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CDW-SDW

Charge density wave dynamics

2000–03 Ph.D. at the Centre de Recherche sur les Très Basses Températures (CNRS), University of Joseph Fourier in Grenoble, France

Supervisor: P. Monceau and R. Currat

Referees: T. Giamarchi, T. Klein and S. Ravy

Subject: Dynamics and structure of a charge density wave

This study follows a theme common in many physical systems: how elastic systems behave when interacting with disorder. I have studied how a charge density waves in one-dimensional materials interact with disorder when it is set in motion and how two charge density waves interact with each other. Most of the results of my thesis have been published (see list of publications). Manuscript (in french) available here [15]

Motional ordering of a CDW

In the early thirty’s, Peierls has demonstrated in quasi-one-dimensional metal, that it exists an electron-phonon coupling which makes unstable the metallic state at low temperature [1]. Peierls considers the quasi-one-dimensional metal such as a chain of equally spaced atoms. Under a critical temperature T_c, the most stable state is a distortion of the atomic lattice associated with an electronic density modulation with the same periodicity: this is the CDW state. The electronic modulation has a wave vector Q equal to two time the Fermi wave vector. Like superconductivity, the modulation is caused by an instability of the Fermi surface due to electron-phonon interaction. Dimerization of the atomic position appears, a gap is opened: the increase of the elastic energy due to the nearness of the atoms (proportional to u²) is compensated by the decrease of the electronic energy (proportional to u²lnu). That’s why ground state at low temperature of the one dimensional system is a modulated lattice. Peierls transition is a metal-insulator transition. At rest the CDW is pinned by randomly distributed impurities. Above a threshold electric field, depinning occurs and the CDW slides with respect to the lattice, thus contributing to the electronic charge transport [2]. CDWs are representative of a general class of dynamical systems exhibiting elastic and plastic properties in the presence of quenched disorder (other examples: moving vortex lattices [3], Wigner crystals formed by a 2D electron gas in a magnetic field [4], solid friction interfaces [5], etc.). Current problems concern the nature of the transient and moving steady states. With this aspect, the correlation lengths of NbSe₃ at rest and in motion have been measured, with and without additional RF excitation (in the mode-locked state), using x-ray synchrotron diffraction (measurements done at the European Synchrotron Radiation Facility [16]). The longitudinal correlation length is increased indicating a better CDW order when the CDW moves.

[1] R. E. Peierls, Quantum theory of solids, (Oxford University Press, 1955).

[2] G. Grüner, Rev. Mod. Phys. 60, 1129 (1988).

[3] G. Blatter, M. V. Feigelman, V. B. Geishkenbein, A. I. Larkin, and V. M. Vinokur, Rev. Mod. Phys. 66, 1145 (1994).

[4] E. Y. Andrei, G. Deville, D. C. Glattli, F. I. B. Williams, E. Paris, and B. Etienne, Phys. Rev. Lett. 60, 2765 (1988).

[5] D. Cule and T. Hwa, Phys. Rev. Lett. 77, 278 (1996).

**Summary of these experiments**
Experimental setup: schematic view of the x-ray scattering geometry used to study sliding CDW deformations in NbSe₃. From R. Danneau et al., Phys. Rev. Lett. 89, 106404 (2002) [1].	These measurement show the increasing of the CDW longitudinal (i.e. the CDW direction) coherent length to when it is set in motion. Longitudinal satellite width (symbols) and normalized differential resistance (solid curve) versus applied current normalized to threshold current I_c. The circles refer to increasing current values (I = 0 to I = 7I_c) and the triangles, to decreasing current values (I = 7I_c to I = 0). Statistical error bars are obtained from Gaussian fits of the experimental satellite profiles. The doted line shows the experimental resolution FWHM. From R. Danneau, PhD Thesis, University of Grenoble (2003) [2] and R. Danneau,et al., Phys. Rev. Lett. 89, 106404 (2002) [3].	Similar measurements but with an additional RF excitation: the system is in the mode-locked state, the CDW is more spatially coherent when the wave fronts are forced by the RF signal to move with the same speed. On the graph one can see the variation of the differential resistance with an applied RF field at 5 Mhz (150 mV) and variation of the longitudinal Full Width at Half Maximum of the satellite (circles) as a function of the sample applied DC current at a central position of the sample at T = 120 K. From R. Danneau et al., J. Phys. IV 12, Pr9-177 (2002).

Here, the measurement of the satellite peak was done transversal to the motion: the peak FWHM is increasing when the CDW is depinned, meaning that the phase coherent length is reduced. The graph shows the transverse satellite width versus applied current normalized by threshold current I_c. Statistical error bars are obtained from Gaussian fits of the experimental satellite profiles. The dashed lines are just a guide for the eye. The dotted line shows the experimental resolution FWHM. From R. Danneau, PhD Thesis, University of Grenoble (2003) [4] and R. Danneau,et al., Phys. Rev. Lett. 89, 106404 (2002) [5].

One can explain the fact that the transverse coherent length of the CDW is reduced by the effect of the surface of the crystal on the wave fronts. When the CDW is set in motion the wave fronts are bent in the vicinity of the surface which changes the Bragg diffraction condition: this is directly visible on the satellite diffraction peak. From R. Danneau, PhD Thesis, University of Grenoble (2003) [6].

Dynamical decoupling of two CDW's

NbSe₃ is a quasi-1D material which undergoes two successive Peierls transitions at T_C1 = 145 K and T_C2 = 59 K, with modulation wave vectors Q₁ (0, 0.241, 0) and Q₂ (0.5, 0.260, 0.5), respectively. These two modulation wave vectors nearly satisfy the relation: 2(Q₁ + Q₂)= (1 1 1) which suggests the possibility of a joint commensurability between the lattice and the two CDWs, effect known as a phase-locking and predicted theoretically [1]. However, no experimental evidences where found on Q₁ in the vicinity of T_C2 [2] and no phase-locking or sign of interaction between the two CDWs were observed previously in the static or the sliding regime. In these experiments. an antagonistic behavior of the two satellite peaks was detected at a much higher current value than the measured threshold. By combining transport, noise and X-ray diffraction measurement, this effect was identified as a sliding induced decoupling by charge transfer between the two CDWs. Finally, these results demonstrate that the switching effects [3] measured by transport measurements 30 K below T_C2 are not linked to a possible phase-locking effect.

[1] R. Bruinsma and S. E. Trullinger, Phys. Rev. B 22, 4543 (1980).

[2] A. H. Moudden, J. D. Axe, P. Monceau, and F. Levy, Phys. Rev. Lett. 65, 223 (1990).

[3] Y. Li, D.Y. Noh, J. H. Price, K. L. Ringland, J. D. Brock, S.G. Lemay, K. Cicak, R. E. Thorne, and M. Sutton, Phys. Rev. B 63, 041103(R) (2001).

**Summary of these experiments**
Depinning curves for different temperatures around the second Peierls transition in a NbSe₃: what happens on the threshold currents? From R. Danneau, PhD Thesis, University of Grenoble (2003) [7].	NbSe₃: longitudinal (bottom) Q₁- and (top) Q₂-satellite profiles versus normalized current at T = 45 K. The origin on the vertical scale coincides with the unshifted (zero-current) peak center and the two vertical dotted lines mark the position of the upper threshold current, I_th2. From R. Danneau, PhD Thesis, University of Grenoble (2003) [8] and A. Ayari et al., Phys. Rev. Lett. 93, 106404 (2004) [9].	Similar maping but on another sample at T = 50 K. From R. Danneau, PhD Thesis, University of Grenoble (2003) [10] and A. Ayari et al., J. Phys. IV 131, 125 (2005).

Broad Band Noise measured with a 1 Hz–1 kHz spectral bandwidth and a gain of 1000 (orange curve) and normalized differential resistance (dark grey curve) versus applied current normalized to the threshold current I_th2 at T = 50 K. From R. Danneau, PhD Thesis, University of Grenoble (2003) [11] and A. Ayari et al., Phys. Rev. Lett. 93, 106404 (2004) [12].

Temperature variation of the depinning threshold fields for the Q₁ (E_th1: down triangle from dV/dI, up triangle from BBN) and Q₂ (E_th2: circle from dV/dI) CDWs. From R. Danneau, PhD Thesis, University of Grenoble (2003) [13] and A. Ayari et al., Phys. Rev. Lett. 93, 106404 (2004) [14].

Spin density wave in quasi-1D organic materials by neutron diffraction

During my PhD, I also studied by neutron diffraction the spin density wave transition (SDW) of two quasi-1D organic deuterated compounds d-(TMTSF) ₂ AsF ₆ and d-(TMTSF) ₂ PF ₆ . Despite several experiments, we haven't been able to detect this transition, certainly due to the small sizes of our crystals. Sometimes things don't work as you wish... These experiments were performed in the Institut Laue Langevin [17]. The quest of the SDW peak remains a serious challenge. For more details see Annexe B of my thesis (this part is in english) [18]

Schematic view of the reciprocal lattice region explored. From R. Danneau, PhD Thesis, University of Grenoble (2003).

Related publications:

A. Ayari, R. Danneau, H. Requardt, L. Ortega, J.E. Lorenzo, P. Monceau, R. Currat, S. Brazovskii and G. Grübel

Switching effects and sliding-induced charge transfer between the coexisting Q₁ and Q₂ charge density waves in NbSe₃

J. Phys. IV 131, 125 (2005)

A. Ayari, R. Danneau, H. Requardt, L. Ortega, J.E. Lorenzo, P. Monceau, R. Currat, S. Brazovskii and G. Grübel

Sliding-induced decoupling and charge transfer between the coexisting Q₁ and Q₂ charge density waves in NbSe₃

Phys. Rev. Lett. 93, 106404 (2004) [19]

R. Danneau, A. Ayari, D. Rideau, H. Requardt, J.E. Lorenzo, L. Ortega, P. Monceau, R. Currat and G. Grübel

Reply to Comment on Motional ordering of a charge density wave in the sliding state

Phys. Rev. Lett. 91, 049704 (2003) [20]

R. Danneau, A. Ayari, D. Rideau, H. Requardt, J.E. Lorenzo, L. Ortega, P. Monceau, R. Currat and G. Grübel

Motional ordering of a charge density wave in the sliding state

Phys. Rev. Lett. 89, 106404 (2002) [21]

R. Danneau, A. Ayari, D. Rideau, H. Requardt, J.E. Lorenzo, L. Ortega, P. Monceau, R. Currat, and G. Grübel

Increase of the charge density wave phase coherence in the sliding and the mode-locked state

J. Phys. IV 12, Pr9-177 (2002)

H. Requardt, D. Rideau, R. Danneau, A. Ayari, F. Ya Nad, J.E. Lorenzo, P. Monceau, R. Currat, C. Detlefs, D. Smilgies and G. Grübel

X-ray diffraction study of the transient structure of sliding charge density waves in NbSe₃

J. Phys. IV 12, Pr9-181 (2002)

H. Requardt, J.E. Lorenzo, R. Danneau, R. Currat and P. Monceau

Studies of the dynamics in charge density wave systems using inelastic X-ray scattering with meV-energy-resolution

J. Phys. IV 12, Pr9-39 (2002)

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