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Univerza v Ljubljani Fakulteta za matematiko in fiziko Oddelek za fiziko Seminar 2.2 Charge density waves in low-dimensional metals Andrej Kocan Date: 9. 10. 2007 Mentor: prof. dr. Albert Prodan Abstract In quasi one- and two-dimensional metals the electrons can organize themselves into regular patterns known as charge density waves (CDW). This phenomenon was theoretically predicted by Peierls in 1950s and first observed in 1970s. Since then CDWs were found in numerous inorganic low-dimensional structures, like the transition-metal chalcogenides TaS2, NbSe3 and TaS3. All these structures undergo a phase transition towards a low-temperature modulated ground state, which can be detected by numerous techniques, i.a. X-ray and electron diffraction and scanning tunneling microscopy. Some of these techniques and the corresponding information about the CDW states will be discussed. A short overview of materials exhibiting CDWs and a theoretical description of a simple 1-D metal will be presented. It will be shown that the transition into the CDW state is connected with a periodic structural distortion and a non-linear electrical conductivity, caused by a partial opening of the Fermi surface. Table of Contents Introduction......................................................................................................................... 2 Theory ................................................................................................................................. 2 Free electron Fermi gas................................................................................................... 2 Nearly free electron model.............................................................................................. 3 Origin of the energy gap ................................................................................................. 4 Charge density wave ........................................................................................................... 5 CDW sliding ................................................................................................................... 7 Low dimensional crystals ................................................................................................... 9 Properties of CDWs .......................................................................................................... 10 CDW peculiarities......................................................................................................... 12 CDW FET ..................................................................................................................... 14 Conclusions....................................................................................................................... 15 Literature:.......................................................................................................................... 16 1 Introduction The nearly free electron model in which electrons are described as propagating through a lattice without interacting with the static ion cores is sufficient in describing many electrical transport properties of solids. When metals are cooled, they often undergo a phase transition into a state exhibiting a new type of order. Metals such as iron and nickel become ferromagnetic below temperatures of several hundred degrees Celsius. Other materials such as lead and aluminum, become superconductors at cryogenic temperatures. The interaction between electrons and the lattice vibrations, or electron-phonon interaction, is essential for a complete description of such transport phenomena. Since the mid-1970s, a wide range of quasi-one-dimensional metals have been discovered that undergo a different type of phase transition, both above and below room temperature: they become charge-density-wave (CDW) conductors. These materials show striking nonlinear and anisotropic electrical properties, gigantic dielectric constants, unusual elastic properties and rich dynamic behavior. Several groups of both organic and inorganic materials are known today and some have been investigated in detail by a wide array of experimental techniques.[1] Because density waves arise in their simplest form in highly anisotropic materials some fundamental aspects of the one-dimensional electron gas will be discussed first. Next chapter focuses on materials, i.e. the various groups of so-called linear chain compounds. Finally, a discussion on the basic experimental observations is given. Theory Free electron Fermi gas We can understand many physical properties of metals in terms of the free electron model – the valence electrons of the constituent atoms become conduction electrons and move freely through the volume of the metal. They form a free electron Fermi gas, which is subject to the Pauli principle. Consider a free electron gas in one dimension (1D), taking account of quantum theory and of the Pauli principle. The wavefunction ψ ( x) of the electron is a solution of the Schroedinger equation Hψ = Eψ ; with the neglect of potential energy we have H = p 2 / 2m where p is the momentum. In quantum theory p may be represented by the operator − i d / dx , so that Hψ = − d 2ψ = Eψ , 2m dx 2 2 where E is the energy of the electron. If the electrons are confined to a linear chain of atoms of length L, the boundary condition is ψ (0) = ψ ( L) = 0 . This condition is satisfied 2 by the sine-like wavefunction of the standing wave ψ n ( x) = A sin(kx) = A sin(π n x / L) , where n is a positive integer. It is convenient to introduce wavefunctions that satisfy periodic boundary conditions – we now require the wavefunctions to be periodic in x with period L, thus ψ ( x + L) = ψ ( x) .[2] Wavefunctions satisfying the free-particle Schroedinger equation and the periodicity condition are of the form of a traveling plane wave ψ k ( x) = exp(ikx) , provided that the values of the wavevector k are of the form 2 n π / L , where n is a positive or negative integer, the energy of the orbital with value k being Ek = 2 2m k2 . To calculate the ground state of N electrons confined to the length L, assuming the electrons don’t interact with each other (independent electron approximation), means first finding the energy levels of a single electron and then filling these levels up in a manner consistent with the Pauli exclusion principle, which permits at most one electron to occupy any single electron level. Electron orbitals with discrete values of k can accommodate two electrons with opposite spins. Nearly free electron model The free electron model of metals gives a good insight into the heat capacity, thermal and electrical conductivity, magnetic susceptibility, and electrodynamics of metals. But the model fails in case of other important questions. One of them is of importance for this discussion: the distinction between metals, semiconductors and insulators. To understand the difference between insulators and conductors, we must extend the free electron model and take into account the periodic lattice of the solid with the weak periodic potential of the ion cores. We know that a Bragg reflection is a characteristic feature of the wave propagation through crystals. At Bragg reflections wavelike solutions of the Schroedinger equation do not exist, as shown in fig 1. Figure 1: (a) Plot of energy E versus wavevector k for a free electron. (b) Plot of energy versus wavevector for an electron in a monatomic linear lattice of lattice constant a. The 3 energy gap E g shown is associated with the first Bragg reflection at k = ±π / a ; other gaps are found at higher energies at ± nπ / a , for integral values of n. [3] The only allowed scattering vectors of an X-ray, a neutron or an electron are the reciprocal lattice vectors {G}.[4] The Bragg condition (k+G)2 = k2 for diffraction of a wave of wavevector k, is given in one dimension k = ± 12 G = ± n π a , where G = 2π n / a is a 1D reciprocal lattice vector and n is an integer. The first reflections and the first energy gap occur at k = ±π / a : At these special values of k the time-independent state is represented by standing waves. The region in k-space between −π / a and π / a is called the first Brillouin zone of this lattice. From the two travelling waves exp(±iπ x / a ) we can form two different standing waves: ψ (+) = exp(iπ x / a ) + exp(−iπ x / a ) = 2 cos(π x / a) ψ (−) = exp(iπ x / a) − exp(−iπ x / a) = 2i sin(π x / a) The standing waves are labeled (+) or (-) according to whether or not they change sign when − x is substituted for x . Both standing waves are composed of equal parts of rightand left-directed travelling waves.[2] Origin of the energy gap The two standing waves pile up electrons in different regions, and therefore the two waves have different values of the potential energy in the vicinity of the ions of the lattice. This is the origin of the energy gap. For a pure travelling wave the charge density is constant, while it is not constant for linear combinations of plane waves. For the standing waves we have 2 ρ (+) = ψ (+ ) ∝ cos 2 π x / a 2 ρ (−) = ψ (−) ∝ sin 2 π x / a The first function piles up electrons on the positive ions centered at x = 0, a, 2a,… , where the potential energy is lowest. The other standing wave concentrates electrons away from the ion cores, as is shown in figure 2. Potential energy of ρ (+) is lower than that of the travelling wave, whereas the potential energy of ρ (−) is higher than the travelling wave. We get an energy gap of width Eg . 4 Figure 2: Distribution of probability density in the periodic potential for standing wave 1 - ρ (−) and 2 - ρ (+ ) . The standing wave piles up charges in the region between the ion cores while standing wave 2 piles up charges around the core points. [3] Charge density wave In 1955, Sir Rudolph Peierls first predicted that at absolute zero temperature a structural modulation ( q = 2k F ) corresponding to a distortion of the crystal lattice and conduction electron density results in splitting of the energy spectrum at the Fermi level and in formation of an additional energy gap (Fig. 3).[5] Figure 3: A quasi 1D metal, as represented in a, can reduce its energy by developing a CDW, as shown in b. A CDW consists of coupled modulations of the conduction electron density and the atomic positions. The modulations have wavelength λC = π / k F and produce an energy gap at the Fermi surface k = ± k F . The modulations are usually quite small; atomic displacements are only about 1% of the interatomic spacing, and the conduction electron density varies by several percent.[1] 5 As a result, the CDW state is the ground state of a 1D metal at T = 0 K. At temperatures above absolute zero, some electrons become thermally excited across the gap. This lowers the reduction in energy associated with the CDW formation and eventually quenches the CDW at a critical temperature TCDW . Above that temperature the metallic state is stable. The charge density of the conduction band exhibits a periodic variation with the same wavevector q as the periodic lattice distortion (PLD). For a crystal with a half filled band, the CDW state corresponds to a periodic structure with a lattice constant that is twice as large as the periodicity of the basic structure. In general however, the fillling of the conduction band is not related to the lattice periodicity of the crystal, and k F can be any fraction of the basis vector of the reciprocal lattice. If the new lattice constant a′ is an integer multiple of the unperturbed lattice constant a, the CDW is referred to as commensurate. Contrary, the CDW is incommensurate if the ratio a′/a is an irrational fraction. If a modulated structure is incommensurate, it lacks translational symmetry and can no longer be considered truly periodic.[5] Since an energy gap forms within the former conduction band, a 1D metal would be expected to become insulating below TCDW. The concept of a 1D metal is of course an idealization and real materials exhibiting CDWs are quasi-1D with coupled adjacent metal atom chains and a 3-D character. Thus the ideal planar Fermi surfaces are deformed and a single CDW wavevector q may fail to remove the entire Fermi surface. As a result, the CDW is generally associated with an anomaly in the electrical resistance which remains however metallic, as shown in fig 4. Figure 4: 2D sections of the Fermi surfaces of quasi-one-dimensional crystals. (a) Noninteracting metallic chains parallel to [001] with (k1 , k 2 , 0) describing k vectors perpendicular to the chains. (b) Weakly interacting metallic chains result in a warping of the Fermi surface. Perfect nesting is obtained for q = (0,1 / 2, q3 ) , where q3 fulfills the nesting condition. Notice that the second half of the Fermi surface is nested by q ′ = q − b * .[5] Interchain interactions thus explain the metallic character of the CDW state (as well as non-zero perpendicular components of the modulation wavevectors). These features are 6 illustrated by NbSe3. NbSe3 exhibits two (independent?) CDW phase transitions, each of which removes part of the Fermi surface as expressed by the anomalous increase of the electrical resistivity on cooling down through the phase transitions (fig 5) (Monceau, 1985).[5] Figure 5: Temperature dependence of the electrical resistance (R) of NbSe3, normalized to the value at 300 K. Data obtained on cooling (squares) and heating (circles) coincide. [5] While commensurate modulations do occur in CDW materials, such as pure Nb3Te4, in general a Peierls transition leads to an incommensurate CDW. For the incommensurate case, the phase of the CDW is independent of the lattice. However, the CDW phases can be “pinned” by defects or impurities in the crystal. In some materials, depinning can also occur as a result of an electric field applied to the crystal. If this field is sufficient to overcome the pinning energy, the incommensurate CDW can effectively carry a current.[1] CDW sliding The simplest model that describes the behavior of density waves is called the classical particle model. The CDW is represented by a single massive particle positioned at its center of mass. The behavior of this particle reflects that of the entire array. When there are no external electric fields, in the commensurate case the particle sits on a ribbed surface, like a marble in a cup of an egg tray. The marble is free to move around the bottom of the eggcup and can therefore readjust its position sensitively in response to applied electric fields. The marble usually adjusts its position so as to reduce the electric field acting on it. Thus, materials with CDWs have a large dielectric constant, so large that they could be called superdielectrics. Measurments on CDWs give values for the dielectric constant more than one million times larger than that of ordinary semiconductors.[6] What happens if a DC voltage is applied to an incommensurate CDW, where the state of the crystal is independent of the phase of the modulation? The CDW becomes mobile in the present of the electric field, but the phase of the modulation is pinned by impurities, 7 lattice defects and the surface. Pinning of the CDW can be overcome by sufficiently large electrical fields. For fields larger than a threshold field ( ET ), the sliding CDW provides a second conduction path next to a single-particle electron conduction. Macroscopically this leads to non-linear electrical conductivity for large fields. Deformations of a sliding CDW have been observed by shifts of the positions of satellite reflections (i.e. changes of the magnitudes of q ) dependent on the distance of the area illuminated by the X-ray beam from the electrodes. Figure 6: Charged-particle model illustrating how the current flow in CDW differs from that in normal metals. In a metal (left ) the particle rests on a flat (electrical potential) surface. If we apply a voltage, the surface tilts, and the particle starts to move: there is a current. For a charge-density wave (right ), the surface is ribbed. If the applied voltage is low - that is, the tilt is small - the particle changes position only slightly, and there is no current. If the tilt is large enough for the particle to cross the barrier, the particle runs down the ribbed surface. The resulting current oscillates as the particle climbs over each barrier. Non-linear conductivity due to this sliding CDW is known as Fröhlich conductivity [6,5]. Figure 7: Current versus voltage is plotted schematically for metals and charge-density waves. In a metal (blue) the current increases linearly with voltage. For a charge-density wave material, there is no current until the voltage increases to a critical value; only then does the current start to flow (red). If in addition to a direct voltage an alternating voltage is applied, the curve shows plateaus (purple). The plateaus correspond to a “mode locking” when the flow of the CDW matches the alternating frequency.[6] 8 Low dimensional crystals Ideal 1D crystals do not exist. Materials occupy a finite volume in space, and thus are 3D by definition. However, 1D electron bands can reside in 3D crystals if the atomic orbitals comprising the conduction band show an appreciable overlap in one direction while they have much smaller overlaps in the perpendicular directions. The crystal then is composed of parallel metallic chains, with weak interchain interactions only. The metallic chains can be embedded in a matrix of non-metallic atoms, such that the crystal structure contains chemical bonds in all three directions of space. Consequently, quasi-1D electronic crystals are 3D as far as the phonons are concerned.[5] Figure 8: (a) Despite widely varying crystal morphologies, CDW materials share a common architecture consisting of weakly coupled molecular chains. Single-crystal whiskers of NbSe3, grown by chemical vapor transport.[1] A large number of organic and inorganic solids have crystal structures in which the fundamental structural units form linear chains. While most of these materials are insulators or semiconductors, several groups have partially filled electron bands, and consequently display metallic behavior at high temperatures[7]; among them, three have been explored in detail: • • • Mixed valence platinum chain compounds are composed of a columnar array of units which incorporate a chain of Pt atoms with strongly overlapping d orbitals; most of the experiments have been performed on the material K2Pt(CN)4Br0.3 3.2H2O, known simply as KCP or Krogmann’s salt. It consists of columnar stacked array of Pt(CN)4 units. Transition metal chalcogenides, MX3 and (MX4)nY: Group IV or V transition metals, Nb or Ta, when combined with chalcogen atoms, S, Se and Te, form a variety of linear chain compounds. Transition metal bronzes. The term bronze is applied to a variety of crystalline phases of the transition metal oxides. Examples are the ternary molybdenum oxides of formula A0.3MoO3, where the alkali metal A can be K, Rb or Tl. 9 Figure 9: The crystal structure of KCP (left) , blue bronze (middle) and NbSe3 (right)[1] Two-dimensional CDWs have been found in several classes of compounds; most extensively studied are the transition-metal dichalcogenids MX2 (M=Nb, Ta; X=S, Se), which are layered compounds known for their polytypism. M atoms are at the nodes of a planar hexagonal lattice, and they are sandwiched between planes of X atoms, with either trigonal prismatic coordination of M (T layers) or octahedral coordination (H layers). [8] Figure 10: Example of stacking NbSe2. In the 2H polytype the stacking is ABAB.[9] Properties of CDWs The CDW of the conduction band and the PLD describing displacements of atoms are different aspects of a single phenomenon. Depending on the property that is studied, either the CDW or the PLD is probed. For example, scanning tunneling microscopy (STM) depends on the density of states (DOS) at the Fermi level, and thus probes the CDW directly.[10] The electrical resistance is determined by electron-phonon coupling, and thus depends on both the CDW and the PLD. X-ray diffraction almost exclusively probes the positions of atoms, because it does not depend on the details of the valenceelectron distribution. 10 The periodicity q of the PLD is responsible for the occurrence of additional Bragg reflections in the X-ray diffraction (satellite reflections) at positions ±mq around each Bragg reflection of the underlying basic structure. Satellite reflections may have equal widths as main reflections, indicating true long range order of the CDW. Alternatively, satellites broader than main reflections indicate a limited order of the CDW. Usually, CDW satellite reflections are weak ( 10 −2 −10 −5 × I main ), and they can be observed for values of m equal to 1 or 2 only.[5] Electron diffraction experiments confirmed the existence of CDW in NbSe3 in 1977.[11] The electron diffraction of NbSe3 was performed with the acceleration voltage of the electron beam of 100kV. Figure 11 left shows the b*-c* plane of the reciprocal space above 140K. The anomaly in diffuse scattering can be clearly seen to be one-dimensional (the chain axis is the b axis). In figure 11 right, which shows the a*-b* plane below 140K, the satellite spots are incommensurate with the underlying lattice. The width of the diffuse scattering, which is oriented perpendicular to the 1D chains, usually increases with increasing temperature, indicating a temperature dependent finite correlation length of the fluctuations along the chains.[11] Figure 11: Electron diffraction pattern of NbSe3. Left: above 140 K showing onedimensional anomaly in diffuse scattering (marked by arrows). Right: below 140 K showing the satellite spots (marked by arrows).. [11] More direct observations of the CDWs can be achieved by STM imaging, as shown in figure 12 for the case of NbSe2 at liquid He temperature, i.e. well below the critical temperature ( TCDW ≈ 35 K ).[9] 11 Figure 12: Image on the left shows a 13 nm2 topografic scan. Inset on the right: Fourier transphorm (FT) showing two sets of reflections. The outer peaks correspond to the hexagonal lattice of the atoms and the inner ones to those of the CDW. The CDW is commensurate with the atomic lattice with the wavelength 3× the atomic lattice. [9] CDW peculiarities Although theory predicts that the CDW state is a ground state for a 1-D metal structure, not all linear chain compounds show the modulation. One example are isostrucural compounds Nb3(S4, Se4, Te4). Of these only Nb3Te4 with the largest lattice constants shows CDWs, which run along the zigzag octahedral columns shown in figure 13.[12] Figure 13: The crystal structure of Nb3S4, Nb3Se4, and Nb3Te4.[12] Furthermore, not all CDW materials exhibit nonlinear conductivity behavior ascribed to CDW sliding. A considerable attention was paid in the past to isostructural compunds, especially NbSe3 and m-TaS3, which show nonlinear transport properties. The peculiarity about these two compounds is that they belong to both, the quasi one- as well as quasi two-dimensional families of structures. The 1D character is provided by the trigonal prismatic columns along their b-direction and 2D character by layers in the b-c plane, held together by very weak van der Waals (vdW) forces as shown in figure 14. As was shown in figure 5, the material shows two peaks in the resistance versus temperature curve. This indicates an onset of two independent CDW states running along two of the 12 three slightly different pairs of trigonal-prismatic chains (named type I, II and III, which form the enlarged unit cell). [13] Figure 14: Crystal structure of NbSe3 viewed perpendicular to the b axis of linear chains. [13] NMR and high resolution X-ray diffraction measurements of NbSe3 show that the HTCDW (Tc = 145K) primarily affects type III chains, while the LT-CDW (Tc = 59K) primarily affects type I chains [14, 15]. However, early STM results at liquid helium temperature (1991) suggest that all three types of chains present a strong CDW modulation [16]. It is still unclear whether all chains are CDW modulated and some efforts were put into theoretical studies of the phenomenon. It is believed that highresolution LT STM experiments can clarify the experimental observations. LT-STM pictures of an in-situ cleaved sample of NbSe3 and the corresponding diffraction patterns are shown in figure 15, 16 and 17. Both the underlying molecular lattice and two CDW superlattices were observed simultaneously at 5K with molecular resolution. The CDW wavevectors extracted from 2D fourier transform are in good agreement with bulk results.[17] Figure 15: 50*50 nm2 STM image of (b,c) plane of NbSe3 at T~62K. Vbias=300mV, It=60pA. Both Q1 and Q2 CDWs are seen above T2 the bulk critical temperature for the Q2 CDW, as confirmed in the FT of this image in Fig 16.[18] Figure 16: 2D FT of Fig.1 showing Q1 and Q2 superlattice spots with C* Bragg spot. The intensity of the LT-CDW(Q2) is stronger than that of HTCDW(Q1).[18] 13 Figure 17: Constant current mode topographical image of the (100) plane of NbSe3 at 5K. Arrows 1, 2 and 3 indicate the observed type I, II and III chains. The image was recorded with negative bias voltage (-100mV, 100 pA); the surface vectors 4b and 2c are indicated by two arrows.[17] CDW FET Recently (1995), a novel electronic device, a CDW field-effect-transistor (CDW FET), has been produced.[19] The structure of this device is similar to that of the metal-oxidesemiconductor field-effect transistor (MOSFET), but the channel layer is made of a CDW material. The CDW FET has the potential to become a useful functional device, because its characteristics are different from those of conventional FETs, therefore a short description of such a device follows. The channel layer of this particular CDW FET, fabricated by Ryo Kurita [20], is made of NbSe3 single crystals grown by the direct chemical-gas-transport reaction. A micrograph and a schematic illustration of the CDW FET are shown in Figs. 18 and 19, respectively. During the fabrication process, a cleaved NbSe3 crystal of 0.2 μm thickness was placed on a glass substrate. The source and drain electrodes made of a of gold paste were formed at both ends of the NbSe3 crystal, and a SiO2 insulator of 100nm thickness was deposited by rf sputtering. Finally, the gate electrode of indium was formed on the SiO2 insulator. In these samples, the gate electrode was formed between the source and drain electrodes. Figure 19: Schematic illustration of CDW FET.[20] Figure 18: Top view of CDW FET. The arrow indicates a thin NbSe3 crystal.[20] 14 Figure 20 shows the I DS − VDS characteristics for several gate voltages at 35K for one of the samples, where I DS and VDS are, respectively, the current and the voltage between the drain and source electrodes. The conduction is ohmic below the threshold voltage VT , but I DS increases nonlinearly above VT and is strongly modulated by the gate bias VG . The nonlinear current is reduced by the positive gate bias and is enhanced by the negative one. Figure 20: I DS − VDS characteristics at 35K for several gate voltages VG for a CDW FET sample. The straight line indicates the ohmic current component. The open arrows indicate the steplike structures.[20] Conclusions In this seminar a simple theoretical description of the origin of the CDW ground state was given. Transition towards a modulated structure requires a strong electron-phonon interaction and the nesting condition of the modulation wavevector q fulfilled for a large part of Fermi surface. From the discovery of low dimensional metals exhibiting CDWs in 1970 on, some of these compounds have been thoroughly studied, but adequate theoretic explanation of CDW properties and their dynamic behavior still remains an open problem. Possible future practical (commercial) CDW abusing devices have been proposed. The opportunities these developments provide should ensure that CDW conductors remain interesting enough for many years to come. 15 Literature: [1] Robert E. Thorne: Charge-Density-Wave conductors. Physics Today, May 1996, 4247 [2] Charles Kittel: Introduction to solid state physics, 8th edition, John Wiley & Sons, Inc., 2005 [3] Energy band formation, last updated: 4.19.2002 [cited 9. 10. 2007]: http://www.mtmi.vu.lt/pfk/funkc_dariniai/quant_mech/bands.htm [4] A. W. Overhauser: Charge-density waves and isotropic metals. Adv. in Phys., 27 (3), 1978 [5] Sander van Smaalen: The Peierls transition in low-dimensional electronic crystals. Acta Cryst. (2005). A61, 51-61 [6] Stuart Brown and George Grüner: Charge and spin density waves. Sci. Am. April (5056), 1994 [7] George Grüner: Density waves in solids, Addison-Wesley, 1994 [8] J. A. Wilson, F. J. Di Salvo and S. Mahajan: Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides. Adv. in Phys., 24 (2), 1975 [9] More on charge density waves, last updated: 2005 [cited 9. 10. 2007] Institut des NanoSciences de Paris: http://www.insp.upmc.fr/axe1/Dispositifs%20quantiques/AxeI2_more/CDW/cdw_more. HTM [10] G. Gammie et al: Scanning tunneling microscopy of NbSe3 and orthorombic TaS3. Phys. Rev. B, 40 (14), 1989 [11] K. Tsutsumi, T. Takagaki: Direct electron diffraction evidence of charge densitywave formation in NbSe3. Phys. Rev. Lett. 39 (1977) [12] F. W. Boswell, J. C. Bennett, A. Prodan: Charge density wave transitions in Nb3Se4 and Nb3S4 by indium intercalation. J. Sol. St. Chem., 144, 454-460 (1999) [13] A. Prodan, N. Jug, H. J. P. van Midden, H. Böhm, F. W. Boswell and J. C. Bennet: An alternative model for the structural modulation in NbSe3 and TaS3. Phys. Rev. B 64, 115423 (2001) [14] F. Devreux, J. Phys. (Paris) 43, 1489 (1982) [15] Van. Smaalen , J. L. de Boer,A. Meetsma, H. Graafsma, H.-S. Sheu, A. Darovskikh and P. Coppens, Phys. Rev. B 45, 3103 (1992) [16] Z. Dai, C.G. Slough and R.V. Coleman, Phys. Rev. Lett. 66, 1318, (1991) [17] C. Brun, Z.Z. Wang and P. Monceau: Charge density waves modulation in different chains of NbSe3. J. Phys. IV France 131 (2005) 225-226 [18] Z.Z. Wang: Surface Charge Density wave in low dimensional system measured by LT-UHV-STM. Workshop „Recent developments in low dimensional charge density wave conductors“ Skradin, Croatia June 29.-July 3. 2006. [19] T. L. Adelman, S. V. Zaitsev-Zotov and R. E. Thorne: Phys. Rev. Lett. 74 (1995) 5264. [20] Ryo Kurita: Current modulation of charge-density-wave field-effect transistors with NbSe3 channel. Journal of the Physical Society of Japan Vol 69, No. 8, August, 2000, pp. 2604-2608 16