Abstract
A charge density wave (CDW) submitted to an electric field displays a strong shear deformation because of pinning at the lateral surfaces of the sample. This CDW transverse pinning was recently observed but has received little attention from a theoretical point of view until now despite important consequences on electrical conductivity properties. Here, we provide a description of this phenomenon by considering a CDW submitted to an external dc electric field and constrained by boundary conditions including both longitudinal pinning due to electrical contacts and transverse surface pinning. A simple formula for the CDW phase is obtained in 3D by using the Green function and image charges method. In addition, an analytical expression of the threshold field dependence on both length and sample cross-section is obtained by considering the phase slip process. We show that the experimental data are well reproduced with this model and that bulk pinning can be neglected. This study shows that the dynamical properties of CDW systems could be mainly driven by boundary effects, despite the comparatively huge sample volumes.
Graphical abstract
Similar content being viewed by others
References
S. Cox, J. Singleton, R.D. McDonald, A. Migliori, P.B. Littlewood, Nat. Mater. 7, 25 (2008)
G. Grüner. Rev. Mod. Phys. 60, 112 (1988)
S. Onishi, M. Jamei, A. Zettl, New J. Phys. 19, 023001 (2017)
D. Le Bolloc’h, V.L.R. Jacques, N. Kirova, J. Dumas, S. Ravy, J. Marcus, F. Livet, Phys. Rev. Lett. 100, 096403 (2008)
V.L.R. Jacques, D. Le Bolloc’h, S. Ravy, J. Dumas, C.V. Colin, C. Mazzoli, Phys. Rev. B 85, 035113 (2012)
A. Rojo-Bravo, V.L.R. Jacques, D. Le Bolloc’h. Phys. Rev. B 94, 201120 (2016)
D. Feinberg, J. Friedel, J. Phys. France 49, 485 (1988)
L.P. Gor’kov, Pis’ma Zh. Eksp. Teor. Fiz. 38, 87 (1983)
L.P. Gor’kov, Sov. Phys. JETP 59, 1057 (1984)
S. Ramakrishna, M.P. Maher, V. Ambegaokar, U. Eckern, Phys. Rev. Lett. 68, 2066 (1992)
J.C. Gill, J. Phys. C 19, 6589 (1986)
J.-M. Duan, Phys. Rev. B 48, 4860 (1993)
K. Maki, Phys. Lett. A 202, 313 (1995)
A.A. Sinchenko, P. Lejay, P. Monceau, Phys. Rev. B 85, 241104 (2012)
N. Ogawa, K, Miyano, Phys. Rev. B 70, 075111 (2004)
V.L.R. Jacques, C. Laulhé, N. Moisan, S. Ravy, D. Le Bolloc’h, Phys. Rev. Lett. 117, 156401 (2016)
M. Prester, Phys. Rev. B 32, 2621 (1985)
P.J. Yetman, J.C. Gill, Solid State Commun. 62, 201 (1987)
G. Mihály, Gy. Hutiray, L. Mihály, Solid State Commun. 48, 203 (1983)
H. Requardt, F.Ya. Nad, P. Monceau, R. Currat, J.E. Lorenzo, S. Brazovskii, N. Kirova, G. Grübel, Ch. Vettier, Phys. Rev. Lett. 80, 5631 (1998)
S. Brazovskii, N. Kirova, H. Requardt, F. Ya. Nad, P. Monceau, R. Currat, J.E. Lorenzo, G. Grübel, Ch. Vettier, Phys. Rev. B 61, 10640 (2000)
S.G. Lemay, M.C. de Lind van Wijngaarden, T.L. Adelman, R.E. Thorne, Phys. Rev. B 57, 12781 (1998)
J. McCarten, D.A. DiCarlo, M.P. Maher, T.L. Adelman, R.E. Thorne, Phys. Rev. B 46, 4456 (1992)
D.V. Borodin, F.Ya. Nad’, Ya.S. Savitskaya, S.V. Zaitsev-Zotov, Physica B + C 143, 73 (1986)
E. Bellec, I. Gonzalez-Vallejo, V.L.R. Jacques, A.A. Sinchenko, A.P. Orlov, P. Monceau, S.J. Leake, D. Le Bolloc’h, Phys. Rev. B 101, 125122 (2020)
I. Batisti’c, A. Bjelis, L.P. Gor’kov, J. Phys. France 45, 1049 (1984)
M. Hayashi, H. Yoshioka, https://arXiv:cond-mat/0010102 (2000)
H. Fukuyama, P.A. Lee, Phys. Rev. B 17, 535 (1978)
G. Gruner,Density Waves in Solids (CRC Press, Boca Raton, 2018)
P.A. Lee, T.M. Rice, Phys. Rev. B 19, 3970 (1979)
K.F. Riley, M.P. Hobson, S.J. Bence,Mathematical Methods for Physics and Engineering: A Comprehensive Guide (Cambridge University Press, Cambridge, 2002)
A. Zettl, G. Grüner, Phys. Rev. B 29, 755 (1984)
J.P. Pouget, B. Hennion, C. Escribe-Filippini, M. Sato, Phys. Rev. B 43, 8421 (1991)
C. Brun, J.C. Girard, Z.Z. Wang, J. Marcus, J. Dumas, C. Schlenker, Phys. Rev. B 72, 235119 (2005)
P. Mallet, K.M. Zimmermann, Ph. Chevalier, J. Marcus, J.Y. Veuillen, J.M. Gomez Rodriguez, Phys. Rev. B 60, 2122 (1999)
G. Gammie, J.S. Hubacek, S.L. Skala, R.T. Brockenbrough, J.R. Tucker, J.W. Lyding, Phys. Rev. B 40, 11965 (1989)
C. Brun, Z.-Z. Wang, P. Monceau, S. Brazovskii, Phys. Rev. Lett. 104, 256403 (2010)
C. Brun, Z.-Z. Wang, P. Monceau, Phys. Rev. B 80, 045423 (2009)
S. Brazovskii, C. Brun, Z.-Z. Wang, P. Monceau, Phys. Rev. Lett. 108, 096801 (2012)
A. Fang, N. Ru, I.R. Fisher, A. Kapitulnik, Phys. Rev. Lett. 99, 046401 (2007)
L. Fu, A.M. Kraft, B. Sharma, M. Singh, P. Walmsley, I.R. Fisher, M.C. Boyer, Phys. Rev. B 94, 205101 (2016)
B. Burk, R.E. Thomson, A. Zettl, J. Clarke, Phys. Rev. Lett. 66, 3040 (1991)
B.M. Murphy, J. Stettner, M. Traving, M. Sprung, I. Grotkopp, M. Müller, C.S. Oglesby, M. Tolan, W. Press, Physica B 336, 103 (2003)
X.-M. Zhu, R. Moret, H. Zabel, I.K. Robinson, E. Vlieg, R.M. Fleming, Phys. Rev. B 42, 8791 (1990)
N. Ru, Charge density wave formation in rare-earth tritellurides, PhD. thesis, Stanford University, 2008
C. Schlenker, Low-dimensional electronic properties of molybdenum bronzes and oxides, inPhysics and Chemistry of Materials with Low-Dimensional Structures (Kluwer Academic Publishers, Berlin, 1989)
E. Bellec, Study of charge density wave materials under current by X-ray diffraction, Ph.D. thesis, Université Paris-Saclay (ComUE), 2019
G. Gammie, J.S. Hubacek, S.L. Skala, R.T. Brockenbrough, J.R. Tucker, J.W. Lyding, Phys. Rev. B 40, 11965 (1989)
C. Brun, J.C. Girard, Z.Z. Wang, J. Marcus, J. Dumas, C. Schlenker, Phys. Rev. B 72, 235119 (2005)
K. Maki, A. Virosztek, Phys. Rev. B 39, 9640 (1989)
S.G. Zybtsev, V.Ya. Pokrovskii, Physica B 460, 34 (2015)
L. Mihaly, G.X. Tessema, Phys. Rev. B 33, 5858 (1986)
L. Mihaly, G. Grüner, Solid State Commun. 50, 807 (1984)
A. Zettl, G. Grüner, Phys. Rev. B 26, 2298 (1982)
R. Danneau, A. Ayari, D. Rideau, H. Requardt, J.E. Lorenzo, L. Ortega, P. Monceau, R. Currat, G. Grübel, Phys. Rev. Lett. 89, 106404 (2002)
M. Peyrard,Physique des solitons (EDP Sciences, Les Ulis, 2012)
F. Pockels, Z. Math. Physik 37, 100 (1892)
M. Lerch, Acta Math. 11, 19 (1887)
A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, inHigher Transcendental Functions (McGraw-Hill Book Company, Inc., New York, Toronto, London, 1953), Vol. I. [Reprinted by R.E. Krieger Publishing Co. Inc., 1981]
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bellec, E., Jacques, V.L.R., Caillaux, J. et al. The essential role of surface pinning in the dynamics of charge density waves submitted to external dc fields. Eur. Phys. J. B 93, 165 (2020). https://doi.org/10.1140/epjb/e2020-10211-6
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2020-10211-6