Physics of quasi one dimensionalconductors

J.-P. PougetLaboratoire 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 chains1D 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

Structural anisotropy : d//<d┴

Anisotropy of overlap of wave functions

Anisotropy of electronic properties:

charge transport σ//(ω)>> σ┴(ω)

d//

d┴

σ t//

π tperp

t//>> t┴

ONE DIMENSIONAL CONDUCTORS

organics: pπ

inorganics: dz²

real 1D conductor: no coherent interchain electronic transfertimply: t┴ <<π kBT (Fermi surface thermal broadening)

InorganicsK2 Pt(CN)4,0.3Br-xH2O

1D overlap of dz² orbitals

Organics(TMTSF)2PF6

1D overlap of pл orbitals

BLUE BRONZE

1D

1D

Tp

Metal Insulator

A0.3MoO3A: K, Rb,Tl

102103

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

Spin degree of freedom

(with kF expressed in reciprocalchain unit: 2π/ d// )

For a free electron gas in D dimensions: kF~(n)1/D

n=ρ/d// is the electron densityin 1D : n= (2/π)kF (with kF expressed in Å-1)

1D metal coupled to the lattice:

PEIERLS’ CHAIN

unstable at 0°K towards a Periodic Lattice Distortion

which provides a new (2kF)-1

lattice periodicity

PLD opens a gap 2Δ at the Fermilevel: insulating ground state

2kF

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 energyEdis = 1/2Ku0² = 1/2K(Δ/g)²because 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 spinn=1 for incommensurate band filling ρn=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-ulThis 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 CDWδρ(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

Periodic Lattice Distortion u(x) (PLD) : observed by diffraction (information in reciprocal space: u(q))

u(x)

Modulation of the electronic density ρ(x) (electronic CDW): observed by STM (information in direct space)

ρ(x)

ρ(x) = - ∂u(x)/∂x

Charge density wave has two components:

coupled by the electron-phonon interaction (g)

Diffraction from a 2kF modulated chain• Direct space Reciprocal space

na x h2π/a Qamplitude 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)]*

Σr Jr(Qu0) Σn expi(Qna+r2kFna+rφ)Σr Jr(Qu0) expirφ Σhδ(Q+r2kF- h2π/a)

If Qu0<<1: Jr(Qu0)~ (Qu0)r only r= ±1 importantICDW(Q) = |A(Q)|²modulation 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 +2kF

h2π/a

2kF modulated chain

-2kF

+ 2kF

Satellite lines:r = ±1

Fourier transform of a modulated chain: diffuse sheets in reciprocal space

Peierls instability of the blue bronze K0.3MoO3

Tp=180K

qCDW=(1, 2kF, 1/2)

2kF=1-qb~3/4 b*(J.P. Pouget et al J. Physique Lettres44, L113 (1983))

6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.60

5000

10000

15000

20000

25000(-1 k 0.5) ESRF-D2AM

Inte

nsity

(cou

nts)

k

+2kF -2kF

+2kF-2kF 2-2kF2+2kF

STM observation of theCDW in the Rb blue bronze

Calculated image

λCDW

2kF=2π/λCDW

Divergence of χ(q) for q such thatEk+q~Ek

q best nesting wave vector of the Fermi surface

CDW instability:driven by the divergence of the

electron-hole response function χ(q)

χ(q)χ(q)

PLD fingerprint of q wave vectors at which χ(q) diverges

∑−−

+

+=

k kqk

kqke

EEEfEf )()(

χ

2kF

2kF

Structural instability at q wave vectorsfor which χ(q) diverges

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)

PLD

CDW

Thermal phase diagram• Mean- field (BCS-like) theory (ignore fluctuations)

2Δ0=3.56kBTPMF

• Isolated chain(because of thermal fluctuations there is no Peierls LRO at finiteT)

• Coupled chainsInterchain coupling restaures a Peierls 3D LRO at finite Tp

TPMFPeierls 1D LRO

0°K 1D Peierls fluctuations

TP~ TPMF/3Peierls 3D LRO

start at ~TPMF

ξ//

ξ//

ξ┴

ξ//-1

ξ┴-1

direct space reciprocal space

T < TP

3D regime

1D regime

satellite reflections

pre-transitional fluctuations diffuse scatteringShort Range Order

3D Long Range Order

TP

T > TP

X-ray diffuse scatteringTF *~TMF

* start of 1D fluctuations

(2kF)-1

2kF

Blue bronze: 2kF CDW superlattice spots

ICDW ~ 10-2 IBragg

ICDW(Q) ~|QuCDW|²; uCDW~0.05Å

qCDW=(1, 2kF, 1/2)

2kF~3/4 b*shifts in T

TP=180K

J. P. Pouget & al J. Physique 46, 1731 (1985)S. Girault & al PRB 39, 4430 (1989)

Peierls transition in a non interacting electron gasopens the same gap in conductivity (charge degrees of freedom) and in magnetism (spin

degrees of freedom)

Δ0(spin)~50meV

(D.C. Johnston PRL 52, 2049 (1984))

TPBlue bronze

TP

Δ0(charge)~40meV(75 meV in optics)

Δ0

1D fluctuation regime:ξ┴ < d┴

±2kF diffuse sheets in reciprocal space

ξ//

ξ//

ξ┴

ξ//-1

ξ┴-1

direct space reciprocal space

T < TP

3D regime

1D regime

satellite reflections

pre-transitional fluctuations diffuse scatteringShort Range Order

3D Long Range Order

TP

T > TP

X-ray diffuse scatteringTF *~TMF

* start of 1D fluctuations

(2kF)-1

2kF

TSF-TCNQ: 2kF CDW instability

2kF

2kF CDW on the TSF stack

Charge transferρ=0.63

a=a’(T–TpMF)/Tp

MF

Thermodynanics of 1D classical systemsExact treatment of 1D fluctuations

Fluctuations controled by the Ginzburg critical region: Δt1D=(bTpMF)2/3/a

For the non interacting Peierls chain: Δt1D= 0.8

Inverse correlation length ξ(T)-1

<u(z)u(0)> ∝ exp(±i2kFz) exp(–z/ ξ)

F[u]= ∫dz [a│u(z)│2 +b│u(z)│4+c│∂u/∂z│2]

( Scalapino et al PRB 6, 3409 (1972); H. J. Schulz)

Complex orderparameteru(z)=uo exp(iφ)exp(±i2kFz)

Δt1D

fluctuations of the phase φ fluctuations of the amplitude uo

J.-P. Pouget, Z. Kristallogr., 219, 711 (2004)

2kF Fluctuations in charge transfert salts

Reduced Ginzburg critical regionof the Peierls chain Δt1D= 0.8

SALT TF (K) ξ0 (nm) ħv*(eVÅ) ħvF(eVÅ) v*/vF

TSF- TCNQ 230 1.5 0.92 1 0.9HMTSF-TCNQ ~300 2.2 1.80 2.3 0.8NMP-TCNQ ~300 1.9 1.54 ~1.35 1.15HMTSF-TNAP ~300 0.9 0.74 0.6 1.2TMTSF- TCNQ ~300 1.1 0.9 1.6 1 0.56TMTSF- DMTCNQ 225 1.0 0.6 1.3 1 0.47HMTTF-TCNQ ~300 0.8 0.65 1.4 1 0.46TTF- TCNQ 150 0.42 0.17 1.21-1.12 0.15start of 2kF fluctuations:TF ~Tp

MF

from plasma edge measurements

Thermal length scale: ξ0= ħv* /π kBTpMF

2kF CDW fluctuations

in 1D: ħvF=2t// d// sin (πρ/2)

interacting electron gascoupled to the lattice

Organic conductorsDensity waves

Charge orderingSpin-Peierls instability

ORGANIC CHARGE TRANSFERT SALTS1D

2 bands crossing atEF

2kF(A)= 2kF(D)Only a single 2kF

critical wavevector!

Acceptor

Donor

D

A

TTF-TCNQ and TSF-TCNQ are isostructural

TSF-TCNQ: 2kF CDW instability

2kF

2kF CDW on the TSF stack

Charge transferρ=0.63

TTF-TCNQ: 2kF and 4kF CDW!

T=60K2kF CDW on the TCNQ stack

4kF CDW on the TTF stack

4kF2kF

J.-P. Pouget et al PRL 37, 873 (1975); S.K. Khanna et al PRB 16, 1468 (1977)

b*

Charge transferρ=0.59

2kF Peierls transition for non interactingFermions

4kF Peierls transition for spinlessFermions (no double occupancy U>>t)

The 4kF CDW can correspond at the first Fourier component of a 1D Wigner lattice of localized charges

(exists whatever the strength of the electron-phonon coupling)

(exists only if the electron-phonon coupling is strongenough)

Τ= 0°Κ ground state in electron-phononcoupled systems

Spin degree of freedomρ = (2) x 2kF

No spin degree of freedomρ = 4kF Average distance between charges: 1/ρ=(4kF)-1

∞

4kF phase diagram for spinless fermions with long range coulomb interactions

J. Hubbard; B. Valenzuela et al PRB 68, 045112 (2003)

organics

4kF=n (d//*)

Extended Hubbard Model:

∑ ∑> ++

↓↑+=

i mi mininmVi

ni

nUHH0,0

1D correlated fermion gas: no quasiparticle

only CDW, SDW and supraconductivity fluctuations

V1

UU/V1~2-3

U~4t//~1eV

ORGANIC CONDUCTORS

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

U>0: Luttinger liquid

2kF « site » CDW - no magnetism

AF

SP

2kF SDW

2kF « bond » CDW(BOW)

(H. Schultz)

PHASE DIAGRAM OF THE 1D INTERACTING ELECTRON GAS

(Kσ =1)Charge sector

Coexistence of 2kF and 4kF CDW!2kF incommensurateno unklapp scattering

Divergence of the CDW electron-hole response function (U,V1≥0 case)

J.E.Hirsch and D.J.Scalapino PRB 29, 5554 (1984)

T 0: χ(2kF) diverges

χ(4kF) diverges

T 0: both χ(2kF) and χ(4kF) diverge

2kF 2kF

2kF

4kF

4kF

V=0

V ~ U/2

U=0U=0

χ(2kF) diverges

U ∞T

T

U

CDW instability on the donor stackKρ

0

1

1/2

1/3

2kF CDW

2kF + 4kF CDW

4kF CDW

S Se

two more cycles

TTF

HMTTF

TSF

HMTSF

Increase of the polarizability of the molecule

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 (groundstates of different symmetry/ energy)case of quarter filled band (ρ=1/2): the 4kF instabilitycorresponds to a lattice dimerization of:- either the bonds: 4kF BOW (Mott dimer)

- or the sites: 4kF CDW (Wigner charge order)I I

I I

I: inversion centeron the bonds

(If BOW+CDW mixture no inversion symmetry: polar chain)

I: inversion centeron the sites

axa

zig-zag chain of TMTCF

P 1

X centrosymmetrical : AsF6, PF6, SbF6, TaF6, Br

X non centro. : BF4, ClO4, ReO4, NO3

:S TMTTF

:Se TMTSF

(TMTCF)2X: BECHGAARD and FABRE SALTS

« C »

Phase diagram of (TMTCF)2X

Loc

O4KFCDW

4KFBOW

Charge ordering transition (4kF CDW)• Charge disproportion at TCO

2 different molecules :NMR line splitting

• Charge disproportion + incipient dimerization (4kF BOW)

ferroelectricity at TCO

Monceau et al PRL 86,4080 (2001)

Chow et al PRL 85, 1698 (2000)

PF6(H12): TCO=69K PF6(D12): TCO=90K AsF6(H12): TCO=100K

Enhancement of the charge localization gap at TCO

Charge ordering transition

4KFBOW +4KF CDW

(4KFBOW)-1

Structural dimerization

Spin susceptibilitymeasurements: (TMTTF)2X

No anomaly at TCO: spin charge decouplingNon magnetic ground state develops below TSP

TCOTSP

Thermal behavior of S=1/2 Heisenberg chain

Ground states of the 1D Heisenberg chain

AF

SP

ρ=1/2 (1 electron per dimer)Case of (TMTTF)2X

2kF SDW

2kF bond « CDW »(BOW)Spin-Peierls

CO state: charge degrees of freedom frozenSpin degrees of freedom remain

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)

PF6(H12): Δ = 79K PF6(D12): Δ = 75K AsF6(H12): Δ = 70K

(C. Coulon)

elastic neutron scattering evidence of SP superlattice reflexions in (TMTTF)2PF6 (D12)

ISP/IB~1

(2.5,0.5,-0.5) reflexion

h= ½ a* : 2a periodicity chain dimerization (pairing into S=0 singlet)

H12 D12

TSP=18+/-1KTSP=13K

Thermal dependence of the (TMTTF)2PF6superlattice peak intensity

(P. Foury-Leylekian et al PRB 70, R180405 (2004))

Analogy between the spin-Peierls transition and the Peierls transition

AF chain: H=Σj[Jxy (SjxSj+1

x+ SjySj+1

y)+ Jz SjzSj+1

z]XY term Ising term

Wigner Jordan transform spins 1/2 into fermionic operators:Sj

+=(Sj–)*= exp(-iπΣl

j-1a+lal) a+

jSj

z=a+jaj-1/2 S+

jS–j+1=a+

jaj+1Vacuum |0>= |↓↓↓↓↓>One pseudo-fermion added: a+

j |0>= |↓↓↓↑↓> spin up in jAF chain: half filled band of pseudo-fermionsIn k space: Fourier transform aj→akH= Σk ε(k)a+

kak+(1/N) Σk,k’,q V(q)a+k+qa+

k’-qak’akwith: 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)

half filled band of spinless fermions coupled to the lattice2kF « 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)

k

E(k)

-kF +kF

EF

2Δ

+a*/4-a*/4

2J

2kF

Phase diagram of the (TMTCF)2X

Loc

O

D. Jérome

ρ

(TMTSF)2PF6: 2kF SDW insulating ground state

TSDW=12K

from NMR: qSDW=(2kF=0.5;0.2±0.05;0.06)

qSDW nests the quasi-1D FSqSDW

magnetization first order metal- insulator transitionm~0.1μΒ

Slater metal –insulator transition to a SDW ground state (1951)

Analogous to a Peierls transition with the Hubbard term playingthe 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

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

[ρ↓(x)]

In fact hybrid 2kF CDW-SDW ground statein (TMTSF)2PF6!

3

1

2

(TMTSF)2PF6

0

100

200

300

10 12 14 16

T(K)In

tens

ity

q1 "2kF"q2 "4kF"

satellit

satellit

satellit

TC

T=10.7KqCDW~(1/2,1/4,1/4)

2kF

4kF

Very weak 2kF and 4kF CDW satellite reflections

J.P. Pouget & S. Ravy J. Phys. I 6, 1501 (1996); Synth. Met. 85, 1523 (1997)

10-5 -10-6 Bragg intensity!

Generalized density waves (Overhauser 1968)

∑ ∑> ++

↓↑+=

i mi mininmVi

ni

nUHH0,0

ρ↑ = ρ↓ expiΘ

2kF CDW Θ=0

2kF SDW Θ=π

U and V weak(4kF CDW for strong U and V)

U> V

Hybrid 2KFSDW/CDW 0<Θ<π

U ~ V

↓≡↑

↓↑

↓ ↑

Case of (TMTSF)2PF6

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 defectsmix -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 electronlifetime due to elastic scattering

(rate of decrease of TC given by the pair breaking Abrikosov-Gorkov equation)

d gap in quasi-1D superconductor

Δk= cte cos(k┴b)

+

+–

––

–Δk node

2kF

Superconducting order parameter in quasi-1Dconductors

warped FSwave vector (rkF, k┴); r=±

SS in 1D

TS in 1Dnodes

–kF +kFf

d,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 pairingAttraction takes place between electrons on neighboring chains as a result of theircoupling 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 groundstates (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 SCfincreases 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 unconventionalbehavior:– 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

Modulation of the charge on site i: CDW

Modulation of the bond length i-i+1: BOW

idem for U

idem for V

« Internal » phonon mode

« External » phonon mode