ЖЭТФ, 2019, том 156, вып. 4 (10), стр. 721-723

MULTI-FLUID HYDRODYNAMICS IN CHARGE DENSITY WAVES

WITH COLLECTIVE, ELECTRONIC, AND SOLITONIC DENSITIES

AND CURRENTS

S. Brazovskiia*, N. Kirova^b

^a LPTMS, CNRS and Université Paris-Sud, Univ. Paris-Saclay

91405, Orsay, France

^b LPS, CNRS and Université Paris-Sud, Univ. Paris-Saclay

91405, Orsay, France

Received May 3, 2019,

revised version May 3, 2019

Accepted for publication June 6, 2019

Contribution for the JETP special issue in honor of I. M. Khalatnikov’s 100th anniversary

DOI: 10.1134/S0044451019100146

the conventional xy model and the complex-field the-

ory but are stabilized by the matter (here the number

Introduction: CDWs and their intrinsic de-

of condensed electrons) conservation law. The mini-

mal dislocation loops are the charge ±2e objects in a

fects. Charge density waves (CDW) are spontaneous

periodic superstructures ∝ Aexp(q₀ · r + ϕ) which are

form of ±2π solitons: in a discrete view of the quasi-1D

system, here the CDW at the defected chain gains or

ubiquitous in quasi-1D electronic systems, see the lat-

looses one period with respect to surrounding chains.

est review [1]. The translational degeneracy of the in-

commensurate CDW ground state allows for formation

These phase solitons were assumed to be seen experi-

mentally as lowest activation charge carriers (see [12]

of topological defects: local ones, like phase and am-

plitude solitons (see [2] for the literature review) and

and Refs. therein).

extended ones, like planes of domain walls as solitonic

Describing the coexistence of electrons and defects

lattices [3,4], lines or loops of dislocations as phase vor-

in static equilibrium, under strains and in the current

tices [5-7]. Experimentally, their presence was identi-

carrying state requires for a general nonlinear hydro-

fied by various methods; just mention the direct visua-

dynamics for two fields — the phase and the electric

lization of solitons by the STM [8,9] and indications on

potential and for two fluids of electrons and defects.

This article suggests a contribution to this request.

dislocations from the coherent X-ray microdiffraction

[10] and from reconstruction in mesa-junctions [11].

Results and conclusions. At presence of topo-

logical defects the local deformations and velocities ω_j ,

The motion of dislocations is allowed only as a mat-

j = x,y,z,t, cannot be derivatives of the same phase

ter conserving glide along the chains, in the direction of

ϕ. Our key observation was an existence of a uniquely

the Burgers vector b = 2π(1, 0, 0). The transversal mo-

defined and allowed for averaging phase χ which deriva-

tion, the non conserving climb, is prohibited whatever

tives are given as

is the driving force coming from the local stress — in

a strong difference with respect to conventional vor-

∂_tχ = 〈ω_t〉 + 2πj_d,

∂_xχ = 〈ω_x〉 - 2πn_d ,

tices. In CDWs, the climb may be allowed by the

(1)

∂_yχ = 〈ω_y〉,

∂_zχ = 〈ω_z〉,

condensation or liberation of normal carriers providing

the conversion between normal and collective currents.

where n_d is the concentration of defects — the mean

Otherwise, even lacking the topological protection, the

area of loops per volume (taking vorticity signs into

pairs or rings of dislocations do not annihilate as in

account).

We employed the local energy functional appropri-

^* E-mail: brazov@lptms.u-psud.fr

ate to CDWs as described in [13]. Equations for the

721

ЖЭТФ, вып. 4 (10)

S. Brazovskii, N. Kirova

ЖЭТФ, том 156, вып. 4 (10), 2019

average phase χ and the potential Φ have been derived

(

)

^Δ-γ∂_t χ = F_pin+E-2γj_d - ∂_x(n_n + 2n_d),

(2)

r²⁰ΔΦ + ∂_xχ + 2n_d + n_n = 0 , E = -∂_xΦ,

(3)

where

Δ = ∂2x + Δ_⊥, Δ_⊥ = α_y∂2y + α_z∂2z, α_y,z are

anisotropy coefficients coming from the interchain cou-

pling of CDWs and r₀ is the Tomas-Fermi radius of

the parent metal; the concentration n_n of normal car-

riers will be neglected from now on. Eq. (2) shows that

the phase χ is driven, in addition to standard forces

F_pin + E, also by the current of defects and by the

longitudinal gradient of the total number of particles.

Eq. (2) shows that the density and the current of de-

fects contribute in the frame of the average phase χ,

contrarily to being obscure in the frame of local inde-

pendent distortions ω_i.

Equations (2), (3) must be complemented by the

laws governing the distribution of defects. It was im-

-1

portant to realize that the force driving the glide of

defect comes only from share strains F_d = 2Δχ. In the

diffusion approximation, we get

(Color online) Distributions around a dislocation centered at

(0, 0). ξ, η are dimensionless rescaled coordinates x, y. Vectors

(

)

and streamlines characterize the phase χ. The color indicates

Δ - γ∂_t χ + 2γb_dnd,tot Δ_⊥χ+

the chemical potential ζ = ZT. Z changes from Z ≈ 0 at

+ 2(γD_d + 1)∂_xn_d = F_pin + E,

(4)

large distances (green color) to a maximal value Z ≈ 2.5 near

the origin (red color) and then drops to zero (blue color)

where b_d and D_d = b_dT are the mobility and the dif-

fusion coefficient of defects. The allowance for defects’

motion contributes additively to the transverse rigidity

∝ Δ_⊥χ of the phase and to the effective field from the

gradient of defects’ concentration.

The derived equations can be generalized to take

Results of a numerical solution of the above equa-

into account isolated dislocation lines embedded to the

tions are illustrated in Figure. The conventional ro-

averaged ensemble of dislocation loops. For a static dis-

tation of the phase following the coordinate angle at

location directed in z and centered at (0, 0), the equa-

large distances becomes near the core a nearly verti-

tions can yield

cal drop indicating the high x-gradient in accordance

with the rapidly growing Z = ζ/T . The enhancement

[r²⁰α∂4y - ∂2x]χ - 2∂_xn_d = ∂_xδ(x)Sgn(y).

(5)

of Z up to Z ≈ 2.5 corresponding to increasing of

solitons’ concentration near the dislocation by a fac-

This equation was derived from Eqs. (2), (3) in the

tor n(core)/n_∞ ≈ 6.

electroneutrality approximation realizing that r₀ is the

In summary, we have derived general equations

smallest length scale. Here, the conventional Lapla-

for the multi-fluid hydrodynamics of plastic flows

cian form for an anisotropic elastic media acquired a

with collective, electronic, and solitonic densities and

nonanalytic dominating contribution of an anomalous

currents. As an application, we presented distributions

elasticity [6] coming from long range Coulomb forces.

of fields around an isolated dislocation line in the

The defects concentration is regulated by their depen-

regime of nonlinear screening by the gas of phase

dence on the chemical potential ζ_d which obeys the

solitons.

equilibrium condition j_d = 0 as

The full text of this paper is published in the English

n_d = n_∞ sh

j_d ∝ α_y∂2yχ - ∂_xζ_d = 0.

(6)

version of JETP.

722

ЖЭТФ, том 156, вып. 4 (10), 2019

Multi-fluid hydrodynamics in charge density...

REFERENCES

7. T. Yi, A. Rojo Bravo, N. Kirova, and S. Brazovskii,

J. Supercond. Nov. Magn. 28, 1343 (2014).

1. P. Monceau, Adv. Phys. 61, 325 (2012).

8. S. Brazovskii, Ch. Brun, Zhao-Zhong Wang, and

2. P. Karpov and S. Brazovskii, Phys. Rev. E 99, 022114

P. Monceau, Phys. Rev. Lett. 108, 096801 (2012).

(2019).

9. D. Cho, Sangmo Cheon, Ki-Seok Kim et al., Nature

3. S. Brazovskii and N. Kirova, Sov. Sci. Rev. A 5, 99,

Comm. 7, 10453 (2016).

ed. by I. M. Khalatnikov, Harwood Acad. Publ., New

10. D. Le Bolloc’h, V. L. R. Jacques, N. Kirova et al,

York (1984).

Phys. Rev. Lett. 100, 096403 (2008).

4. S. Brazovskii, J. Supercond. Nov. Magn. 20,

489

11. Yu. I. Latyshev, P. Monceau, S. Brazovskii et al.,

(2007).

Phys. Rev. Lett. 96, 116402 (2006).

5. D. Feinberg and J. Friedel, J. Phys. France, 49, 485

12. S. V. Zaitsev-Zotov and V. E. Minakova, Phys. Rev.

(1988).

Lett. 97, 266404 (2006).

6. S. Brazovskii and S. Matveenko, Sov. Phys. JETP

13. S. Brazovskii and N. Kirova, Ann. Phys. 403, 184

72, 864 (1992).

(2019).

723

10*