Outline. Part II - Electronic Properties of Solids Lecture 13: The Electron Gas Continued (Kittel Ch. 6) E. Electron Gas in 3 dimensions

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Eleanore Golden
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1 Part II - lectronic Properties of Solids Lecture 13: The lectron Gas Continued (Kittel Ch. 6) quilibrium - no field Physics 460 F 2006 Lect 13 1 Outline From last time: Success of quantum mechanics Pauli xclusion Principle, Fermi Statistics nergy leels in 1 and 3 dimensions Density of States, Heat Capacity Today: Fermi surface Transport lectrical conductiity and Ohm s law Impurity, phonon scattering Hall ffect Thermal conductiity Metallic inding (Read Kittel Ch 6) Physics 460 F 2006 Lect 13 2 lectron Gas in 3 dimensions Recall from last lecture: nergy s k (k) = ((h 2 /2m) (k x2 + k y2 + k z2 ) = (h 2 /2m) k 2 Density of states D() = (1/2π 2 ) 1/2 (h 2 /2m) -3/2 ~ 1/2 F mpty states Filled states k D() Filled F mpty lectrons obey exclusion Principle: The lowest energy possible is for all states filled up to the Fermi momentum and Fermi energy F = (h 2 /2m) 2 gien by = (3π 2 N elec /V ) 1/3 and F = (h 2 /2m)(3π 2 N elec /V ) 2/3 Physics 460 F 2006 Lect 13 3 Fermi Distribution At finite temperature, electrons are not all in the lowest energy states. Thermal energy causes states to be partially occupied. Fermi Distribution (Kittel appendix) f() = 1/[exp((-µ)/k T) + 1] 1 1/2 f() µ D() k T For typical metals the Fermi energy is much greater than ordinary temperatures. xample: For Al, F = 11.6 ev, i.e., T F = F /k = 13.5 x10 4 K At ordinary temperature, the only change in the occupation of the states is ery near the chemical potential µ. States are filled for states with << µ, and empty for states with >> µ. Heat capacity C = du/dt ~ N elec k (T/ T F ) Physics 460 F 2006 Lect 13 4 Chemical potential for electrons = Fermi energy at T=0 lectrical Conductiity & Ohm s Law The filling of the states is described by the Fermi surface the surface in k-space that separates filled from empty states For the electron gas this is a sphere of radius. lectrical Conductiity & Ohm s Law Consider electrons in an external field. They experience a force F = -e Now F = dp/dt = h dk/dt, since p = h k Thus in the presence of an electric field all the electrons accelerate and the k points shift, i.e., the entire Fermi surface shifts empty filled Lowest energy state filled for states with k <, i.e., < F Physics 460 F 2006 Lect 13 5 quilibrium - no field Physics 460 F 2006 Lect 13 6

2 lectrical Conductiity & Ohm s Law What limits the acceleration of the electrons? Scattering increases as the electrons deiate more from equilibrium After field is applied a new equilbrium results as a balance of acceleration by field and scattering quilibrium - no field Physics 460 F 2006 Lect 13 7 lectrical Conductiity and Resitiity The conductiity σ is defined by = σ, where = current density How to find σ? From before F = dp/dt = m d/dt = h dk/dt quilibrium is established when the rate that k increases due to equals the rate of decrease due to scattering, then dk/dt = 0 If we define a scattering time τ and scattering rate1/τ h ( dk/dt + k /τ ) = F= q (q = charge) Now = n q (where n = density) so that = n q (h k/m) = (n q 2 /m) τ σ = (n q 2 /m) τ Resistance: ρ = 1/ σ m/(n q 2 τ) Note: sign of charge does not matter Physics 460 F 2006 Lect 13 8 Scattering mechanisms Impurities - wrong atoms, missing atoms, extra atoms,. Proportional to concentration Lattice ibrations - atoms out of their ideal places Proportional to mean square displacement lectrical Resitiity Resistiity ρ is due to scattering: Scattering rate inersely proportional to scattering time τ ρ scattering rate 1/τ Matthiesson s rule - scattering rates add ρ = ρ ibration + ρ impurity 1/τ ibration + 1/τ impurity This also applies to a crystal (not ust the electron gas) using the fact that there is no scattering in a perfect crystal as discussed in the next lectures Temperature dependent <u 2 > Temperature independent - sample dependent Physics 460 F 2006 Lect 13 9 Physics 460 F 2006 Lect lectrical Resitiity Consider relatie resistance R(T)/R(T=300K) Typical behaior (here for potassium) Relatie resistence Increase as T 2 Inpurity scattering dominates at low T in a metal (Sample dependent) Phonons dominate at high T because mean square displacements <u 2 > T Leads to R T (Sample independent) T Physics 460 F 2006 Lect Interpretation of Ohm s law lectrons act like a gas A electron is a particle - like a molecule. lectrons come to equilibrium by scattering like molecules (electron scattering is due to defects, phonons, and electron-electron scattering). lectrical conductiity occurs because the electrons are charged, and it shows the electrons moe and equilibrate What is different from usual molecules? lectrons obey the exclusion principle. This limits the allowed scattering which means that electrons act like a weakly interacting gas. Physics 460 F 2006 Lect 13 12

3 Hall ffect I lectrons moing in an electric and a perpendicular magnetic field Now we must carefully specify the ector force F = q( + (1/c) x ) (note: c 1 for SI units) (q = -e for electrons) F F Vector directions shown for positie q Physics 460 F 2006 Lect Hall ffect II Releant situation: current = σ = nq flowing along a long sample due to the field ut NO current flowing in the perpendicular direction This means there must be a Hall field Hall in the perpendicular direction so the net force = 0 = q( Hall + (1/c) x ) = 0 Hall y Physics 460 F 2006 Lect z x Hall ffect III Since = q( Hall + (1/c) x ) = 0 and = /nq then defining = () x, Hall = ( Hall ) y, = ( ) z, Hall = - (1/c) (/nq) (- ) and the Hall coefficient is R Hall = Hall / = 1/(nqc) or R Hall = 1/(nq) in SI Sign from cross product Hall ffect IV Finally, define the Hall resistance as ρ Hall = R Hall = Hall / which has the same units as ordinary resistiity R Hall = Hall / = 1/(nq) ach of these quantities can be measured directly Note: R Hall determines sign of charge q Since magnitude of charge is known R Hall determines density n Hall Physics 460 F 2006 Lect Hall Physics 460 F 2006 Lect Heat Transport due to lectrons A electron is a particle that carries energy - ust like a molecule. lectrical conductiity shows the electrons moe, scatter, and equilibrate What is different from usual molecules? lectrons obey the exclusion principle. This limits scattering and helps them act like weakly interacting gas. hot Heat Flow cold Physics 460 F 2006 Lect Heat Transport due to lectrons Definition (ust as for phonons): thermal = heat flow (energy per unit area per unit time ) = - K dt/dx If an electron moes from a region with local temperature T to one with local temperature T - T, it supplies excess energy c T, where c = heat capacity per electron. (Note T can be positie or negatie). On the aerage for a thermal : T = (dt/dx) x τ, where τ = mean time between collisions Then = - n x c x τ dt/dx = - n c x2 τ dt/dx Density Flux Physics 460 F 2006 Lect 13 18

4 lectron Heat Transport - continued Just as for phonons: Aeraging oer directions gies ( x 2 ) aerage = (1/3) 2 and = - (1/3) n c 2 τ dt/dx Finally we can define the mean free path L = τ and C = nc = total heat capacity, Then = - (1/3) C L dt/dx and K = (1/3) C L = (1/3) C 2 τ = thermal conductiity (ust like an ordinary gas!) Physics 460 F 2006 Lect lectron Heat Transport - continued What is the appropriate? The elocity at the Fermi surface = F What is the appropriate τ? Same as for conductiity (almost). Results using our preious expressions for C: K = (π 2 /3) (n/m) τ k 2 T Relation of K and σ -- From our expressions: K / σ = (π 2 /3) (k /e) 2 T This ustifies the Weidemann-Franz Law that K / σ T Physics 460 F 2006 Lect lectron Heat Transport - continued K σt Recall σ constant as T 0, σ 1/T as T large Thermal conductiity K W/cm K 50 0 Low T -- K increases as heat capacity increases ( and L are ~ constant) T Approaches high T limit: K fi constant Physics 460 F 2006 Lect lectron Heat Transport - continued Comparison to Phonons lectrons dominate in good metal crystals Comparable in poor metals like alloys Phonons dominate in non-metals Physics 460 F 2006 Lect Metallic inding (Treated only in problems in Kittel) lectron gas kinetic energy is positie, i.e., replusie. See homework for, pressure, bulk modulus Key point: kinetic (1/V) 2/3 What is the attraction that holds metals together? Coulomb attraction for the nuclei NOT included in gas so far - must be added nergy of point nuclei in uniform electron gas: Key point: Coulomb (1/V) 1/3 Approximate expressions in Kittel problem 8 nergy per electron: Coulomb 1.80/r s Ryd, where (4π/3)r 3 s = V Net effect is metallic binding Physics 460 F 2006 Lect Where can the electron gas be found? In semiconductors! More later - in doped semiconductors, the extra electrons (or missing electrons) can act like an electron gas in a background Where can 1d or 2d gas be found? In semiconductor structures! Layers of GaAs and AlAS can make nearly Ideal 2d gasses 1d wires can also be made More later Physics 460 F 2006 Lect 13 24

5 Summary lectrical Conductiity - Ohm s Law σ = (n q 2 /m) τ ρ = 1/σ Hall ffect ρ Hall = R Hall = Hall / ρ and ρ Hall determine n and the charge of the carriers Thermal Conductiity K = (π 2 /3) (n/m) τ k 2 T Weidemann-Franz Law: K / σ = (π 2 /3) (k /e) 2 T Metallic inding Kinetic repulsion Coulomb attraction to nuclei (not included in gas model - must be added) Physics 460 F 2006 Lect Next time XAM Wednesday, October 11 Next week: lectrons in crystals nergy ands We will use many ideas from the understanding of crystals and lattice ibrations to describe electron waes in a periodic crystal! (Read Kittel Ch 7) Physics 460 F 2006 Lect Comments on xam Three types of problems: Short answer questions Order of Magnitudes ssay question Quantitatie problems Physics 460 F 2006 Lect 13 27

Lecture 22 Metals - Superconductivity

Lecture 22 Metals - Superconductivity Lecture 22: Metals (Review and Kittel Ch. 9) and Superconductivity I (Kittel Ch. 1) Resistence Ω Leiden, Netherlands - 1911.1 4.6 K g sample < 1-5 Ω Outline metals Recall properties (From lectures 12,