Evidence for weak collective pinning and δl pinning in topological superconductor Cu_xBi₂Se₃

Figure 1(a) shows the isothermal MHLs at 1.8 K for $B\parallel ab$ and $B\parallel c$. When the magnetic field approaches the upper critical field, a peak effect can be observed in both field configurations, but it is more remarkable for $B\parallel c$ than $B\parallel ab$ due to the superconducting anisotropy caused by its layered-structure. The observation of the peak effect close to ${{B}_{{\rm c}2}}$ indicates a very weak pinning in Cu_0.10Bi₂Se₃, that can be regarded as a nearly clean crystal. In cuprates, the peak effect can only be seen in ultra-clean crystals with a low level of pinning centers, for instance, in YBa₂Cu₃O_7−δ [24]. With increasing weak point disorders in clean YBa₂Cu₃O_7−δ by electron irradiation [24], the ${{J}_{{\rm c}}}(B)$ peak broadens accompanying with the shift of the peak position towards the intermediate fields away from ${{B}_{{\rm c}2}}$, which is often assigned to the second magnetization effect or fishtail effect that distinguishes the peak effect. The field dependence of ${{J}_{{\rm c}}}$ was estimated according to the extended Bean critical model, as shown in figures 1(b) and (c). Below 3.0 K, a series of pronounced ${{J}_{{\rm c}}}$ peaks appear in the high field region under $B\parallel c$. The magnitude of ${{J}_{{\rm c}}}$ is suppressed by the external field until a critical field of ${{B}^{*}}$  =  0.31 T, above which ${{J}_{{\rm c}}}$ rises again. By further increasing the field, the peak effect emerges at the $B_{{\rm p}}^{{\rm onset}}$ and reaches a cusp at ${{B}_{{\rm P}}}$. The same feature also exists for $B\parallel ab$ as shown in figure 1(c), but the ${{B}^{*}}$, $B_{{\rm p}}^{{\rm onset}}$, and ${{B}_{{\rm P}}}$ shift to higher field locations. Below 0.14 T, the ${{J}_{{\rm c}}}(B)$ curves obey a power-law relationship ${{J}_{{\rm c}}}\propto {{B}^{-n}}$ at 1.8 K, which is a signature of collective pinning [25, 26]. Above 0.14 T, the ${{J}_{{\rm c}}}$ deviates from the power law scaling, and the peak effect occurs at the $B_{{\rm p}}^{{\rm onset}}$. In a superconducting crystal, the quenched disorder can destroy the long-range order of the Abrikosov vortex lattices, forming a quasi-long range ordered Bragg glass in the case of weak disorders [22, 27]. The power-law behavior of ${{J}_{{\rm c}}}(B)$ implies the existence of a fairly dense distribution of weak pins in Cu_0.10Bi₂Se₃ [23]. In a weakly pinned state, the power-law behavior actually reflects the increase of the vortex lattice rigidity with the increasing field and is typical for a weakly-disordered vortex lattice in Larkin–Ovchinnikov' collective pinning theory. In high-${{T}_{{\rm c}}}$ cuprates, it should be noted that a similar power-law dependence with $n$  =  1/2 follows from the single-vortex pinning on large normal particles (dimension  >  $\xi$) [28] while $n$  =  −1 from the thermodynamic analysis of MHL scaling in a TmBa₂Cu₃O_6.8 crystal [29]. Based on the strong pinning theory [30], it is predicted that there is a crossover field ${{B}_{{\rm cr}}}$ in the ${{J}_{{\rm c}}}\left(B \right)$ curve. For $B<{{B}_{{\rm cr}}}$, the ${{J}_{{\rm c}}}\left(B \right)$ is nearly constant, while ${{J}_{{\rm c}}}\left(B \right)\propto {{B}^{-5/8}}$ for $B>{{B}_{{\rm cr}}}$ showing a temperature independent exponent. However, we observed a considerable change of $n$ as a function of temperature and no plateau was observed in the low field region, which is inconsistent with the power-law behavior of cuprates.

Moreover, the exponent $n$ tends to decrease with an increase in the temperature, e.g. $n$ changes from 0.88 to 0.40 within the temperature span of 1.8 K–3.0 K, which may be related to the flux creep effect. To elucidate this, the normalized relaxation/creep rate is determined either in the form of the dynamical relaxation rate [18], $Q={\rm dln}({{J}_{{\rm s}}})/{\rm dln}\left({\rm d}B/{\rm d}t \right)$, or the conventional magnetic relaxation rate, $S={\rm dln}(M)/{\rm dln}(t)$, where ${{J}_{{\rm s}}}$ is the superconducting current density (normally ${{J}_{{\rm s}}}\leqslant {{J}_{{\rm c}}}$) and $t$ is time. Both relaxation rates are fundamentally identical, i.e. $Q=S$, meaning the dependence of the MHL amplitude on the sweep rate contains the same information of the magnetization versus time during relaxation. As seen in figure SM2 (see supplementary materials for details, available at stacks.iop.org/JPhysCM/30/31LT01/mmedia), the temperature depend- ence of $S$ was obtained within the order of 10⁻²–10⁻³ for Cu_0.10Bi₂Se₃, which is much smaller than that of high-${{T}_{{\rm c}}}$ cuprates [18, 31]. This is understandable since the Ginzburg number ${{G}_{{\rm i}}}$, quantifying the effect of thermal fluctuations, is estimated to be ~10⁻⁵ for Cu_0.10Bi₂Se₃ using the reported penetration depth value [14], which is three orders of magnitude smaller than that of high ${{T}_{{\rm c}}}$ cuprates (10⁻²) [18], indicating quite a narrow critical fluctuation region. Meanwhile, the $S$ firstly tends to decrease up to 3.3 K before a small increase is observed. This seems to be consistent with the decrease of $n$ when increasing the temperature, which may be caused by a diminution of flux creep.

Figure 2 shows the temperature dependence of ${{J}_{{\rm c}}}$ at different temperatures. We fitted the ${{J}_{{\rm c}}}(T)$ by a widely adopted phenomenological formula ${{J}_{{\rm c}}}(T)={{J}_{{\rm c}}}\left(0 \right){{(1-T/{{T}_{{\rm c}}})}^{n}}$ with the exponent $n$  =  1–5/2 [32]. The best fitting result yields ${{J}_{{\rm c}}}\left(0 \right)$  =  1.89  ×  10² A cm⁻² at 0 K and $n$  =  1.70  ±  0.10. Note that the fitting exponent is very close to $n$  =  3/2 as predicted by single band GL theory [33]. As a result, this provides the critical current ratio of Cu_0.10Bi₂Se₃ in an order of magnitude of ${{J}_{{\rm c}}}/{{J}_{0}}$ ~ 10⁻⁵ in the limit of $B\to$ 0 T, which further demonstrates a very weak pinning regime in Cu_0.10Bi₂Se₃. Compared with other superconductors, the ratio of ${{J}_{{\rm c}}}/{{J}_{0}}$ for Cu_0.10Bi₂Se₃ is 3–4 orders of magnitude smaller than high-${{T}_{{\rm c}}}$ cuprates (~10⁻³–10⁻² for YBa₂Cu₃O_7−δ) and low-${{T}_{{\rm c}}}$ superconductors (~10⁻²–10⁻¹) with strong pinning [18]. According to the relation ${{J}_{{\rm c}}}\cong {{J}_{0}}{{(\xi /{{L}_{{\rm c}}})}^{2}}$, the length scale of ${{L}_{{\rm c}}}$ should be much larger than the coherence length $\xi$, ensuring the validity of elasticity theory. This allows us to discuss the influence of quenched disorder in terms of the weak collective pinning theory [23].

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In fact, the quenched disorder for vortex pinning can arise from various types of atomic defects in Cu_0.10Bi₂Se₃, as identified by scanning tunneling microscopy (STM) measurements [15–17]. Besides Se vacancies and Bi_Se antisites in pristine Bi₂Se₃, a number of new defects can be introduced by Cu doping [17], including type A: Cu_Bi substitution sites (in the second atomic layer in QtL), type B and C: Cu intercalation sites (H₃&T₁, T₄ sites between QtLs), and type D: Cu_Bi/Cu_Se substitution sites (the fourth/fifth atomic layer in QtL). Since the defects counts of type B and C determined by high-resolution topographs are significantly larger than the D defect count, they are more likely to occur in Cu_xBi₂Se₃. For superconducting Cu_xBi₂Se₃, the distance between the internal atomic defects may be within the superconducting coherence length (e.g. ${{\xi }_{ab}}\left(0 \right)$  =  13.96 nm for Cu_0.10Bi₂Se₃), hinting that a group of related defects is required to pin one single flux-lattice line collectively. On the other hand, these atomic defects mainly act as point pins because they have smaller dimensions than the vortex lattice constant ${{a}_{0}}=1.07{{\left({{\Phi }_{0}}/B \right)}^{1/2}}$, e.g. ${{a}_{0}}$  =  37.46 nm at $B_{{\rm c}2}^{\parallel c}(0)$  =  1.69 T. Importantly, the atomic defects are typically below 1 nm size, which is far less than ${{\xi }_{ab}}(0)$ and also confines them as weak point pins.

Since the Cu_xBi₂Se₃ is classified as one of the extremely large $\kappa$ superconductors [15], one can safely assume that its core interaction is dominant in vortex pinning dynamics. The source of vortex pinning is usually divided into two categories, $\boldsymbol{\delta l}$ pinning resulting from the spatial variations of charge carrier mean free path near normal particles or lattice defects and $\boldsymbol{\delta }{{\boldsymbol{T}}_{{\bf c}}}$ pinning from the spatial variations of the GL coefficient associated with fluctuations of the ${{T}_{{\rm c}}}$. On the basis of collective pinning theory [34], the temperature dependence of normalized ${{J}_{{\rm c}}}$ in a single vortex pinning regime can be described by $J_{{\rm c}}^{\delta l}\left(t \right)/J_{{\rm c}}^{\delta l}\left(0 \right)={{(1-{{t}^{2}})}^{5/2}}{{(1+{{t}^{2}})}^{-1/2}}$ for $\delta l$ pinning, and $J_{{\rm c}}^{\delta {{T}_{{\rm c}}}}\left(t \right)/J_{{\rm c}}^{\delta {{T}_{{\rm c}}}}\left(0 \right)={{(1-{{t}^{2}})}^{7/6}}{{(1+{{t}^{2}})}^{5/6}}$ for $\delta {{T}_{{\rm c}}}$ pinning, where $t=T/{{T}_{{\rm c}}}$. Correspondingly, the $g\left(t \right)=U(t)/U(0)$ functions characterizing the temperature dependence of the pinning energy are given by ${{g}^{\delta l}}\left(t \right)=1-{{t}^{4}}$ for $\delta l$ pinning, and ${{g}^{\delta {{T}_{{\rm c}}}}}\left(t \right)={{(1-{{t}^{2}})}^{1/3}}{{(1+{{t}^{2}})}^{5/3}}$ for $\delta {{T}_{{\rm c}}}$ pinning. Due to the presence of the giant flux creep in high-${{T}_{{\rm c}}}$ cuprates [31], the measured ${{J}_{{\rm s}}}$ values can differ markedly from the true ${{J}_{{\rm c}}}$ corresponding to that without thermally activated flux flow. Therefore, the ${{J}_{{\rm c}}}(T)$ should be first derived by the general inversion scheme (GIS) before applying the pinning regime analysis [31, 34]. Following this procedure, the randomly distributed weak pinning centers of the oxygen vacancies induced $\delta l$ pinning in YBa₂Cu₃O₇ & YBa₂Cu₄O₈ films [34] and a DyBa₂Cu₃O_7−δ single crystal [31]. Accordingly, to evaluate the relaxation effect on ${{J}_{{\rm c}}}$, we attempted to derive the dynamical relaxation rate $Q$ by measuring MHLs with different sweeping rates at 2 K. The ${{J}_{{\rm s}}}(B)$ curves are plotted in figure SM3. However, all the curves overlapped and meaningful field dependence of $Q$ was not obtained, implying a negligible relaxation effect on the true ${{J}_{{\rm c}}}$ values, i.e. ${{J}_{{\rm s}}}\cong {{J}_{{\rm c}}}$ satisfies in the experimental temperature region at least. This also assures that the previously yielded small ${{J}_{{\rm c}}}/{{J}_{0}}$ ratio sounds reasonable given the small relaxation effect in this material.

In figures 3(a) and (b), we plot the normalized ${{J}_{{\rm c}}}\left(t \right)$ and $g\left(t \right)$ data obtained at $B$  =  0.005 T, 0.01 T, 0.02 T, and 0.03 T, together with the theoretical curves of the $\delta l$ and $\delta {{T}_{{\rm c}}}$ pinning regimes. The experimental data follows the theoretical $\delta l$ pinning well, evidencing that the vortex pinning in Cu_0.10Bi₂Se₃ mainly originates from the spatial variations in the charge-carrier mean free path by lattice defects. As the most abundant atomic defects [15–17], the intercalated Cu atoms are probably more relevant to the weak pins. Here, the pinning regime in Cu_0.10Bi₂Se₃ is distinguished from the Co/Mn intercalated 2H–NbSe₂ [35], for which the effective pinning is suggested to come from the fluctuations of local ${{T}_{{\rm c}}}$ caused by Co/Mn magnetic impurities. However, no direct evidence of the magnetic order of the doped Cu atoms in Cu_xBi₂Se₃ has been reported yet [36].

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The collective-pinning theory has been successfully applied for pinning analysis in low ${{T}_{c}}$ type-II superconductors with randomly distributed weak pinning centers [25, 37, 38]. According to collective pinning theory [23], the volume pinning force ${{F}_{{\rm p}}}$ is written as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{F}_{{\rm p}}}=\left| {{J}_{{\rm c}}}\times B \right|={{\left(W/{{V}_{{\rm c}}} \right)}^{1/2}}\nonumber \end{align} \tag{ 1 }$$

where $W={{n}_{{\rm p}}}\left\langle\,f_{{\rm p}}^{2} \right\rangle$ is the collective pinning parameter as a measure of the pinning and ${{V}_{{\rm c}}}=R_{{\rm c}}^{2}{{L}_{{\rm c}}}$ is the correlation volume (${{n}_{{\rm p}}}$ is the volume density of the pinning centers, ${{f}_{{\rm p}}}$ is the elementary pinning interaction with an interaction range ${{r}_{f}}$). The longitudinal correlation length (${{L}_{{\rm c}}}$) and transverse correlation length (${{R}_{{\rm c}}}$) are defined by ${{L}_{{\rm c}}}={{({{c}_{44}}/{{c}_{66}})}^{1/2}}{{R}_{{\rm c}}}$ and ${{R}_{{\rm c}}}=(c_{44}^{1/2}c_{66}^{3/2}r_{f}^{2})/W$, where ${{c}_{44}}$ is the tilt modulus, and ${{c}_{66}}$ is the shear modulus. To characterize the configuration of the vortex matter [18], it is convenient to use the ratio ${{L}_{{\rm c}}}/d$ (where ${{L}_{{\rm c}}}={{\xi }_{ab}}{{\left({{J}_{0}}/{{J}_{{\rm c}}} \right)}^{1/2}}/ \Gamma$ is the Larkin scale, $~ \Gamma$ the anisotropy ratio, and $d$ sample thickness), i.e. an isotropic vortex structure is expected for ${{L}_{{\rm c}}}/d\gg 1$ with 2D collective pinning, whereas an anisotropic structure is expected for ${{L}_{{\rm c}}}/d\ll 1$ with 3D collective pinning [18, 23]. For Cu_0.10Bi₂Se₃, the Larkin scale is estimated to be ${{L}_{{\rm c}}}$  =  3.40 µm at 1.8 K, which is much smaller than the sample thickness. According to weak collective pinning theory [23], no peak effect appears for a 2D collective pinning case unless a dimensional crossover occurs to trigger it [37, 38]. So far, 2D collective pinning has been only demonstrated in micrometer-thin amorphous films or crystals [37, 38]. Hence, it is reasonable to suppose that 3D collective pinning applies in Cu_0.10Bi₂Se₃, which means it has an anisotropic vortex structure with flexible flux-lattice line configurations that favors a rapid relaxation of diamagnetic magnetization [21].

Calculating the values of ${{L}_{{\rm c}}}$ and ${{R}_{{\rm c}}}$ as a function of the magnetic field gives a deeper insight into vortex pinning dynamics. For anisotropic superconductors with $B\parallel c$ [39], the shear modulus is given by: $\newcommand{\e}{{\rm e}} {{c}_{66}}=\frac{B_{{\rm c}}^{2}}{{{\mu }_{0}}}\frac{b{{\left(1-b \right)}^{2}}}{4}\left(1-0.29b \right)\left[ \exp \left(\frac{b-1}{3{{\kappa }^{2}}b} \right) \right]\frac{\left(2{{\kappa }^{2}}-1 \right)2{{\kappa }^{2}}\beta _{0}^{2}}{{{\left[ \left(2{{\kappa }^{2}}-1 \right){{\beta }_{0}}+1 \right]}^{2}}}$ with $b=B/{{B}_{{\rm c}2}}$, where ${{B}_{{\rm c}}}(T)={{B}_{{\rm c}}}(0)\left(1-{{t}^{2}} \right)$ is thermodynamic critical field and ${{\beta }_{0}}=1.16$ for the triangular lattice. The dispersive tilt modulus is ${{\widehat{c}}_{44}}=\frac{{{B}^{2}}}{{{\mu }_{0}}}\left(\frac{1}{1+k_{z}^{2}{{\lambda }^{\prime 2}}+k_{\bot }^{2}{{\Gamma }^{2}}{{\lambda }^{\prime 2}}}+\frac{1}{k_{{\rm BZ}}^{2}{{\Gamma }^{2}}{{\lambda }^{\prime 2}}} \right)$ [40], where ${{k}_{\bot }}={{\left(k_{x}^{2}+k_{y}^{2} \right)}^{1/2}}\approx \pi /{{R}_{{\rm c}}}$ and ${{k}_{z}}\approx \pi /{{L}_{{\rm c}}}$ describe the deformation field normal and parallel to the magnetic field direction, $k_{{\rm BZ}}^{2}=4\pi B/{{\Phi }_{0}}$ with BZ denotes the Brillouin Zone of the vortex lattice of $\pi k_{{\rm BZ}}^{2}$, ${{\lambda }^{\prime }}={{\lambda }_{ab}}/{{(1-b)}^{1/2}}$ is the magnetic penetration depth, and ${{\Gamma }^{2}}={{M}_{z}}/M$ is the mass ratio where ${{M}_{z}}$ is the quasiparticle effective mass along the $c$-axis and $M$ is in the $ab$-plane. For extreme type-II superconductors, where $k_{{\rm BZ}}^{2}{{\lambda }^{\prime 2}}=2b{{\kappa }^{2}}/(1-b)\gg 1$, the shear ${{c}_{66}}$ and tilt ${{\widehat{c}}_{44}}$ moduli can be further simplified as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{c}_{66}}\approx \frac{B_{{\rm c}2}^{2}}{{{\mu }_{0}}}\frac{b{{\left(1-b \right)}^{2}}}{8{{\kappa }^{2}}}\left(1-0.29b \right)\nonumber \end{align} \tag{ 2 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\widehat{c}}_{44}}\approx \frac{{{B}^{2}}}{{{\mu }_{0}}}\frac{1}{1+{{\left(\frac{\pi \kappa {{\xi }_{ab}}}{{{L}_{{\rm c}}}} \right)}^{2}}\left(\frac{1}{1-b} \right)+{{\left(\frac{\pi \kappa {{\xi }_{ab}} \Gamma }{{{R}_{{\rm c}}}} \right)}^{2}}\left(\frac{1}{1-b} \right)}.\nonumber \end{align} \tag{ 3 }$$

Generally, the longitudinal correlation length ${{L}_{{\rm c}}}$ can be derived when the dispersion of ${{c}_{44}}$ is taken into account:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle &{{L}_{{\rm c}}}\approx {{\left(\frac{{{\widehat{c}}_{44}}}{{{c}_{66}}} \right)}^{1/2}}{{R}_{{\rm c}}} ={{\left\{\frac{8b{{\kappa }^{2}}}{\left[ 1+{{\left(\frac{\pi \kappa {{\xi }_{ab}}}{{{L}_{{\rm c}}}} \right)}^{2}}\left(\frac{1}{1-b} \right)+{{\left(\frac{\pi \kappa {{\xi }_{ab}} \Gamma }{{{R}_{{\rm c}}}} \right)}^{2}}\left(\frac{1}{1-b} \right) \right]{{\left(1-b \right)}^{2}}\left(1-0.29b \right)} \right\}}^{1/2}}{{R}_{{\rm c}}}.\nonumber \end{align} \tag{ 4 }$$

Next, we consider the derived expressions of the pinning parameter $W$, as defined by $W={{W}_{0}}\left(T \right){{b}^{n}}{{(1-b)}^{2}}$ [41], where $n$  =  1 for $\delta {{T}_{{\rm c}}}$ pinning, while $n$  =  3 for $\delta l$ pinning. In a single-vortex pinning regime [38], the parameter ${{W}_{0}}\left(T \right)$ at $T$  =  1.8 K can be determined by ${{W}_{0}}\left(T \right)\approx {{L}_{{\rm c}}}(0)J_{{\rm c}}^{2}(0){{\Phi }_{0}}{{B}_{{\rm c}2}}$  =  1.77  ×  10⁻⁹ N² m⁻³ for Cu_0.10Bi₂Se₃. In general⁴, we compute ${{L}_{{\rm c}}}$ and ${{R}_{{\rm c}}}$ by combining equation (1) with equation (4) to obtain the field dependence of ${{L}_{{\rm c}}}$ and ${{R}_{{\rm c}}}$.

We present the computed results of the field dependence of ${{L}_{{\rm c}}}$ and ${{R}_{{\rm c}}}$ and the ratios of ${{L}_{{\rm c}}}/d$, ${{R}_{{\rm c}}}/{{a}_{0}}$ in figures 4(a) and (b). Both ${{L}_{{\rm c}}}$ and ${{R}_{{\rm c}}}$ first tend to increase with the rising field. However, once reaching a maximum at ${{B}^{{\rm *}}}$  =  0.31 T, a rapid decrease of ${{L}_{{\rm c}}}$ and ${{R}_{{\rm c}}}$ subsequently follows with the further increasing field. We note that a curvature change of ${{L}_{{\rm c}}}(B)$ and ${{R}_{{\rm c}}}(B)$ appears at the $B_{{\rm p}}^{{\rm onset}}$. Above ${{B}^{{\rm *}}}$, the ${{V}_{{\rm c}}}$ start to collapse dramatically while the $W$ actually tends to increase in a broad range of 0–${{B}_{{\rm p}}}$ (see supplementary materials for details). A simultaneous faster decay of the elastic energy (${{E}_{{\rm el}}}$) than the pinning energy (${{E}_{{\rm p}}}$) appears, which results in a significant increase of ${{F}_{{\rm p}}}$ and ${{J}_{{\rm c}}}$ near ${{B}_{{\rm c}2}}$, creating the peak effect. In general, the ${{J}_{{\rm c}}}$ value of weak pinning superconductors decreases with a field increase up to the $B_{{\rm p}}^{{\rm onset}}$, above which ${{J}_{{\rm c}}}$ shows a significant increase due to the enhanced pinning force. Surprisingly, the collapse of ${{L}_{{\rm c}}}$ and ${{R}_{{\rm c}}}$ starts at ${{B}^{{\rm *}}}$, corresponding to the minimum of ${{J}_{{\rm c}}}(B)$, instead of the $B_{{\rm p}}^{{\rm onset}}$. This differs from the behavior of ${{L}_{{\rm c}}}$ and ${{R}_{{\rm c}}}$ in Ca₃Rh₄Sn₁₃ and 2H–NbSe₂ [26, 38], for which the drastic decline of ${{L}_{{\rm c}}}$ and ${{R}_{{\rm c}}}$ occurs at the $B_{{\rm p}}^{{\rm onset}}$. Figure 4(b) shows the magnetic field dependence of ${{L}_{{\rm c}}}/d$ and ${{R}_{{\rm c}}}/{{a}_{0}}$ at 1.8 K. Both ${{L}_{{\rm c}}}/d$ and ${{R}_{{\rm c}}}/{{a}_{0}}$ begin to decrease between ${{B}^{*}}$ and ${{B}_{{\rm p}}}$, and ${{R}_{{\rm c}}}/{{a}_{0}}$  →  1.85 when it approaches ${{B}_{{\rm p}}}$. It should be highlighted that ${{L}_{{\rm c}}}/d$ starts to decline exactly at ${{B}^{*}}$ since $d$ is constant. Nevertheless, due to the field dependence of ${{a}_{0}}$, ${{R}_{{\rm c}}}/{{a}_{0}}$ starts decreasing at a slightly higher field than ${{B}^{*}}$. Our results indicate that the peak effect of Cu_0.10Bi₂Se₃ can be interpreted as the softening of the elastic moduli in the framework of 3D collective pinning theory.

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Finally, we will discuss some technical application implications drawn from our first observation of the peak effect in the topological superconductor Cu_xBi₂Se₃. Firstly, the normal core of vortices in Cu_xBi₂Se₃ can host MFs applied in fault-tolerant quantum computing [2, 3, 11]. To detect the surface MFs in Cu_xBi₂Se₃ [11], a sample with a Cu doping content ${{x}_{{\rm c}}}$  <  0.12 is preferred as it can shift the chemical potential below a critical ${{\mu }_{{\rm c}}}$ ~ 0.24 eV for $B\parallel c$. This makes the present Cu_0.10Bi₂Se₃ single crystal a promising host candidate for MFs in the vortices core when the external field is fixed to or tilted from the $B\parallel c$ configuration. However, the complex vortex pinning dynamics reported here put constraints on the detection of MFs in the various vortex phases controlled by external fields [42]. In this context, whether the MFs remain robust in various magnetic fields strengths needs further theoretical and experimental exploration. Secondly, as a weakly pinned type-II superconductor Cu_xBi₂Se₃, like 2H–NbSe₂ [43], can in principle provide the basis of a binary device used as a magnetic memory cell. Thirdly, our work indicates that the atomic defects (e.g. Cu intercalants) result in a $\delta l$ pinning regime. The atomic defects can induce a local point potential to pin vortices at sites, and this vortex pinning effect is both theoretically and practically interesting. Alternatively, one can utilize the movable atomic-scale tip of an atomic force microscope to generate the point potential [13]. When the tip approaches the vortex core and subsequently moves slowly away from the vortex center, the vortex can be dragged along with the point potential, which offers an effective way of controlling vortices in a topological superconductor. This implies a possible means of braiding MFs by manipulating the pinning centers. Thus, Cu_xBi₂Se₃ is a promising candidate material for examining the theoretical scheme.