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The quantum mechanics of superconductors Subir Sachdev Talk online: http://pantheon.yale.edu/~subir The quantum mechanics of •Insulators •Metals •Semiconductors •Superconductors Technology in the 19th century Technology of the 20th century Technology of the 21st century ? Electrical resistivity of Cu at 273 K: 1.56 10 Electrical resistivity of C (diamond) at 273 K: 10 Ohm meter 9 1021 Ohm meter size of atom ~ 1010 meters Ratio is comparable to the ratio size of galaxy ~ 1021 meters Resistance of Hg measured by Kammerlingh Onnes in 1911 6 108 Ohm meter K Quantum theory of metals Valence electrons occupy plane wave states which extend across the entire material k2 k 2 m* i2 H * 2 m i 2 e e 2 k e e e kF k 1 4 3 n 2 kF 3 2 3 kF Fermi velocity vF * ~ 108 cm/sec m Fermi surface of copper Electrical conduction occurs by acceleration of electrons near the Fermi surface Quantum theory of metals i2 H VQ cos Qxi * 2m i i 2 e e e e Quantum theory of metals i2 H VQ cos Qxi * 2m i i 2 Electrons can scatter off VQ and change their momentum by Q k Q Q k Fermi surface of copper Deviation in shape from sphere is caused by non-resonant scattering off periodic potential Quantum theory of insulators i2 H VQ cos Qxi * 2m i i 2 Electrons can scatter off VQ and change their momentum by Q Resonant scattering of electrons between momenta k Q / 2 k Q Q 2 Q 2 k Quantum theory of insulators i2 H VQ cos Qxi * 2m i i 2 Electrons can scatter off VQ and change their momentum by Q Resonant scattering of electrons between momenta k Q / 2 Energy gap: k No electron states in a certain range of energies ! Q 2 Q 2 k Quantum theory of insulators Insulators have their Fermi level exactly at the edge of the band gap. Electrical conduction requires excitation of electrons across the band gap. k Q 2 Q 2 k Quantum theory of semiconductors Semiconductors are insulators with a small energy gap Si k Q 2 Q 2 k Quantum theory of semiconductors Doping with a small concentration of P introduces mobile charge carriers Si k P Q 2 Q 2 k We have so far considered the quantum theory of a large number of fermions moving in a periodic potential and found ground states which are metals, insulators and semiconductors Now consider the quantum theory of a large number of bosons moving in a periodic potential 87Rb bosonic atoms in a magnetic trap and an optical lattice potential The strength of the period potential can be varied in the experiment M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Strong periodic potential “Eggs in an egg carton” Tunneling between neighboring minima is negligible and atoms remain localized in a well. However, the total wavefunction must be symmetric between exchange Strong periodic potential “Eggs in an egg carton” Tunneling between neighboring minima is negligible and atoms remain localized in a well. However, the total wavefunction must be symmetric between exchange Strong periodic potential “Eggs in an egg carton” Tunneling between neighboring minima is negligible and atoms remain localized in a well. However, the total wavefunction must be symmetric between exchange Insulator = + + + + + Weak periodic potential A single atom can tunnel easily between neighboring minima Weak periodic potential A single atom can tunnel easily between neighboring minima Weak periodic potential A single atom can tunnel easily between neighboring minima Weak periodic potential A single atom can tunnel easily between neighboring minima Weak periodic potential G = e i + + The ground state of a single particle is a zero momentum state, which is a quantum superposition of states with different particle locations. The Bose-Einstein condensate in a weak periodic potential Lowest energy state for many atoms BEC = G G G = e3i + ( + + + + + + ....27 terms Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation) ) The Bose-Einstein condensate in a weak periodic potential Lowest energy state for many atoms BEC = G G G = e3i + ( + + + + + + ....27 terms Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation) ) The Bose-Einstein condensate in a weak periodic potential Lowest energy state for many atoms BEC = G G G = e3i + ( + + + + + + ....27 terms Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation) ) The Bose-Einstein condensate in a weak periodic potential Lowest energy state for many atoms BEC = G G G = e3i + ( + + + + + + ....27 terms Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation) ) The Bose-Einstein condensate in a weak periodic potential Lowest energy state for many atoms BEC = G G G = e3i + ( + + + + + + ....27 terms Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation) ) The Bose-Einstein condensate in a weak periodic potential Lowest energy state for many atoms BEC = G G G = e3i + ( + + + + + + ....27 terms Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation) ) The Bose-Einstein condensate in a weak periodic potential Lowest energy state for many atoms BEC = G G G = e3i + ( + + + + + + ....27 terms Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation) ) 87Rb bosonic atoms in a magnetic trap and an optical lattice potential The strength of the period potential can be varied in the experiment M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Superfluid-insulator transition V0=0Er V0=13Er V0=3Er V0=7Er V0=10Er V0=14Er V0=16Er V0=20Er In any subvolume of the superfluid, there are large fluctuations in the number of atoms but a definite phase Physical meaning of Consider two superfluids connected by a weak link Phase 1 ; iN11 e w N1 Phase 2 ; H w N1 1, N 2 1 N1 , N 2 N1 1, N 2 1 N1 , N 2 (Super)current in weak link = = = iw N ,N 1 2w 2 eiN22 N2 d N1 i N 1, H dt N1 1, N 2 1 N1 , N 2 N1 1, N 2 1 sin 1 2 A superfluid state is characterized by the Ginzburg-Landau complex “order parameter” Y(x) Y x ~ e i ( x ) Supercurrent ~ Persistent currents C C dx 2 n No local change of the wavefunction can change the value of n Supercurrent flows “forever” How do metals form superconductors ? Electrons form (Cooper) pairs at low temperatures, and these pairs act like bosons Pair wavefunction ky kx Y k S 0 (Bose-Einstein) condensation of Cooper pairs The quantum mechanics of •Insulators •Metals •Semiconductors •Superconductors

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