Classical and Quantum impurities in superconductors 100 years old - PowerPoint PPT Presentation

Classical and Quantum impurities in superconductors 100 years old and still dirty I lya Vekhter Louisiana St at e Univers rsit y, USA MPIPKS, Dresden 6/3/2011

Superconductivity review I Simplest (well understood) correlated system: often even when emerges from a strange normal state pairing of electrons pairing amplitude Nontrivial object: near the Fermi surfc + ( ) Ψ α β ∝ ψ ψ , ; , ( ) ( ) r r r r α β 1 2 1 2 Bose-condensation of Cooper pairs Superco conducti ctivity ty Ψ α β = χ ϕ − ( , ; ) ( ) α , β =↑ , ↓ r r r r Simplest case: αβ 1 2 1 2 Coop ooper p pai airs have w well-defined sp spin (si singlet o t or tr triplet t pairs) MPIPKS, Dresden 6/3/2011

Why impurities? Kondo impurity: local singlet + electrons Ψ α β = χ ϕ − ( , ; ) ( ) Simplest superconductor: no spin-orbit r r r r αβ 1 2 1 2 singlet S=0   0 1   χ s = χ = σ y   ↑↓ − ↓↑ ( ) i , αβ − αβ αβ , s  1 0  triplet S=1 − +   d id d   χ = x y z χ = σ d ⋅ σ y [( )( )] ↑↓ + ↓↑ ↑↑ ↓↓   i , , + αβ αβ t , t d d id   z x y Competition of energy scales: impurities vs pairing MPIPKS, Dresden 6/3/2011

Superconductors vs Kondo metals supercond Kondo No resistance minimum: superconductivity From D. MacDonald et al. 1962 H. Kamerlingh-Onnes 1911 J. Cooper and M. Miljak’ 1976 No susceptibility: Mn in Al Meissner effect NB: sometimes NMR MPIPKS, Dresden 6/3/2011

Superconductivity review II = ∑ + + + ξ − ∆ ( ) H BCS c c k c c BCS Hamiltonian: α α αβ α − β k k k k k band pairing, “anomalous” singlet/triplet; ∑ ′ ∆ = ( ) ( , ) Order parameter: isotropic/anisotropic; k V k k c c ′ ′ αβ αβ γδ − δ γ , k k unitary or not… MPIPKS, Dresden 6/3/2011

Superconductivity review II = ∑ + + + ξ − ∆ ( ) H BCS c c k c c BCS Hamiltonian: α α αβ α − β k k k k k band pairing, “anomalous” singlet/triplet; ∑ ′ ∆ = ( ) ( , ) Order parameter: isotropic/anisotropic; k V k k c c ′ ′ αβ αβ γδ − δ γ , k k unitary or not… ξ ∆     ( ) ( ) k c α +     = k k Matrix form: H BCS c c     singlet α − α + ∆ − ξ k k *  ( )    k c − α − k k MPIPKS, Dresden 6/3/2011

Superconductivity review II = ∑ + + + ξ − ∆ ( ) H BCS c c k c c BCS Hamiltonian: α α αβ α − β k k k k k band pairing, “anomalous” singlet/triplet; ∑ ′ ∆ = ( ) ( , ) Order parameter: isotropic/anisotropic; k V k k c c ′ ′ αβ αβ γδ − δ γ , k k unitary or not… ξ ∆     ( ) ( ) k c α +     = k k Matrix form: H BCS c c     singlet α − α + ∆ − ξ k k *  ( )    k c − α − k k ∑ = γ + σ γ ( ) k H BCS E Bogoliubov transformation σ k k Excitation energies = ξ + ∆ 2 2 energy gap ( ) | ( ) | E k k k 2 electron u k γ k σ = u k c k σ − σ v k * c − k σ † Eigenstates 2 hole v k MPIPKS, Dresden 6/3/2011

Isotropic vs anisotropic superconductors ∆ ∝ Ψ α β = χ ϕ − ( , ) ( , ; ) ( ) r r r r r r Connection to pair wave function αβ αβ 1 2 1 2 1 2 Ψ α β = − Ψ β α Ψ α β = χ ϕ − ( , ; ) ( ) ( , ; ) ( ; , ) r r r r Fermion exchange r r r r αβ 1 2 2 1 1 2 1 2 Spatial part: angular momentum l Spin part: 2x2 matrix singlet S=0 d xy = χ = σ = − χ 0 , 2 , 4 ... l y ( ) i αβ αβ βα , , s s s, d… wave triplet S=1 = 1 , 3 , 5 ... χ = σ ⋅ = χ l σ y [( )( )] i d αβ αβ βα t , t , p, f… wave non-s-wave (anisotropic) states favored by strong Coulomb repulsion MPIPKS, Dresden 6/3/2011

Anisotropic superconductors density of states s- wave No excitations at low T ∆ Activated behavior e - ∆ /T Isotropic gap T Al, Be, Nb 3 Sn ∆ k = ∆ < 1978 ( ) 0 d- wave Density of qp ∝ T d 2 − 2 x y Specific heat C(T) ∝ T 2 -1 ∝ T 3 NMR T 1 ∆ Gap with κ /T universal zeroes (nodes) T ∆ k = ∆ φ ( ) cos 2 0 Power laws Cuprates, heavy fermions, >1979 MPIPKS, Dresden 6/3/2011

Pure and impure superconductors Pure superconductor: What is the effect of: density of states 1) an isolated impurity (STM spectra) 2) ensemble of impurities Isotropic gap ( T c , planar junctions) Al, Be, Nb 3 Sn < 1978 How is this picture modified by impurities: Gap with 1) locally zeroes (nodes ) 2) globally Cuprates, heavy fermions, >1979 MPIPKS, Dresden 6/3/2011

Classical and quantum impurities 1. Potential scatterers 2. Spin scattering = 2a. Classical spin [ , ] 0 S i S j ≠ [ , ] 0 2b. Quantum spin S i S j ∑ 3. Anderson impurity: interpolate + = + + + + ( ) . . H E n n Un n Vd c h c σ σ ↑ ↓ ↑ ↓ 0 imp k i i i i between the two regimes k 4. Single Impurity vs. many impurities 5. Conventional vs unconventional superconductors MPIPKS, Dresden 6/3/2011

Single Impurities MPIPKS, Dresden 6/3/2011

Single Impurity Problem We are solving a scattering problem (drop spin indices) + ∫ ∫ = ρ = Ψ Ψ ( ) ( ) ( ) ( ) ( ) H U r r d r d r U r r r σ σ imp ∑ ′ ∑ + − + ∫ = = ( ) i k k r ( ) d r U r c c e U c c ′ ′ ′ σ σ σ σ k k k k k k ′ ′ k k k k U k ′ k k ′ k For classical impurities ( U is a function) this can be solved exactly MPIPKS, Dresden 6/3/2011

Reminder: Green’s functions Prescription: + r ′ ′ ω τ τ = − ψ τ ψ τ ω = π + ( , ; ) ( , ; , ) ( , ) ( , ) G r r 2 ( 1 / 2 ) G r r T r T n αβ αβ τ α β 1 2 1 2 1 2 n n Matsubara ω → ω + δ ω R i i ( , ; ) • obtain retarded Green’s function G r r n 1 2 ω = − π − ω 1 R ( , ) Im ( , ; ) • poles=excitation energies N r G r r ∫ − = − π ω 1 R Im ( ; ) d k G k • density of states=Im part [ ] [ ] Example: normal metal − − ω = ω − ξ → ω − ξ + δ 1 1 ( , ) G k i i n n k k = ∫ d ω δ ω − ξ ( ) ( ) N k k MPIPKS, Dresden 6/3/2011

Nambu formalism and matrices I • Mix particles/holes, spin up/down Nambu-Gor’kov 4x4 matrix • BCS hamiltonian σ , i τ • Matrices in spin and particle-hole space respectively i • Matrix structure of the impurity scattering: ˆ ⇒ τ Potential: e.g attracts electrons/repels holes ( ) ( ) U r U r 3 [ ] 2 ( ) ( ) = + τ + − τ σ σ α σ σ Magnetic: ⋅ ⇒ ⋅ σ α 1 1 / S S 3 3 3 3 • Pure BCS MPIPKS, Dresden 6/3/2011

Nambu formalism and matrices II − 1 ω − ξ ∆ “anomalous” Green’s   Green’s function of i ˆ   ω = n k k ( ; ) function, ODLRO G k   a superconductor ∆ ω + ξ 0 *   i k n k “normal” particle & hole propagators MPIPKS, Dresden 6/3/2011

Nambu formalism and matrices II − 1 ω − ξ ∆ “anomalous” Green’s   Green’s function of i ˆ   ω = n k k ( ; ) function, ODLRO G k   a superconductor ∆ ω + ξ 0 *   i k n k “normal” particle & hole propagators − ω = − π ω 1 Density of states R “normal” part ( , ) [Im ] ( , ; ) r r r N G 11 poles: energies → energy gap Self-consistency “anomalous” part ∑∫ ′ ′ ′ ∆ = ω ) ( , ) ( , T d k V k k G 12 k condition on the k n highest T with sol’n → ω order parameter n transition temperature Not important for single impurity Crucial for multiple impurities MPIPKS, Dresden 6/3/2011

Single impurity ∑ + = + • Key: multiple scattering . c . H imp U c c h ′ ′ σ σ k k k k ′ k k change of momentum/spin at each scattering event can include all the scattering events … in principle MPIPKS, Dresden 6/3/2011

T-matrix solution U = + + … k ′ k MPIPKS, Dresden 6/3/2011

T-matrix solution U = + + … k ′ k structure is especially simple for isotropic scatterers, T-matrix = U U ′ k , k depends on ω only. = ω ( ) T T ′ , k k − 1  −  ∑ ˆ ˆ ω = ˆ + ˆ ω ˆ ω ∑ ω = ˆ ω ˆ ˆ ( , ) ( ) ( ) T U U G k T ( ) 1 ( , ) T U G k U   0 0   k k T-ma matrix in includ ludes a all ll the effects o of mult multiple scattering on sc on a si single i e impu purity MPIPKS, Dresden 6/3/2011