After collaborations with : C. Brun & Z.Z. Wang Marcoussis France N. Kirova Orsay France Yu. Latyshev M

Preface 1: STRONGLY CORRELATED ELECTRONIC SYSTEMS AS UNCONVENTIONAL SEMICONDUCTORS IN UNCONVENTIONAL CONDITIONSComparisons: Charge Density WavesConducting and optically active polymersSemiconducting and superconducting oxidesOrganic conductors with charge ordering and ferroelectricity After collaborations with : C. Brun & Z.Z. Wang Marcoussis France N. Kirova Orsay France Yu. Latyshev Moscow Russia P. Monceau Grenoble France

Preface 2: STRONGLY CORRELATED ELECTRONIC SYSTEMS AS UNCONVENTIONAL SEMICONDUCTORS IN UNCONVENTIONAL CONDITIONS Fundamental intrigue of many nonstandard conductors:The energy gap is formed spontaneously by electronic correlations through various symmetry breakings: Charge/Spin Density Wave, Antiferromagnetism, Charge Ordering Charge Disproportionation, Orbital Ordering. These are volatile states, locallyaffected by electrons’ injection. Help from CDWs: very small electric field deforms the ground state quite moderate field reaches the microscopic energy scale.

Real time dynamics for reconstruction of a CDW state under applied field in restricted geometry Simulations of CDW vortices under increasing voltage – the CDW amplitude Collaboration: S. Brazovskii LPTMS, CNRS, Orsay, France N. Kirova LPS, CNRS, Orsay, France Y. Luo Université Paris-Sud, France MS A. Rojo-Bravo Boston University, USA PD Tianyou Yi LPTMS, CNRS, Orsay, France PhD

Outline • CDW as an electronic crystal and its deformable ground state. • Experiments on nano-junctions. • Dislocations – the CDW vortices. • Time dependent GL approach to numerical modeling. • Results for realistic experimental geometries and parameters. • Stationary and dynamic multi-vortex configurations. • Conclusion Vortices – the CDW phase

Zeros of the CDW amplitude at a microscopic scale identified by the STM as amplitude solitons C.Brun, Z.Z.Wang, P.Monceau and S.Brazovski 2011 Profiles along the defected chain vsits nearest neighbor Defected chain vs theory

Can be viewed also as branching of stripes at field-effect dopping. Topological defects in a CDW. Solid lines: maxima of the charge density. Dashed lines: chains of the host crystal. From left to right: dislocations of opposite signs and their pairs of opposite polarities. Embracing only one chain of atoms, the pairs become a vacancy or an interstitial or ±2psolitons in CDW language. Bypassing each of these defects, the phase changes by 2p- far from the defect the lattice is not perturbed.

source drain v~I v=0 Formation of new planes in the electronic crystal Elimination of additional planes Dynamic origin of dislocations CDW sliding in the applied external electric field – collective motion of electronic crystal . To set it in motion at different velocities: Transfers flow of vortices – thick channels, Ong and Maki Coherent phase slip - thin cannel Gorkov; Bielis et al. Direct access to the current conversion via dislocations: Cornell – Grenoble, late1990’s Another reason for dislocations – static equilibrium structures due to applied transverse voltage or current

Dislocation in CDW versus vortex in SC Equivalence of given Ey and Hz upon the order parameters. Dislocations in CDW appear as vortices in SC. Reverse effect of order parameters upon the fields are opposite: CDW – electric field is screened via dislocations. SC - magnetic field enters via vortices

 Intra-plain elasticity (∂x)2+ Coulomb energy (y)∂x force to shift the equilibrium CDW charge density ∂x-(y), i.e. the CDW wave number =Qx -(y)x Breaking of inter-plane correlation. Resolution : dislocation lines allow to bring new periods in a smooth way, except in a vortex core.

Minimal model: Interlayer decoupling as an incommensurability effect. Only two layer 1,2 kept at potentials ±V/2 Decoupling threshold: arrays of solitons or dislocations.Discommensurations in a two layers model. Lattice of discommensurations (solitons in phase difference ) Develops from the isolated discommensuration = the 2πsoliton in ∆. Critical voltage = the energy necessary to create the first discommensuration: J1/2

In reality : CDW junction as an array of dislocation lines DLs. A bulk of many planes, voltage difference monitored at its sides. Lattice of discommensurations => sequence of DLs - vortices of the ICDW phase Critical voltage - DL entry energy, like Hc1 in superconductors.(Old theories by Feinberg-Friedel, S.B.-Matveenko) Recall a parallel topics – plastic flows of ICDW with multiple generation of DLs within the current conversion area of a junction. Space-resolved X-ray experiments of Cornell and Grenoble, theory by N.Kirova and S.B.

Phase slips Electrons  Amplitude solitons 2 phase solitons D-loops/lines Excitations or stable perturbations Pair-breaking energy Eg=2(to be précised) One-particle adding energy ∆1(≠ in general) In 2D,3D ordered phases, T<Tc<< : Dislocation = 2 vortex of  Addatoms/vacancies = particles with charges ±2e = minimal pairs of dislocations (2D) /dislocation loop (3D) on one chain = 2 solitons Phase slips: Microscopically – a self-trapping of electrons into solitons with their subsequent aggregation Macroscopically – the edge dislocation line proliferating/expanding across the sample. Low T: the energetics of dislocation lines/loops is determined by the Coulomb forces and by screening facilities of the free carriers.

Motivation for our modeling : Experiment on tunnel junctions YuriiLatyshevtechnology of mesa-structures: fabrication by focused ion beams. All elements – leads, the junction – are pieces of the same single crystal whiskerNbSe3 Overlap junction forms a tunneling bridge of 200A width -- only 20-30 atomic plains of a layered material.

Direct observation of solitons and their arrays in tunneling on NbSe3 Y. Latyshev, P. Monceau, A. Orlov, S.B., et al, PRLs 2005 and 2006 creation of solitons at ≈2D/3: Es=2D/p ! peak 2Dfor inter-gap creation of e-h pairs absolutethreshold at low Vt≈0.2 All features scale with the gap (T) ! oscillating fine structure First degree puzzle: Why the voltage is not multiplied by N~20-30 - number of layers in the junction - It seems to be concentrated at just one elementary interval. In similar devices for superconductors the peak appears at V=2*N

Junction reconstruction by entering of dislocations Fine structure is not a noise ! It is :sequential entering into the junction area of dislocation lines = CDW vortices = solitons‘ aggregates. Need a complex modeling for intricate distributions of the order parameter (amplitude and the phase), electric potential, normal density and normal current. Recall a new science: field effect transformationsin strongly correlated materials Their symmetry broken phases will be subject to reconstruction.

Known from analytic static solutions for an infinite CDW media: Potential distribution in a DL vicinity.Notice concentration of potential Ф(x,y)drop facilitating the tunneling. 3d and contour plots ±y(x) for surfaces Ф(x,y)±∆ where the tunnelling takes place.

What does tunnel at these low subgap voltages ? ±2π phase solitonsstretching/squeezing of a chain by one period with respect to the surrounding ones: elementary particles with the charge ±2e and the energy E~Tp 3D ordering temperature Tp. Outcome : pair of 2p solitons can be created by tunneling almost exclusively within the dislocation core, The process can be interpreted as a excitation of the dislocation line as a quantumstring.

Before the reconstruction: Distribution of potentials (values in colours, equipotential lines in black) and currents (arrows) for moderate conductivity anisotropy (||/=100).

Junction reconstruction with cross-sections of dislocations. Junction scheme with crossections of dislocations The very low sharp threshold voltage Vt ~ 0.2 Δ can be provided only by the low energy phase channel, and the experiment also indicates that the voltage applied to the whole stacked junction drops mostly at a single elementary interlayer spacing. It can happen when the electric field in this junction exceeds a threshold value for phase decoupling in neighboring CDW layers. This decoupling is expected to proceed via the successive development of dislocation lines entering to the junction area.

EF -PF GL – like model H=HCDW+Hel Only extrinsic carriers n are taken explicitly. Intrinsic ones, in the gap region, are hidden in the CDW amplitude A. PF

Equations Boundaryconditions CDW stress vanishes at the boundaries: Natural for sides, for drain/source boundaries the no-sliding is implied Normal electric field is zero at all boundaries: total electro neutrality and confinement of the electric potential within the sample No normal current flow at the boundaries except for the two source/drain boundaries left for the applied voltage. There, the chemical potentials are applied:

Simplified rectangular geometry assuming passive role of other parts Amplitude V=7meV, 9 meV, 11 meV; t~10-8 sec Phase: wider sample, higher V

Many vortices appear temporarily in the course of the evolution. For that run, only two will be left. Time unit – 10-13 sec given by the inverse CDW conductivity. Here, t~100ps – 10GHz

Real geometry: initial short time fast dynamics, t=3.4x10-10 sec Unexpected result: long living traces of the amplitude reduction following fleshes of vortices. Amplitude A W()+W(A) 1 W()=0 Phase  A Phase deformations energy - cannot relax fast enough following the rapidly moving vortex. Composite energy W()+W(A) All these 5 flashes are the phase-slip processes serving to redistribute the CDW collective charge

Real geometry: final stationary state at the first threshold voltage A V=7-8 meV, t=10-7 sec  Strong drop of the electric potential (with inversion !) and of the current concentration. Perturbations are concentrated near the vortex core – the location of tunneling processes. F z

Movie for a full multi-vortex evolution to the junction stationary state with just one remnant vortex

Distributions of the chemical potential zeta, electric potential phi, electro-chemical potential zeta+phi; lines of current.

I(V) and dI/dV, rectangular geometry I(V) and dI/dV, slits geometry

Conclusions. • Modeling of stationary states and of their transient dynamic for the CDW in restricted geometries is reachable. • Model takes into account multiple fields in mutual nonlinear interactions: the complex order parameter Aexp(i) of the CDW, and distributions of the electric field, the density and the current of normal carriers. • Vortices are formed in the junction when the voltage across, or the current through, exceed a threshold; their number increases step-wise - in agreement with experiments. A much greater number appears in transient processes • The vortex core concentrates the total voltage drop, working as a self-tuned microscopic tunnelling junction, which might give rise to observed peaks of the inter-layer tunneling . • Parameters need to be adjusted – e.g. the conductivity increased. • The reconstruction in junctions of the CDW can be relevant to modern efforts of the field-effect transformations in strongly correlated material which also show a spontaneous symmetry breaking.

Conclusions II:Specifics of strongly correlated electronic systems : inorganic CDW, organicsemiconductors, conjugatedpolymers, conductingoxides, etc… Electronic processes, in junctions at least, are governed by solitons or more complex nonlinear configurations. As proved by presented experiments and recent oneson charge ordered states, they can lead to : • Conversion of a single electron into a spin solitons • Conversion of electrons pair into the 2p phase slip • Pair creation of solitons (tunneling and optics) • Arrays of solitons aggregates – dislocation lines, stripes, walls of discommensurations – reconstruct the junction state and provide self-assembled micro-channels for tunneling;

After collaborations with : C. Brun & Z.Z. Wang Marcoussis France N. Kirova Orsay France Yu. Latyshev M