Sliding Charge Density Wave Induces Electron Transport P. Monceau and M. Renard, Grenoble (Centre de Recherches sur les Très Basses Températures)

Over the past fifteen years, intensive ions. The local electron charge density is experimental and theoretical work has partially neutralized by a concomitant Fig. 1 — Charge density waves on a surface been undertaken to understand the phy­ displacement of each ion to a new equili­ of 1T-TaS2 below the Peierls transition detec­ sical properties of systems with restric­ brium position, the displacement of the ted by a tunnelling microscope. The spacing ted dimensionality. A low-dimensional nth ion, initially at nro, being between the mounds which form a hexa­ conductor has an electrical conductivity un = u0 sin (nqro + Φ) (2) gonal array is the CDW wavelength (from re­ that is strongly anisotropic with respect Since a gap, A, is opened at the Fermi ference 2). to the crystal structure — a specimen level, the CDW state has an energy lower made up of infinite parallel chains for than the metallic state. example, will exhibit a high conductivity CDW formation has also been obser­ Very recently the use of a scanning tun­ only in the chain direction. Interest in ved in two-dimensional layered com­ nelling microscope on a cleaved surface this type of phenomenon has been pounds, namely transition metal dichal- of the two-dimensional 1TTaS2 CDW greatly stimulated by the recently-acqui­ cogenides, the Fermi surface of which compound has revealed that the real red possibilities of synthesizing new approximates to a cylinder with nearly space CDW structure is formed by hex­ families of inorganic as well as organic parallel faces. Thus a large fraction of agonal arrays of mounds with the CDW low-dimensional conductors. In a great states on the Fermi surface are con­ wavelength spacing 2) (Fig. 1). number of these compounds the inter­ nected by the same vector q = 2kF (the The opening of a gap below the Peierls action between ions and electrons, the nesting condition). Nevertheless, the transition temperature is reminiscent of so-called electron-phonon interaction, low-temperature ground state remains semiconductors, but the essential fea­ can cause a modulated collective defor­ metallic. In contrast, for a strictly one­ ture of a CDW is that its wavelength, mation of the electronic charge density dimensional conductor, the Fermi sur­ λcdw = 2π/2kF, is controlled by the to give a lower energy state at low tem­ face consists of two parallel planes, so Fermi surface dimensions and is gene­ perature, as was first pointed out by that all states are connected by the rally unrelated to the undistorted lattice Peierls 1). same q. The energy gap removes the periodicities, i.e. the CDW is incommen­ whole Fermi surface, and the low tempe­ surate with the lattice. Consequently Peierls Transition and Fröhlich Conduc­ rature ground state is insulating. the crystal no longer has a translation tivity In the strictly one-dimensional case group and in contrast to semiconduc­ As is well known from band theory, no long range order can be established tors, the phase, Φ, of the lattice distor­ every Brillouin zone constitutes a locus because of fluctuations and there is no tion is not fixed relative to the lattice but of discontinuity for the electronic ener­ phase transition at any temperature. In is able to slide along q. This phenome­ gy. If, in a one-dimensional electronic practice, however, with pseudo one­ non is easy to understand if we reco­ system with a Fermi vector of kF, a pe­ dimensional conductors, we can iden­ gnize that if the lattice is regular, no posi­ riodic lattice distortion of wave-vector tify a characteristic temperature, called tion is energetically favoured and no 2kF is introduced, the band structure the Peierls transition temperature below locking results. In more theoretical will be modified because of the new which a lattice distortion occurs and the terms: if we think of the CDW as resul­ periodicity. A new Brillouin zone appears condensed state can be described by an ting from an electronic interaction via at | kf | and so, each occupied electronic order parameter. The latter can be defin­ the lattice phonons, this interaction is energy for|k| < kF decreases, giving rise ed either in terms of the electron density the same in every galilean frame, provi­ to a new ground state of the system modulation as pq exp (iΦ) (see equation ded that the frame velocity is small com­ characterized by a charge density wave 1) or in terms of the lattice distortion pared with the sound velocity (in which (CDW) with wave vector q = 2kF. The which is proportional to pq. case the interaction would be strongly occupied electronic states are Bloch Modulation of the ion positions can be modified). CDW condensation may thus wave functions with the superlattice pe­ detected by X-ray, neutron or electron arise in any set of galilean frames with riodicity: diffraction measurements: superlattice uniform velocity, v, giving in the labora­ ψk = exp(ikr) E Vk n exp (i n q r) spots appear near the main Bragg spots tory frame an electronic current density, and consequently the electronic density that correspond to the unmodulated J = - noev (3) has Fourier components with wave-vec­ structure. Measurements of the inverse where no is of the order of the electron tors ± nq, especially for the fundamen­ separation of these superlattice spots number density condensed in the band tal ones ± q: give the CDW wave length. In real space, below the CDW gap. Pel = P0 + 2pq cos (qr + Φ) + ... (1) images of CDWs have been obtained This model of a sliding CDW was pro­ where po is the uniform electron density using high resolution electronic diffrac­ posed by Fröhlich in 1954 3) as a mecha­ and 2pq the charge modulation ampli­ tion. This method is very well suited to nism which could lead to a supercon­ tude. The phase, Φ, specifies the posi­ study defects in the CDW lattice indu­ ducting state. This Fröhlich mode is a tion of the CDW relative to the lattice ced, for instance, by electron irradiation. direct consequence of translation inva- 99 riance. In practice, as shown by Lee, Rice and Anderson 4) this translation inva­ riance is broken because the phase, Φ, can in fact be pinned to the lattice, for example by impurities or by a long- period commensurability between the CDW wavelength and the lattice. Oscil­ lations of the pinned CDW are expected to produce a large low-frequency AC conductivity and a large dielectric cons­ tant. An applied DC electric field, how­ Fig. 2 — Variation of the non­ ever, can supply the CDW with an ener­ linear electrical conductivity gy sufficient to overcome the pinning, so (normalized to the Ohmic value) that above a threshold field, the CDW as a function of the reduced can slide and carry a current. Damping electric field for an orthorhom­ prevents superconductivity. This extra bic TaS3 sample. The insert conductivity associated with the collec­ shows the V(l) characteristics. tive CDW motion, called Fröhlich con­ ductivity, has recently been observed 5). follows: tends to the value σ a + σbb which is of — The DC electrical conductivity in­ the order of the metallic conductivity creases above a threshold field ET. extrapolated from above the critical tem­ Materials — The conductivity is strongly frequen­ perature. ET is also observed to increase Up to now three families of inorganic cy-dependent in the range of 100 MHz - significantly when the crystals are compounds have been found to exhibit few GHz. doped with impurities. non-linear transport properties at tem­ — Above the threshold field, a time- The low-field AC conductivity shows peratures below the Peierls transition, dependent voltage is generated in the a strong increase in the range of 100 namely: the transition metal trichalco- crystal which can be analysed as the MHz-1 GHz and then saturates at a value genides such as NbSe3, NbS3, TaS3 combination of a periodic component close to the DC infinite field limit. This with monoclinic or orthorhombic struc­ and a broad band noise following a 1/f behaviour can be described in terms of a tures; the molybdenum oxides variation. harmonic oscillator response of the K030MoO3 and Rb030MoO3 called blue — Interference effects occur between pinned CDW mode. For NbSe3 and TaS3 bronzes and halogened transition metal the AC voltage generated in the crystal the response is overdamped. Recent tetrachalcogenides such as (NbSe4)2l, in the non-linear state and an external RF measurements in the range of 10-100 (NbSe4)10I3, (TaSe4)2l. Without going field. GHz have revealed an inertial term that into detail, the structure of these com­ — Hysteresis and memory effects are has led to an estimation being made of pounds can be described as comprising : observed, principally at low tempera­ the CDW effective mass, the pinning fre­ chains of trigonal prisms stacked on top ture. quency and the damping constant. of each other with a cross-section close However at low temperatures the single to an isosceles triangle in the case of Fig. 2 shows the typical variation of oscillator description fails and a distribu­ NbSe3, layers in the case of K03MoO3, the electrical conductivity σ (normalized tion of pinning frequencies must be and parallel (TaSe4) chains with iodine to the ohmic value) as a function of elec­ taken into account. atoms lying between them in the case of tric field. The corresponding V(l) charac­ When E is above ET, a time-dependent (TaSe4)2l. Whilst NbSe3 remains metal­ teristic is drawn in the inset; a deviation voltage is generated in the crystal which lic at low temperature, all the other com­ from Ohm's law is observed above a can be studied with a spectrum analy­ pounds exhibit a semiconducting beha­ critical current IT which leads to a thres­ ser. Besides a broadband noise with a 1/f viour below the Peierls transition tempe­ hold field defined as ET = RIT/l where R frequency dependence, Fourier-trans­ rature which, depending on the com­ is the resistance of the sample and l the formed voltage spectra such as that pound, lies between 330 K (for NbS3) distance between voltage contacts. shown in Fig. 3 for NbSe3, reveal a fun­ and 59 K (for NbSe3). The wavelength ET varies typically froma few mV/cm in damental frequency and many har­ of the CDW distortion appears to be in­ NbSe3 to a few tenths of a V/cm in the monics. The fundamental frequency commensurate, but is very often near other compounds; moreover ET increa­ appears at ET and increases with the four lattice distances along the chain ses strongly when T is lowered. A num­ current applied to the sample. direction. A temperature dependence of ber of phenomenological laws have Steps can be observed in the DC V(l) the CDW wavelength has only been been derived to fit the conductivity characteristics if an RF current is super­ detected in orthorhombic TaS3 and blue variation σ(E), amongst which one that posed on a DC current exceeding I/T. bronze with an apparent commensura­ has been widely used is reminiscent of a Such a synchronization effect is expec­ bility setting at low temperature. kind of interband tunnelling process, ted in non-linear oscillators and has in originally envisaged by Zener: fact been observed in Josephson junc­ σ(E) = σa + σb [1-ET/E] exp(-E0/E) (4) tions (Shapiro steps) for frequencies in Properties of the CDW Current-carrying where σa is the ohmic conductivity and the immediate vicinity of the harmonic State E0 = CET with C between 2 and 5. Such or subharmonics of the DC-current Since the first observation in 1976 6) a low activation field E0 precludes a driven fundamental frequency. It is to be of non-linear transport properties when single electron process because the noticed that during synchronization, the a DC or a microwave field was applied to bandgap which can be derived from sample nearly recovers its ohmic DC dif­ NbSe3, the properties of this new cur­ Zener's theory is several orders of ma­ ferential conductivity implying that near rent-carrying state have been widely gnitude smaller than the thermal energy coherent oscillations are occurring over studied; they can be summarized as kBT. When E >> ET the conductivity the whole sample. 100 Because of the strong interaction of the CDW with impurities, it is unlikely that the CDW can be described by a uni­ que ground state. Rather as in glasses, many metastable states have to be taken into account. Deformation of the CDW phase can be induced by a current and by temperature. The time scale for metastable states to relax to lower lying ones can be very broad depending on Fig. 3 — Fourier the material and the temperature; it is transformed voltage found that this decay time increases spectra as a func­ tion of frequency for strongly at low temperature. a NbSe3 sample (T = 42 K) in the non­ Current Models for CDW Transport linear state. Bardeen 7) was the first to interpret the non-linear conductivity in the mate­ rials described above as the Fröhlich root dependence on the impurity con­ measurement is concerned with many conduction induced by the CDW mo­ tent with ET going to zero in the thermo­ domains. If it can be assumed that the tion. The major part of the theoretical dynamic limit. In contrast, some elastici­ domains act independently, then the work has been carried out for the incom­ ty allows for a deformation of the phase, total pinning force will be proportional to mensurate case, considering the extra and a finite second order effect. the sample length and ET will be propor­ conductivity as coming from a collective A very popular model by Lee and Rice tional to the square of the impurity con­ motion of the CDW phase. 8) shows that the phase coherence bet­ centration, and independent of length. It The existence of a threshold field, ET, ween distant points tends to zero with is far from clear what the thermodyna­ creates a problem as it can only be ex­ distance if an arbitrarily small elasticity mic limit of ET means, but since experi­ plained if we allow for a finite static de­ is introduced together with random pin­ ments give finite values for relatively formation of the phase. Physically, a ran­ ning centres owing to the accumulation macroscopic samples, theories that dom distribution of impurities or disloca­ of small phase disturbances. They defi­ consider domain motion have been de­ tions is the most probable cause of the ned a "domain size" such that the phase veloped. pinning force, but if we consider a com­ deviation from the ideal is of the order of A coherent domain is characterized by pletely rigid lattice, summation over indi­ π, but in actual samples, domains are an equivalent mass M, some dissipative vidual pinning forces will give a square only a few microns across and any mechanism (thermalization of the phase Professor in Computational Solid State Physics and leader of the Group "Electronic Structure of Materials” (ESM) The successful candidate will be leader of should actively participate in studies of the a group with a strong tradition in studies of relationship between the electronic the Electronic Structure of Materials structure of solids and their experimental (including band structure calculations) in properties through the calculation of these relation to experimental results. He or she properties. should be capable of continuing the The candidate is expected to fulfill normal extensive - national and international - teaching duties (introductory general cooperation and should interact both with courses in physics and specialized experimentalists and with theoretists in courses in solid state physics) within the related fields. department of Physics - coordinated by the RIM - and the administrative The extraordinary professorship leads to a obligations of a chair-holder and group full-time appointment jointly financed by leader. the Dutch National Science Foundation (FOM) and the Catholic University of Please send Curriculum Vitae including a Nijmegen (KUN). The group ESM which list of publications and names of forms part of the interdisciplinary references to: Research Institute of Materials (RIM) of ms. drs. C.M. van den Heuvel (committee the Science faculty also represents a joint secretary), Algemeen Stafbureau venture of FOM and KUN. Katholieke Universiteit, Toernooiveld, 6525 ED Nijmegen, The Netherlands. The candidate should have a strong and Those wishing to draw attention to a wide background in solid state physics potential candidate are welcome to contact and modern computational methods. He the committee.

101 motion by the phonon bath), a net charge Q, and an effective pinning force which must be periodic in 0, since if 0 is increased by 2n, each impurity sees the same charge distribution in its vicinity. This leads to an equation of motion MΦ + ΓΦ+ FosinΦ = QE (5) This simple equation found in other phy­ sical contexts, e.g. Josephson contacts, dislocation dynamics, leads to at least qualitative explanations of many CDW phenomena: — A threshold field defined by ET = F0/Q — If E = E0 coscot and E0<< ET, lineari­ zation of the sine term leads to a good agreement with the complex conduc­ tivity σ(ω) measured in low fields (li­ near response of an overdamped os­ cillator). Fig. 4 — Variation of the fundamental noise frequency, v, measured in the Fourier-trans­ For a DC field E > ET, the sine term formed voltage as a function of E-ET at different temperatures for an orthorhombic TaS3 gives rise to a velocity modulation at a sample. The Peierls transition temperature is Tp = 215 K. The lines are fits to the expression fundamental frequency, v, the har­ (E-E)Twith γ = 1.5. monics of which can be considered as the origin of the periodic AC voltage the v/JCDW slope is, for any CDW com­ vortex motion and Josephson effects in generated in these systems. It has to be pound, of the order of the electron con­ superconductors have provided useful noted that the assumed periodicity for centration in the bands affected by the insights. the force, connected with the CDW CDW condensation as can be calculated wavelength, λCDW implies that the fun­ from band structures or from chemical Conclusions damental frequency is linked to the bonds. This result is thought to be the Describing one-dimensional systems mean CDW velocity by proof of the Fröhlich conductivity: when was for a long time of interest to theore­ Vcdw =λ cdw v (6) the field overcomes the threshold, the ticians only, but in the 70's inorganic as according to Eq. 3 the extra current den­ electrons, which were trapped below well as organic chemists succeded in sity carried by the CDW in motion is the CDW gap, coherently participate in growing families of compounds relevant therefore given by: the electrical conductivity. to this area of physics. Non-linear pro­ J CDW = n oeVCDW = nOeλCDW V A quantum approach to the same pro­ perties associated with the Peierls tran­ A consequence of the classical equation blem has been put forward by Bardeen sition were also considered a mere of motion (Eq. 5) is that for E slightly 7) and an alternative explanation has curiosity so long as NbSe3 was the only higher than ET, the extra DC current also been suggested: the motion of a material to exhibit such properties. Now, varies as: discommensurate lattice. Discommen- because of the availability of other com­ JCDW ~ (E-ET)1/2 (8) surations are thought of as bearing a pounds, it can be said that this However, experimental results show a charge so that their collective motion behaviour is in fact a general property of nearly 3/2 power law as shown in Fig. 4, will carry a current. pseudo one-dimensional systems as where the variation of the fundamental Another controversial subject is the described above. frequency is drawn for an orthorhombic mechanism ensuring the condensation The sliding of the CDW nearly ex­ TaS3 sample as a function of (E—ET) at of electrons that must take place in the plains every feature at least qualitatively. different temperatures. Attempts have vicinity of electrical contacts supplying However some problems remain to be been made to explain the regime near ET the DC current required to drive the solved. Among them is the increase of by establishing some analogy between CDW motion. Here also analogies with the threshold field at low temperature: the vicinity of ET and the critical beha­ viour at a second order phase transition, leading to the 3/2 exponent. According to Eq. 7 the slope of JCDW/v is a measure of the number of electrons condensed below the CDW gap. The Fig. 5 — Variation of the extra-current JCDW is measured from the current JCDW carried by non linear V(I) characteristics. When v is the CDW as a function of plotted as a function of JCDW, all the the fundamental fre­ curves drawn in Fig. 4 collapse in a uni­ quency measured in the que straight line (except for tempera­ Fourier-transformed vol­ tage for an orthorhombic tures near the Peierls transition tempera­ TaS3 sample at T = 127 ture). Fig. 5 shows the linear relationship K. The slope JCDW/v = between JCDW and v for an orthorhom­ neλCDw leads to the bic sample which is still valid with a number of electrons con­ CDW current density of 30000 A/cm2. densed below the CDW The number of electrons deduced from gap. 102 the only obvious energy scale is the gap so one would expect that the low tempe­ The Pennsylvania State University rature regime would be reached a few degrees below the Peierls transition Experimental Surface Physics temperature, yet as shown in Fig. 4 the The Department of Physics and the Materials Research Laboratory apparent viscosity of the CDW in­ (MRL) are seeking candidates for a tenure-track faculty position in creases strongly as the temperature is Experimental Surface Physics. Special preference will be given to lowered. Further, the mechanism for the individuals with experience in or related to scanning tunnelling CDW damping and its temperature de­ microscopy techniques. Candidates should have a Ph.D. in Physics, pendence remain to be clarified. an established record of research accomplishments and expect to Definitive evidence for the sliding set up a research program to complement the active surface phy­ CDW would be a direct observation of its sics program which exists in the Department. This appointment velocity. Recent NMR measurements 9) will be made jointly between the Department and MRL, allowing show a motional narrowing of the NMR the full utilization of existing technologies in both units and the ap­ lineshape when the CDW is activated, plication of results to important materials problems. A desire and which gives microscopic proof of the aptitude for teaching of undergraduate and graduate students is motion, but further experiments with essential. the help of the tunnelling microscope Send applications, including a curriculum vitae and names of at could be even more convincing. least four references, to Professor Gerald A. Smith, Head, Department of Physics, BIBLIOGRAPHY Box S, The Pennsylvania State University, 1. Peierls R.F., Quantum Theory of Solids University Park, PA 16802 (Oxford University Press) 1955, p. 108. by October 15, 1986, or until a suitable pool of applicants is iden­ 2. Coleman R.V., Drake B., Hansma P.K. and tified . An affirmative action/equal opportunity employer. Slough G., Phys. Rev. Lett. 55 (1985) 394. 3. Fröhlich H., Proc. Royal Soc. A 223 (1954) 296. 4. Lee P.A., Rice T.M. and Anderson P.W., Computational Scientists Solid State Commun. 14(1974) 703. Solid State Physics and Surface Science 5. For a review see Proceedings of the Inter­ national Conference on Charge Density The Theory and Computational Science Division of the Waves in Solids, held in Budapest in August SERC's Daresbury Laboratory has two vacancies for 84, Lecture Notes in Physics Vol. 217, eds. Gy Hutiray and J. Solyom (Springer-Verlag, work in computational solid state and surface physics. Berlin) 1985; Electronic Properties of Inor­ ganic Quasi One-Dimensional Compounds The Laboratory is located in rural North Cheshire, and provides major facilities for Parts I and II, ed. P. Monceau (D. Reidel, Dor­ scientific research by university groups. The present experimental facilities are cen­ drecht) 1985; Crystal Chemistry and Proper­ tred on a 2 GeV Synchrotron Radiation Source, and a 20 MV van de Graaff accelera­ ties of Materials with Quasi One-Dimensio­ tor. Computing facilities include an AS-7000 and FPS-164 attached processor on site, nal Structures, ed. J. Rouxel (Reidel, Dor­ with access to the Cray 1-S in London, the CDC Cyber-205 in Manchester and a Cray drecht) 1986. X-MP at the Atlas Centre from early 1987. 6. Monceau P., Ong N.P., Portis A., Meer- These posts are to support the development of theoretical and computational schaut A. and Rouxel J., Phys. Rev. Lett. 37 (1976) 602. methods for the Collaborative Computational Projects on Band Structure Theory 7. Bardeen J., Phys. Rev. Lett. 42 (1979) (CCP9) and Surface Science (CCP3). These projects involve large and active groups 1498 and 45 (1980) 1978. of university collaborators with whom the successful candidates will interact. The 8. Lee P.A. and Rice T.M., Phys. Rev. B 19 major work of CCP9 over the next few years will be on highly accurate self-consistent (1979) 3970. methods for calculating the electronic structure of solids; the interests of CCP3 will 9. Ross J.H. Jr., Wang Z. and Slichter C.P., centre on electronic properties of surfaces and atom-surface interactions. There is Phys. Rev. Lett. 56 (1986) 663. other work in progress in the Theory and Computational Science Division on solid — Segransan P., Janossy A., Berthier C., state and surface physics, atomic and molecular physics, quantum chemistry, Marcus J. and Butaud P, to be published. molecular dynamics and computer simulation of solids. The successful candidates will have recently obtained a Ph.D. or expect to obtain one Meetings Issue before taking up the appointment, preferably in theoretical or computational con­ densed matter physics. An appointment will be made at a salary (under review) bet­ The October issue of Europhysics ween £7701 and £11781 according to age, ability and experience. The post will be News will comprise a calendar of mee­ available for a fixed term of three years and will be superannuable. There is flexibility tings and schools of interests to phy­ in the starting date. sicists in Europe starting from January 1987. To guard against errors, orga­ CLOSING DATE: 15 September 1986 nisers are requested to send their no­ Further information may be obtained from tices to the EPS Secretariat by 19 Sept. Dr. J.E. Inglesfield (Tel. Warrington (925) 60 31 21); 1986 at the latest. All entries are inclu­ Dr. B.L. Gyorffy for CCP9 (University of Bristol, (272) 30 30 30 Ext. 3677) or ded free of charge. Professor J.B. Pendry FRS for CCP3 (Imperial College London, (1) 589 51 11 Ext. Organisers who believe that their 6901). event is eligible for EPS sponsorship Application forms may be obtained from: should ask for an application form from The Personnel Officer, Daresbury Laboratory, Warrington WA4 4AD, Cheshire. the Secretariat and return it completed Ref: DL/963: CCP3; DL/964: CCP9 Tel. (925) 60 34 67 as soon as possible. 103