Impurity scattering and frequency dependent conductivity in spin density waves

– The quasiparticle contribution to the frequency dependent electric conductivity in the presence of randomly distributed impurities is calculated within mean field theory for spin density waves as formed in quasi one dimensional conductors like Bechgaard salts. Interchain hopping is taken into account and the effects of imperfect nesting are considered. In case of an electric field perpendicular to the chain direction there is no collective contribution to the conductivity and our results are directly applicable to recent measurements on both (TMTSF)2PF6 and quenched (TMTSF)2ClO4. The experimental data are well described by the theory in its clean limit, with somewhat larger scattering in the ClO4 salt. Spin density wave (SDW) is one of the ground states of quasi one dimensional electronic systems, and commonly found at low temperature in Bechgaard salts (TMTSF)2X with X=PF6, ClO4, etc[1, 2]. Both the appearance of SDW in Bechgaard salts with the pressure dependence of the SDW transition temperature Tc, and the electronic properties of the SDW are well described in terms of the standard model, where the approximate nesting of the quasi one dimensional Fermi surface (i.e. the imperfect nesting), and the repulsive Coulomb interaction between electrons are the crucial ingredients[3]. In particular, the enhanced gap to Tc ratio compared to the weak coupling BCS value is interpreted as the consequence of imperfect nesting[4]. Within this model the quasi particle energy gap edges depend on the quasi particle momentum perpendicular to the conducting chains (the b direction) as ∆±(py) = ±∆+ ε0 cos(2bpy), (1) where ∆ is the order parameter and ε0 is the ”unnesting” parameter characterizing the deviation from one dimensionality. Indeed, the above momentum dependent energy gap is inferred from the magnetoresistance data[5] in the SDW state of (TMTSF)2PF6, and it offers Typeset using EURO-TEX 2 EUROPHYSICS LETTERS an explanation of the difference between the energy gap as observed by transport and optical measurements for a number of density wave materials[6]. While the optical (pair breaking) gap is still given by 2∆, dc transport is sensitive to the smaller gap in the density of states[4]. The frequency dependent electric conductivity of the SDW in Bechgaard salts has been studied over decades[2]. Unfortunately however, the clear understanding of the physical situation still appears to be lacking. Earlier experiments were mostly done in the geometry E ‖ a, i.e. the electric field parallel to the chain direction[7]. Under these circumstances it is expected that the phason couples strongly to the electromagnetic field in analogy to the mean field treatment of CDW by Lee, Rice and Anderson[8, 9]. In SDW, most of the optical weight is shifted to the phason mode and gives rise to low energy absorption[10] for ω ≪ 2∆, since in the absence of pinning the transverse phason is naturally coupled to the transverse phonon[11] as well as the photon. However, in general the phason is pinned by impurities or crystalline defects as evidenced by the existence of the threshold electric field in the non Ohmic dc conductivity in (TMTSF)2NO3 and (TMTSF)2PF6[12, 13], and therefore the longitudinal phason gets mixed in as well. On the other hand we have shown[10] that the longitudinal phason is almost completely removed from the low frequency range in an SDW due to the Anderson-Higgs mechanism[14]. We believe that this is the main reason for the lack of a clear optical energy gap in the data taken in the geometry E ‖ a, since first of all the optical weight at the gap region ω ≥ 2∆ is shifted down to the pinned mode at ω ≪ 2∆, which in turn is depleted due to the admixture with the longitudinal phason which is nothing but the plasmon in a SDW. In this situation the frequency dependent conductivity perpendicular to the chains is ideal to measure the optical energy gap, since in the geometry E ⊥ a the electromagnetic field does not couple to the phason[11]. Within mean field theory we only need to consider the so called quasiparticle contribution to the conductivity since no collective contribution is present in this case. We obtain an expression for the perpendicular electric conductivity which is very similar to the one derived by Mattis and Bardeen[15], and by Abrikosov et.al.[16] for s-wave superconductors except one crucial point. The authors of both papers[15, 16] introduced a simplification which is valid either in the anomalous skin limit or in the dirty limit. Unfortunately, for organic conductors and also for high Tc cuprate superconductors none of the above approximations applies, since they are usually in the clean limit and the penetration depth of the ac field is much larger than 10 μm. An expression for the SDW conductivity appropriate for the above conditions has already been obtained in the extreme clean (or collisionless) limit characterized by infinite quasiparticle mean free path[17], but for a more complete description the effect of finite mean free path should certainly be taken into account. In this report we incorporate the quasiparticle mean free path in terms of impurity scattering, characterized by a forward (Γ1) and backscattering (Γ2) rate[18]. These two rates stand for the amplitude of scattering processes involving electrons originating from and arriving to the same or different Fermi sheets of the quasi one dimensional Fermi surface respectively. Then we compare the theoretical results with recent experimental data on the SDW of (TMTSF)2PF6 and (TMTSF)2ClO4[19, 20]. The agreement looks rather reasonable as long as we use 2∆(T = 0) ≈ 70cm−1 ≈ 100K for both compound, where ∆(T = 0) is the SDW order parameter at zero temperature. The order parameter thus determined appears to be about a factor two larger than the ones determined from transport measurements, for example ∆(T = 0) = 21K was deduced in Ref.[5]. This discrepancy however may be understood by invoking the effect of imperfect nesting[6] expressed by Eq.(1). On the other hand, the SDW transition temperature Tc can be reduced from its weak coupling BCS value of 28K for at least two reasons. First is the effect of imperfect nesting on Tc which is still within the realm A. VIROSZTEK et al.: FREQUENCY DEPENDENT CONDUCTIVITY IN SPIN DENSITY WAVES 3 of mean field theory, and is suggested in particular by the fact that the same energy gap has been observed in both (TMTSF)2PF6 and (TMTSF)2ClO4. Second is the suppressing effect of fluctuations, although in our case they should be at least two dimensional. Indeed, the observed frequency dependence of the conductivity just above Tc in Bechgaard salts[20] suggests strongly the pseudogap phenomenon. Our calculation of the frequency dependent conductivity involves standard diagrammatic treatment of impurity scattering in density waves[18]. In order to evaluate the quasiparticle contribution we dress a single loop of current correlation function by impurity lines. Self energy corrections are taken into account in the non crossing approximation, while vertex corrections will vanish due to the constant scattering rates Γ1 and Γ2, and to the structure of the transverse current vy(p) = √ 2vy sin(bpy), (2) where vy = √ 2btb is the relevant electron velocity perpendicular to the chains and tb is the hopping integral in the b direction. The thermal product corresponding to the current correlation function is written down, and analytic continuation to the real frequency axis leads directly to σyy(ω). For simplicity, we cite here only the results for perfect nesting, leaving the detailed treatment of imperfect nesting effects to a forthcoming publication. The frequency dependent conductivity of a SDW in the b direction consists of two contributions: Re[σyy(ω)] = e v yN0 Im[In(ω) + Ip(ω)] ω , (3) where N0 is the total density of states (including spin degeneracy) of the electron system, and the ”normal” contribution (involving intraband processes) is given by