Lecture 22 Metals - Superconductivity

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1 Lecture 22: Metals (Review and Kittel Ch. 9) and Superconductivity I (Kittel Ch. 1) Resistence Ω Leiden, Netherlands K g sample < 1-5 Ω Outline metals Recall properties (From lectures 12, 13; Kittel ch. 6) Superconductivity - xperimental Facts ZRO resistance at nonzero temperature xclusion of magnetic fields eat Capacity shows there is a gap Isotope effect (Kittel Ch 1 ) Physics 46 F 26 Lect 22 1 Physics 46 F 26 Lect 22 2 What is special about electrons? Fermions - obey exclusion principle Fermions have spin s = 1/2 - two electrons (spin up and spin down) can occupy each state lectron Gas Kinetic energy = ( p 2 /2m ) = ( h 2 /2m ) k 2 Fermi Surface is the surface in reciprocal space that is the boundary between the filled and empty states For the electron gas this is a sphere of radius k F where N elec /V = (1/3π 2 ) k F 3 he Fermi energy is F = ( h 2 /2m ) k F 2 k F Recall - lectron Gas Density of States 3 dimensions D() = (1/2π 2 ) 1/2 (2m / h 2 ) 3/2 ~ 1/2 Fermi Surface Physics 46 F 26 Lect 22 3 Physics 46 F 26 Lect 22 4 D() Filled F mpty Fermi Distribution At finite temperature, electrons are not all in the lowest energy states Applying the fundamental law of statistics to this case (occcupation of any state and spin only can be or 1) leads to the Fermi Distribution (Kittel appendix) 1 1/2 f() = 1/[exp((-µ)/k B ) + 1] f() µ k B Chemical potential for electrons = Fermi energy at = D() Physics 46 F 26 Lect 22 5 ypical values for electrons ere we count only valence electrons (see Kittel table) lement N elec /atom F F = F /k B Li ev 5.5 x1 4 K Na eV 3.75 x1 4 K Al ev 13.5 x1 4 K Conclusion: For typical metals the Fermi energy (or the Fermi temperature) is much greater than ordinary temperatures Physics 46 F 26 Lect

eat Capacity for lectrons Just as for phonons the definition of heat capacity is C = du/d where U = total internal energy For << F = F /k B it is easy to see that roughly U ~ U + N elec (/ F ) k B so

2 eat Capacity for lectrons Just as for phonons the definition of heat capacity is C = du/d where U = total internal energy For << F = F /k B it is easy to see that roughly U ~ U + N elec (/ F ) k B so that C = du/d ~ N elec k B (/ F ) 1 1/2 f() µ Chemical potential for electrons D() Physics 46 F 26 Lect 22 7 eat capacity Comparison of electrons in a metal with phonons eat Capacity C 3 lectrons dominate at low in a metal Phonons approach classical limit C ~ 3 N atom k B lectrons have C ~ N elec k B (/ F ) Phonons dominate at high because of reduction factor (/ F ) Physics 46 F 26 Lect 22 8 What about a real metal? In a crystal the energies are not = ( h 2 /2m ) k 2 Instead the energy is n (k), where k is the wavevector in the Brillouin Zone, and n = 1,2,3, labels the bands he energy n (k) is different for k in different directions he concepts still apply he states are filled for n (k) < Fermi he states are empty for n (k) > Fermi his defines the Fermi surface: the surface in k- space where n (k) < Fermi the boundary between filled and empty states Physics 46 F 26 Lect 22 9 he Fermi surface in copper Cu has the fcc crystal structure he figure shows the Brillouin Zone and the Fermi Surface Note that the Fermi surface is nearly spherical! he Fermi surface is very different from a sphere in many crystals but the idea is still the same! See Kittel ch. 9, Fig 29 for the same figure Figure from Nara Women s University Physics 46 F 26 Lect 22 1 eat capacity xperimental results for metals C/ = γ + A 2 +. It is most informative to find the ratio γ / γ(free) where γ(free) = (π 2 /2) (N elec / F ) k B2 is the free electron gas result. quivalently since F 1/m, we can consider the ratio γ / γ(free) = m(free)/m th *, where m th * is an thermal effective mass for electrons in the metal Metal m th */ m(free) Li 2.18 Na 1.26 K 1.25 Al 1.48 Cu 1.38 m th * close to m(free) is the good, simple metals! Physics 46 F 26 Lect lectrical Conductivity & Ohm s Law Consider electrons in an external field. hey experience a force F = -e Now F = dp/dt = h dk/dt, since p = h k hus in the presence of an electric field all the electrons accelerate and the k points shift, i.e., the entire Fermi surface shifts quilibrium - no field he same ideas apply to real metals with non-spherical Fermi surfaces With applied field Physics 46 F 26 Lect

3 lectrical Conductivity & Ohm s Law What limits the acceleration of the electrons? Scattering increases as the electrons deviate more from equilibrium After field is applied a new equilibrium results as a balance between acceleration by field and scattering quilibrium - no field With applied field Physics 46 F 26 Lect lectrical Conductivity and Resistivity he conductivity σ is defined by j = σ, where j = current density ow to find σ? From before F = dp/dt = m dv/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 = If we define a scattering time τ and scattering rate1/τ h ( dk/dt + k /τ ) = F= q (q = charge) Now j = n q v (where n = density) so that j = 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) Physics 46 F 26 Lect Scattering mechanisms Impurities - wrong atoms, missing atoms, extra atoms,. Proportional to concentration Lattice vibrations - atoms out of their ideal places Proportional to mean square displacement lectrical Resistivity Resistivity ρ is due to scattering: Scattering rate inversely proportional to scattering time τ ρ scattering rate 1/τ Matthiesson s rule - scattering rates add ρ = ρ vibration + ρ impurity 1/τ vibration + 1/τ impurity (Really these conclusions depend upon ideas from the next section that there is no scattering in a perfect crystal.) emperature dependent <u 2 > emperature independent - sample dependent Physics 46 F 26 Lect Physics 46 F 26 Lect lectrical Resitivity Consider relative resistance R()/R(=3K) ypical behavior (here for potassium) Relative resistence.5.1 Increase as 2 Inpurity scattering dominates at low in a metal (Sample dependent) Phonons dominate at high because mean square displacements <u 2 > Leads to R (Sample independent) Physics 46 F 26 Lect Laboratory of Kamerling Onnes in Leiden Why there? Why then? elium had just been liquified in Onnes lab - making possible experiments at temperatures around 4.2K and below Resistence Ω K g sample < 1-5 Ω Physics 46 F 26 Lect

elements NO the magnetic 3d transition and 4f rare earth elements - NO the best metals - like Cu, Ag, Na transition ransition is VRY narrow - < 1-4 K Reversible (unlike magnet) ransition emperatures

4 elements NO the magnetic 3d transition and 4f rare earth elements - NO the best metals - like Cu, Ag, Na transition ransition is VRY narrow - < 1-4 K Reversible (unlike magnet) ransition emperatures c Al 1.2 K g 4.6 K Pb 7.2 K Super conducting Au <.1 K - not found to be superconducting! Na 3 C 6 4 K (199) YBa 2 Cu 3 O 7 93 K (1987) Record today 14 K Physics 46 F 26 Lect Physics 46 F 26 Lect 22 2 Is Resistance Really ZRO?? Currents have been flowing in rings in laboratories with no detectable loss for > 5 years! heory says the current can continue for > age of universe ffect of a Magnetic Field Magnetic fields tend to destroy superconductivity c Note: = external applied field B = internal field B = + µ M M = Magnetization Phase ransition SUPRCONDUCING SA IS A NW PAS OF MAR c Physics 46 F 26 Lect Physics 46 F 26 Lect Not just a perfect conductor! A superconductor is NO just a perfect conductor A perfect conductor would do the following: Zero Field Cooled Field Cooled Meisner ffect (1934) A superconductor can actively push out a magnetic field - the Meisner effect xcludes Magnetic Field Zero Field Cooled Field Cooled > c < c A superconductor is different! > c < c rapped Field Physics 46 F 26 Lect > c < c > c < c he superconductor can exclude a magnetic field up to a critical field c Physics 46 F 26 Lect

5 Meisner ffect Magnetic field B is excluded for fields less than a critical field c where is the external applied field he total internal field is B = + µ M For type I superconductors B= for < c Perfect Diamagnetism! B - µ M ype I vs ype II Magnetic field B is excluded only up to a critical field c1 For type II superconductors, at higher fields there is penetration of the field coexisting with superconductivity up to = c2 - µ M c c c1 c c2 Physics 46 F 26 Lect Physics 46 F 26 Lect ype II For type II superconductors, at higher fields there is penetration of the field in lines of normal material coexisting with superconductivity in surrounding material up for c1 < < c2 Flux Lattice of quantized units of flux (more later) Magnetic flux penetrates through the superconductor by creating small regions normal metal applied Physics 46 F 26 Lect eat capacity (Specific eat) Comparison of electrons in a superconductor and a normal metal Phase ransition eat Capacity C Superconductor metal lectrons have C ~ N elec k B (/ F ) Shows there is an energy gap in the superconductor! (Specific heat is like an insulator!) Physics 46 F 26 Lect Isotope ffect (195) For materials made from the same elements - but different isotopes - c changes! xperiment - c ~ 1/ M 1/2 MUS be connected to MOION of the nuclei Physics 46 F 26 Lect Summary metal - Recall properties No special magnetic properties for nonmagnetic metals, µ 1, B Resistance vs eat capacity vs Superconductivity - xperimental Facts ZRO resistance at nonzero temperature NW PAS OF MAR - Meisner ffect (expulsion of magnetic fields) - shows a superconductor is not just a perfect conductor eat Capacity - shows there is a phase transition - below c a gap, like an insulator! Isotope effect - something to do with MOION of nuclei Physics 46 F 26 Lect

6 Next time Superconductivity - theory Basic ideas and phenomena Bardeen- Cooper-Schrieffer heory (Nobel Prize for work done in UIUC Physics) (Kittel parts of Ch 1) Physics 46 F 26 Lect

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

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