Residual Meissner effect and other pre-pairing phenomena in the cuprate superconductors. T. Domański

Wrocław, 2 Oct. 2014 Residual Meissner effect and other pre-pairing phenomena in the cuprate superconductors T. Domański M. Curie-Skłodowska University, Lublin, Poland http://kft.umcs.lublin.pl/doman/lectures

Outline

Outline Preliminaries / pairing vs coherence /

Outline Preliminaries / pairing vs coherence / Pre-pairing / experimental evidence /

Outline Preliminaries / pairing vs coherence / Pre-pairing / experimental evidence / Selected results

Outline Preliminaries / pairing vs coherence / Pre-pairing / experimental evidence / Selected results Bogoliubov quasiparticles above T c

Outline Preliminaries / pairing vs coherence / Pre-pairing / experimental evidence / Selected results - Bogoliubov quasiparticles above T c Diamagnetism above T c

Outline Preliminaries / pairing vs coherence / Pre-pairing / experimental evidence / Selected results - Bogoliubov quasiparticles above T c - Diamagnetism above T c Summary

1. Preliminaries

Superconducting state properties

Superconducting state properties ideal d.c. conductance

Superconducting state properties ideal d.c. conductance Normal conductors: resistance R = ρ l S where ρ 1/σ and σ = ne2 τ m τ(t) relaxation time

Superconducting state properties ideal d.c. conductance a.c. conductance Drude conductance σ(ω) = ne2 τ m 1 1 iωτ obeys the f-sum rule ne2 Re σ(ω) = π m σ N Re σ(ω) 0 1 2 ωτ 3 4 5

Superconducting state properties ideal d.c. conductance a.c. conductance This f-sum rule ne2 Re σ(ω) = π m must be obeyed also below T c, where n = n n + n s n s superfluid density

Superconducting state properties (continued)

Superconducting state properties (continued) ideal diamagnetism /perfect screening of d.c. magnetic field/ T > T c T < T c

Superconducting state properties (continued) ideal diamagnetism /perfect screening of d.c. magnetic field/ Meissner effect is described by the Londons equation T > T c T < T c j = e2 n s (T) mc 2 A where the coefficient e 2 n s (T) mc 2 ρ s (T) = 1 λ 2 ρ s (T) superfluid stiffness λ(t) penetration depth B(x) = B 0 e x/λ x

Superconducting state basic concepts

Superconducting state basic concepts ideal d.c. conductance

Superconducting state basic concepts ideal d.c. conductance ideal diamagnetism (Meissner effect)

Superconducting state basic concepts ideal d.c. conductance ideal diamagnetism (Meissner effect) can be regarded as two sides of the same coin.

Superconducting state basic concepts ideal d.c. conductance ideal diamagnetism (Meissner effect) can be regarded as two sides of the same coin. Both effects are caused by the superfluid fraction n s (T)

Such superfluid fraction marks the coherence between paired electrons.

Such superfluid fraction marks the coherence between paired electrons. Pairing mechanisms can be driven for instance by:

Such superfluid fraction marks the coherence between paired electrons. Pairing mechanisms can be driven for instance by: 1. exchange of phonons / classical superconductors, MgB 2,... /

Such superfluid fraction marks the coherence between paired electrons. Pairing mechanisms can be driven for instance by: 1. exchange of phonons / classical superconductors, MgB 2,... / 2. exchange of magnons / heavy fermion compounds / 3. strong correlations / exchange coupling 2t2 ij U in the high T c superconductors /.. other exotic processes / ultracold atoms, nuclei, gluon-quark plasma /

Formal issues generalities

Formal issues generalities The order parameter χ ĉ ( r i )ĉ ( r j )

Formal issues generalities The order parameter χ ĉ ( r i )ĉ ( r j ) is a complex quantity χ = χ e iθ

Formal issues generalities The order parameter χ ĉ ( r i )ĉ ( r j ) is a complex quantity χ = χ e iθ It has the following physical implications:

Formal issues generalities The order parameter χ ĉ ( r i )ĉ ( r j ) is a complex quantity χ = χ e iθ It has the following physical implications: χ 0 amplitude causes the energy gap

Formal issues generalities The order parameter χ ĉ ( r i )ĉ ( r j ) is a complex quantity χ = χ e iθ It has the following physical implications: χ 0 amplitude causes the energy gap θ 0 phase slippage induces supercurrents

Critical temperature classification

Critical temperature classification The complex order parameter χ = χ e iθ

Critical temperature classification The complex order parameter χ = χ e iθ can vanish at T T c by:

Critical temperature classification The complex order parameter χ = χ e iθ can vanish at T T c by: 1. closing the gap................................. [conventional BCS superconductors ] lim T Tc χ = 0 2. disordering the phase.............................. [ HTSC compounds, URh 2 Si 2 (?) ] lim T Tc θ = 0

Amplitude vs phase driven transition scenario # 1

Amplitude vs phase driven transition scenario # 1 amplitude transition / classical superconductors / k B T c (0) 1.76

Amplitude vs phase driven transition scenario # 1 amplitude transition / classical superconductors / k B T c (0) 1.76 (0) (T) Electrons pairing is responsible for the energy gap (T) in a single particle spectrum 0 Temperature T c (T c ) = 0 Appearance of the electron pairs is simultaneous with onset of their coherence

Amplitude vs phase driven transition scenario # 2

Amplitude vs phase driven transition scenario # 2 phase-driven transition / high T c cuprate oxides / T c / (0)

Amplitude vs phase driven transition scenario # 2 phase-driven transition / high T c cuprate oxides / T c / (0) Early experiments using the muon-spin relaxation indicated that in HTSC T c ρ s (0) / Uemura scaling / The superfluid stiffness ρ s (T) is here defined by ρ s (T) 1 λ 2 (T) = 4πe2 m c 2 n s (T) Y.J. Uemura et al, Phys. Rev. Lett. 62, 2317 (1989).

Amplitude vs phase driven transition scenario # 2 phase-driven transition / high T c cuprate oxides / T c / (0) Recently such scaling has been updated from transport measurements 1 8 ρ s = 4.4σ dc T c / Homes scaling / This new relation is valid for all samples ranging from the underdoped to overdoped region. C.C. Homes, Phys. Rev. B 80, 180509(R) (2009). / ab - plane /

2. Pre-pairing /experimental evidence/

Incoherent pairs above T c experimental fact # 1

Incoherent pairs above T c experimental fact # 1 140 120 La 2-x Sr x CuO 4 T (K) 100 80 60 40 10 20 50 100 500 T onset 20 T c0 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Sr content x Phase slippage detected in the large Nernst effect. Y. Wang et al, Science 299, 86 (2003).

Incoherent pairs above T c experimental fact # 2

Incoherent pairs above T c experimental fact # 2 M d (10 3 A/m ) 0.3 0.2 0.1 0.0-0.1 29 30 31 32 33 3435 36 37 38 K Bi2201 La:0.44 T c = 30 K 30 31 32 34 33 35 38 0.0-0.1-0.2 M d (10 3 A/m ) Van Vleck background (A + BT)H is subracted -0.2 (A) -0.3-2 -1 0 1 2 0 H ( T ) (B) 29 K 0.0 0.2 0.4 0.6 0 H ( T ) -0.3 T c = 30K Enhanced diamagnetic response revealed above T c by the high precission torque magnetometry. L. Li,... and N.P. Ong, Phys. Rev. B 81, 054510 (2010). / Princeton University /

Incoherent pairs above T c experimental fact # 1 + 2

Incoherent pairs above T c experimental fact # 1 + 2 150 ( A ) La 2-x Sr x CuO 4 ( B ) Bi 2 Sr 2-y La y CuO 6 150 T ν onset T onset, T C ( K ) 100 50 T M onset T M onset T ν onset 100 50 T onset, T C ( K ) T c T c 0 0.0 0.1 0.2 0.3 Sr content x 0.6 0.4 0.2 La content y 0 T ν T M onset of the Nernst effect onset of the diamagnetism L. Li,... and N.P. Ong, Phys. Rev. B 81, 054510 (2010).

Incoherent pairs above T c experimental fact # 1 + 2 G. Yu et al, arxiv://1210.6942v1 (preprint). T c T c onset of the sc fluctuations onset of the superconductivity

Incoherent pairs above T c experimental fact # 3

Incoherent pairs above T c experimental fact # 3 G. Drachuck et al, Phys. Rev. B 85, 184518 (2012). / Technion Group / Magnetization measurements for the needle shape La 2 x Sr x CuO 4 single crystals.

Incoherent pairs above T c experimental fact # 4

Incoherent pairs above T c experimental fact # 4 Transfer of the spectral weight in the single particle spectrum. T. Kondo, R. Khasanov, T. Takeuchi, J. Schmalian & A. Kamiński, Nature 457, 296 (2009).

Incoherent pairs above T c experimental fact # 4 A. Kamiński, T. Kondo, T. Takeuchi, and G. Gu, arxiv://1403.0492 (2014) preprint.

Incoherent pairs above T c experimental fact # 5

Incoherent pairs above T c experimental fact # 5 Josephson current in YBaCuO - LaSrCuO - YBaCuO junction with LaSrCuO being in the pseudogap state well above T c T. Kirzhner and G. Koren, Scientific Reports 4, 6244 (2014).

Incoherent pairs above T c experimental fact # 5 Phase diagram for the coherence length ξ(t, x) appearing in the empirical law I c exp [ d/ξ] for Josephson current I c. T. Kirzhner and G. Koren, Scientific Reports 4, 6244 (2014).

Incoherent pairs above T c experimental fact # 6

Incoherent pairs above T c experimental fact # 6 a) STM tip thin normal metal 20 b) 90 nm LSCO x=0.10 SrTiO 3 R (Ω) 10 Au/0.08 Au/0.10 Au/0.18 0 0 50 100 T (K) Pseudogap induced well above T c in ultrathin metallic slab deposited on La 2 x Sr x CuO 4. c) 600 nm O. Yuli et al, Phys. Rev. Lett. 103, 197003 (2009).

Incoherent pairs above T c experimental fact # 6 Measured DOS 1.0 0.5 a) 8 K 15 30 70 80 x = 0.08 90 K Proximity induced pseudogap observed di/dv 0.0 b) 4 K 1.0 10 0.5 15 20 30 45 x = 0.10 far above T c. 0.0 50 90 K 1.0 c) 8 K 10 15 x = 0.18 18 O. Yuli et al, PRL 103, 197003 (2009). 0.5 0.0 22 27 33 K -50 0 50 Sample Bias (mv)

Incoherent pairs above T c experimental fact # 7

Incoherent pairs above T c experimental fact # 7 S.J. Moon et al, Phys. Rev. B 90, 014503 (2014). Schematic view of the optical conductance in the pseudogap state of YBaCuO superconductor compared with conventional BCS materials.

Incoherent pairs above T c experimental fact # 7 A. Dubroka et al, Phys. Rev. Lett. 106, 047006 (2011). Onset of the superfulid fraction observed in the c-axis optical measurements Reσ c (ω).

Incoherent pairs above T c... continued

Incoherent pairs above T c... continued Josephson-like features seen above T c in the tunneling N. Bergeal et al, Nature Phys. 4, 608 (2008).

Incoherent pairs above T c... continued Josephson-like features seen above T c in the tunneling N. Bergeal et al, Nature Phys. 4, 608 (2008). Bogoliubov quasiparticles detected above T c in ARPES measurements for YBaCuO and LaSrCuO compounds Argonne (2008), Villigen (2009). Bogoliubov-type interference patterns in pseudogap state as the fingerprint of phase incoherent d-wave superconductivity preserved up to 1.5 T c J. Lee,... and J.C. Davis, Science 325, 1099 (2009).

3. Selected results / below and above T c /

Boson and fermion degrees of freedom [ spatial representation ] Ĥ = (t ij µ δ i,j ) ĉ iσĉjσ + i,j,σ l + ] g ij [ˆb l ĉi, ĉ j, + h.c. i,j ( E (B) l 2µ )ˆb lˆb l

Boson and fermion degrees of freedom [ spatial representation ] Ĥ = (t ij µ δ i,j ) ĉ iσĉjσ + i,j,σ l + ] g ij [ˆb l ĉi, ĉ j, + h.c. i,j ( E (B) l 2µ )ˆb lˆb l R l = ( r i + r j )/2

Boson and fermion degrees of freedom [ spatial representation ] Ĥ = (t ij µ δ i,j ) ĉ iσĉjσ + i,j,σ l + ] g ij [ˆb l ĉi, ĉ j, + h.c. i,j ( E (B) l 2µ )ˆb lˆb l R l = ( r i + r j )/2 This Hamiltonian describes a two-component system consisting of: ĉ ( ) iσ itinerant fermions..................................(e.g. holes near the Mott insulator) ˆb( ) l immobile local pairs................................ (RVB defines them on the bonds)

Phenomenological arguments supporting BF scenario

Phenomenological arguments supporting BF scenario 3 paired & unpaired carriers 2 1 0-1 -2-3 -3-2 -1 0 1 2 3 V.G. Geshkenbein, L.B. Ioffe, A.I. Larkin, Phys. Rev. B 55, 3173 (1997).

Phenomenological arguments supporting BF scenario K.-Y. Yang, E. Kozik, X. Wang, and M. Troyer, Phys. Rev. B 83, 214516 (2011). cooperons & holes

Phenomenological arguments supporting BF scenario nodal antinodal dichotomy T. Kondo, R. Khasanov, T. Takeuchi, J. Schmalian & A. Kamiński, Nature 457, 296 (2009).

Bogoliubov quasiparticles below and above T c

BCS physics The BCS ground state:

BCS physics The BCS ground state: BCS = Π k ( uk + v k e iθk ĉ k ĉ k ) vac

BCS physics The BCS ground state: BCS = Π k ( uk + v k e iθk ĉ k ĉ k ) vac v k, u k θ k coherence factors, phase

BCS physics The BCS ground state: BCS = Π k ( uk + v k e iθk ĉ k ĉ k ) vac v k, u k θ k where coherence factors, phase v k e iθ k 2 = 1 2 [ 1 ] ε k µ (εk µ) 2 + 2 = 1 u k 2

BCS excitation spectrum

BCS excitation spectrum The effective (Bogoliubov) quasiparticles : ˆγ k = u k ĉ k + v k ĉ k ˆγ k = v k ĉ k + u k ĉ k

BCS excitation spectrum The effective (Bogoliubov) quasiparticles : ˆγ k = u k ĉ k + v k ĉ k ˆγ k = v k ĉ k + u k ĉ k represent a coherent superposition of the particles and holes

BCS excitation spectrum The effective (Bogoliubov) quasiparticles : ˆγ k = u k ĉ k + v k ĉ k ˆγ k = v k ĉ k + u k ĉ k represent a coherent superposition of the particles and holes A(k, ω) = u k 2 δ(ω E k ) + v k 2 δ(ω + E k ) Occupancy of the momentum k is given by n k = u k 2 f F D (E k ) + v k 2 f F D ( E k ) }{{} 1 f F D (E k )

(a) Energy T c < T Intensity E F k F Wave vector ε k Intensity (b) v k 2 u k 2 E F Energy T <T c k F Wave vector -E k 2 E k Single particle spectrum of conventional superconductors consists of two Bogoliubov branches gaped around E F (no fluctuation effects are here taken into account).

Experimental data for cuprates below T c H. Matsui, T. Sato, and T. Takahashi et al, Phys. Rev. Lett. 90, 217002 (2003).

Beyond the BCS approximation

Beyond the BCS approximation We have generalized the Bogoliubov ansatz taking into account the non-condensed (preformed) pairs

Beyond the BCS approximation We have generalized the Bogoliubov ansatz taking into account the non-condensed (preformed) pairs ĉ k (l) = u k (l) ĉ k + v k (l) ĉ k + 1 [ ] uk,q (l) ˆb qĉ q+k + v k,q (l) ˆb q ĉ q k, N q 0 ĉ k (l) = v k (l) ĉ k + u k (l) ĉ k 1 [ ] v k,q (l) ˆb qĉ q+k + u k,q (l) ˆb q ĉ q k, N q 0 (1) (2) with the boundary conditions u k (0)=1 and v k (0)=v k,q (0)=u k,q (l)=0

Effective spectrum above T c T c < T < T A(k,ω) 1.0 0.5 0-0.1-0.05 k - k F 0.0 0.05 0.1-0.05 0 ω 0.05 T. Domański and J. Ranninger, Phys. Rev. Lett. 91, 255301 (2003).

Evidence for Bogoliubov QPs above Tc

Evidence for Bogoliubov QPs above T c J. Campuzano group (Chicago, USA) Results for: Bi 2 Sr 2 CaCu 2 O 8 A. Kanigel et al, Phys. Rev. Lett. 101, 137002 (2008).

Evidence for Bogoliubov QPs above T c PSI group (Villigen, Switzerland) Max Energy [ev] Cut 1 Cut 1 Intensity [arb. units] Energy [mev] Cut 2 Cut 2 Min Energy [ev] Energy [ev] k - k F [ /a] Results for: La 1.895 Sr 0.105 CuO 4 M. Shi et al, Eur. Phys. Lett. 88, 27008 (2009).

Evidence for Bogoliubov QPs above T c D. Jin group (Boulder, USA) T/Tc = 0.74 T/Tc = 1.24 Spectral Weight (arb.) k = 0.10 0.25 0.40 0.55 0.70 0.85 1.00 1.15 1.30 1.45 T/Tc = 1.47 T/Tc = 2.06-4 -2 0 2-4 -2 0 2 Single-Particle Energy (E S/E F) Results for: ultracold 40 K atoms J.P. Gaebler et al, Nature Phys. 6, 569 (2010).

Bogoliubov QPs above Tc

Bogoliubov QPs above T c Reproducing experimental line-shapes of the ultra-cold fermion atoms A(k,ω) 0.3 0.2 0.1 0 T = 0.76 T c k / k F 1 2 3-4 -2 0 2 4 6 ω / T. Domański, Phys. Rev. A 84, 023634 (2011).

Bogoliubov QPs above T c Reproducing experimental line-shapes of the ultra-cold fermion atoms A(k,ω) 0.3 0.2 0.1 0 T = 1.24 T c k / k F 1 2 3-4 -2 0 2 4 6 ω / T. Domański, Phys. Rev. A 84, 023634 (2011).

Bogoliubov QPs above T c Reproducing experimental line-shapes of the ultra-cold fermion atoms A(k,ω) 0.3 0.2 0.1 0 T = 1.47 T c k / k F 1 2 3-4 -2 0 2 4 6 ω / T. Domański, Phys. Rev. A 84, 023634 (2011).

Bogoliubov QPs above T c Reproducing experimental line-shapes of the ultra-cold fermion atoms A(k,ω) 0.3 0.2 0.1 0 T = 2.06 T c k / k F 1 2 3-4 -2 0 2 4 6 ω / T. Domański, Phys. Rev. A 84, 023634 (2011).

Diamagnetism its origin above T c

Diamagnetic response above Tc

Diamagnetic response above T c The anomalous contributions k - q k + q q 1 q 1 q 1 + k q 1 - k

Diamagnetic response above T c The anomalous contributions k - q k + q q 1 q 1 q 1 + k q 1 - k resemble the Aslamasov-Larkin diagram

Diamagnetic response above T c The anomalous contributions k - q k + q q 1 q 1 q 1 + k q 1 - k resemble the Aslamasov-Larkin diagram enhancing the conductance/diamagnetism above T c.

Onset of diamagnetism above Tc

Onset of diamagnetism above T c Onset of the diamagnetism coincides with appearance of the collective features in the fermion/boson spectrum.

Onset of diamagnetism above T c Onset of the diamagnetism coincides with appearance of the collective features in the fermion/boson spectrum. -0.05 k B T / D = 0.08-0.10 ~ E q / D -0.05-0.10 k B T / D = 0.04-0.05 k B T / D = 0.02-0.10 q x a -1 0 1 Qulitative changes of the preformed pairs spectrum.

Onset of diamagnetism above T c Onset of the diamagnetism coincides with appearance of the collective features in the fermion/boson spectrum. m B / m 0 F 100 10 1 0.1 T c g = 0.2 D T * 0.01 0 0.05 0.1 0.15 0.2 k B T / D Effective mass m B of the preformed pairs M. Zapalska and T. Domański, Phys. Rev. B 84, 174520 (2011).

Onset of diamagnetism above T c Onset of the diamagnetism coincides with appearance of the collective features in the fermion/boson spectrum. M. Zapalska and T. Domański (2014). 100 80 theory expt M d (T) T * M (A/m) 60 40 20 0 0.05 0.1 0.15 T / D 0 0 1 2 3 4 5 6 T/T c K.-Y. Yang,... and M. Troyer, Phys. Rev. B 83, 214516 (2011).

Onset of diamagnetism above T c Onset of the diamagnetism coincides with appearance of the collective features in the fermion/boson spectrum. n s (T) / n 1 0.8 0.6 0.4 fluctations T * 0.2 0 T c 0 0.02 0.04 0.06 0.08 0.1 k B T / D J x (q 0, 0) = e2 n s (T) m A x(q 0, 0) M. Zapalska and T. Domański, Phys. Rev. B 84, 174520 (2011). G. Drachuck et al, Phys. Rev. B 85, 184518 (2012).

Diamagnetic response above Tc

Diamagnetic response above T c Residual diamagnetism originates from collective behavior of the pre-formed pairs. Pair susceptibility is enhanced at T sc and it ultimately diverges at some lower T c.

Diamagnetic response above T c Residual diamagnetism originates from collective behavior of the pre-formed pairs. Pair susceptibility is enhanced at T sc and it ultimately diverges at some lower T c. Cartoon illustrating the vortex liquid above T c.

Summary

Summary A number of experimental data provide a clear evidence for the pre-existing pairs above T c

Summary A number of experimental data provide a clear evidence for the pre-existing pairs above T c residual diamagnetism / torque magnetometry, magnetization of needle samples /

Summary A number of experimental data provide a clear evidence for the pre-existing pairs above T c residual diamagnetism / torque magnetometry, magnetization of needle samples / Bogoliubov quasiparticles / ARPES, Josephson effect, FT-STM / Short-range superconducting correlations