K and 59K, remains metallic down to T = 0 • 3 •

• 3 types of chains, 2 successive Peierls trasitions NbS e : Layered structure, can be grown to extremely high quality A superstructure with 3a • 2 •

• CDW state below 33K while being metallic, Superconductor below 7.2K • Multiband material

5. Jahn-Teller Theorem: Non-linear degenerate molecules cannot be stable. Change of electronic structure MIT • • ⇒ Energy Gain: Electronic Energy Cost: Elastic, Hund's rule coupling • • Octahedral complexes of the transition metals: Elongation along z, Lower dz2 , Upper dx2 y2 • − K C Monolayers: MIT From K C to K C Electrons in the LUMO orbital interact with certain • x 60 3 60 4 60 • vibrational mode of the C60 molecule and cause a permanent molecular distortion

5 Low Dimensional Electron Systems

1. The Motion of Microscopic Degrees-of-freedom

2. Examples: Quasi-2D: Giant magnetoresistance in multilayer magnetic films High T in • • • c layered copper oxides 2D: QHE in MOSFET Fractional QHE in semiconductor heterostructures • • • Quasi-1D: CDW in TTF-TCNQ and NbS e • • 3 1D: Luttinger liquid in semiconducter nanowires Carbon nanotubes • • • 0D: Quantum dots Molecular electronics and magnetism • • • 3. Affect Propagation of waves and Formation of ordered phases

7 4. Susceptible to defects and thermal fluctuations

5. Electron DOS shows quantized behavior

6. Surface and Interface: Surface is a special type of interface • Why? : Break the periodicity in one dimension Ideal for low-dimensional systems Information • • • •

and electronics industries heavily rely on • Surface catalysis can greatly increase the rate Surface structure: Techniques (Surface sensitive): LEED, RHEED, SXD Surface relaxation: May • • • extend several layers Surface reconstruction: Si (100) -- dimer rows , Si (111) -- 7 7 superstructure , • × Au (111) -- 22 √3 superlattice (hcp & f cc) Mechanism: Minimization of surface free energy Semi- × conductors: Surface healing process Reduce dangling bonds Metals: Formation of denser packing → → Maximize the surface metallic bonding Surface Electronic Structure: Parallel: Bloch waves Perpendicular: Extended Bloch wave within • • •

the crystal, Exponentially decaying tail outside the surface • Surface state energy may lies within the bulk gap Surface electronic state, for both semiconductors and conductors (specific directions) Energy: ⇒ • !2k2 Es = E0 + || 2m∗ Applications: Surface catalysis • Creation: Cleaving method: Materials with Van der Waals force between layers, In ultrahigh vacuum • •

(UHV) environment • Cleaning method: Repeated ion sputtering (cleaning) & High T annealing (heal- ➊ ➋ ing) Usually for metal • Growing method: Thin films on substrate Thermal evaporation, Pulsed laser deposition, Magnetron sputtering deposition, MOCVD, MBE MBE: Slow deposition rate (typically < 1 µm/h) (controlled by T of source) Organize sufficiently • ⇒ and grow epitaxially ➊ Substrate T : Too low: Without arranging properly poor quality ; Too high: Desorb readily low ⇒ ⇒ 8 growth rate & poor quality ➋ UHV: For impurity < 10− Pa ➌ MBE system: Mass spectrometer: Analyze residual gas; RHEED: in-situ check layer structure and growth rate; Effusion cell: Evaporator source, accu- rate T control and fast shutter; Substrate: accurate T control and rotation ➍ RHEED oscillation: Frequency Growth rate; Intensity Quality ⇒ ⇒ 7. STM/STS: Power: Atomic level real space resolution Single atom / molecule manipulation High resolution • • • •

spectroscopy (< 1 meV) • Both occupied and unoccupied spectroscopy • Used in extreme conditions Limits: No use for bulk property No use for insulators No direct k space information • • • • − 8. ARPES:

8 p = !k = √2mEkin sin θ • || || Components: Light source (E, δν, I) Sample (Cleanness, Flatness, Rotator) Electron analyzer (E, • • • • P resolution) Synchrotron ARPES: ➊ Variable E ➋ High Flux density ➌ High Monochromacity (< 5 meV) • • ➊ ➋ • He discharge lamp ARPES: Convenient, low cost Fixed E, low I ➊ ➋ ➌ • Laser ARPES: High I Better resolution Lack of suitable nonlinear crystals produce UV photons Power: Direct mapping of electronic structure Can extract many crucial microscopic parameters • • • Limits: Can't be used in a magnetic field Difficult to do at ultra low temperatures Can't measure • • • •

unoccupied states • Hard to provide real-space information • UHV • Mainly a surface probe

6 Quantum Hall Effect (QHE)

1. MOSFET: Gate voltage V n Bias voltage V 2D electron gas • G ⇒ e • sd ⇒ Depletion layer, Inversion layer, Triangular potential •

h Ey 2. QHE: The von Klitzing constant: RK = 2 Hall angle: tan θ = = ωcτ • e • Ex 2D materials at low T, high B, ω τ 1 Classical motion in B and E: Along the equi-potential • ⇒ c ≫ • contours defined by the total potential V Hall angle = 90◦ ⇒

σxx σxy 1 ωcτ σ0 3. Conductivity Tensor: J = σ E = E = 2 E ←→ 1+(ωcτ) σyx σyy  ωcτ 1    −      1 ωcτ    ρ Similarly, ρ ρ − ρ σ  σ σ  ρxx σ σ xy E = ←→J = 0 ←→ ←→ = ←→I xx = yy = ρ2 +ρ2 xy = yx = ρ2 +ρ2 • ω τ  • · • xx xy • − xx xy  c 1      4. Landau Levels: D = m∗ S !ω = B S 2e/h For each spin-polarized LL: D = B S e/h n = ieB • π!2 · · c · · • · · • h For integer filling factor, E lies in the E • F g 5. Impurities: B destroy the phase coherence between 2 loops Delocalization • ⇒ Hall plateau: Only extended states carry current • 6. Sample Edge: Edge state Infinite potential wall at edge Each LL crosses E at edge • • ⇒ F “Protected” edge state Cannot backscatter from one edge to another • ⇒ Quantum transport coherent over the whole sample • ⇒

7. Current in 1D: I = 2e ∞ f (ε , µ )v(k) dk = 2e ∞ f (E, µ )dE • L 0 k L 2π h 0 L 2e µL 2e dE = (µ"l µR) , (µl µR = eV)"is large I = h µR h − − 2 ∂ ( ,µ) •  V "2e ∞ f E dE , eV µ  h 0 ∂E  · − ≪  "  9 2 2 G = i e I = Vi e • h • h Vanishing longitudinal resistance, the whole upper edge has the same potential as the source electrode, • the whole lower edge has the same potential as the drain electrode.

7 Magnetism

1. From: Spin (spin magnetic moments) & Charge (orbital magnetic moment, induced electrical current loops of the core electrons) degree of freedom

2. Diamagenetic All non-metallic elemental compounds (except Oxygen) • 2 N e µ0 2 χ = i r • − V 6me i ! 1 2 3. Paramagnetic Most metals • Pauli Paramagnetism: ➊ χ = µ2 D(E ) ➋ Caused by free electrons, Near E • B F F nµ µ2 Curie Paramagnetism: ➊ Partially filled inner electronic shell ➋ χ = 0 e f f C • 3kBT ∼ T

23 4 4. Magnetic Dipolar Interaction: Usually ignored E 10− J 10− eV 1K • ∼ ∼ ∼ 5. Exchange Energy: Overall wavefunction must be anti-symmetric when two electrons are exchanged • E E = 2 Ψ∗ (r ) Ψ∗ (r ) V (r , r ) Ψ (r ) Ψ (r ) dr dr • S − T 1 1 2 2 I 1 2 2 1 1 2 1 2 H = 1 (E + E" ) (E E )S S JS S • 4 s T − s − T 1 · 2 ∼ − 1 · 2 Short range Microscopic mechanism of Hund's first rule • • Superexchange: La CuO AF order • 2 4 ⇒ Double exchange: ➊ Magnetic ions show mixed valency ➋ Strong Hund's rule coupling, eg ∥ t2g • ⇒ Ferromagnetic

6. Measurements: Conventional: ➊ Vibrating sample magnetometer (VSM) ➋ SQUID ➌ Micro-Cantilever • torque magnetometer: Very high sensitivity in high field Neutron Scattering: nuclear scattering and magnetic scattering • 7. Geometrical Frustration: Triangular lattice: Na CoO , Curie-Weiss metal at low T • x 2 Absence of long-range magnetic ordering in the ground state • Residual entropy: ➊ E.g. Water ice: "two-close-two-far", S = R ln 3 ➋ Spin ice: "two-in-two-out" • m 2 8. Magnetic Monopoles: Experimental: Spin ice, rather as emergent particles •

10 8 Superconductor

1. Properties: Zero resistance and Perfect diamagnetism

2 2 ∂Js nse nse 2. The London equations: = E , Js = B ∂t m∗ ∇ × − m∗c

1 3. Key experimental results: The isotope effect: T M− 2 Lattice is involved • C ∼ ⇒ T 1.5 C Heat capacity: Below T , C e− T ∆E 1.5k T • C es ∼ ⇒ ∼ B C 2 4. : ∆E = 2 ε !ω e− g0U0 F − C 3 4 5. BSC theory: Conventional superconductors ∆ = 1.764k T McMillan limitation: T < 40K • • B C • C 6. high T : Cuprates: ➊ LaBaCuO System: 35 K ➋ YBa Cu O : 93 K C • 2 3 7 Structure: ➊ Perovskite-like ➋ The CuO plane • 2 The parent compound: Cuprate without charge dopants ➊ La CuO ➋ Mott insulator ➌ AF ordering • 2 4 ⇐ Superexchange 146 meV ∼ The hole doped: ➊ Metallic Superconducting ➋ holes , T first , then , last no superconductivity • → ↑ C ↑ ↓ states ➌ Optimally doped Charge carriers are electron pairs Flux quantum Φ = h Energy gap: V-shaped, ⇐ 0 2e 2∆ 10 s-wave: ∆(k) = constant, Isotropic d-Wave: Anisotropic, ∆dx2 y2 = ∆0 (cos kx cos ky) kBTC ∼ − − phase •

9 Others

1. Topological Insulator: Single electron model Mainly spin-orbit interaction • • 2. Superfluidity: Liquid He-4 & He-3: Low boiling T Weak van der Waals force & low atomic mass • ⇐ He-4 (Boson) < 2.17 K (BEC), He-3 (Fermion) < 2.49 mK (BCS) •

3. Diamond: Indirect : Eg = 5.5 eV, good insulator

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