Physics of quasi one dimensional conductors
J.-P. Pouget Laboratoire de Physique des Solides, CNRS-UMR 8502, Université Paris-sud 91405 Orsay
SFB Colloquium April 24, 2008 1D systems • Allow to obtain exact solutions of N body problem – AF chain (Ising, Hülten, Bethe, Griffiths, De Cloizeau & Pearson, Haldane…) – 1D interacting electron gas (Lieb &Wu, Mattis,Tomonaga & Luttinger (TL), Luther & Emery (LE)…) • In 1D: enhanced quantum and thermal fluctuations – in 1D no long range order at finite T Ising chain (Landau) XY and Heisenberg chains (Mermin and Wagner) – 1D interacting electron gas: Collective (non Fermi liquid) behavior; quasi–order at T=0°K (TL and LE liquids) – Exact treatment of thermal fluctuations (matrix transfer method) • Since ~1970 experimental results allow to test exact calculations 1D experimental systems
• magnetic (AF) chains and ladders • 1D inorganic and organic electronic conductors • edge states in 2D conductors • quantum wires • carbon nanotubes
Real materials made of weakly coupled chains
1D behavior for: πkB T > J┴ inter-chain coupling outlook • One dimensional conductors
• Peierls/charge density wave (CDW) instability – Inorganic conductors: Mo blue bronze – Organic charge transfer salts
• Effect of electron-electron interactions (organic conductors: charge transfer salts, Fabre and Bechgaard salts)
–4kF CDW instability/charge ordering – spin Peierls instability – Spin density waves and generalized density waves – Unconventional superconductivity ONE DIMENSIONAL CONDUCTORS
Structural anisotropy : d// d// d┴ Anisotropy of overlap of wave functions π organics: pπ σ tperp t t >> t inorganics: dz² // // ┴ Anisotropy of electronic properties: charge transport σ//(ω)>> σ┴(ω) real 1D conductor: no coherent interchain electronic transfert imply: t┴ <<π kBT (Fermi surface thermal broadening) Inorganics K2 Pt(CN)4,0.3Br-xH2O 1D overlap of dz² orbitals Organics (TMTSF)2PF6 1D overlap of pл orbitals BLUE BRONZE A0.3MoO3 1D A: K, Rb,Tl 102 103 1D Tp Metal Insulator Anisotropic optical conductivity σ(ω) // ┴ B.P. Gorshunov et al PRL 73, 308 (1994) 1D non interacting electron gas No orbital degree of freedom ρ electrons per atom (per unit cell of length d// ) ρ = 2 x 2kF (with kF expressed in reciprocal chain unit: 2π/ d ) Spin degree of freedom // 1/D For a free electron gas in D dimensions: kF~(n) n=ρ/d// is the electron density -1 in 1D : n= (2/π)kF (with kF expressed in Å ) PEIERLS’ CHAIN 1D metal coupled to the lattice: unstable at 0°K towards a Periodic Lattice Distortion -1 which provides a new (2kF) lattice periodicity 2kF PLD opens a gap 2Δ at the Fermi level: insulating ground state Peierls’ instability (basic) • In the static (adiabatic) limit: set a 2kF static PLD: u(x)=u0 sin (2kFx+φ) at T=0°K a gap 2Δ is open at ±kF (Δ=gu g: electron-phonon coupling) gain of electronic energy: Eel ≈Δ²N(EF) ln (Δ/εF) cost of elastic distortion energy Edis =1/2Ku0² = 1/2K(Δ/g)² because of ln (Δ/εF): Eel+ Edis<0 whatever Δ Long range PLD always energetically favorable 1D metallic state not stable at 0°K (Peierls « theorem » 1955) Peierls ground state • Gap at 0°K (in the adiabatic limit) Δ0=2εF exp(-1/λ) with the reduced electron-phonon coupling: λ=n(2s+1)N(EF)g²/K* N(EF) density of states per spin direction • Complex order parameter (ρ incommensurate) u0 /Δ0 expiφ (exp ±2ikF ) amplitude phase new spatial periodicity * s=0 for spinless fermions s=1/2 for fermions with spin n=1 for incommensurate band filling ρ n=2 for half-band filling (ρ=1): real order parameter Electron-phonon coupling (basic) • PLD displacement of site l: ul [in the continuum limit: ul →u(x)=u0 sin (2kFx+φ)] for the bond (l, l+1): d0→d=d0+ ul+1-ul This induces a modulation of the intersite transfert integral t*: t(d)= t(d0)+ ∂t/∂d (ul+1-ul) → t(d0)-st∂u(x)/∂x which modulates the charge on the bonds: ρ(x)= ρ0 + δρ(x) with δρ(x) ≈ -∂u(x)/∂x • electronic CDW δρ(x)= -[2N(EF)g u0/λ] cos(2kFx+φ) π/2 phase shift between the PLD and the CDW *t varies ~ exponentially with the intersite distance d: ∂t/∂d ≈ -st Charge density wave ground state Charge density wave has two components: Periodic Lattice Distortion u(x) (PLD) : observed by diffraction (information in reciprocal space: u(q)) Modulation of the electronic density ρ(x) (electronic CDW): observed by STM (information in direct space) u(x) coupled by the electron- ρ(x) phonon interaction (g) ρ(x) = - ∂u(x)/∂x Diffraction from a 2kF modulated chain • Direct space Reciprocal space na x h2π/a Q amplitude diffracted: A(Q)= Σn expiQna = Σhδ(Q- h2π/a) modulated lattice: atoms located in na+u0 sin(2kFna+φ) A(Q)= Σn expiQna exp[iQu0sin(2kFna+φ)]* Σ J (Qu ) Σ expi(Qna+r2k na+rφ) r r 0 n F h2π/a Σr Jr(Qu0) expirφ Σhδ(Q+r2kF-h2π/a) r -2k +2k If Qu0<<1: Jr(Qu0)~ (Qu0) only r= ±1 important F F ICDW(Q) = |A(Q)|² modulation of the phase of the scattered wave: side band satellite reflections in: Q= h2π/a -r2kF *exp(izsinθ)= Σr Jr(z) exp(irθ) with z=Qu0 and θ=2kFna+φ 2kF modulated chain -2kF Satellite lines: r = ±1 + 2kF Fourier transform of a modulated chain: diffuse sheets in reciprocal space Peierls instability of the blue bronze K0.3MoO3 qCDW=(1, 2kF, 1/2) (J.P. Pouget et al J. Physique Lettres 2kF=1-qb~3/4 b* 44, L113 (1983)) Tp=180K +2kF -2kF 25000 (-1 k 0.5) ESRF-D2AM 20000 15000 10000 Intensity (counts) 5000 2+2k -2kF +2kF 2-2kF F 0 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.6 k λCDW STM observation of the CDW in the Rb blue bronze Calculated image 2kF=2π/λCDW CDW instability: driven by the divergence of the electron-hole response function χ(q) Divergence of χ(q) for q such that f (Ek + q) − f (Ek ) χe = ∑ Ek+q~Ek k Ek + q − Ek q best nesting wave vector of the Fermi surface 2kF χ(q) χ(q) 2kF PLD fingerprint of q wave vectors at which χ(q) diverges Structural instability at q wave vectors for which χ(q) diverges CDW PLD screening of the force constant K by setting ρq: Keff=K-2g²χ(q) max. screening for q=2kF (in 1D) set at T=0°K a 2kF PLD (Peierls instability) accompanied by a 2kF modulation of the electron density ρ(2kF) vanisking of Keff induces a phonon softening (Kohn anomaly) Thermal phase diagram • Mean- field (BCS-like) theory (ignore fluctuations) MF Peierls 1D LRO TP MF 2Δ0=3.56kBTP • Isolated chain (because of thermal fluctuations there is no Peierls LRO at finiteT) MF 0°K 1D Peierls fluctuations startat~TP • Coupled chains Interchain coupling restaures a Peierls 3D LRO at finite Tp MF Peierls 3D LRO TP~ TP /3 X-ray diffuse scattering T *~T direct space reciprocal space F MF 2k ξ// F T > TP 1D regime -1 -1 Short Range Order (2kF) ξ// pre-transitional fluctuations Æ diffuse scattering ξ// -1 3D regime ξ┴ ξ┴ TP T < TP 3D Long Range Order satellite reflections * start of 1D fluctuations Blue bronze: 2kF CDW superlattice spots qCDW=(1, 2kF, 1/2) -2 ICDW ~ 10 IBragg 2kF~3/4 b* shifts in T TP=180K J. P. Pouget & al J. Physique 46, ICDW(Q)~|QuCDW|²; uCDW~0.05Å 1731 (1985) S. Girault & al PRB 39, 4430 (1989) Peierls transition in a non interacting electron gas opens the same gap in conductivity (charge degrees of freedom) and in magnetism (spin degrees of freedom) Blue bronze TP Δ0 (D.C. Johnston PRL 52, 2049 (1984)) TP Δ0(spin)~50meV Δ0(charge)~40meV (75 meV in optics) 1D fluctuation regime: ξ┴ < d┴ ±2kF diffuse sheets in reciprocal space X-ray diffuse scattering T *~T direct space reciprocal space F MF 2k ξ// F T > TP 1D regime -1 -1 Short Range Order (2kF) ξ// pre-transitional fluctuations Æ diffuse scattering ξ// -1 3D regime ξ┴ ξ┴ TP T < TP 3D Long Range Order satellite reflections * start of 1D fluctuations TSF-TCNQ: 2kF CDW instability Charge transfer 2kF ρ=0.63 2kF CDW on the TSF stack Thermodynanics of 1D classical systems Exact treatment of 1D fluctuations F[u]= ∫dz [a│u(z)│2 +b│u(z)│4+c│∂u/∂z│2] MF MF a=a’(T–Tp )/Tp Complex order Δt1D parameter u(z)=uo exp(iφ) exp(±i2kFz) ( Scalapino et al PRB 6, 3409 (1972); H. J. Schulz) fluctuations of the phase φ fluctuations of the amplitude uo MF 2/3 Fluctuations controled by the Ginzburg critical region: Δt1D=(bTp ) /a For the non interacting Peierls chain: Δt1D= 0.8 2kF Fluctuations in charge transfert salts Reduced Ginzburg critical region of the Peierls chain Δt1D= 0.8 J.-P. Pouget, Z. Kristallogr., 219, 711 (2004) 2kF CDW fluctuations MF Thermal length scale: ξ0= ħv* /π kBTp SALT TF (K) ξ0 (nm) ħv*(eVÅ) ħvF(eVÅ) v*/vF TSF- TCNQ 230 1.5 0.92 1 0.9 HMTSF-TCNQ ~300 2.2 1.80 2.3 0.8 NMP-TCNQ ~300 1.9 1.54 ~1.35 1.15 HMTSF-TNAP ~300 0.9 0.74 0.6 1.2 TMTSF- TCNQ ~300 1.1 0.9 1.6 1 0.56 TMTSF- DMTCNQ 225 1.0 0.6 1.3 1 0.47 HMTTF-TCNQ ~300 0.8 0.65 1.4 1 0.46 1 2 TTF- TCNQ 150 0.42 0.17 1.2 -1.1 0.15 MF start of 2kF fluctuations:TF ~Tp from plasma edge measurements in 1D: ħvF=2t// d// sin (πρ/2) interacting electron gas coupled to the lattice Organic conductors Density waves Charge ordering Spin-Peierls instability ORGANIC CHARGE TRANSFERT SALTS 1D Donor A 2 bands crossing at EF 2kF(A)= 2kF(D) Acceptor D Only a single 2kF critical wave TTF-TCNQ and TSF-TCNQ are isostructural vector! TSF-TCNQ: 2kF CDW instability Charge transfer 2kF ρ=0.63 2kF CDW on the TSF stack TTF-TCNQ: 2kF and 4kF CDW! b* Charge transfer ρ=0.59 2kF 4kF 2kF CDW on the TCNQ stack T=60K 4kF CDW on the TTF stack J.-P. Pouget et al PRL 37, 873 (1975); S.K. Khanna et al PRB 16, 1468 (1977) Τ= 0°Κ ground state in electron-phonon coupled systems 2kF Peierls transition for non interacting Fermions (exists whatever the strength of the electron- Spin degree of freedom phonon coupling) ρ = (2) x 2kF 4kF Peierls transition for spinless (no double occupancy U>>t) ∞ Fermions (exists only if the electron-phonon coupling is strong enough) No spin degree of freedom ρ = 4kF -1 Average distance between charges: 1/ρ=(4kF) The 4kF CDW can correspond at the first Fourier component of a 1D Wigner lattice of localized charges 4kF phase diagram for spinless fermions with long range coulomb interactions 4kF=n (d//*) organics J. Hubbard; B. Valenzuela et al PRB 68, 045112 (2003) ORGANIC CONDUCTORS U U/V1~2-3 V 1 U~4t//~1eV Extended Hubbard Model: H = H + U ∑ n n + ∑ V m n n 0 i i↑ i↓ i,m > 0 i i + m 1D correlated fermion gas: no quasiparticle only CDW, SDW and supraconductivity fluctuations The most diverging fluctuation at low T depends upon: The strength and range of Coulomb interactions: Kρ (>0 ) Kρ<1 density waves / Kρ>1 superconductivity The sign of the 2kF Fourier transform of the Coulomb interaction: g1 (= U+2V1cosπρ) If repulsion(g1>0): Luttinger Liquid If attraction (g1<0): Luther Emery Liquid Ground states with Kρ<1: illustration in the commensurate case of ρ=1 U<0: Luther Emery liquid 2kF « site » CDW - no magnetism U>0: Luttinger liquid AF 2kF SDW SP 2kF «bond»CDW (BOW) PHASE DIAGRAM OF THE 1D INTERACTING ELECTRON GAS Charge sector (Kσ =1) 2kF incommensurate Coexistence of 2kF and 4kF CDW! (H. Schultz) no unklapp scattering Divergence of the CDW electron-hole response function (U,V1≥0 case) χ(2kF) U=0 2kF 2kF diverges U=0 V=0 4kF T χ(4kF) U U ∞ diverges T 0: χ(2kF) diverges 2kF 4kF V ~ U/2 T 0: both χ(2kF) and χ(4kF) diverge T J.E.Hirsch and D.J.Scalapino PRB 29, 5554 (1984) Kρ CDW instability on the donor stack 1 Increase of the polarizability of the molecule HMTSF 2kF CDW TSF 1/2 SSe 2kF + 4kF CDW HMTTF two more cycles 1/3 4kF CDW TTF 0 4kF charge localization • In 1D for strong enough coulomb repulsion (U,V) 4kF charge localization (4kF CDW) •For commensurate band filling the electrons can be localized either on the sites or on the bonds (ground states of different symmetry/ energy) case of quarter filled band (ρ=1/2): the 4kF instability corresponds to a lattice dimerization of: - either the bonds: 4kF BOW (Mott dimer) I I I: inversion center on the bonds - or the sites: 4kF CDW (Wigner charge order) II I: inversion center on the sites (If BOW+CDW mixture no inversion symmetry: polar chain) (TMTCF)2X: BECHGAARD and FABRE SALTS :S TMTTF «C» :Se TMTSF P 1 a a x X centrosymmetrical : AsF6, PF6, SbF6, TaF6, Br zig-zag chain of TMTCF X non centro. : BF4, ClO4, ReO4, NO3 Phase diagram of (TMTCF)2X 4KFBOW Loc 4KF O CDW Charge ordering transition (4kF CDW) • Charge disproportion at TCO 2 different molecules :NMR line splitting Chow et al PRL 85, 1698 (2000) • Charge disproportion + incipient dimerization (4kF BOW) ferroelectricity at TCO Monceau et al PRL 86,4080 (2001) Charge ordering transition Enhancement of the charge localization gap at TCO +4KF CDW 4KFBOW -1 (4KFBOW) Structural dimerization PF6(H12): TCO=69K PF6(D12): TCO=90K AsF6(H12): TCO=100K Spin susceptibility measurements: (TMTTF)2X Thermal TCO behavior of TSP S=1/2 Heisenberg chain No anomaly at TCO: spin charge decoupling Non magnetic ground state develops below TSP CO state: charge degrees of freedom frozen Spin degrees of freedom remain Ground states of the 1D Heisenberg chain AF 2kF SDW SP 2kF bond « CDW » Spin-Peierls (BOW) ρ=1/2 (1 electron per dimer) Case of (TMTTF)2X Phase diagram of the (TMTCF)2X Loc O D. Jérome Spin-Peierls ground state: Singlet-Triplet splitting gap χΤ∼exp-Δ/T (ground state is a singlet S=0) (C. Coulon) PF6(H12): Δ = 79K PF6(D12): Δ = 75K AsF6(H12): Δ = 70K elastic neutron scattering evidence of SP superlattice reflexions in (TMTTF)2PF6 (D12) (2.5,0.5,-0.5) reflexion ISP/IB~1 h= ½ a* : 2a periodicity chaindimerization(pairingintoS=0 singlet) Thermal dependence of the (TMTTF)2PF6 superlattice peak intensity H12 D12 TSP=18+/-1K (P. Foury-Leylekian et al PRB 70, R180405 (2004)) TSP=13K Analogy between the spin-Peierls transition and the Peierls transition x x y y z z AF chain: H=Σj[Jxy (Sj Sj+1 + Sj Sj+1 )+ Jz Sj Sj+1 ] XY term Ising term Wigner Jordan transform spins 1/2 into fermionic operators: + – j-1 + + Sj =(Sj )*= exp(-iπΣl a lal) a j z + + – + Sj =a jaj-1/2 S jS j+1=a jaj+1 Vacuum |0>= |↓↓↓↓↓> + One pseudo-fermion added: a j |0>= |↓↓↓↑↓> spin up in j AF chain: half filled band of pseudo-fermions In k space: Fourier transform aj→ak + + + H= Σk ε(k)a kak+(1/N) Σk,k’,q V(q)a k+qa k’-qak’ak with: ε(k)=Jxy cos(ka) -Jz V(q)=Jz cos(qa) Half filled band of spinless fermions: - free fermions for XY chain (Jz=0) - interacting fermions for Heisenberg chain (Jxy=Jz=J) E(k) 2J 2Δ EF k -k +k -a*/4F 2kF +a*/4F half filled band of spinless fermions coupled to the lattice 2kF « Peierls »instability leads to a lattice dimerization (because 2kF=a*/2 → λCDW=2a) Spin-Peierls transition in the Heisenberg chain equivalent to the Peierls (CDW) transition in a Luther-Emery liquid with: V(q=2kF) = -V(q=0) = -Jz <0 « Peierls » transition opens a gap Δ in the pseudo-fermion dispersion (gap opened in the spin degrees of freedom: spin-Peierls transition) Phase diagram of the (TMTCF)2X Loc O D. Jérome (TMTSF)2PF6: 2kF SDW insulating ground state ρ magnetization first order metal- insulator transition m~0.1μΒ TSDW=12K from NMR: qSDW=(2k =0.5;0.2±0.05;0.06) F qSDW qSDW nests the quasi-1D FS Slater metal –insulator transition to a SDW ground state (1951) Analogous to a Peierls transition with the Hubbard term playing the role of the electron phonon coupling Uni↑ni↓ set a potential Uρ↑(x) to the charge density ρ↓(x) If this potential is modulated at 2kF : a gap Δ is opened in E[ρ↓(x)] at ±kF [ρ↓(x)] Gain of electronic energy as for the Peierls transition. Loss of energy by double occupancy of site i. Loss minimized if: ρ↑(x) and ρ↓(x) are out of phase this leads to the stabilization of a 2kF Spin Density Wave In fact hybrid 2kF CDW-SDW ground state in (TMTSF)2PF6! -5 -6 Very weak 2kF and 4kF CDW satellite reflections 10 -10 Bragg intensity! (TMTSF) 2PF6 300 2 satellit q1 "2kF" 2kF 1 q2 "4kF" 200 satellit Intensity Intensity 100 3 4kF satellit T(K) 0 10 12TC 14 16 T=10.7K qCDW~(1/2,1/4,1/4) J.P. Pouget & S. Ravy J. Phys. I 6, 1501 (1996); Synth. Met. 85, 1523 (1997) Generalized density waves (Overhauser 1968) ρ↑ = ρ↓ expiΘ 2kF CDW ↓≡↑ Θ=0 U and V weak (4kF CDW for strong U and V) 2k SDW ↑ Θ=π F ↓ U> V ↓↑ Hybrid 2KFSDW/CDW 0<Θ<π U ~ V Case of (TMTSF)2PF6 H = H +U∑n n + ∑ Vmn n 0 i i↑ i↓ i,m>0 i i+m Phase diagram of the (TMTCF)2X Loc O D. Jérome Tc=1.4K Singlet superconductor at low H at high H transition to: - inhomogeneous FFLO state or - triplet paired state Drastic effect of a non magnetic disorder on the superconductivity of (TMTSF)2ClO4/ReO4 nR: substitionnal disorder ReO4/ClO4 nQ: orientational disorder of ClO4 Depairing of Cooper pairs ≈τ-1 ≈α N. Joo et al EPJB 40, 43 (2004) Elastic collisions on non magnetic ponctual defects mix -kF to +kF states (scattering momentum 2kF) Elastic process: not allowed if the gap is always >0 (s superconductor) allowed if elastic scattering link gap regions of opposite sign (unconventional superconductor) unconventional gap symmetry averaged out by finite electron lifetime due to elastic scattering (rate of decrease of TC given by the pair breaking Abrikosov-Gorkov equation) – + – d gap in quasi-1D superconductor 2kF Δk= cte cos(k┴b) Δk node – – + Superconducting order parameter in quasi-1D conductors warped FS wave vector (rkF, k┴); r=± SS in 1D TS in 1D nodes d,f –k +k F f F Increasing number of nodes Purely intrachain repulsive interactions leads to unconventional superconductivity (only interchain hopping t┴ and nesting deviation t┴2 ) Interference between SDW instability and Cooper pairing Attraction takes place between electrons on neighboring chains as a result of their coupling to spin fluctuations d-wave superconductivity: interchain singlet pairing, nodes Duprat et al EPJB 21, 219 (2001); Fusaya et al JPSJ 74, 1263 (2005) Extended quasi-1D electron gas (interchain coulomb interactions g┴) Stabilization with increasing g┴ of: CDW and f triplet SC (r cosk┴) Interference between density wave and pairing Boundary between SDW and CDW ground states (hybrid SDW/CDW state?) Nickel et al PRL 95,247001 (2005) & PRB 73,165126 (2006) Impact of electron-phonon interactions (retardation effects) For weak coulomb interactions SCd robust Electron-phonon interactions: enlarge the g┴ interval for SDW to SCf increases the SDW gap and CDW correlations Bakrim and Bourbonnais ISCOM 2007 Main messages • 1D (quasi-1D) systems allow to test exact calculations of the many body problem • 1D conductors exhibits unconventional behavior: – Large regime of fuctuations – Collective behavior – Exotic ground states Thank you for your attention Documents supplémentaires N. Joo et al Europhys. Lett. 72, 645 (2005) Modulation of the charge on site i Modulation of the bond length i-i+1 « Internal » phonon mode Modulation of the charge on site i: CDW idem for U « External » phonon mode Modulation of the bond length i-i+1: BOW idem for V