Dynamic properties of weakly coupled linear chains in two-dimension charge-density-waves (CDW’s) system: A numerical study

Results in Physics 13 (2019) 102270

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Dynamic properties of weakly coupled linear chains in two-dimension charge-density-waves (CDW’s) system: A numerical study

T

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R. Essajaia, , N. Hassanaina, A. Mzerda, M. Qjanib a b

Group of Semiconductors and Environmental Sensor Technologies-Energy Research Center, Faculty of Science, Mohammed V University, B.P. 1014, Rabat, Morocco LCMP, Faculty of Sciences, Chouaïb Doukkali University, B.P. 20, El Jadida, Morocco

A R T I C LE I N FO

A B S T R A C T

Keywords: Charge density waves Non-linear transport Generalized Fukuyama-Lee-Rice model Inter-chain coupling Numerical simulation

In the present letter, the numerical study on the dynamic properties of incommensurate charge-density-waves (CDW’s) in the two-dimensional (2-D) system has been performed by means of the generalized model of Fukuyama-Lee-Rice (FLR). The results showed that the weak interchain interaction effect affects the threshold field and the nonlinear excess current transported by the sliding CDW’s as well as the associated nonlinear excess conductivity. All findings obtained in this work are discussed in the context of damping mechanisms of the collective motion of CDW-condensates.

Introduction The interest in low-dimensional conductors lies in the anisotropy of the Fermi surface, which, associated with some electron-phonon coupling, leads to instabilities of the Peierls type [1]. The energy gap opening at the Fermi surface will reduce the electronic energy, this is associated with the coupled lattice distortion, the so-called Peierls distorted state will be favored which takes place below the Peierls critical temperature Tp . At low temperature, the modulation of the electron gas density or charge density wave (CDW) coexists with the lattice distortion. In ideal 1-D conductors, the Fermi surface is made of two parallel planes and gap opening is expected to be complete, leading to a semiconducting CDW state. The situation is more complex in real materials. However, an important parameter governing the low dimensional electronic properties is the transverse coupling between chains in the 1-D case or between slabs in the 2-D one. This depends critically on the crystal structure and chemistry of the compounds. In quasi one-dimensional conductors, there are a lot of coupling mechanisms between the 1-D conductors, such as electron-phonon interaction [2], the interchain coulomb interaction [3], the electron tunneling [4], the Van Der Waals interactions [5], the interchain elastic coupling [6], etc. The effect of these kinds of interchain coupling on the properties of CDW compounds has generated much interest by several authors. Balistic et al. [7] suggested that, in the weak coupling case, the interchain interaction stabilizes the local lattice strain or solitons. Takusagawa [8] studied the vibration modes in the incommensurable systems and showed that the effect of interchain coupling between

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chains courses the lifetime of a phason which is proportional to the momentum along the longitudinal chain. Further, within the generalized Ginzburg Landau theory [9], Dieterich [10] has developed a theory to describe the fluctuations in a system of weakly coupled linear chains system. They found that in the limit of weak interchain coupling, the critical temperature varies logarithmically. In addition, Wei [11] showed theoretically that the interchain coupling increases the transfer of electrons between chains, which results in that the electrons tend to be distributed uniformly between the chains and, therefore, the onset of CDW state is suppressed due to the interchain coupling. Besides, the physical properties of real compounds are reflected in electronic properties as theoretically predicted by Soda et al. [12], where they showed that electronic conductivity is not regular in any space direction. Moreover, it is worth underlining that the local effect of interchain coupling in coupled linear chains has been widely studied on the ground state of CDW, however, the dynamic properties of the CDWcarrying linear chains under the influence of the inter-chain coupling have not been explored yet and, hence, it intensive research is needed to fill this void in the literature. In the present work, the effect of interchain coupling on the dynamic properties of the 2-D array of the CDWcarrying linear chains weakly pinned by randomly distributed impurities; has been investigated. To this end, the numerical simulation study based on generalized Fukuyama–Lee–Rice (FLR) model which consider the interaction between CDWs as was proposed by Nakajima and Okabe was employed. This paper is ordered as follows: Section “Model and numerical implementation” presents a short description of

Corresponding author. E-mail address: [email protected] (R. Essajai).

https://doi.org/10.1016/j.rinp.2019.102270 Received 7 December 2018; Received in revised form 12 March 2019; Accepted 4 April 2019 Available online 06 April 2019 2211-3797/ © 2019 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

Results in Physics 13 (2019) 102270

R. Essajai, et al.

The analytical solution of the motion equation (Eq. (4)) is difficult to achieve because the nonlinearity of the problem comes both from the CDW-impurities interaction on different column and from coupling between CDW’s and, therefore, it is needed to use the numerical simulation technique which may provide an analytical tool to overcome complexity of the motion equation of the CDWs. Before beginning the numerical simulations, the some approximations are needed to simplify the problems associated with the dynamics of CDWs along the direction of the chains. Firstly, the damping force is not acting only at the impurities positions along the considered chain j [18], but also it is depending on the amplitude of inter-chain coupling effect. Secondly, along the single-chain j , we have chosen a short-range interaction between CDW phases ϕj (x ) and impurity potential Vi (x − ri ) located at site impurity ri , as following Vi (x − ri ) = V0 δ (x − ri ) [20–22]. Where, the index i runs over the impurities whose concentration is cj and V0 is the pinning potential considered identical for all the impurities. Following three approximations listed above, at each single chain, we rescale the unit of time by the factor kcj / γ and the length by cj . In these dimensionless units, the dynamic behavior of the CDW on each chain j is specified by

Fig. 1. Representative schema of two-dimensional-array of the CDW-carrying linear chains weakly pinned by randomly distributed impurities.

the theoretical model and the computational method as well as the calculation procedure. Our results and discussions are exposed in Section “Results and discussion”. Finally, the conclusions arising from this work are summarized in Section “Conclusion ”.

dϕj (x , t ) dt

= Δϕj (x , t ) + ξ + ε ∑ sin(θi + ϕj (x i , t )) δ (x − x i ) j′ = j + 1

+η

Model and numerical implementation

∑j (H‖j + H⊥j)

dϕ

2

∫ k2 ⎛ dxj ⎞ dx − ⎜

⎟

(1)

⎝

⎠

ρc E π

⎛ 1 ϕj, i (t + 1) = ϕj, i (t ) + δt Δϕj, i (t ) + ξ (x i + 1 − x i − 1) + ε sin(θi + ϕj, i (t )) ⎜ 2 ⎝

∫ dxϕj (x, t ) + ∑ ∫ dxρj (x, t ) Vi (x − ri)

j′ = j + 1

+η

i

(2) where ρj (x , t ) = ρ + ρ0 cos (Qx + ϕj (x )) , Q = 2KF , KF is the wave vector c at the Fermi energy; ρc and ρ0 are respectively the concentration of uncondensed electrons and the amplitude of the CDW, x is the position and ϕj (x ) is the CDW phase along longitudinal chain j ; K is the elastic constant of the CDW, Vi is the interaction potential of the CDW with impurity at position ri .The first term in Eq. (2) represents the elasticity of the CDW, the second term describes the polarization energy of the CDW in a uniform applied electric field E and the third term corresponds to the CDW interaction energy with impurities. The second term H⊥, j of Eq. (1) is the Hamiltonian of the interaction between CDWs which was proposed by Nakajima and Okabe, it is to be parameterized as [13,14,19]

H⊥ j = −

1 2

∑ j′ λjj′ cos(ϕj (x ) − ϕ j′ (x ))

⎞ sin(ϕj (t ) − ϕ j′ (t )) ⎟ j′ = j − 1 ⎠

∑

(6)

where δt is the time step, ϕj (x i , t ) = ϕj, i (t ) . The parameter Δϕj, i (t ) is the discrete second derivative which is described by the following equation:

Δϕj, i (t ) =

ϕj, i + 1 (t ) − ϕj, i (t ) xi+1 − xi

−

ϕj, i (t ) − ϕj, i − 1 (t ) xi − xi−1

(7)

For numerical work, we have dedicated a Fortron program to generate an initial distribution of the system, where we choose the initial values of the phase randomly ϕj, i taken in the interval [0; 2π ] and an area sample s = 800 μm2 containing randomly distributed impurities nT . The dimensionless parameter ε = 0.315[a. u] is taken (weak pinning ε < 1). Due to limiting computer capacity, a cutoff radius (R cutt − off ) is used in the numerical simulation, i.e., the phase ϕj, i situated on the considered chain j is assumed to interact only with phases of CDW on the first adjacent chain within R cutt − off equal to 2 Γ , where Γ is the interchain distance. Exploiting the configuration chosen and the input parameters, the numerical algorithm of the solution of nonlinear Eq. (6) is carried out by using periodic boundary conditions and the zero applied electric field to relax the system (all the local velocities are near zero). For a given applied electric field along the chain directions, we assume that the system responds instantaneously between the impurities sites in all systems; this neglects the dynamic degree of freedom on short time scales, in agreement with the fact that in this Letter we shall be interested only in the long time response of the system

(3)

The index j and j′ are respectively related to a given chain and its nearest neighbours. λj,j′ is the phenomenological parameter of the interchain coupling and non-zero only for nearest-neighbour (λj,j′ = λ). At zero temperature, the dynamic behavior of the CDWs is specified by the over-damped equation of motion [16].

dϕ (x ) ⎞ δH γ⎛ =− δϕ (x ) ⎝ dt ⎠

(5)

is a dimensionless electric field where x i = cj ri ,θi = Qx i . ξ = along the chain direction. The parameter ε = ρ0 V0/ kcj is a dimensionless interaction strength which describes the competition between the elastic energy and the pinning energy on single chain. The parameter η = λ / kcj is related to the interchain coupling constant. Thirdly, we may now integrate between the impurities sites along each longitudinal chain and, then, Eq. (5) is discretized using a standard finite-difference method on a 2-D grid, obtaining

The first term H‖j of Eq. (1) is the Fakuyama-Lee-Rice (FLR) Hamiltonian for a one dimensional incommensurate CDW, written as [16–18].

H‖, j =

sin(ϕj (x , t ) − ϕ j′ (x , t ))

ρc E / Qkc j2

Taking into account the interchain interaction, the phenomenological Hamiltonian of a two-dimensional CDW system consisting of linear chains weakly pinned by impurities (parallel to the x-axis (see Fig. 1)) can be written as [13–15]:

H=

∑ j′ = j − 1

(4)

where γ is a friction coefficient. 2

Results in Physics 13 (2019) 102270

R. Essajai, et al.

Fig. 3. Excess current vs. applied electric field for various η values for different values of the electric field, for number of randomly distributed impurities is equal to nT = 800 and ε is fixed at0.315[a. u]. Solid lines are a fit according to Eq. (11).

Fig. 2. Threshold field versus inter-chain coupling parameter for various number of randomly distributed impurities and ε is fixed at0.315[a. u]. Solid lines are fit according to Eq. (8).

The CDWs are not destroyed by the applied field [26], showing that the sliding motion of the CDWs occurs beyond the critical value, ''threshold field''(ξT ) . The excess current associated with the sliding of 2D CDW system can be investigated by means of a steady-state timeaveraged velocity (υCDW ) of the CDWs, which is specified by the following equation [27]:

Results and discussion It is well known that the competition between pinning forces and the external applied field is responsible for nonlinear phenomena. The sliding state of CDW-condensates provides an additional contribution to the conductivity resulting in a non-linear current. The onset of nonlinearity may be marked by an abrupt switching from the linear state to non-linear one [23] and varies from one sample to another for the same type of compound. The sliding state of the 2-D-array of the CDW-carrying linear chains is affected by interaction between CDW’s (Fig. 2), the threshold field (ξT ) increases when increasing amplitude of the intechain coupling parameter (η) . This behavior can be ascribed to the fact that in addition to impurities, characterized by the parameterε [24], the effect of interchain coupling is an integral part of the damping force, which in turn prevents the sliding motion of the CDWs along different chains. For high amplitude η, ξT seems to reach saturation, indicating that the interchain coupling has so a limited effect; which is in agreement with the unidimensional nature of our system, high anisotropy along the chains [25]. For various number of randomly distributed impurities (nT ), the variation of threshold field (ξT ) according to the parameter η is well fitted by Eq. (8), ξT varies logarithmically with η.

ξT (η) = ξT (0)(1 + βηδ )

where υ¯ (t ) is a time dependence of the spatially averaged velocity (Eq. (10)) [20,27].

υ¯ (t ) =

1 Nimp

Nimp

∑ i

dϕi (t ) dt

(10)

whereNimp is the impurity number. Excess current υCDW plotted versus the applied electric field for various values of interchain coupling amplitude η (Fig. 3) shows that the excess current not only depends on the amplitude of the applied field, but also on the weak interchain coupling amplitude. Investigating the field-dependent excess current density at different parameter of interchain coupling η indicates that υCDW are best fitted with a power law Eq. (11).

υCDW (ξ ) = υ0 (

(8)

where ξT (0) , β and δ are respectively, the threshold field without interchain coupling, a constant and an exponent, such that 0 < δ < 0 . By fitting the data to Eq. (8) we have extracted the exponent δ as a function of the number of randomly distributed impurities nT for η is fixed at 0315[a. u] and the results shown in Table.1. The present result indicated that the interchain coupling has a nonnegligible effect in the depinning phenomenon of the 2-D array of the CDW-carrying linear chains. This finding is in conformity with what has been observed by the technique of neutron inelastic scattering [19]. For more definiteness, we will reinforce our work by exploring this local effect carefully on other quantities below-mentioned.

ξ − 1)α ξT

(11)

Experimentally, the field-dependent electric conductivity of CDWs is strong nonlinear just above the threshold field (ET). This behavior was first observed in Niobium Triselenide (NbSe3) [28]. Fig. 4 presents the excess conductivity (σCDW ) transported by the sliding of 2-D CDW system, which can be defined by following Eq. (12) [20].

σCDW (ξ ) =

υCDW (ξ ) ξ

(12)

It was shown that the conductivity is non–ohmic when the applied field exceeds ξT and it seems to reach a limiting value for high fields (Fig. 4), indicating that the conductivity is saturated for high fields. Investigating the field-dependent conductivity for various values of η indicates that, in agreement with experimental results [25,29,30], σCDW (ξ ) curves can be fitted using Eq. (13):

Table 1 Exponent δ depend on to number of randomly distributed impurities (nT ) , for ε is fixed at0.315[a. u]. nT = 1000 nT = 900 nT = 800

(9)

υCDW = < υ¯ (t ) >t

σCDW (ξ ) = σ0 (

δ = 0.68 δ = 0.66 δ = 0.65

ξT ξ )( − 1)α ξ ξT

(13)

a is the dynamic critical exponent. Experimentally a is near 1.4 in the molybdenum blue bronzes [31]. 3

Results in Physics 13 (2019) 102270

R. Essajai, et al.

resulting from the fact that the interaction between CDW’s not allows the system to overcome the potential barriers due to the defects. In quasi 1-D materials, the interchain interaction between conductors has often been ascribed to several sources, mainly from an interaction between the modes which do not contribute to the CDW condensate (including phonons and free carriers) [19]. In this context, the increase in the damping effect with respect to interaction amplitude between CDWs which obtained through the analysis of the data presented in Figs. 3 and 4 can be attributed to the increase in local electric field screening and to diminish the non-homogenous response of the CDWs along different chains due to a strong increase in the electronphonon interactions. Conclusion Within the method of numerical simulations using the generalized FLR model at zero temperature, the interchain coupling effect on the dynamic properties of 2-D CDW system weakly pinned by randomly distributed impurities have been studied. It was found that the depinning threshold field ξT increases but the nonlinear excess current υCDW (ξ ) and the associated nonlinear excess conductivity σCDW (ξ ) decreases. The behavior of these quantities can be a consequence of the damping mechanism of collective motion of the CDW’s.

Fig. 4. Excess conductivity vs. applied electric field for various η values for different values of the electric field, for number of randomly distributed impurities is equal to nT = 800 and ε is fixed at0.315[a. u]. Solid lines are a fit according to Eq. (13).

Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.rinp.2019.102270. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Fig. 5. Dependence of a on the amplitude of η for number of randomly distributed impurities is equal to 800 and ε is fixed at.0.315[a. u].

[13] [14] [15]

Similar value is obtained by our numerical simulation in the system whose weak interchain coupling effect is higher than η = 0.01[a. u],for the number of randomly distributed impurities equal to nT = 800 and ε is fixed at 0.315[a. u],as shown in Fig. 5. Through the analysis of the data presented in Figs. 3 and 4, it was noted that the inter-chain interaction does not alter the behavior of both the excess conductivity and the excess current as a function of the applied electric field; the changes in υCDW (ξ ) and σCDW (ξ ) as a function of ξ are governed by the law Eqs. (11) and (13) respectively at different values of the parameter η and the nonlinearity becomes more pronounced as η increases (Fig. 4). Furthermore, just above the threshold field, it was found that υCDW (ξ ) and σCDW (ξ ) are very sensitive to the amplitude of the weak interchain coupling. Their values decrease when the parameter η increases, suggesting that the CDW-condensates display a slow dynamic when increasing interchain coupling effect. This finding further proved that the dynamic of 2-D CDW system is damped by the effect of interchain interaction between CDW’s on parallel linear chains

[16] [17] [18] [19] [20] [21] [22] [23] [24]

[25] [26] [27] [28] [29] [30] [31]

4

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