Evidence of Charge Density Wave transverse pinning by x-ray micro-diffraction

TL;DR

This study uses in-situ x-ray micro-diffraction to visualize charge density wave deformations in NbSe3, revealing large transverse deformations and shear effects that influence soliton nucleation and CDW dynamics.

Contribution

It provides the first direct imaging of CDW deformation at the local scale, demonstrating transverse pinning and shear effects in the sliding regime.

Findings

CDW dissociates from the lattice in the sliding regime

Large transverse deformations are observed, ten times larger than longitudinal ones

Shear effects significantly influence soliton nucleation and CDW dynamics

Abstract

Incommensurate charge density waves (CDW) have the extraordinary ability to display non-Ohmic behavior when submitted to an external field. The mechanism leading to this non trivial dynamics is still not well understood, although recent experimental studies tend to prove that it is due to solitonic transport. Solitons could come from the relaxation of the strained CDW within an elastic-to-plastic transition. However, the nucleation process and the transport of these charged topological objects have never been observed at the local scale until now. In this letter, we use in-situ scanning x-ray micro-diffraction with micrometer resolution of a NbSe$_3$ sample designed to have sliding and non-sliding areas. Direct imaging of the charge density wave deformation is obtained using an analytical approach based on the phase gradient to disentangle the transverse from the longitudinal components over a large surface of a hundred microns size. We show that the CDW dissociates itself from the host lattice in the sliding regime and displays a large transverse deformation, ten times larger than the longitudinal one and strongly dependent on the amplitude and the direction of the applied currents. This deformation continuously extends across the macroscopic sample dimensions, over a distance 10 000 times greater than the CDW wavelength despite the presence of strong defects while remaining strongly pinned by the lateral surfaces. This 2D quantitative study highlights the prominent role of shear effect that should play a significant role in the nucleation of solitons.

Equations

$$\delta q_i(\vec{r})=\frac{\partial \varphi }{\partial i}(\vec{r}),i=x,y,z$$

$$\begin{array}{rl}{F}_{\varphi }=& \int d^3\vec{r}[\frac{1}{2}{A}^2(K_x(\frac{\partial \varphi }{\partial x}{)}^2+K_y(\frac{\partial \varphi }{\partial y}{)}^2+K_z(\frac{\partial \varphi }{\partial z}{)}^2)\\ & -\frac{e{\rho }_s}{2k_f}U\frac{\partial \varphi }{\partial x}]\end{array}$$

Full text

Topological objects carrying charges also play an important role in Charge Density Wave (CDW) systems. Indeed, when a sufficiently large current is applied to a sample displaying an incommensurate CDW, a collective charge transport is observedPhysRevB.85.241104 ; Monceau_review . Those charges are carried by topological objects known as solitons, traveling through the sample over macroscopic distances, regularly spaced from each otherPhysRevLett.52.663 ; PhysRevB.32.6582 ; PhysRevB.94.201120 ; PhysRevLett.100.096403 ; PhysRevB.85.035113 and periodically created in time PhysRevB.35.6348 .

This collective transport of charges is a complex phenomenon that requires a periodic creation and annihilation of solitons. This takes place close to the two electrodes, where the strain is maximum, and is based on elasticity and pinning of the CDW at the two edges of the sample: as a consequence, under an applied field, the CDW is compressed at one edge and expanded at the other along the CDW wave vector. This longitudinal CDW deformation has been observed by several indirect methods, like electromodulated IR transmissionPhysRevB.52.R11545 and local conductivity measurementsItkis_1993 ; PhysRevB.53.1833 ; PhysRevB.57.12781 . The most direct measurement of this compression-dilatation phenomenon was obtained by x-ray diffraction, also showing that beyond 100$\mu m$ from the two electrical contacts, the CDW wave vector varies little and linearlyPhysRevLett.70.845 ; PhysRevLett.80.5631 . However, the transverse CDW deformations were not measurable in those two studies because the x-ray beam was larger than the sample width.

Even though the longitudinal deformation is directly associated to the nucleation of solitons, the transverse one also affects the collective current. The first theoretical study dealing with shear deformations was reported inElastic_Feinberg where an elastic CDW submitted to an applied field and pinned by lateral surfaces display a quadratic phase behavior. From an experimental point of view, shear deformation influences transport properties: the threshold current increases with smaller sample widths and smaller cross sections small_o_TaS3 ; PhysRevB.46.4456 ; Size_dependent_threshold_fields . The influence of shear on the collective current can be also observed through the response to current pulses PhysRevLett.114.016404 . From x-rays diffraction experiments, a transverse deformation around a surface step has been observed by topography in NbSe3PhysRevLett.83.3514 , as well as a local transverse deformation of the CDW by coherent diffraction in TbTe3 PhysRevB.93.165124 and in NbSe3PhysRevLett.109.256402 . However, no qualitative and direct study dealing with both longitudinal and transverse deformations over a large area of the sample was reported despite its importance in the nucleation process.

In order to simultaneously measure the longitudinal and the transverse deformation, x-ray micro-diffraction experiment has been performed in an NbSe3 compound submitted to external $dc$ currents. Four gold contacts were evaporated on a 39$\mu m$ $\times$ 3$\mu m$ $\times$2.25$mm$ single crystal glued on a sapphire substrate to perform in-situ four point resistivity measurements. In addition, a L-shaped cut was made through the sample by a focused ion beam (FIB) so that the current could only flow in the upper half part of the sample, (see Fig. 1a). This geometry allows to simultaneously observe the sliding area above the cut submitted to the currents and the non-sliding area from a single sample.

The connected sample was inserted into a cryostat mounted on the ID01 diffractometer at the ESRF and cooled to 120K, below the first CDW transition (Tc1=145K PhysRevB.18.5265 ; Hodeau_1978 ). The satellite reflections associated with the CDW are located at $\pm\vec{q}_{cdw}$ of each Bragg reflection with an incommensurate wave vector $q_{cdw}=0.243\pm 0.001(\times\frac{2\pi}{b}$). The threshold current, measured in situ, was equal to Is = 0.5 mA (see Fig. 2). The 8 keV X-ray beam was focused by a Fresnel zone plate down to 200nm$\times$300nm on the sample (see Fig 1b)) and the diffracted intensity was recorded with a 2D Maxipix Detector with 55$\mu$m$\times$55$\mu$m pixel size, located 70 cm downstream. The Fresnel zone plate was mounted on a piezo stage to continuously map a 40$\mu$m$\times$90$\mu$m area, through the whole sample width, with a step size of 1 $\mu$m (see the probed area in Fig. 1 (a) and Chahine ; StevenLeake for more technical details). Rocking curves were performed at each position for the $Q_{020}$ Bragg and (0 $\vec{q}_{cdw}$ 0) satellite reflections for 6 currents, below and above the threshold and in the two opposite directions. We thus obtained a 5D intensity matrix containing the three reciprocal coordinates of the wave vector and the two spatial coordinates of the beam position.

The $Q_{020}$ Bragg reflection gives information on the host crystal lattice while the $Q_{s}=(0,1,0)+\vec{q}_{cdw}$ satellite reflection contains information from both the host crystal lattice and the CDW modulation. It is therefore crucial to measure the two types of reflections to dissociate the CDW from the host lattice. The correlation between the two is particularly visible on the maps displaying the integrated intensity by summing over the full rocking curve (see Fig.2 a) and b)). The FIB cut is obviously visible on both maps, as well as disturbed areas of the host lattice reflecting on the CDW (see Fig. 4 in Supp. Mat.). These perturbed areas remain weak, in the $\delta q=10^{-3}\AA^{-3}-10^{-4}\AA^{-3}$ range and are most probably due to constraints induced by the FIB at the line cut and at few specific places at the surface, leading to rocking curves extending beyond our measurement window (see Fig. 2(a) and (b)).

The most interesting information comes from the wave vector orientation. The 3 coordinates of the two wave vectors are obtained independently from Eq.1 in Supp. Mat. from the maximum of each fitted rocking curve ($\varphi$ angle) and the location of the maximum on the detector ($\delta$, $\gamma$). Along the direction perpendicular to the sample surface (called the $y$ direction), the variations of the satellite $Q^{y}_{s}$ and the Bragg $Q^{y}_{020}$ are similar and small (see Fig.4 in Supp. Mat.). Let us first consider the longitudinal $x$ deformation parallel to $\vec{q}_{cdw}$. As discussed in the introduction, a compression-dilatation of the CDW under current is expected but small far from electrical contacts. We indeed observed this small effect by averaging over the more homogeneous areas. Although $q_{x}$ is dominated by the host lattice variations, a small compression and dilation of the CDW period is well observed in the $10^{-4}\mathring{A}^{-1}$ range as expected, in agreement with PhysRevLett.70.845 ; PhysRevLett.80.5631 . Although weak, this variation is robust since the $q_{x}$ variation changes sign when the current is reversed in the sliding area while remaining constant in the non-sliding one (see Fig. 6 in Supp. Mat.).

The most surprising maps is the one displaying the coordinate $q_{z}$, the direction transverse to the CDW (see Fig. 2 c-f). Unlike $q_{y}$ and $q_{x}$, the $q_{z}$ map is strongly different between the two reflections (see Fig. 4 in Supp. Mat.) and displays an unambiguous evolution with applied currents. At I=1mA, above the current threshold $I_{s}$, a clear distortion is visible in the sliding area. In this region, the amplitude of the shear deformation is 10 times larger ($\delta q_{z}\sim 10^{-3}\mathring{A}^{-1}$) than the longitudinal one ($\delta q_{x}\sim 10^{-4}\mathring{A}^{-1}$). Furthermore, the sign of $q_{z}$ changes with inverse currents: at I=-1mA, the deformation is opposite to the one at I=1mA (compare Fig. 2 d) and f)), proving that the distortion is not due to sample heating. Going back to 0mA, the CDW does not relax to its ground state as in Fig. 2(c)). Hysteresis effects are a common feature observed by resistivity measurements in several CDW compoundsPhysRevB.33.5858 ; MIHALY1984807 including NbSe3PhysRevB.26.2298 . In contrast, the non-sliding area located below the line cut remains constant for all currents.

The CDW phase has been recovered from the diffraction patterns by using a phase gradient approach. This analytical method was preferred to Bragg ptychography despite a weaker resolutioncha:hal-01397101 because it is more appropriate to map large areas from weak satellite reflection by using a weakly coherent beam, about 10 times more intense than a fully coherent beam, without overlap between steps. The CDW is described by a periodic lattice modulation $\rho(\vec{r})=A\cos(2k_{f}x+\phi(\vec{r}))$ where A is the CDW amplitude, $x$ is the direction parallel to the CDW and to the current direction (see Fig. 1). $\phi(\vec{r})$ is the phase that describes space-dependent CDW distortions. If we assume that the phase varies linearly between two steps, i.e. over 1$\mu m$, the phase at $\vec{r}=(x,z)$ can be locally expanded to the first order $\phi(\vec{r}+\delta\vec{r})\approx\phi(\vec{r})+\delta x\frac{\partial\phi}{\partial x}|_{\vec{r}}+\delta y\frac{\partial\phi}{\partial y}|_{\vec{r}}+\delta z\frac{\partial\phi}{\partial z}|_{\vec{r}}$. In that case, $q_{cdw}$ is nothing else but the phase gradientPhysRevB.84.144109 .

[TABLE]

This first order approximation is justified if an elastic model of CDW deformation is considered where abrupt variations at the micrometer scale can not appear. It is also validated by the relevance of the results obtained afterwards. The estimation of the error made by this integration method is described in the Supp. Mat. It is however obvious that abrupt CDW phase shifts like CDW dislocations can not be observed.

The phase $\phi(\vec{r})$ is obtained analytically by spatial integration of the measured $\delta q_{z}$. To avoid contribution of the crystal lattice distortion in the phase reconstruction (Fig. 3 and 4), we subtracted all 5D matrices with the one obtained at I = 0.15mA considered as the reference below the threshold.

Since $\delta q_{z}>>\delta q_{x}$, we can focus on the phase shear deformation $\phi(x,z)\approx\phi(z)$. The inset of Fig. 3 shows the variation profile $\delta q_{z}$ versus currents averaged over the $x$ direction. As expected from $q_{z}$ in Fig. 2., the CDW phase $\phi(z)$ remains constant for all currents in the non-sliding area ($z\in[0,17]$). In contrast, the sliding area ($z\in[19,39]$) displays continuous and large variations. Despite the presence of strong defects like the surface step (see Fig. 1(a)), the phase displays a remarkable continuity over 20$\mu m$, i.e. over 1.4 $10^{4}$ times the CDW wavelength ($\lambda=14\mbox{\AA}$). The $\phi(z)$ amplitude is maximum in the central part of the sliding region, reaching 20 CDW periods for I=-1mA. This CDW wavefront curvature seems impressive but it extends for 20$\mu m$, that is a phase shift $\Delta\phi(z)=2.8\ 10^{-3}(2\pi)$ per CDW period in average. The sign of the curvature depends of the current direction and the amplitude curvature increases with increasing currents. Furthermore, $\phi(z)$ converges to a constant value for all currents at the upper edge (z=39$\mu$m) showing CDW pinning at the two lateral surfaces. Note that this constant value is obtained without any constraint in the integration procedure.

The map of $\phi(z)$ as well as the corresponding CDW modulation is displayed in Fig.4 for two opposite currents and for all currents in Fig.7 in Supp. Mat. It emerges from this measurement 3 unexpected points: 1. The large shear effect, ten times larger than the longitudinal one. 2. The pinning from the the two lateral surfaces and 3. the continuous deformation of $\phi(z)$ over the whole sliding area despite the presence of strong defects, especially the surface step visible on the sample image (Fig.1a). This collective transverse deformation is even more striking at the the extreme left part of the map for x=30$\mu$m, where it starts to display a continuous deformation over the whole sample width, i.e. over 40 $\mu m$.

In order to explain the appearance of the CDW shear, we introduce the phase part of the free energy containing the interaction with an applied electric field PhysRevLett.68.2066 :

[TABLE]

where A is the CDW’s amplitude, $K_{i}$ are the anisotropic elastic constants, $\rho_{s}$ is the condensate density and $U$ the applied potential ($E=-\vec{\nabla}U$). Since the CDW in NbSe3 is incommensurate, the free energy does not depend on $\phi$ but only on its derivatives. The last term is related to the fact that any phase deformation along x induces a local charge density $-\frac{e\rho_{s}}{2k_{f}}\frac{\partial\phi}{\partial x}$ gruner_book . This last term differs in PhysRevLett.68.2066 .

When a positive electric field is applied (negative $U$ in Eq. (2)), $\partial\phi/\partial x<0$ is favorable corresponding to a compression of the CDW wavelength as observed by diffraction PhysRevB.61.10640 ; 0295-5075-56-2-289 . However, shear effects ($\partial\phi/\partial z\neq 0$) are not energetically favorable in Eq. (2)), unless one takes surface pinning into account. The only way to compress or expand a CDW ($\partial\phi/\partial x\neq 0$) while keeping the phase pinned on lateral surfaces is to add a shear deformation along z. The shear observed here is therefore nothing else but a consequence of the longitudinal CDW deformation and lateral surface pinning.

The strong shear effect measured here explains why several resistivity measurementssmall_o_TaS3 ; PhysRevB.46.4456 ; Size_dependent_threshold_fields showed that the threshold current strongly depends on the sample cross section. Finally, the nature of surface pinning remains unclear. A possible pinning mechanism may be due to small steps on the surface as proposed in Elastic_Feinberg but a commensurabity of the CDW at surfaces can not be excluded neither.

In conclusion, the understanding of solitons nucleation needs the observation of the CDW behavior under currents over a large enough area and a small enough resolution. For this, the CDW modulation, with ten angstroms period, has to be probed over hundred of micrometers with micrometer resolution. This has been achieved by coupling fast scanning x-ray microdiffraction and phase gradient method revealing that the CDW becomes independent of the host lattice, with a large CDW shear and a strong lateral pinning centers, while keeping its continuity over tens of micrometers. This dominant shear effect should plays an important role in the dynamics of solitons.