The optical conductivity of the quasi one-dimensional organic conductors: the role of forward scattering by impurities Peter Kopietz1 and Guillermo E. Castilla2 1Institut fu¨r Theoretische Physik, Universit¨at G¨ottingen, Bunsenstrasse 9, D-37073 G¨ottingen, Germany 2Department of Physics, University of California, Riverside, California 92521 (December17, 1998) 9 9 9 dimensionalbehaviorathigherenergiestoaregimechar- We calculate the average conductivity σ(ω) of interact- 1 acterized by three-dimensional phase coherence: this is ing electrons in one dimension in the presence of a long- a consequence of the finite coupling between the chains n range random potential (forward scattering disorder). Tak- [4]. Experimental evidence for the existence of such a a ing the curvature of the energy dispersion into account, we J crossover in the organic conductors has also been given 5 show that weak diso2rder leads to a transport scattering rate byMoseretal.[5]viaDCtransportmeasurements. How- that vanishes as ω for small frequency ω. This implies Imσ(ω) ∼ D˜c/ω and Reσ(ω) ∼ D˜cτ for ω → 0, where D˜c ever,theauthorsofRef.[1]emphasizedthattheinterpre- ] tationof the frequency-dependence in Eqs.(2,3)in terms l is the renormalized charge stiffness and the time τ is pro- e of Fermi liquid theory leads to an anomalously small portional to the strength of the impurity potential. These - value of 1/α. Moreover,the Fermi surface of the organic r non-trivial effects due to forward scattering disorder are lost t conductors is nested, so that one should expect a scat- s within the usual bosonization approach, which relies on the . linearization of the energy dispersion. We discuss our result tering rate linear in ω [6], in disagreement with the ex- t a in thelight of a recent experiment. perimental result [1]. In this note we would like to point m out that there exists an alternative (and in our opinion - PACS numbers: 78.20.-e, 71.10.Pm, 75.30.Fv physically more plausible) explanation for the quadratic d frequency-dependence of the scattering rate and the fi- n nite effective mass renormalization in the organic con- o In a recent measurement of the frequency-dependent ductors, namely forward scattering by impurities. As we c [ conductivity σ(ω) of quasi one-dimensional organic con- shallshowbelow,thisinterpretationofthedataalsoleads ductors of the (TMTSF)2X-series, Schwartz et al. [1] to a natural explanation of the anomalously small value 2 foundthatforsmallfrequenciesωthedatacouldbefitted of 1/α seen in the experiment [1]. v by As discussed in Ref. [7], in the quasi one-dimensional 3 2 D organicconductorsit is naturalto expect that the disor- c 2 σ(ω)= , (1) der potential seen by the electrons on the chains is weak 7 Γ(ω)−iωmm∗(bω) andslowlyvaryingalongthe chain. Suchapotentialcan 0 be modelled by a Gaussian random potential U(x) with 8 where Dc is the bare charge stiffness, and where the zero average and long-range correlator 9 transportscattering rateΓ(ω) andthe effective mass en- at/ hancement m∗(ω)/mb are given by U(x)U(x′)=γ0C(x−x′), (4) m λ αω2 Γ(ω)=Γ + 0 , (2) where the overline denotes averaging over the disorder. - 0 1+α2ω2 Here γ is a measure of the strengthof the disorder,and d 0 C(x) is assumed to have a maximal range ξ that is large n o m∗(ω) λ0 compared with (2kF)−1, where kF is the Fermi wave- =1+ . (3) c m 1+α2ω2 vector. In other words, we assume that C(x) is a finite : b positive constant for x < ξ, and vanishes for x ξ. v i Thequadraticfrequency-dependenceinEqs.(2)and(3)is For convenience we no|rm|∼alize C(x) such that its| F|o≫urier X characteristic of Fermi liquids in three dimensions. Ex- transform r pressions of this type were used by Sulewski et al. [2] a ∞ in their study of the compound UPt3, which is a three- C˜(q)= dxeiqxC(x) (5) dimensional Fermi liquid with heavy mass. The exper- Z−∞ imentally obtained values [1,3] for a sample consisting of (TMTSF) PF are Γ /(2πc) = 0.56cm−1, λ = 1, is dimensionless. The above properties of C(x) imply 2 6 0 0 and 1/(2πcα) = 1cm−1. Schwartz et al. [1] specu- that C˜(q) vanishes if q ξ 1. The inverse of ξ can be | | ≫ latedthatthephysicaloriginofthequadraticfrequency- identified with the maximal possible momentum trans- dependence of the second term in Eq.(2) is inelastic fer between two electrons due to scattering by the im- electron-electron scattering in a clean three-dimensional purity potential. The requirement ξ−1 2kF means ≪ Fermi liquid. The physical picture is that at suffi- that impurities do not give rise to backward scattering, ciently low temperatures, there is a crossover from one- i.e., the random potential is dominated by the forward 1 scattering. If this problem is treated by means of the ThememoryfunctionapproachhasalsobeenusedbyGi- usual bosonization approach(with linearized energy dis- amarchito study the effect of umklappscattering onthe persion), one finds that forward scattering disorder does conductivityofone-dimensionalinteractingfermions[15]. not affect the conductivity at all [8,9]. It is also possi- FortheTomonaga-Luttingermodelwithshort-rangedis- ble to confirm this result by directly expanding the av- order Eq.(8) has been evaluated by Luther and Peschel erage conductivity in powers of the impurity potential. [16]. In this case the anomalous scaling of Π(q,ω) for For linearized energy dispersion one easily verifies that momenta close to 2k dominates the conductivity. In F all impurity corrections cancel. This is a consequence of contrast, in our case we need Π(q,ω) only for small mo- the closed-loop theorem [10,11], which is well known in menta, q < ξ−1 2k . Observe that the leading effect F the context of the Tomonaga-Luttinger model [12]. The of the n|o|n∼-lineari≪ty in the energy dispersion is already closedlooptheoremimpliesthatatlongwave-lengthsall contained in the prefactor 1/m in Eq.(8); thus, we can b closed fermion loops with more than two external legs calculateΠ(q,ω) onthe right-handside ofEq.(8)forlin- vanishafter symmetrization[10,11]. For this cancelation earized energy dispersion. For simplicity we substitute to take place, it is irrelevant whether the external legs forΠ(q,ω)thedensity-densitycorrelationfunctionofthe representthe dynamic Coulombinteractionor static im- Tomonaga-Luttinger model [12] with interaction param- purity lines. However, the closed-loop theorem is only eters g =g =πv F, 2 4 F valid if the energy dispersion ǫ(k) is linearized close to 2ω qthueadFreartmiciapnodinhtisgh±ekrFte,rwmhsicinhtahmeoeuxnptasnstioonignoring the Π(q,ω)=Zqωq2−qω2 , (9) q2 ǫ( kF +q) ǫ( kF)= vFq+ +O(q3). (6) q ± − ± ± 2mb Zq = | | , ωq =v˜F q , (10) π√1+F | | Here v is the Fermi velocity and m is the band mass. F b Clearly,inordertocalculatetheleadingeffectofforward where v˜F = √1+FvF. Using Eq.(8) and assuming scattering disorder on the conductivity it is insufficient C˜(q) = Θ(1 q ξ), we obtain after a simple calculation −| | to work with linearized energy dispersion, as it is done for ω v˜ /ξ F | |≪ usually in the bosonization approach. Wenowpresentasimplecalculationoftheeffectoffor- ImM(ω+i0+)=aω2 , (11) ward scattering disorder on the conductivity of an inter- ReM(ω+i0+)=bω[1+O(ω2)], (12) actingelectrongasinonedimension. Weassumethatthe electrongasismetallicandfocusonenergyscalessmaller where than possible spin gaps, so that we can ignore backward γ 0 and umklapp scattering. In principle, one could try to a= , (13) πnm v˜4√1+F treatthenon-lineartermsintheenergydispersionwithin b F 2γ theframeworkofbosonization,asitwasrecentlydonein 0 b= . (14) adifferentcontextinRef. [13]. However,there isa much π2nmbv˜F3ξ√1+F simpler and physically more transparent solution to our problem. AccordingtoG¨otzeandW¨olfle[14],intheper- Notethataandbbothvanishfor1/mb 0,correspond- → turbativecalculationoftheconductivityitisoftenuseful ing to the linearization of the energy dispersion (6). We to introduce the memory function M(ω) by setting conclude that forwardscattering disorderleads to a con- ductivity of the form given in Eq.(1), with iD c σ(ω)= , (7) Γ(ω)=aω2 , (15) ω+M(ω) m∗(ω) =1+b+O(ω2). (16) and calculating M(ω) instead of σ(ω) in powers of the m b impurity potential. In this way one implicitly takes into account vertex corrections to all orders in perturbation In the absence of other scattering mechanisms this im- theory. To leading order in the strength of the impurity plies forthe real-andimaginarypartofthe conductivity potential one finds in one dimension [14] at frequencies ω v˜F/ξ, | |≪ γ ∞ dq D D˜ τ M(ω)= 0 q2C˜(q) Reσ(ω)=Re c = c , (17) nmb Z−∞ 2π (cid:20)aω2−i(1+b)ω(cid:21) 1+(ωτ)2 Π(q,ω+i0+) Π(q,i0+) D˜ − , (8) Imσ(ω)= c , (18) ×(cid:20) ω+i0+ (cid:21) ω[1+(ωτ)2] where Π(q,ω) is the density-density correlation function where the renormalizedcharge stiffness D˜ and the time c and n is the density of the one-dimensional electron gas. τ are given by 2 D a D˜ = c , τ = . (19) by identifying ℓtr = v˜F/Γ0. Using the measured value c 1+b 1+b Γ 2πc 0.56cm−1, we obtain ℓ 9 10−5cm. The 0 tr ≈ × ≈ × importanceoflong-rangedisorderinthe organicconduc- Keeping in mind that for the derivation of Eqs.(17,18) tors has been pointed out by Gorkov [7] long time ago: wehaveassumed ω v˜ /ξ,itiseasytoseethatinthe F | |≪ because defects damage only a small fraction of the big regime where Eqs.(17,18)are valid the parameter ω τ is | | planar molecules, it is natural to expect that the imper- alwayssmallerthanunity. Hence,for ω v˜ /ξwemay F | |≪ fectionpotentialofthedefects,asseenbytheelectronson approximate Reσ(ω) D˜ τ and Imσ(ω) D˜ /ω. Note ≈ c ≈ c theconductingchains,isslowlyvaryingalongthechains. that the imaginary part of the conductivity exhibits at An inelastic single-particle scattering rate Γ (ω) ω2 small frequencies the usual 1/ω behavior of a clean sys- 1 ∝ due to electron-electron interactions is one of the hall- tem(butwithrenormalizedchargestiffness),whereasthe marksofa three-dimensionalFermiliquid. However,the Drude peak πD δ(ω) in the real part of the conductiv- c transport scattering rate Γ(ω) in Eq.(15) is defined in ity of the clean system is completely destroyed by the terms of a two-particle Green’s function, and it is obvi- disorder, and is replaced by a constant D˜ τ. In this c ously not related to Γ (ω) of a three-dimensional Fermi respect the effect of forward scattering disorder in one 1 liquid. This shows that it is crucial to distinguish be- dimensionissimilartotheeffectofweakshort-rangedis- tween transport and single-particle scattering rates in orderinthreedimensions. Note,however,thataccording theinterpretationofconductivitymeasurementsofquasi to Eqs.(13,19) the effective lifetime τ is proportional to one-dimensionalconductors. To understandwhythe ω2- the strength γ of the impurity potential. In contrast, 0 behavior of the transport scattering rate Γ(ω) does not for short-range impurity scattering one finds within the imply a similar behavior of the single-particle scattering Born approximation that the inverse lifetime is propor- rateΓ (ω),oneshouldkeepinmindthattheabovemen- tionalto the impurity strength. These non-trivialeffects 1 tioned cancelations between self-energy and vertex cor- associatedwith long-rangedisorderinone dimensionare rections, which are responsible for the ω2-correction to missed within the usual bosonization approach, which the transport scattering rate, do not occur in the calcu- predicts a Drude peak πD δ(ω) in the real part of the c lation of the single-particle Green’s function. In fact, for conductivity, even in the presence of long-range disorder non-Fermiliquidsinarbitrarydimensionsthereisnorea- [8,9]. sontoexpectthatthe transportandsingle-particlescat- KeepinginmindthatEqs.(11)and(12)aretheleading tering rates are simply related [19]. Although Eqs.(15) terms for small ω, Eqs.(15) and (16) are consistent with and (16) look like conventional Fermi liquid behavior, the frequency-dependent partof the experimentally seen the single-particle Green’s function of the system ex- behavior given in Eqs.(2) and (3) [17]. The constant hibitsLuttingerliquidbehavior. Forexample,thesingle- part Γ of the scattering rate in Eq.(2) must be due to 0 particle density of states vanishes with a non-universal some other (short-range)impurity scatteringmechanism power-law [10,12]. which we havenot takeninto accountin our calculation. So far we have assumed that the system is strictly AsimilarassumptionconcerningtheoriginofΓ mustbe 0 one-dimensional. However, realistic experimental sys- made if one interpretsthe frequency-dependence ofΓ(ω) temshaveafinitehoppingt betweenthechains,sothat to be due to inelastic scattering in a three-dimensional ⊥ it is importantto estimate the modification ofthe above Fermi liquid [1]. Obviously we may identify λ =b, and 0 result due to a finite value of t . For simplicity, let us ⊥ 1 b 2v˜F assume that the interchain hopping ty = t⊥ in one di- = = . (20) rectiontransversetothechains(the y-direction)ismuch α a πξ larger than in the other transverse direction. To a first Note that α is independent of γ0 and mb; it depends approximationwe may thenignorethe hopping tz in the only on the renormalized Fermi velocity v˜F and on the z-direction. In view of the anisotropy of the interchain correlation length ξ of the impurities. Using the exper- hopping in the organic conductors [18], this approxima- imental value [1] 2πcα = 1cm we obtain from Eq.(20) tion is not unreasonable. The transport scattering rate ξ = π−2(v˜F/c)cm. Given a bare interchain hopping due to forwardscattering by impurities canthen be esti- tk 250meV [1], a reasonable estimate for the Fermi mated from the two-dimensional analog of Eq.(8), ≈ velocity is v˜ 107cm/s. Then we obtain for the im- purity correlFat≈ion length ξ 3 10−5cm. This is cer- M(ω)= γ0 ∞ dqxdqyq2C˜(q) tainly larger than (2kF)−1, ≈so th×at our interpretation of n2mb Z−∞ (2π)2 x the low-frequency behavior of the conductivity data [1] Π(q,ω+i0+) Π(q,i0+) in terms of long-range disorder is internally consistent. − , (21) ×(cid:20) ω+i0+ (cid:21) Thelargevalueofξ explainstheanomalouslysmallvalue of α−1 seen in the experiment [1]. The impurity corre- where n is the two-dimensional density, m is the ef- 2 b lation length ξ should not be confused with the trans- fective band mass in the x-direction, C˜(q) is the two- port mean free path ℓ , although in the experiment [1] dimensionalFourier transformofthe disordercorrelator, tr both length scales have the same order of magnitude. A and Π(q,ω) is the two-dimensional density-density cor- rough estimate for ℓ can be obtained from Eqs.(1,2) relation function. Keeping in mind that by assumption tr 3 C˜(q)isdominatedbysmallwave-vectors,wemayusethe transfers. The data therefore do not necessarily im- random-phase approximation to calculate the density- ply that at small energy scales the organic conductors densitycorrelationfunction. Forsmallt itcanbeshown become conventional three-dimensional Fermi liquids. ⊥ [20] that Π(q,ω) is dominated by the plasmon pole, so Note that the impurity scattering mechanism proposed that Π(q,ω) can be written as in Eq.(9). Defining the here leads for temperatures T E to a temperature- F ≪ dimensionless parameter independent scattering rate. This can be easily veri- fied from Eq.(8), which is also valid at finite temper- t θ = | ⊥| , (22) atures if we use the finite temperature density-density EF correlation function. Keeping in mind that electron- electron or electron-phonon scattering in general lead where E is the Fermi energy, the residue and the plas- F to temperature-dependent scattering rates, it should be mon dispersion are for small θ given by [20] possible to distinguish the impurity scattering mecha- nismproposedherefromothermechanismsbymeasuring ν v q2+θ2q2 2 F x y Z = q , ω =v˜ q2+θ2q2 , (23) the temperature-dependence of the scattering rates [21]. q 2√1+F q Fq x y Note that in several experiments on organic conductors a T2-behavior of the resistivity has been observed [22]. where ν is the two-dimensional density of states. As- 2 This cannotbe explained by invokingforwardscattering suming for simplicity C˜(q) = Θ(1 q ξ)Θ(1 q ξ), −| x| −| y| by impurities. If the quadratic frequency-dependence of the low-frequency behavior of M(ω) is easily calculated. thescatteringrateseenintheexperiment[1]isreallydue Wefindthatthestrictlyone-dimensionalresultforM(ω) toimpurities,thenEqs.(15,16)and(20)provideasimple given in Eqs.(11,12)remains valid as long as way to estimate the impurity correlation length ξ from v˜ θ the low-frequency data for the conductivity. ω > F ω . (24) ⊥ We would like to thank Ward Beyermann, Martin | |∼ ξ ≡ Dressel, George Gru¨ner, and Kurt Scho¨nhammer for For ω ω⊥ we find that Eqs.(11,12) should be multi- comments on the manuscript. PK acknowledges finan- | | ≪ pliedby anextrafactoroforder ω /ω⊥ 1. Hence, the cialsupportedfromtheHeisenbergprogramoftheDFG. | | ≪ finiteness of the interchain hopping induces a crossover in the transport scattering rate due to forward scatter- ing by disorder from a ω2-behaviorat higher frequencies to a ω 3-law at frequencies ω ω . Note, however, ⊥ | | | | ≪ that the crossover frequency ω is of order t /(k ξ), ⊥ ⊥ F | | so that for large ξ and small t the regime where the ⊥ ω 3-behavioris visible isverysmallandprobablyexper- [1] A. Schwartz, M. Dressel, G. Gru¨ner, V. Vescoli, L. De- i|m|entally irrelevant. giorgi, andT. Giamarchi, Phys.Rev.B 58,1261 (1998). Somecautionaryremarksareinorder: Inthisworkwe [2] P. E. Sulewski et al.,Phys.Rev. B 38, 5338 (1988). have assumed that the scattering due to the impurities [3] Throughout this work we shall use units where ¯h = 1. is forward. It is clear, however, that in the organic con- AlthoughEqs.(2)and(3)seemtofitthedataratherwell, ductorsalsobackwardscatteringbyimpuritiesispresent oneshouldkeepinmindthattheprecisedeterminationof which, in a purely one-dimensional system, should lead thefrequency-dependenceofthescatteringrateisrather sensitive to the normalization and extrapolation of the to localization. Because this implies a vanishing con- data. ductivity, atsufficiently smallfrequenciesReσ(ω) should [4] Anupperlimit forthecrossovertemperatureisgivenby become smaller than D˜ τ and eventually vanish. How- c the hoppingintegral t⊥ between thechains: T ∼< t⊥. ever, in the experimental systems the finite interchain [5] J. Moser, M. Gabay, P. Auban-Senzier, D. J´erome, K. couplingmightstabilizeametallicphase. Moreover,from Bechgaard,andJ.M.Fabre,Eur.Phys.J.B1,39(1998). theabovecalculationitisclearthattheconstantpartΓ0 [6] J. Ruvalds and A. Virosztek, Phys. Rev. B 43, 5498 of the scattering rate in Eq.(2) cannot be explained by (1991). invokinganykindofforwardscatteringdisorder. We are [7] L. P. Gorkov, in Electron-Electron Interactions in Dis- awareofthe factthatourexplanationofthe constantΓ0 ordered Systems, edited by A. L. Efros and M. Pollak in Eq.(2) is based on a plausible argument rather than (North-Holland, Amsterdam, 1985). See the discussion on a microscopic calculation. Such a calculation should on defects in quasi one-dimensional conductors on page treat both forward and backward scattering by impuri- 634. tiesandtakeintoaccountalsotheweakcouplingbetween [8] A. A. Abrikosov and J. A. Ryzhkin,Adv.Phys. 27, 147 the chains. (1978). In summary, we have shown that the frequency- [9] T. Giamarchi and H. J. Schulz, Phys. Rev. B 37, 325 dependent part of the scattering rate seen in the (1988). low-frequency conductivity data [1] of the quasi one- [10] T.Bohr,Norditapreprint81/4,LecturesontheLuttinger dimensional organic conductors (TMTSF) X can be ex- Model, 1981, unpublished. 2 plained by impurity scattering with small momentum [11] I.E.DzyaloshinskiiandA.I.Larkin,Zh.Eksp.Teor.Fiz. 4 65, 411 (1973) [Sov.Phys. JETP 38, 202 (1974)]. [12] S. Tomonaga, Prog. Theor. Phys. 5, 544 (1950); J. M. Luttinger, J. Math. Phys. 4, 1154 (1963); D. C. Mattis and E. H. Lieb, J. Math. Phys. 6, 304 (1965). [13] R. Fazio, F. W. J. Hekking, and D. E. Khmelnitskii, Phys.Rev.Lett. 80, 5611 (1998). [14] W. G¨otze and P. W¨olfle, Phys. Rev.B 6, 1226 (1972). [15] T. Giamarchi, Phys. Rev.B 44, 2905 (1991). [16] A.LutherandI.Peschel,Phys.Rev.Lett.32,992(1974). [17] Note that Eqs.(15) and (16) are the leading behavior for small ω, so that it would be incorrect to perform a Kramers-Kronig analysis on these expressions. In con- trast, the phenomenological Eqs.(2) and (3) interpolate from small to large frequencies. [18] D.J´erome, Science 252, 1509 (1991). [19] L. Bartosch and P. Kopietz, cond-mat/9805005, to ap- pear in Phys. Rev.B (Feb.1999). [20] P.Kopietz,V.Meden,andK.Sch¨onhammer,Phys.Rev. B 56, 7232 (1997). [21] We would like to thank George Gru¨ner for pointing this out to us. [22] K. Bechgaard et al., Solid State Commun. 33, 1119 (1980); M. Weger et al., J. Phys. Cond. Mat. 5, 8569 (1993). 5