Odd-Parity Superconductors Nematic, Chiral and Topological

Transcription

1 Odd-Parity Superconductors Nematic, Chiral and Topological Liang Fu Nordita, Stockholm July 7, 2016

2 Collaboration Vlad Kozii Jorn Venderbos Erez Berg

3 Odd-Parity Superconductors: Δ k = Δ( k) p-wave superfluid He-3 T-breaking A phase: point node in 3D; full gap in 2D T-invariant B phase: full gap Topological superfluid (Candidate) Odd-Parity SCs: UPt 3 and Sr 2 RuO 4 Unconventional pairing symmetry Odd- vs even-parity debated

4 Odd-Parity Pairing and Spin-Orbit Coupling Without SOC: odd-parity = spin triplet, meditated by spin fluctuations SOC in inversion-symmetric materials: bands are spin-orbit-entangled & doubly degenerate (pseudospin) odd-parity = pseudospin triplet Odd-parity superconductivity with strong SOC (λ SOC ~ E F ): does not require strong electron correlation can be meditated by electron-phonon interaction leads to new & topological phases absent in He-3

5 ional wisdom: odd-parity = p-wave spin-triplet, requires strong correlation. lk: odd-parity pairing susceptibility is parametrically enhanced in materials Cu x Bi 2 Se 3 trong spin-orbit coupling (λ so of order E F ), and can be realized in electronon superconductors. Bulk SC in a doped topological insulator: Cava et al., PRL (2010) sal for odd-parity pairing in Cu-doped Bi 2 Se 3 LF & Berg, (PRL 10) doped semiconductor, carrier density ~10 20 cm -3 carrier density ~10 20 Cu Tc ~ 3K, type-ii superconductor zero resistivity Hor et al, (PRL 10) Ando et al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specific heat: full gap or point node

6 Normal State Electronic Structure ARPES and quantum oscillations: Dagan & Kanigel et al. PRB 13 Lu et al, PRB 14 Single pocket: evolves from closed to open as electron density increases.

7 Spin-Orbit Coupling Spin-Orbit Spin-Orbit Coupling Coupling internal E field σ z =1 σ z =-1 conduction conduction μ valence valence bonding bonding anti-bonding anti-bonding two p z orbitals (σ z ) + electron spin (s z ) two p z orbitals (σ z ) + electron spin (s z ) Low-energy Hamiltonian: orbital (σ) and spin (s) k.p Hamiltonian dictated by symmetries: k.p Hamiltonian dictated by symmetries: LF & Berg, PRL 10 inter-layer, spin-independent intra-layer, bilayer Rashba SOC from inter-layer, spin-independent intra-layer, bilayer Rashba SOC from local electric fields with opposite signs local electric fields with opposite signs staggered Rashba SOC from opposite E fields on top and bottom layers with inversion symmetry, SOC necessarily involves multiple orbitals SOC strength SOC strength depends depends on doping: on doping: comparable comparable to E F to E energy bands energy NOT bands spin-split; NOT spin-split; Bloch wavefunctions Bloch wavefunctions are spin-orbit are spin-orbit mixed. mixed.

8 Pairing Order parameters and pairing symmetries: LF & Berg, PRL 10 A 1g c + 1 c c c 2 A 1u A 2u E u c + 1 c c c 2 c + 1 c + 1 c c 2 odd-parity (c + 1 c + 2, c + 1 c + 2 ) order parameters classified by representations of D 3d point group spin is locked to lattice by spin-orbit coupling

9 Pairing Order parameters and pairing symmetries: LF & Berg, PRL 10 A 1g A 1u A 2u E u c + 1 c c c 2 c + 1 c c c 2 c + 1 c + 1 c c 2 odd-parity (c + 1 c + 2, c + 1 c + 2 ) A 2u : sign-changing spin-singlet + - odd-parity spin triplet SOC is crucial for SC gap order parameters classified by representations of D 3d point group spin is locked to lattice by spin-orbit coupling

10 Pairing Order parameters and pairing symmetries: LF & Berg, PRL 10 A 1g c + 1 c c c 2 A 1u E u A 1u c + 1 c c c 2 A 2u E u c + 1 c + 1 c c 2 odd-parity (c + 1 c + 2, c + 1 c + 2 ) inter-orbital triplet easy axis vs. plane order parameters classified by representations of D 3d point group spin is locked to lattice by spin-orbit coupling

11 Gap Functions in Band Basis A 1g c + 1 c c c 2 Projection to band basis* Δ αβ k I A 1u c + 1 c c c 2 Δ αβ k c + + k,α ε ββ c k,β k τ A 2u c + 1 c + 1 c c 2 k x τ y k y τ x E u (c + 1 c + 2, c + 1 c + 2 ) (k x τ z k z τ x, k y τ z k z τ y ) Yip, PRB 13; Kozii & LF, PRL 15 Odd-parity pairing within SOC leads to p-wave gap functions distinct from He-3. * d-vectors depend on band basis; use manifestly covariant Bloch basis LF, PRL 15

12 here n Hamiltonian[22]: i (x) = α=, c iα (x)c iα(x) is electron density here orbital n Hamiltonian[22]: i (x) i. U= and V are intra-orbital and inter-orbital n teractions orbital i. respectively. U and V are We intra-orbital assume that and at inter-orbital least one nteractions them is attractive, respectively. responsible We assume for superconductivity. that at least one f perconductivity king them the is attractive, band is structure responsible likely phonon-meditated. and pairing for superconductivity. interaction tother, uperconductivity aking we the introduce band is structure likely thephonon-meditated. following and pairing two-orbital interaction U V toether, odel we introduce the following two-orbital U V two henomenological for p z orbitals Cu x Bi(σ 2 Se z ) + Hamiltonian: 3, electron spin (s z ) Phenomenological odel for Cu x Bi 2 Se Hamiltonian: 3, k.p Hamiltonian dictated by symmetries: density-density = dkc k (H interactions: 0(k) µ)c k + dxh int (x). (8) H = dkc 0 (k)=mσk x (H + 0(k) v z k z σµ)c y + k v(k + x σdxh z s y int k(x). y σ z s x )(8) k = 0 iswell described by thefollowing continuum k p Model α=, Model c iα Study Study (x)c iα(x) is electron density bonding of of Paring Paring Symmetry anti-bonding Symmetry H 0 (k)=m x + v(k x z s y k y z s x )+v z k z y, (6) H 0 (k)=m x + v(k x z s y k y z s x )+v z k z y, (6) valence Odd-Parity Pairing from Density Interaction where z = ±1denotesthetwoorbitals; s z = ±1denotes where electron spin z = ±1denotesthetwoorbitals; s parallel/ anti-parallel to the z = ±1denotes direction (caxis). As for thepairing mechanism, very little is known electron spin parallel/ anti-parallel to the z direction (caxis). As for thepairing mechanism, very little is known so far. For simplicity we consider short-range electron so far. For simplicity we consider short-range electron density-density interactions: uetocu H int (x)= U[n doping, thefermi 2 1(x)+n energy 2 2(x)] µ in 2Vn Cu 1 x (x)n Bi 2 Se 2 (x), 3 lies (7) uetocu H int (x)= doping, U[n thefermi 2 theconduction band about 1(x)+n energy 2 0.4eV 2(x)] local µ electric in 2Vn Cu 1 fields abovethemiddleof x (x)n Bi 2 with Se 2 (x), 3 opposite lies (7) signs ntheconduction e where band energy U: intra-orbital gap[21], nbands i (x) NOT band = which interaction spin-split; about α= leads, c Bloch V: iα (x)c 0.4eV inter-orbital iα(x) isinteraction electron density where n to a small Fermi surface. i (x) = α=, We in orbital i. and V are intra-orbital and inter-orbital short now range determine density-density the superconducting c iα (x)c wavefunctions abovethemiddleof are spin-orbit SOC mixed. he U: intra-orbital interaction V: inter-orbital iα(x) isinteraction electron density band gap[21], which leads to a small Fermi surface. SOC strength depends on doping: comparable to interaction mean E in field ase interactions orbital i. U and V are intra-orbital and F We short now U diagram range determine & V are phenomenological of respectively. density-density the superconducting the U V We model. interaction mean inter-orbital field parameters, assume Since that assume theat pairing at least one hase Berg, interactions (PRL is attractive teraction of them U diagram 10) see & V involves is are attractive, phenomenological of respectively. also the tight-binding U Vmodel We model. in two orbitals responsible parameters, assume Hsieh Since & LF that (PRL and for islocal superconductivity. assume the 12) at pairing in x, at least the least one one is attractive nteraction of them Density involves is attractive, interaction two orbitals responsible leads to and pairing for islocal superconductivity. in even- in x, and the odd-parity channels, because ean-field Taking the pairing band potential structure isorbital-dependent and pairing interaction but k- together, Δ1 and Δ2 are fully gapped & TR-invariant. Mean-field: ean-field Taking dependent. strong the pairing we SOC band In introduce Table (λpotential structure SOC ~E I, the F we ) makes isorbital-dependent and following classify pairing pairing all two-orbital possible interaction but k- pair- pseudospin-dependent. together, we In introduce Table I, the we following classify all two-orbital possible pair- U Δ2 can win when SOC is strong. ndependent. Mean-field: V V ng gmodel potentials for Cuaccording x Bi 2 Se intra-orbital, 3, to the spin representation singlet of the model potentials U(nfor Cu n 2 according 2) x Bi 2 Se intra-orbital, 3, to the spin representation singlet of the inter-orbital, spin singlet or triplet Vn = 1 n 2 H = dkc dkc k (H inter-orbital, spin singlet or triplet Proof of principle: 0(k) µ)c k + dxh int (x). (8) k (H 0(k) odd-parity µ)c k + pairing dxh int electron-phonon (x). LF (8) & Berg, (PRL 10) superconductors LF LF & & Berg, Berg, (PRL (PRL 10) 10) inter-layer, spin-independent intra-layer, bilayer Rashba SOC from DuetoCu DuetoCudoping, thefermi thefermi energy energy µ µ in incu Cu x Bi x Bi 2 Se 2 Se 3 3 lies lies intheconduction band band about about 0.4eV 0.4eV abovethemiddleof V V U U Phase Diagra doping (m/µ) determines effective spin-orbit streng Spin-orbit coupling favors odd-parity pairing, even wit U

13 We next study the pairing symmetry of the newly discovered superconductor Cu x Bi 2 Se 3 within a two-orbital model, and find that a novel spin-triplet pairing with odd parity is favored by strong spinorbit coupling. Based on our criterion, we propose that Cu x Bi 2 Se 3 is a good candidate for a topological superconductor. Support from electron-phonon calculations: Wan & Savrasov, Nat. Commun. 14

14

15 NMR Knight Shift Zheng et al, Nat. Phys. (2016) Knight shift shows uniaxial anisotropy in SC state SC states spontaneously breaks three-fold rotational symmetry first clear observation of rotational symmetry breaking in a SC

16 Rotation Symmetry Breaking from E u Pairing A 1g A 1u A 2u E u c + 1 c c c 2 c + 1 c c c 2 c + 1 c + 1 c c 2 (c + 1 c + 2, c + 1 c + 2 ) We find four different pairing symmetries in the A 1g, A 1u, A 2u, and E u representations of the D 3d group. The three A representations are one dimensional and the E representation is two dimensional. LF & Berg, PRL 10 +e iθ Rotation symmetry breaking is only compatible with E u pairing Anisotropic Knight shift requires locking of spin to the lattice by SOC.

17 We calculate the temperature dependence of the specific heat and spin susceptibility for four promising superconducting pairings proposed by Fu and Berg we obtain wide variations of the temperature dependence of spin susceptibility for each pairing, reflecting the spin structure of the Cooper pair. SOC in the two-orbital model is crucial for spin susceptibility

18 Odd-Parity Nematic Superconductor Basis functions for Δ 4 pairing LF, PRB 14 (Ψ 1, Ψ 2 ) ~ (x,y) Real vs. complex order parameters: Ψ = cosθ Ψ 1 + sinθ Ψ 2 is T invariant and rotation breaking nematic superconductor : new SC phase distinct from He3 Ψ and -Ψ correspond to the same superconducting state => odd-parity nematic SC Ψ 1 ± i Ψ 2 is rotation invariant and T breaking => chiral SC

19 Pinning the Two-Component Order Parameter Ginzburg-Landau free energy quartic terms: Subsidiary nematic order: couples linearly to uniaxial strain D 3d point group allows a sixth order term and selects easy axes in ab plane. pinning by symmetry-breaking field, e.g. uniaxial strain

20 Anisotropic Gap Structures Including crystal anisotropy: LF, PRB 14 (hexagonal warping) Quasi-particle gap k y k y k x Ψ 2 Ψ 1 k x T-invariant, full gap => topological SC Point nodes protected by mirror topological nodal SC

21 arxiv:

22 Specific heat shows two-fold oscillation, down to H/Hc 2 = 1.5%

23 Field-Direction Dependent Specific Heat k y Ψ 2 k x From these agreements between experiment and theory, we conclude that the SC gap of Cu x Bi 2 Se 3 is 4y, possessing gap minima or nodes lying along the k x direction

24 Detecting Nematic Superconductivity Basal plane anisotropy in thermodynamic, transport and spectroscopy gap anisotropy: directional tunneling, penetration depth, thermal conductivity in magnetic field (Nagai, PRB 12) vortex lattice anisotropy: neutron scattering, STM upper critical field anisotropy (Venderbos, Kozii, LF, arxiv: )

25 Sr x Bi 2 Se 3 & Nb x Bi 2 Se 3 Nb x Bi 2 Se 3 Zhang et al, J. Am. Chem. Soc. 15 Hor et al, arxiv 15 Figure 1. (a) A representative energy dispersive x-ray spectroscopy pattern showing the existence of Sr, Bi, and Se. (b) Powder x-ray diffraction patterns of the Bi 2 Se 3, Sr Bi 2 Se 3, Bi 1.85 Sr 0.15 Se 3 samples. (c) Temperature dependence of resistivity of the Sr Bi 2 Se 3 sample and the Bi 1.85 Sr 0.15 Se 3 sample. (d) Magnetic field dependence of transverse resistivity of the Sr Bi Se sample measured at different temperatures.

26 arxiv: In-plane upper critical field shows twofold anisotropy Doped topological insulators/semimetals with strong SOC (λ SOC ~ E F ) may offer a new fertile ground for discovering odd-parity superconductivity.

27 Odd-Parity Chiral Superconductors Ψ 1 ± i Ψ 2 is a chiral state, with non-unitary gap function due to SOC Δ k = (k x +i k y ) s z k z s + (distinct from He-3 A phase) 3D closed Fermi surface: c-axis point nodes for spin ; full gap for spin nodal quasiparticles are 3D Majorana fermions Ψ q + = open Fermi surface: full gap, stack of 2D p + i p SC Venderbos, Kozii, LF, arxiv:

28 3D Majorana Fermions in Chiral Superconductors Classification of MFs with different locations, dispersions & topological charges Kozii, Venderbos, LF, to arxiv:1607

29 Detecting Majorana Fermions by Anisotropic Spin Relaxation For quadratic Majorana nodes on rotation axis: Spin relaxation rate has different T-dependence for parallel and perpendicular nuclear spin components because gapless states near Majorana nodes are spin-polarized.

30 Majorana Surface Arcs 3-fold symmetry Linear Majorana nodes 6-fold symmetry Quadratic Majorana nodes

31 New Mechanism for Odd-Parity SC Odd-parity pairing meditated by parity fluctuations in spin-orbit-coupled systems temperature Inversion breaking odd-parity SC pressure, doping, superconductivity near inversion breaking structural distortion Kozii & LF, PRL 15;

32 Inversion Breaking Coupled to Fermi Surface Consider normal state with time-reversal (T) and inversion (P) symmetry q=0 symmetry-breaking OPs: nematic + + ferromagnet - + current - - P-breaking orders + - T P generalized spin current Spin-orbit interaction enables coupling between P-breaking orders and electrons on Fermi surface, with k- and (pseudospin)-dependent form factor. LF, PRL 115, (2015)

33 Inversion-Breaking Particle-Hole Order Parameter classified by representations of crystal point group symmetry operation acts jointly on momentum and spin gyrotropic ferroelectric multipolar Consequences: P-breaking structural transition induces Fermi surface spin splitting LF, PRL 115, (2015)

34 SC meditated by Parity Fluctuations Parity fluctuations generate k- and spin-dependent effective interaction: rewritten in pairing channel: S denotes s-wave channel F s denote p-wave channels (form factor identical to particle-hole orders) Vlad Kozii & LF, PRL 15

35 SC meditated by Parity Fluctuations Attraction in both s-wave and the odd-parity pairing channel with the same symmetry as the fluctuating particle-hole order (a n >0 is attractive) Types of parity fluctuations Odd-parity pairing competes with s-wave; comparable pairing strengths. Coulomb interaction or Zeeman field suppresses s-wave, promotes odd-parity. see also Wang, Cho, Hughes, Fradkin PRB 16

36 Summary and Outlook Theoretical proposal: Cu x Bi 2 Se 3 may be an odd-parity SC with strong SOC. 2016: NMR and field-dependent specific heat experiments discovered rotational symmetry breaking, providing strong evidence for odd-parity Δ 4 pairing with E u symmetry. Search for odd-parity chiral SCs with nonunitary gap & 3D Majorana fermions Search for Majorana surface states---hallmark of topology Microscopic pairing mechanisms