Topological Superconductivity
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Topological Superconductivity Annica Black-Schaffer [email protected] Materials Theory 6th MaNEP winter school, Saas Fee, January 2015 Uppsala Open PD and PhD positions! Ångström laboratory Contents • Introduction to superconductivity – BCS theory – BdG formulation • Unconventional superconductivity • Topological matter • Topological superconductivity – Chiral superconductors – “Spinless” superconductors – Majorana fermions Introduction to Superconductivity What is it? How do we describe it? What is Superconductivity? • What is superconductivity? – Electric transport without resistance – Meissner effect Superconductor Magnet History of Superconductivity Superconductivity in Hg: Heike Kammerlingh Onnes Tc Superconductors Department of Energy Nobel Prizes How? • But how do the electrons move without resistance? – The electrons are all in a coherent quantum state with a fixed phase (condensate) i' = ∆0e – One of the few macroscopic manifestations of QM – Spontaneous gauge symmetry breaking – Higgs mechanism (discovered by P. W. Anderson before Higgs et al.) Higgs Mechanism • In the Standard Model: – Interactions with the Higgs field (Higgs bosons) give masses to the W+,- and Z bosons due to electroweak (gauge) symmetry breaking • In superconductors: – The superconducting pairs give mass to the electromagnetic field (photons) due to spontaneous symmetry breaking of the gauge symmetry (fixed phase on Ψ) – Explored by P.W. Anderson for superconductivity 2 years before Higgs et al. Meissner effect = The “Higgs mass” of the electromagnetic field expels it from the superconductor But Wait! • All electrons in one state: a superconducting condensate – Pauli exclusion principle?? How do electrons end up in the same state? Statistics • Fermions (half-integer spin): – Electrons, quarks, neutrinos – Only one electron per quantum state • Bosons (integer spin): – Photons, gauge, and Higgs boson – As many particles as you wish in one quantum state (Bose-Einstein condensate, BEC) Maybe two electrons can form an electron pair!? Sort of, but not the whole story…. Electron Pairing • Electrons are negatively charged, how can they possibly pair? – In fact, they can pair pretty easily (at least at very, very low temperatures) • The only instability of a Fermi Liquid is to attractive interactions – Any attractive interaction destroys the Fermi Liquid and creates electron pairs: Cooper pairs Attractive Interaction • Where is attractive interaction coming from? Normal state At Tc In the superconducting state Electron-lattice vibration (phonon) interactions gives an effective attractive potential between two electrons Conventional Superconductors • Superconductivity from electron-phonon interactions – Superconductivity in “normal” metals – Low temperatures, Tc < 40 K BCS Theory • How do the Cooper pairs condense (superconduct)? • Bardeen-Cooper-Schrieffer (BCS) theory: – Condensation of Cooper pairs (with a fermionic wave function) – Truly many-body state (not a two-body bound state condensing) (Higher temperatures than BECs) Cooper Pairs Pair of electrons above the Fermi surface (FS) with no net momentum: Fourier transform ( ): spatial property of the Cooper pair attractive For the energy of a Cooper pair is < 0 ! FS is unstable to Cooper pair formation for any V<0 (see e.g. Tinkham: Introduction to superconductivity) BCS Wave Function Coherent state of Cooper pairs (ansatz): with • Ill-defined number of particles: phase of order parameter: fixed (but same energy for all phases) • Breaks gauge invariance: BCS Hamiltonian Pairing Hamiltonian: kinetic (band) energy We can determine uk, vk in ψBCS by minimizing <ψBCS|H-µN|ψBCS> but it is a bit clumsy (see e.g. Tinkham: Introduction to superconductivity) Instead we use mean-field theory with the order parameter pair amplitude at k (off-diagonal long-range order) Mean-Field Treatment ignore fluctuations (mean-field approximation) Set the order parameter ! Matrix Formulation (BdG) Define the Nambu spinor ! TRS: εk= εk constant ! Bogliubov -de Gennes (BdG) formulation: 2x2 matrix problem ! Solve by finding eigenvalues and vectors ! ( ) Eigenstates = Quasiparticles QP energy (eigenvalue): Bogoliubov QP operators (eigenvector): tranformation (Same coefficients as in the BCS wave fcn) Diagonal Hamiltonian: QP excitations Condensation energy Quasiparticle Properties • Excitations out of the condensate, always positive energy Band structure Density of states (DOS) P. Coleman: Introduction to Many Body Physics Quasiparticle Properties II • Excitation creates electron with momentum k and spin up, destroys electron with momentum –k and spin down • At E = 0: ! QP is equal parts electron and hole ! Can become its own antiparticle (Majorana fermion) • annihilates ground state puts 1 electron in state –k, and prevents a Cooper pair at momentum k Additional formulas January 8, 2015 u 2 + v 2 =1 | k| | k| Fk = c k ck h − # "i u 2 + v 2 =1 | k| | k| 1 u 2 = v 2 = | k| | k| 2 ak BCS =0 "| i † † † † a k BCS = c k (uk0 + vk0 ck c k ) 0 − #| i − # 0" − 0# | i k =k Y06 E /kB T 1 a† ak σ =(1+e k0 )− h k0σ 0 i † † † † † † Vk,k0 ck c k c k0 ck0 = Vk,k0 [(ck c k Fk)+Fk][(c k0 ck0 Fk0 )+Fk0 " − # − # " " − # − − # " − ⇡ Xk,k0 Self-ConsistentXk,k0 Equation † † † † (c c F )(cc F ) Vk,k0 (Fk0 ck c k + Fkc k0 ck0 FkFk0 ] † † † " − # − # " − − − Xk,k0 Having found the diagonal Hamiltonian we can determine the superconducting order parameter self-consistently: ∆ = V F = V c c k k,k0 k0 kE,k0/kB Tk0 1k0 a† a =(1+e k0 h − )# "i ∆k = Vk,k c k ck = Vk,k uk v⇤ a† akkσ k+0σa† ak 1 − 0 0 0 0 0 k0 kk0h0 00 i k0 k0 0 h − # "i h X" " − X# # − i k,k0 k,k0 X X Fermi-Dirac E /kB T 1 † k0 ak σak0σ =(1+1 e ∆)k−0 distribution 0 h ∆k i=(c†c† F †V)(kcc,k0 F ) −2− −Ek0 1 Xk0 (c c F )(cc F ) 1 † † †∆ ∆ = V− F k=0 − V c c ∆k =k Vkk,k,k0 k0 tanh[Ek,kk0/(2k0BTk)]0 0 0 h − # "i −2 k0 Ek0 k0 Xk0 X X ∆k = Vk,k Fk = Vk,k c k ck 0 0 0 − 0# 0" 1 h ∆k0 i At T =k0 0: ∆ = k0 V X Hk =(σX k) kzˆ,k0 −2⇥ · Ek0 Xk0 1 ∆ ∆ = V k0 k 2 k,k0 E 1 − ∆k0 k0 ∆ = H = Vvk0 (σ ktanh[) zˆE /(2k T )] k ~XkF,k0 k0 B −2 E⇥k0 · Xk0 1 ∆k0 ∆k = Vk,k0 tanh[Ek0 /(2kBT )] −2 Ek k UHc⇢(=(E0 Fσ)=1k) zˆ X0 ⇥ · H =(σ k) zˆ ⇥ · HVpot= ~=vFV(σδ(r)k) zˆ ⇥ · H = ~vF (σ k) zˆ ⇥ 1 · EUc⇢(EF )=1 res / V Uc⇢(EF )=1 Vpot = V δ(r) !1 V = V δ(r) pot 1 Eres2 / V 1 E res / V V !1 V !12 2 Summary BCS Theory BCS wave function: BCS = (uk + vkck† c† k ) φ0 | i " − # | i k Y Cooper pair Hamiltonian with kinetic energy εk and a pairing potential Vkl: Quasiparticle energy: E = "2 + ∆ 2 k k | k| Energy gap = q order parameter Quasiparticles: Order parameter (Ψ before): Introduction to Superconductivity What is it? A charged superfluid of Cooper pairs (2 electrons) with fermionic character Cooper pairs formed by effective attractive interaction How do we describe it? BCS theory for the condensation BdG matrix formalism Unconventional Superconductivity What is unconventional? How do we handle unconventional superconductivity? Conventional Superconductors • Superconductivity from electron-phonon interactions – Superconductivity in “normal” metals – Low temperatures, Tc < 40 K 1 ∆l – Isotropic pairing ( ∆ k = V kl = Δ) −2 El Xl Superconducting Symmetries Superconducting order parameter is fermionic i.e. odd under particle exchange: (k)= ( k) ↵ − ↵ − i' ↵(k)=∆e ⌘(k)χ↵ orbital spin ! η even function in k ! η odd function in k Spatial Symmetries y • Conventional superconductors: x – Spin-singlet, s-wave (η k-independent) y • Cuprate (high-Tc) superconductors: 2 2 – Spin-singlet, d-wave (η = kx - ky ) UNCONVENTIONAL • Exotic superconductors: 3 i – Spin-triplet, p-wave ( He, Sr2RuO4?) x – Topological “spinless” superconductors kx + ik y with Majorana fermions i Superconducting Symmetries Vk,k’ (and the band structure) determines the pairing symmetry, but it often very hard to determine • Lattice fluctations: spin-singlet s-wave • Antiferromagnetic spin fluctuations: spin-singlet d-wave (extended s-wave) • Ferromagnetic spin fluctuations: spin-triplet p-wave • Strong on-site repulsion (Heisenberg interaction): spin-singlet d-wave Is there a way to determine the possible pairing symmetries in a material without knowledge of Vk,k’? Yes, we can do a general symmetry group analysis See e.g. Sigrist and Ueda, RMP 63, 239 (1991) General Pairing Hamiltonian General pairing Hamiltonian: Mean-field order parameter: ! E /kB T 1 a† ak σ =(1+e k0 )− h k0σ 0 i (c†c† F †)(cc F ) − − ∆k = Vk,k Fk = Vk,k c k ck 0 0 0 h − 0# 0"i Xk0 Xk0 E /kB T 1 a† ak σ =(1+e k0 )− h k0σ 0 i 1 ∆k0 ∆k = Vk,k0 −2 Ek /kBET 1 (c ca† aFk σ)(cc=(1+F ) e 0 )k−0 † †k0σ †0 k0 h − i − E /kB T 1 a† ak σ =(1+X e k0 )− h k0σ 0 i ∆k = Vk,k Fk (c=†c† VFk†,k)(ccc k Fck) 1 0 0 −∆k 0 h −−0# 0"i XkMatrix0 (c†cXk† 0 F †0)(Formulationcc F ) ∆k = Vk,k0 − tanh[− Ek0 /(2kBT )] −2 Ek0 k0 1 ∆k ∆∆k == Vk,k FVk = 0 Vk,k c k ck T Xk 0 k0,k0 0 h − 0# 0"i 4-componentk− notation2 Ek k0(Nambu): ∆k =X0 Vkk,0k Fk =X0 Vk,k c k ck X0 0 0 h − 0# 0"i Xk0 Xk0 1 ∆ 1 ∆k0 ˆ k0 ∆ = V ∆"k(k=)tanh[σ0 E V/k(2∆,kk(kT )]) ˜k −2 k,k0 E −2 k0 0 EB ! = a† k0 1 ∆kk0 ak kXk0 ∆ˆ = k0 V 0 H ∆k †(k)X k",k(0 k)σ0 −2 Ek ✓ 0 ◆ Xk0 − 1 ˆ∆ ˜ "(k)σ0 ∆(kk)0 =∆akk†= Vk,k0 tanh[akEk0 /(2kBT )] H −∆ˆ2 (k) "(Ekk)σ 1† k ∆ 0 0 ψ even ✓ X0 − k0 ◆ Spin-singlet∆k = pairing:Vk,k 0 tanh[Ek0 /(2kBT )] −2 Ek0 function of k H =(k0 σ k) zˆ X ˆ (k)[ck† c† k "(kck†)σ⇥c0† k ]∆(·k) ˜ = "a†− # − " − # a H k ∆ˆ (k) "(k)σ k ✓ "(†k)σ − ∆ˆ (k)0 ◆ ˜ 0 = ak† ak H ∆ˆ †(k) "(k)σ0 dz(k)[ck† c† ✓k + ck† c† k −] ◆ " − # " − # Spin-triplet pairing: (k)[ck† c † k ck† c† k ] H = ~vF" (−σ# − k" )− #zˆ d vectorial odd (k)[ck† c† k ⇥ck† c†