- Email: [email protected]
Numerical based study on charge density wave dynamics in a one-dimensional conductor in the low temperature limit
Accepted Manuscript Numerical based study on charge density wave dynamics in a one-dimensional conductor in the low temperature limit R. Essajai, O. Idrissi, A. Mortadi, N. Hassanain, A. Mzerd PII: DOI: Reference:
S2211-3797(18)32128-4 https://doi.org/10.1016/j.rinp.2019.01.086 RINP 2052
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
9 September 2018 25 January 2019 30 January 2019
Please cite this article as: Essajai, R., Idrissi, O., Mortadi, A., Hassanain, N., Mzerd, A., Numerical based study on charge density wave dynamics in a one-dimensional conductor in the low temperature limit, Results in Physics (2019), doi: https://doi.org/10.1016/j.rinp.2019.01.086
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Numerical based study on charge density wave dynamics in a onedimensional conductor in the low temperature limit.
a
b
Group of Semiconductors and Environmental Sensor Technologies- Energy Research Center , Faculty of Science, Mohammed V University, B. P. 1014, Rabat, Morocco.
Laboratory Physics of Condensed Matter (LPMC), Faculty of Sciences, Chouaib Doukkali University, B.P. 20, 2400, El-Jadida, Morocco.
*Corresponding author: E-mail address: [email protected] (Rida Essajai)
___________________________________________________________________________
Abstract Understanding how the temperature affects in a wide range of quasi-one-dimensional conductors is one of the most significant scientific challenges in solid state physics. In this paper, the temperature-dependent charge density wave (CDW) dynamic properties of a onedimensional conductor in the weak pinning limit was studied within the temperature limit of 0 to 1K. To do so, a model which incorporates the essential feature of the Fukuyama–Lee–Rice (FLR) model of CDW’s has been considered The simulation results have shown that the threshold field and nonlinear excess current exhibit non-monotonous behavior as a function of the temperature. The mechanisms leading to this finding is interpreted in terms of several phenomena like thermal softening, damping mechanisms of the collective mode as well as the generation of the CDW dislocation. Besides, it was found that the CDW phase profile variations are more pronounced when the temperature increases. This finding was discussed in the context of thermal softening behavior.
Keywords:
Charge density waves, Thermal fluctuations, Non-linear transport, Fukuyama-
Lee-Rice model, Numerical simulation.
1- Introduction
In the low-dimensional conductors, the electronic crystal of the charge density wave (CDW) is the only one of its kind which is originated from some electron-phonon coupling, leading to instabilities of the Peierls type [1]. At low electric field, the collective CDW mode is pinned by randomly distributed impurities in the underlying network. When an applied electric field exceeds a well defined threshold value, the CDW’s overcome the pinning forces, leading to an onset of nonlinear electrical conduction, resulting from the sliding of the CDW condensates. The onset of non-linearity is also accompanied by some unusual physical properties which are not yet fully understood, such as broad-band noise [2], phase-slip phenomenon [3,4], and nucleation of CDW-phase dislocations [5]. The temperature dependence of CDW dynamics has been investigated by several organizations around the world. The obtained results from some studies showed that the temperature strongly affects the CDW dynamics [6,7]. Besides, others reports indicated that the non-Ohmic and Ohmic conductivity in a given temperature range exhibit the same activated temperature dependences [8,9]. Experimentally [10,11,12], the thermal variation of both the non-Ohmic conductivity and threshold field ET show a non-monotonous variation. This latter depends on assumptions according to its origin, such as the nucleation of CDWphase dislocations. Artemenko et al. [13] found that the nonlinear current density is nonmonotonous function of the temperature and they showed that the change of JCDW may occur near 100 K. However, theoretically, the temperature effect leads to an exponential decrease of the threshold field of the onset of nonlinear conductivity as predicted by Kazumi Maki [14]. In real compounds, the CDW-phase dislocations in the CDW system were viewed as plastic defects [15]. In the past decade, they have been directly observed by X-ray diffraction
methods close to sample contacts [16] or in the bulk [17]. At low temperature, these intrinsic defects may be nucleated and moved when external forces are applied to the crystal (high electric fields, mechanical stress or thermal gradient), thus contribute to the CDW conductivity [18,19]. Increasing these forces should induce sliding of these CDW defects, controlled by the existence of pinning centers [20]. A model of the macroscopic proprieties of CDW compounds was given some time ago [21,22]. The ground state of the CDW is characterized by a Periodic Lattice Distortion (PLD) coupled to a periodically modulated charge density and CDW,
; where
are respectively the concentration of uncondensed electrons and the amplitude of the is the position and
is the CDW phase. At zero temperature, impurities induce
fluctuations in the CDW phase, while the CDW amplitude can be ignored. In this context, a Fukuyama-Lee-Rice (FLR) model of an incommensurate CDW
has been proposed to
describe the macroscopic CDW condensate as a classical deformable medium interacting with randomly distributed impurities and under of diving electric field. Moreover, this model has been successful in explaining experimental results related to the sliding of the CDW both qualitatively and quantitatively, such as the threshold field for non-Ohmic conductivity above a threshold field, mode locking, existence of metastable states [23–27], etc. However, it is difficult to obtain experimental results for the temperature dependent CDW dynamics based on FLR model because all the parameters of this model are strongly temperature dependent [28], especially, the CDW amplitude. Despite this limitation, the finite temperature effect on the CDW dynamics by means of FLR model can be considered with neglecting fluctuations in the CDW amplitude which is legitimated only at finite temperatures, as it was signaled by Ref [29]. However, to our knowledge, investigating the new phenomena that take place at dynamic properties of low-temperature limit in the one-dimensional incommensurate system based on FLR model still needs intensive research in order to fill this void in the existing
literature. In this letter, the numerical simulation of the effect of finite thermal fluctuations, represented by the Gaussian random force [29-33] on the dynamics of a one-dimensional incommensurate charge density waves system in the weak pinning limit has been reported. This will be done within the context of model which incorporates the essential features of the FLR model of CDW’s. This work is organized as follows: Section 2 presents a short description of the computational method and the calculation procedure. Our results and discussions are exposed in Section 3. Finally, the important conclusions arising from this work are summarized in Section 4.
2- Model for numerical simulation At zero temperature, the dynamics of CDW phase fluctuations given by the following overdamped equation of motion, as [23].
H d ( x) ( x) dt
where is the phenomenological damping constant and
(1)
is the FLR Hamiltonian for the
phase pinning model of one dimensional incommensurate CDW. For more details on this Hamiltonian see Refs [23,27,30,34,35]. The expression of FLR Hamiltonian is given by the following formula: k d c E H dx ( x, t ) dx ( x, t )Vi ( x ri ) dx 2 dx i 2
(2)
It is well known that the nonlinearity of the problem which comes from CDW-impurities interaction makes it difficult to handle Eq.1 using an analytical method [27]. In this limit, it is recommended to use the numerical method which may provide an analytical tool in order to overcome complicity of the motion equation of the CDW’s.
Before using the numerical simulation, we make some approximations to simplify the problems associated with the incommensurate CDW dynamics along the direction of the chain. Firstly, the damping force is assumed to act only at the impurities positions along the considered chain [35]. Secondly, we have chosen a short-range interaction between the phases of CDW and impurity potential
located at site impurity , as following
[36,37] where, the index runs over the impurities whose concentration is and
is the pinning potential considered identical for all the impurities.
With the power of two approximations listed above, we rescale the unit of time by the factor
and the length by . In these dimensionless units, Eq.1 reduces to [23,27,30,36]: d ( x, t ) ( x, t ) sin(i ( x, t )) ( x xi ) dt
where
,
. The parameters
(3)
and
are respectively
the dimensionless electric field along the chain direction and the dimensionless interaction strength. Finally, we may now integrate between the impurities sites along longitudinal chain and, then, Eq.3 is discretized using of a standard finite-difference method on a 1-D grid, obtaining
1 2
i (t 1) i (t ) t i (t ) ( xi1 xi1 ) sin(i i (t ))
where δt is the time step size and
. The parameter
(4)
is the discrete
second derivative (It is defined in Refs [27,30]). In non-zero temperature effect, the dynamics properties of CDW system in the weak pinning limit may be specified by the following equation.
1 i (t 1) i (t ) t i (t ) ( xi1 xi1 ) sin(i i (t )) i(t ) 2
(5)
where
is an expression which represents the thermal fluctuations [29-33].
i
(t )
(t ) 0 and (t ') 2 K BT i ,i ' (t t ') i' i
(6)
K B is the Boltzmann constant and K BT is the thermal energy.
In order to ignore the random problem, we must first dedicates a program to generate the initial distribution of our system, where we choose the initial values of the phase randomly taken in the interval
and the sample of length
containing randomly
distributed impurities, except where noted. From the previous configuration chosen, the numerical algorithm of solution of nonlinear Eq.5 is carried out using a periodic boundary condition; the applied electric field and temperature are zero to relax our system in order to reach a static configuration equilibrium characterized by a given relaxation time where all the local velocities are near zero. Then, for a given applied electric field along the direction chain, we assume that the system responds instantaneously between the impurities sites; this neglects the dynamic degree of freedom on short time scales.
3- Results and discussions. A variety of transport phenomena observed in CDW’s materials has been attributed to the inhomogeneous plastic response of the CDW, when set into motion by an external drive. Variations of the CDW local phase are observed for an applied continuous electric field above the threshold field
, as shown in Fig.1. It was found that variations of the
CDW phase are very sensitive to the temperature. Their amplitudes increase when the temperature increases. This behavior can be related to the change of the elastic properties of CDW due to the thermal softening behavior, in other words, the increase in temperature
leads to reduce the CDW stiffness parameter and, hence, enhancing the non-homogeneous response of the CDW. A similar behaviour has been previously obtained by N. Habiballah et al [33]. They have attributed the phenomenon to the fact that the generation of structural defects such as CDW dislocations may be make the system less rigid and more affected by the external field.
3,85
c ppm
=0K
a,u
=0,025K
=0,1K
j
3,50
T
3,15
=0,2K
( ) ( )
2,80 2,45 2,10 1,75 1,40 1,05 0,70 0,35 20
40
60
80
100
120
140
160
180
Position [a,u]
Fig.1 CDW phase profile for various values of temperatures at
and
.
Understanding how the temperature affects in a wide range of quasi-one-dimensional conductors is one of the most significant scientific challenges in solid state physics. So, the temperature effect on the collective CDW dynamics is explored using a steady-state timeaveraged velocity (
) of the CDW (excess current), which is described by the following
equation [38]:
CDW (t ) t where,
is a time dependence of the spatially averaged velocity (Eq.8) [27,36,38]:
(7)
(t )
where
1
N
N imp d (t )
i
imp
(8)
i
dt
is the impurity number.
For a one-dimensional incommensurate CDW system without temperature effect, the steadystate time-averaged velocity of the CDW, varies depending on the electric field as follows
CDW ( ) 0 (
1) T
(9)
is the dynamic critical exponent. The thermal fluctuations, does not alter the behavior of field-dependent current density, the change in
as a function of the applied electric field is governed by the same power law
Eq.9. Furthermore, for a given electric field above the threshold one, it was observed that the variation of
is non-monotonous under the temperature effect (Fig.2), characterized by
the existence of two crossover temperature
around
and
, leading us to
distinguish three temperature regions (Fig.3). Such behavior has been experimentally observed in the K0.3MoO3 compound [14]. For T ranging from
to
,
increases when increasing the temperature, suggesting that the CDW motion exhibits a very fast dynamic under temperature variations. This can be attributed to a decrease in the rigidity of the CDW as the temperature is increased. For T ranging from
to
,
decreases when increasing the temperature. This is a direct consequence of the fact that the collective motion of CDW is damped; resulting from the increase in the carrier concentration with increasing temperature, as it was theoretically showed by L.Sneddon [39].
0,018
T=0K T=0,2K T=0,5K
0,015
T=0,6K Fit with Eq:9 =0,215[a,u]
0,012
c ppm j
CDW
0,009
0,006
T 0,003
0,00
0,01
0,02
0,03
0,04
0,05
0,06
a,u
Fig.2 Nonlinear excess currant vs applied electric field for various temperatures.
For
,
increase for
shows a minimum more or less pronounced around
and
, which suggests that the CDW motion displays a very fast dynamic
when temperature increases. This behavior can be related to the fact that the variation with temperature in the structure (edge or screw) or mobility of CDW-phase dislocations generated through the electronic crystal can occur around
, which in turn induces a change in
. A similar behavior has been theoretically and numerically predicted by several
authors [13]. The presence of CDW-phase dislocations tends to make the CDW less rigid [40], causing plastic deformations [41]. Recent result obtained from a numerical simulation for phase dislocations effect on the CDW dynamics has shown that the multiplication of dislocations contribute to the CDW collective transport, where just above threshold field, the current density increases when the dislocations number increases [42].
0,16
=1,25
c =200ppm
T
0,14
=1,5
T
=2
j
=0,215[a,u]
T
0,12
T
cs2
=0,55K
CDW / 0
0,10
0,08
0,06
0,04
T
0,02
0,00
0,07
0,14
cs1
0,21
=0,25K
0,28
0,35
0,42
0,49
0,56
0,63
T(K)
Fig.3. Nonlinear-excess-currant as a function of temperature for various applied electric field.
Fig.4 shows the temperature dependence of the threshold field. It was established that the threshold field
is not a monotonic function of the temperature, where decreases when
the temperature increase from 0K to∼0.25K then shows a rise in the temperature interval and finally decreases as T is away from observed in various CDW materials [10,12].
. The same behavior was
c =200ppm j
0,0035
=0,215[a,u]
0,0030
T [a,u]
0,0025
T
cs1
=0,25K
T
cs2
=0,55K
0,0020
0,0015
0,0010 0,00
0,07
0,14
0,21
0,28
0,35
0,42
0,49
0,56
T(K)
Fig.4 Temperature dependence of threshold field ET.
The decreases of the threshold field from
to
can be attributed to the fact
that CDW is softened. This result is qualitatively in agreement with both the theoretical work reported in [14] and numerical simulations by Miyashita and Takayama [39]. Then, the behaviour of shows a minimum less or more pronounced around the temperature increases from
to
and increase when
(Fig.4). This is the direct consequence of the
damping phenomenon, originating from the CDW interacting more strongly with the normal carriers when the temperature is increased. Finally, managed to access a broad maximum which appears near
and decrease for
. It can be suggested that a change in
with temperature may be generated by the onset of the CDW-phase dislocations. Among
the numerous results signalled in Ref [42], the most important is that the threshold field decreases when increasing the dislocation number. This last has been explained in terms of an increase in the number of CDW dislocations leading to a softening of CDW.
4- Conclusion Within the context of a model which incorporates the essential features of the FLR elastic model, the temperature effect on a one-dimensional incommensurate CDW dynamics was studied. With a small change in the low-temperature limit, the non-linear excess current for a given electric field above the threshold one shows the existence of temperature crossovers around 0.25K and 0.55K, leading us to distinguish three observed behavior of
versus
temperature. Besides, the temperature dependence of the threshold field is found nonmonotonous, shows a minimum near 0.25 K and a maximum near 0.55K. Such behavior has been obtained in the preceding experimental studies. The temperature affects drastically the local dynamical properties of the CDW: the CDW phase profile variations are more pronounced when the temperature increases, indicating that the CDW is deformed and more affected by the applied electrical field. This may be related to the change of the elastic properties of CDW due to the thermal softening behavior.
References
[1] R.E. Peierls, Theory of Solids, Oxford University Press, London, 1955. [2] G. Gruner, Rev. Mod. Phys. 60, 1129 (1988). [3] M. Qjani, A. Arbaoui, A. Ayadi, Y. Boughaleb, J. Dumas, Ann.Chim. Sci., Mater. V 23 (1998) p. 237-240. [4]
N. Habiballah, A.Arbaouiz, M.Qjani , Rida Essajai,13th International Conference on
Frontiers of Polymers and Advanced Materials Marrakesh, Morocco 29 March - 02 April, 2015. [5] D. LeBolloc’h, S. Ravy, J. Dumas, J. Marcus, F. Livet, C. Detlefs, F. Yakhou, L.Paolasini, Phys. Rev. Lett. 95 (2005) 116401. [6] G. Mihaly, J. Marcus, J. Dumas, C. Schlenker, Phys. Rev. B 37 (1988) 1047. [7] J. Dumas, C. Schlenker, Int. J. Mod. Phys. 7 (1993) 4045. [8] R.M. Fleming, R.J. Cava, L.F. Schneemeyer, E.A. Rietman, R.G. Dunn, Phys. Rev. B 33 (1986) 5450. [9] A. Janossy, C. Berthier, P. Segransan, P. Butaud, Phys. Rev. Lett. 59 (1987) 2348. [10] J. Dumas, C. Schlenker, J. Marcus, R. Buder, Phys. Rev. Lett. 50 (1983) 757. [11] J. Dumas, J. Marcus, Phys. Lett. A 373 (2009) 4189. [12] D. Moses and R. M. Boysel , Phys. Rev. B 31 (1985) 3202. [13] S.N. Artemenko, F. Gleisberg, W. Wonneberger, J. Phys. IV 9 (1999). [14] K. Maki, Phys. Rev. B 33 (1986) 2852. [15] P.A. Lee, T.M. Rice, Physical Review B 19 (1979). [16] S. Brazovskii, N. Kirova, H. Requardt, F. Nad, Europhysics Letters 56 (2001) 289. [17] (a) D. LeBolloc’h, S. Ravy, J. Dumas, J. Marcus, F. Livet, C. Detlefs, F. Yakhou, L.Paolosini, Physical Review Letters 95 (2005) 116401;
(c) D. LeBolloc’h, V.L.R. Jacques, N. Kirova, J. Dumas, S. Ravy, J. Marcus, F. Livet, Physical Review Letters 100 (2008) 096403; (b) V.L.R. Jacques, D. LeBolloc’h, S. Ravy, J. Dumas, C.V. Colin, C. Mazzoli, Physical Review B 85 (2012) 035113. [18] (a) D. Feinberg, J. Friedel, Journal de Physique (France) 49 (1988) 485; (b) J. Dumas, D. Feinberg, Europhysics Letters 2 (1986) 555. [19] T. Csiba, G. Kriza, A. Janossy, Europhysics Letters 2 (1989) 555. [20] C. Schlenker, J. Dumas, C. Escribe-Filippini, M. Boujida, Physica Scripta. Vol. T29, 55-61, 1989 [21] H. Fukuyama and P. A. Lee, Phys. Rev. 8 17, 535 (1977). [22] P. A. Lee and T. M. Rice, Phys. Rev. 8 19, 3970 (1979) [23] P.B. Littlewood, Phys. Rev. B 33 (1986) 6694. [24] C.R. Myers, J.P. Sethna, Phys. Rev. B 47 (1993) 11171. [25] C.R. Myers, J.P. Sethna, Phys. Rev. B 47 (1993) 11194. [26] M. Qjani, A. Arbaoui, A. Ayadi, Y. Boughaleb, J. Dumas, Ann. Chim. Sci., Mater. V 23 (1998) 435. [27] M. Qjani, A. Arbaoui, A. Ayadi, Y. Boughaleb, J. Dumas, Synth. Met. 96 (2) (1998) 137. [28] Kazumi Maki and Attila Uirosztek , Phys. Rev. B 39 (1989) 9640. [29] T. Miyashita, H Takayama , Jpn. J. Appl. Phys. 26 (Suppl. 26-3) (1987), 603. [30] N. Habiballah, A. Arbaoui, M. Qjani, J. Duma, Synthetic Metals 182 (2013) 56– 59. [31] A.A Middleton , Phys Rev,B 45, (1992) 9465 [32] D.S. Fisher, Phys. Rev. B. 31 (1985) 1396. [33] N. Habiballah, M.Qjani, A.Arbaoui, J.Dumas. Journal of Physics and Chemistry of Solids 01 (2014); 75:153-156.
[34] N. Teranishi, R. Kubo, J. Phys. Soc. Japan 47 (1979) 720. [35] L. Pietronero, S. Strassler, Phys. Rev. B 28 (1983) 5863. [36] M. Qjani, A. Arbaoui, Y. Boughaleb, J. Dumas, Physics Letters A 348 (2005) 71–76. [37] N.Habiballah, M.Zouadi, A.Arbaoui, M.Qjani,J.Dumas. Results in Physics, 7 (2017), 3277. [38] S. N. Coppersmith and Daniel S. Fisher’, Phys. Rev. A 38 (1988) 6338. [39] L. Sneddon, Phys. Rev. B 29 (1984) 719. [40] M. Hayashi, Physica B 324 (2002) 82. [41] F.R.N. Nabarro, Theory of Crystal Dislocations, Oxford University Press, London, 1967. [42] N. Habiballah, M. Qjani, A. Arbaoui, J. Dumas, Synth. Metals 164 (2013) 52.
Highlights
o
The temperature plays a crucial role on the CDW dynamic properties.
o
The current density and the threshold field of CDW are non-monotonous
when the temperature increases. o
The FLR model is not sufficient to explain all results observed
experimentally in real systems.
S2211-3797(18)32128-4 https://doi.org/10.1016/j.rinp.2019.01.086 RINP 2052
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
9 September 2018 25 January 2019 30 January 2019
Please cite this article as: Essajai, R., Idrissi, O., Mortadi, A., Hassanain, N., Mzerd, A., Numerical based study on charge density wave dynamics in a one-dimensional conductor in the low temperature limit, Results in Physics (2019), doi: https://doi.org/10.1016/j.rinp.2019.01.086
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Numerical based study on charge density wave dynamics in a onedimensional conductor in the low temperature limit.
a
b
Group of Semiconductors and Environmental Sensor Technologies- Energy Research Center , Faculty of Science, Mohammed V University, B. P. 1014, Rabat, Morocco.
Laboratory Physics of Condensed Matter (LPMC), Faculty of Sciences, Chouaib Doukkali University, B.P. 20, 2400, El-Jadida, Morocco.
*Corresponding author: E-mail address: [email protected] (Rida Essajai)
___________________________________________________________________________
Abstract Understanding how the temperature affects in a wide range of quasi-one-dimensional conductors is one of the most significant scientific challenges in solid state physics. In this paper, the temperature-dependent charge density wave (CDW) dynamic properties of a onedimensional conductor in the weak pinning limit was studied within the temperature limit of 0 to 1K. To do so, a model which incorporates the essential feature of the Fukuyama–Lee–Rice (FLR) model of CDW’s has been considered The simulation results have shown that the threshold field and nonlinear excess current exhibit non-monotonous behavior as a function of the temperature. The mechanisms leading to this finding is interpreted in terms of several phenomena like thermal softening, damping mechanisms of the collective mode as well as the generation of the CDW dislocation. Besides, it was found that the CDW phase profile variations are more pronounced when the temperature increases. This finding was discussed in the context of thermal softening behavior.
Keywords:
Charge density waves, Thermal fluctuations, Non-linear transport, Fukuyama-
Lee-Rice model, Numerical simulation.
1- Introduction
In the low-dimensional conductors, the electronic crystal of the charge density wave (CDW) is the only one of its kind which is originated from some electron-phonon coupling, leading to instabilities of the Peierls type [1]. At low electric field, the collective CDW mode is pinned by randomly distributed impurities in the underlying network. When an applied electric field exceeds a well defined threshold value, the CDW’s overcome the pinning forces, leading to an onset of nonlinear electrical conduction, resulting from the sliding of the CDW condensates. The onset of non-linearity is also accompanied by some unusual physical properties which are not yet fully understood, such as broad-band noise [2], phase-slip phenomenon [3,4], and nucleation of CDW-phase dislocations [5]. The temperature dependence of CDW dynamics has been investigated by several organizations around the world. The obtained results from some studies showed that the temperature strongly affects the CDW dynamics [6,7]. Besides, others reports indicated that the non-Ohmic and Ohmic conductivity in a given temperature range exhibit the same activated temperature dependences [8,9]. Experimentally [10,11,12], the thermal variation of both the non-Ohmic conductivity and threshold field ET show a non-monotonous variation. This latter depends on assumptions according to its origin, such as the nucleation of CDWphase dislocations. Artemenko et al. [13] found that the nonlinear current density is nonmonotonous function of the temperature and they showed that the change of JCDW may occur near 100 K. However, theoretically, the temperature effect leads to an exponential decrease of the threshold field of the onset of nonlinear conductivity as predicted by Kazumi Maki [14]. In real compounds, the CDW-phase dislocations in the CDW system were viewed as plastic defects [15]. In the past decade, they have been directly observed by X-ray diffraction
methods close to sample contacts [16] or in the bulk [17]. At low temperature, these intrinsic defects may be nucleated and moved when external forces are applied to the crystal (high electric fields, mechanical stress or thermal gradient), thus contribute to the CDW conductivity [18,19]. Increasing these forces should induce sliding of these CDW defects, controlled by the existence of pinning centers [20]. A model of the macroscopic proprieties of CDW compounds was given some time ago [21,22]. The ground state of the CDW is characterized by a Periodic Lattice Distortion (PLD) coupled to a periodically modulated charge density and CDW,
; where
are respectively the concentration of uncondensed electrons and the amplitude of the is the position and
is the CDW phase. At zero temperature, impurities induce
fluctuations in the CDW phase, while the CDW amplitude can be ignored. In this context, a Fukuyama-Lee-Rice (FLR) model of an incommensurate CDW
has been proposed to
describe the macroscopic CDW condensate as a classical deformable medium interacting with randomly distributed impurities and under of diving electric field. Moreover, this model has been successful in explaining experimental results related to the sliding of the CDW both qualitatively and quantitatively, such as the threshold field for non-Ohmic conductivity above a threshold field, mode locking, existence of metastable states [23–27], etc. However, it is difficult to obtain experimental results for the temperature dependent CDW dynamics based on FLR model because all the parameters of this model are strongly temperature dependent [28], especially, the CDW amplitude. Despite this limitation, the finite temperature effect on the CDW dynamics by means of FLR model can be considered with neglecting fluctuations in the CDW amplitude which is legitimated only at finite temperatures, as it was signaled by Ref [29]. However, to our knowledge, investigating the new phenomena that take place at dynamic properties of low-temperature limit in the one-dimensional incommensurate system based on FLR model still needs intensive research in order to fill this void in the existing
literature. In this letter, the numerical simulation of the effect of finite thermal fluctuations, represented by the Gaussian random force [29-33] on the dynamics of a one-dimensional incommensurate charge density waves system in the weak pinning limit has been reported. This will be done within the context of model which incorporates the essential features of the FLR model of CDW’s. This work is organized as follows: Section 2 presents a short description of the computational method and the calculation procedure. Our results and discussions are exposed in Section 3. Finally, the important conclusions arising from this work are summarized in Section 4.
2- Model for numerical simulation At zero temperature, the dynamics of CDW phase fluctuations given by the following overdamped equation of motion, as [23].
H d ( x) ( x) dt
where is the phenomenological damping constant and
(1)
is the FLR Hamiltonian for the
phase pinning model of one dimensional incommensurate CDW. For more details on this Hamiltonian see Refs [23,27,30,34,35]. The expression of FLR Hamiltonian is given by the following formula: k d c E H dx ( x, t ) dx ( x, t )Vi ( x ri ) dx 2 dx i 2
(2)
It is well known that the nonlinearity of the problem which comes from CDW-impurities interaction makes it difficult to handle Eq.1 using an analytical method [27]. In this limit, it is recommended to use the numerical method which may provide an analytical tool in order to overcome complicity of the motion equation of the CDW’s.
Before using the numerical simulation, we make some approximations to simplify the problems associated with the incommensurate CDW dynamics along the direction of the chain. Firstly, the damping force is assumed to act only at the impurities positions along the considered chain [35]. Secondly, we have chosen a short-range interaction between the phases of CDW and impurity potential
located at site impurity , as following
[36,37] where, the index runs over the impurities whose concentration is and
is the pinning potential considered identical for all the impurities.
With the power of two approximations listed above, we rescale the unit of time by the factor
and the length by . In these dimensionless units, Eq.1 reduces to [23,27,30,36]: d ( x, t ) ( x, t ) sin(i ( x, t )) ( x xi ) dt
where
,
. The parameters
(3)
and
are respectively
the dimensionless electric field along the chain direction and the dimensionless interaction strength. Finally, we may now integrate between the impurities sites along longitudinal chain and, then, Eq.3 is discretized using of a standard finite-difference method on a 1-D grid, obtaining
1 2
i (t 1) i (t ) t i (t ) ( xi1 xi1 ) sin(i i (t ))
where δt is the time step size and
. The parameter
(4)
is the discrete
second derivative (It is defined in Refs [27,30]). In non-zero temperature effect, the dynamics properties of CDW system in the weak pinning limit may be specified by the following equation.
1 i (t 1) i (t ) t i (t ) ( xi1 xi1 ) sin(i i (t )) i(t ) 2
(5)
where
is an expression which represents the thermal fluctuations [29-33].
i
(t )
(t ) 0 and (t ') 2 K BT i ,i ' (t t ') i' i
(6)
K B is the Boltzmann constant and K BT is the thermal energy.
In order to ignore the random problem, we must first dedicates a program to generate the initial distribution of our system, where we choose the initial values of the phase randomly taken in the interval
and the sample of length
containing randomly
distributed impurities, except where noted. From the previous configuration chosen, the numerical algorithm of solution of nonlinear Eq.5 is carried out using a periodic boundary condition; the applied electric field and temperature are zero to relax our system in order to reach a static configuration equilibrium characterized by a given relaxation time where all the local velocities are near zero. Then, for a given applied electric field along the direction chain, we assume that the system responds instantaneously between the impurities sites; this neglects the dynamic degree of freedom on short time scales.
3- Results and discussions. A variety of transport phenomena observed in CDW’s materials has been attributed to the inhomogeneous plastic response of the CDW, when set into motion by an external drive. Variations of the CDW local phase are observed for an applied continuous electric field above the threshold field
, as shown in Fig.1. It was found that variations of the
CDW phase are very sensitive to the temperature. Their amplitudes increase when the temperature increases. This behavior can be related to the change of the elastic properties of CDW due to the thermal softening behavior, in other words, the increase in temperature
leads to reduce the CDW stiffness parameter and, hence, enhancing the non-homogeneous response of the CDW. A similar behaviour has been previously obtained by N. Habiballah et al [33]. They have attributed the phenomenon to the fact that the generation of structural defects such as CDW dislocations may be make the system less rigid and more affected by the external field.
3,85
c ppm
=0K
a,u
=0,025K
=0,1K
j
3,50
T
3,15
=0,2K
( ) ( )
2,80 2,45 2,10 1,75 1,40 1,05 0,70 0,35 20
40
60
80
100
120
140
160
180
Position [a,u]
Fig.1 CDW phase profile for various values of temperatures at
and
.
Understanding how the temperature affects in a wide range of quasi-one-dimensional conductors is one of the most significant scientific challenges in solid state physics. So, the temperature effect on the collective CDW dynamics is explored using a steady-state timeaveraged velocity (
) of the CDW (excess current), which is described by the following
equation [38]:
CDW (t ) t where,
is a time dependence of the spatially averaged velocity (Eq.8) [27,36,38]:
(7)
(t )
where
1
N
N imp d (t )
i
imp
(8)
i
dt
is the impurity number.
For a one-dimensional incommensurate CDW system without temperature effect, the steadystate time-averaged velocity of the CDW, varies depending on the electric field as follows
CDW ( ) 0 (
1) T
(9)
is the dynamic critical exponent. The thermal fluctuations, does not alter the behavior of field-dependent current density, the change in
as a function of the applied electric field is governed by the same power law
Eq.9. Furthermore, for a given electric field above the threshold one, it was observed that the variation of
is non-monotonous under the temperature effect (Fig.2), characterized by
the existence of two crossover temperature
around
and
, leading us to
distinguish three temperature regions (Fig.3). Such behavior has been experimentally observed in the K0.3MoO3 compound [14]. For T ranging from
to
,
increases when increasing the temperature, suggesting that the CDW motion exhibits a very fast dynamic under temperature variations. This can be attributed to a decrease in the rigidity of the CDW as the temperature is increased. For T ranging from
to
,
decreases when increasing the temperature. This is a direct consequence of the fact that the collective motion of CDW is damped; resulting from the increase in the carrier concentration with increasing temperature, as it was theoretically showed by L.Sneddon [39].
0,018
T=0K T=0,2K T=0,5K
0,015
T=0,6K Fit with Eq:9 =0,215[a,u]
0,012
c ppm j
CDW
0,009
0,006
T 0,003
0,00
0,01
0,02
0,03
0,04
0,05
0,06
a,u
Fig.2 Nonlinear excess currant vs applied electric field for various temperatures.
For
,
increase for
shows a minimum more or less pronounced around
and
, which suggests that the CDW motion displays a very fast dynamic
when temperature increases. This behavior can be related to the fact that the variation with temperature in the structure (edge or screw) or mobility of CDW-phase dislocations generated through the electronic crystal can occur around
, which in turn induces a change in
. A similar behavior has been theoretically and numerically predicted by several
authors [13]. The presence of CDW-phase dislocations tends to make the CDW less rigid [40], causing plastic deformations [41]. Recent result obtained from a numerical simulation for phase dislocations effect on the CDW dynamics has shown that the multiplication of dislocations contribute to the CDW collective transport, where just above threshold field, the current density increases when the dislocations number increases [42].
0,16
=1,25
c =200ppm
T
0,14
=1,5
T
=2
j
=0,215[a,u]
T
0,12
T
cs2
=0,55K
CDW / 0
0,10
0,08
0,06
0,04
T
0,02
0,00
0,07
0,14
cs1
0,21
=0,25K
0,28
0,35
0,42
0,49
0,56
0,63
T(K)
Fig.3. Nonlinear-excess-currant as a function of temperature for various applied electric field.
Fig.4 shows the temperature dependence of the threshold field. It was established that the threshold field
is not a monotonic function of the temperature, where decreases when
the temperature increase from 0K to∼0.25K then shows a rise in the temperature interval and finally decreases as T is away from observed in various CDW materials [10,12].
. The same behavior was
c =200ppm j
0,0035
=0,215[a,u]
0,0030
T [a,u]
0,0025
T
cs1
=0,25K
T
cs2
=0,55K
0,0020
0,0015
0,0010 0,00
0,07
0,14
0,21
0,28
0,35
0,42
0,49
0,56
T(K)
Fig.4 Temperature dependence of threshold field ET.
The decreases of the threshold field from
to
can be attributed to the fact
that CDW is softened. This result is qualitatively in agreement with both the theoretical work reported in [14] and numerical simulations by Miyashita and Takayama [39]. Then, the behaviour of shows a minimum less or more pronounced around the temperature increases from
to
and increase when
(Fig.4). This is the direct consequence of the
damping phenomenon, originating from the CDW interacting more strongly with the normal carriers when the temperature is increased. Finally, managed to access a broad maximum which appears near
and decrease for
. It can be suggested that a change in
with temperature may be generated by the onset of the CDW-phase dislocations. Among
the numerous results signalled in Ref [42], the most important is that the threshold field decreases when increasing the dislocation number. This last has been explained in terms of an increase in the number of CDW dislocations leading to a softening of CDW.
4- Conclusion Within the context of a model which incorporates the essential features of the FLR elastic model, the temperature effect on a one-dimensional incommensurate CDW dynamics was studied. With a small change in the low-temperature limit, the non-linear excess current for a given electric field above the threshold one shows the existence of temperature crossovers around 0.25K and 0.55K, leading us to distinguish three observed behavior of
versus
temperature. Besides, the temperature dependence of the threshold field is found nonmonotonous, shows a minimum near 0.25 K and a maximum near 0.55K. Such behavior has been obtained in the preceding experimental studies. The temperature affects drastically the local dynamical properties of the CDW: the CDW phase profile variations are more pronounced when the temperature increases, indicating that the CDW is deformed and more affected by the applied electrical field. This may be related to the change of the elastic properties of CDW due to the thermal softening behavior.
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Highlights
o
The temperature plays a crucial role on the CDW dynamic properties.
o
The current density and the threshold field of CDW are non-monotonous
when the temperature increases. o
The FLR model is not sufficient to explain all results observed
experimentally in real systems.