1 Unconventional Superconductivity Introduction
2 Superconductivity Field expulsion (1933) Electrical resistance (1911) Meissner-Ochsenfeld effect resistivity B B=0 B T c T<T c T>T c temperature Superconductivity as a thermodynamic phase B London theory (1935) density of superconducting electrons r � � � 2 r j = � B � 2 r � 2 r � 2 = 4 � e 2 n s r r B = � B � B = 4 � mc 2 j � � c x � London penetration depth
2’ Superconductivity Field expulsion (1933) Electrical resistance (1911) Meissner-Ochsenfeld effect resistivity B B=0 B T c T<T c T>T c temperature Superconductivity as a thermodynamic phase � r ( ) = � r i � r ( ) r Order parameter: ( ) e r r condensate with a broken U(1)- gauge symmetry r r r r � � + 1 2 + b � 4 + K 2 2 [ ] = ( ) � d 3 r a ( T ) � F � , A D A � � � � � � 8 � � r r r � + i 2 e Ginzburg-Landau theory (1950) minimal coupling D = A h c
3 Conventional superconductivity � r ( ) = � r i � r structureless complex ( ) r ( ) e r r Order parameter condensate wave function Microscopic origin: Coherent state of Cooper pairs B ardeen- C ooper- S chrieffer (1957) r k � violation of U(1)- gauge symmetry r � e i � �� r k � � k e i 2 � c r � � c r k � � r k k r pairs of electrons independent of Conventional � r k = � k diametral on Fermi surface; vanishing total momentum
4 The unsteady rise of T c High-temperature Temperature superconductors 150 K HgBa 2 Ca 2 Cu 3 O 9 133.5K -173 C K.A. Müller =100 K YBa 2 Cu 3 O 7 liquid Nitrogen (77K) 1986 La 2-x Sr x CuO 4 50 K MgB 2 1911 La 2-x Ba x CuO 4 Nb 3 Ge liquid Hydrogen (20K) J.G. Bednorz Nb 3 Sn -273 C Hg Pb NbO liquid Helium (4.2K) = 0 K novel 1900 1950 2010 H. Kamerlingh-Onnes 1910 1940 1920 1930 2000 1960 1970 1980 1990 year low-temperature superconductors
5 The novel superconductors Heavy Fermion superconductors: CeIn 3 CeCu 2 Si 2 Steglich et al. (1979) Mathur et al. (1998) U 1-x Th x Be 13 Ott et al. (1983) 1 normal B T (K) A A+B Quantum A T-violating 0 Critical point CeRhIn 5 Thompson et al. (2001) 4 0 2 x(%) 50 40 CeRhIn T Max UPt 3 Stewart et al. (1984) 5 30 AF PM 20 T (K) T N magnetic field normal 4 C 3 T c 2 B 1 A 0 0 0.5 1.0 1.5 2.0 2.5 3.0 0 0.5 T(K) P (GPa)
6 The novel superconductors High-temperature superconductors Organic superconductors Jerome, Bechtgard et al (1980) Layered perovskite cooper-oxides (TMTSF) 2 M (M=PF 6 , SbF 6 , ReO 4 ,…) T c ~ 1K Müller & Bednorz (1986) La 2-x Sr x CuO 4 T c =45K T c ~ 10K (BEDT-TTF) 2 M ….. YBa 2 Cu 3 O 6+ � T c =92K HgBa 2 Ca 2 Cu 3 O 9 T c =133.5K T T N T* AF T c SC x
7 The novel superconductors Ferromagnetic superconductors: Sr 2 RuO 4 UGe 2 Saxena et a. (2000) RuO 2 plane some similarities with high-T c superconductors, ZrZn 2 Pfleiderer et al. (2001) but T c = 1.5 K Superconductivity within spin-triplet superconductor the ferromagnetic phase
8 The novel superconductors - under extreme conditions Iron under pressure Hydrated Na x CoO 4 � � � Layered structure: triangular Superconductivity in a T c ~ 5 K frustrated electron system Shimizu et al. Nature 412, 316 (2001) Takada et al., Nature 422, 53 (2003)
9 The novel superconductors Time-dependent superconductivity Skutterudite 0.0 500 -0.2 400 ) C/T (mJ/mol K 2 -0.4 300 �� 200 4 -0.6 100 0 5 10 15 20 -0.8 T (K) -1.0 5 10 15 20 T (K) T c = 1.8 K PuCoGa 5 PrOs 4 Sb 12 T c = 18 K Bauer et al. PRB 65, R100506 (2002) Thompson et al. (Los Alamos) Multiple phases
10 The novel superconductors - no inversion symmetry No paramagnetic limiting Ferromagnetic quantum phase transition T c =0.8 K CePt 3 Si UIr T c =0.15 K H c2 exceeds drastically the paramagnetic limit Akazawa et al. J.Phys. Condens. Bauer et al. PRL 92, 027003 (2004) Matter 16, L29 (2004)
Bardeen-Cooper-Schrieffer Microscopic theory of superconductivity
11 BCS mean field theory simple model: pairing interaction band energy band energy: attractive contact pairing interaction: ' Interaction g < 0 k k ’ consider only scattering between zero-momentum - k ’ electron pairs of opposite spin (spin singlet) - k
12 BCS mean field theory simple model: decoupling of interaction term by means of mean fields: particle density spin density BCS - “off diagonal”
13 BCS mean field theory simple model: , replace: mean field Hamiltonian: with
14 BCS mean field theory find quasiparticle states with Bogolyubov-transformation quasiparticle energy
15 Quasiparticle Spectrum hole-like electron-like E E k � quasiparticle excitation gap: � k k F � k condensation energy gain due to gap hole-like electron-like Self-consistence equation: Fermi distribution function solution only for g < 0 attractive
16 critical temperature linearized gap equation continuous transition (2 nd order) Interaction with characteristic energy scale N( � ): electron density of states cutoff constant density of states between �� c and + � c
17 Zero-temperature Gap at T=0: E cond = E s - E n Condensation energy at T=0: energy gain relative to normal state depends on density of states at the Fermi surface and the gap magnitude weak-coupling approach
18 Zero-temperature Gap at T=0: E cond = E s - E n Condensation energy at T=0: energy gain relative to normal state hole-like electron-like E cond = � 1 E 2 2 N (0) � E k � modification of the quasiparticle k spectrum k F � k hole-like
19 Pairing interaction Cooper pair formation (bound state of 2 electrons) needs attractive interaction k’ k k k’ - k -k’ - k’ - k electrons polarize their environment electron phonon interaction: renormalized Coulomb interaction
20 electron-phonon versus Coulomb interaction Polarization effects: with Thomas-Fermi screening length renorm.Coulomb electron-phonon phonon spectrum V � � D �� q � q Debye frequency: � characteristic energy scale repulsive attractive repulsive q
21 Poor-man’s model N(0)V Anderson & Morel (1962) V � µ �� q � q � - � D + � D -W + W � poor man’s interaction: poor man’s electron band: V k , k ’ = V( � , � ’) = V C + V ep N ( � ) repulsive part µ | � , � ’ | < W � N(0)V C = 0 otherwise + W -W attractive part band width: 2W �� | � , � ’ | < � D N(0)V ep = constant density of states: N ( � ) = N(0) 0 otherwise
22 Poor-man’s model N(0)V Anderson & Morel (1962) linearized self-consistent gap equation: � µ - � D + � D � -W + W � -W +W poor man’s interaction: - � D + � D � V k , k ’ = V( � , � ’) = V C + V ep repulsive part µ | � , � ’ | < W N(0)V C = 0 otherwise attractive part �� | � , � ’ | < � D N(0)V ep = 0 otherwise
23 Poor-man’s model N(0)V Anderson & Morel (1962) linearized self-consistent gap equation: � µ - � D + � D � -W + W � -W +W transition temperature T c - � D + � D � renormalized Coulomb repulsion Retardation effect: W Coulomb fast � D electron-phonon slow T c = 0 even if � < µ
24 Retardation effect: � << 1 strong-coupling regime � > 1 weak-coupling regime Eliashberg, McMillan (68) renormalized Coulomb repulsion Metallic strongly correlated electron systems small energy scales: T F small band widths: W T F W � D ~ Important: T D >> 1 strong effect of Coulomb repulsion handy-cap for electron-phonon mediated superconductivity
When Coulomb repulsion is too strong for electron-phonon induced pairing Alternative ways to superconductivity
25 Alternative ways to Cooper pairing Coulomb and electron-phonon interaction very short-ranged ( � TF ) “contact interaction” Bound Cooper pair wavefunction: How to avoid Coulomb repulsion? higher-angular momentum pairing l > 0 with relative angular momentum l=0 “contact interaction” not effective important for “contact interaction” r r Symmetry of pairs of identical electrons: ) = � ˆ k s ˆ � ss ' ( k c c k s ' � = � ( k ) � ( s , s ') r r � spin orbital wave function totally antisymmetric under particle exchange l =0,2,4,… , S=0 singlet even parity: r even odd k s r k s ' � l = 1,3,5,… , S=1 triplet odd parity: r r odd even k k s s ' � � �
26 Requirements for the formation of Cooper pairs Anderson’s Theorems (1959,1984) Cooper pair formation with P=0 relies on symmetries which guarantee degenerate partner electrons Spin singlet pairing: time reversal symmetry r r r harmful: k � T k � = � k � magnetic impurities ferromagnetism paramagnetic limiting Spin triplet pairing: time reversal & inversion symmetry r r r r r r r k � I k � = � k � T k � = � k � IT k � = k � harmful: crystal structure without inversion center
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