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Spin-Peierls Transition

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Spin-Peierls Transition 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// 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 <u(z)u(0)> ∝ exp(±i2kFz) exp(–z/ ξ) Inverse correlation length ξ(T)-1 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.
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