⇦Chinese Physics Letters, 2023, Vol. 40, No. 3, Article code 037403 Critical Current Density, Vortex Pinning, and Phase Diagram in the NaCl-Type Superconductors InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2) Linchao Yu (于林超)1†, Song Huang (黄嵩)1†, Xiangzhuo Xing (邢相灼)1,2*, Xiaolei Yi (易晓磊)3, Yan Meng (孟炎)4, Nan Zhou (周楠)5, Zhixiang Shi (施智祥)6*, and Xiaobing Liu (刘晓兵)1,2* Affiliations 1School of Physics and Physical Engineering, Qufu Normal University, Qufu 273165, China 2Advanced Research Institute of Multidisciplinary Sciences, Qufu Normal University, Qufu 273165, China 3College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000, China 4Department of Physics, Jining University, Qufu 273155, China 5Key Laboratory of Materials Physics, Institute of Solid State Physics, HFIPS, Chinese Academy of Sciences, Hefei 230031, China 6School of Physics, Southeast University, Nanjing 211189, China Received 3 February 2023; accepted manuscript online 22 February 2023; published online 7 March 2023 †These authors contributed equally to this work.
*Corresponding authors. Email: xzxing@qfnu.edu.cn; zxshi@seu.edu.cn; xiaobing.phy@qfnu.edu.cn Citation Text: Yu L C, Huang S, Xing X Z et al. 2023 Chin. Phys. Lett. 40 037403    Abstract Research of vortex properties in type-II superconductors is of great importance for potential applications and fundamental physics. Here, we present a comprehensive study of the critical current density $J_{\rm c}$, vortex pinning, and phase diagram of NaCl-type InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2) superconductors synthesized by high-pressure technique. Our studies reveal that the values of $J_{\rm c}$ calculated by the Bean model exceed $10^{4}$ A/cm$^{2}$ in the InTe$_{1- x}$Se$_{x}$ system, signifying good potential for applications. The magnetic hysteresis loops (MHLs) show an asymmetric characteristic at various degrees, which is associated with the surface barrier. Intriguingly, a rare phenomenon in which the second magnetization peak in the MHLs occurs only in the field-descending branch is detected in InTe$_{0.9}$Se$_{0.1}$. Such an anomalous behavior has not been observed before and can be described by considering the respective roles of the surface barrier and bulk pinning in the field-ascending and field-descending branches. By analyzing the pinning force density versus reduced field, the pinning mechanisms are studied in detail in the framework of the Dew-Hughes model. Finally, combining the results of resistivity and magnetization measurements, the vortex phase diagrams are constructed and discussed.
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 DOI:10.1088/0256-307X/40/3/037403 © 2023 Chinese Physics Society Article Text Topological superconductors have attracted extensive research interest because of the their potential applications of exploiting the non-Abelian statistics of Majorana state for quantum computation devices.[1] Among the candidate materials, a typical example is Sn$_{1- x}$In$_{x}$Te[2-5] that derives from the topological crystalline insulator SnTe with a NaCl-type structure.[6,7] By elemental In doping, the superconducting transition temperature $T_{\rm c}$ is significantly increased from 0.3 K in pristine SnTe to 4.5 K in Sn$_{1- x}$In$_{x}$Te with $x \sim 0.45$;[3] while for $x > 0.5$, In is no longer soluble in the NaCl-type SnTe structure, and amounts of InTe with tetragonal structure are formed instead.[3] So far, various measurements[8-12] have performed to detect the nature of topological superconductivity, but no consensus has been reached in Sn$_{1-x}$In$_{x}$Te. It is known that research of the vortex properties in type-II superconductors is always of great importance for potential applications and fundamental physics. To assess the potential applications of Sn$_{1-x}$In$_{x}$Te, Wang et al. initially studied the critical current density $J_{\rm c}$ and vortex phase diagram of bulk Sn$_{1-x}$In$_{x}$Te with $x \sim 0.45$,[13] indicative of the potential excellent performance in the application of superconducting electronics like single-photon detectors. Unfortunately, the knowledge of the vortex properties in the solubility limit region ($x>0.5$) is still lacking until now, due to difficulties in synthesizing samples under ambient conditions. Recently, Kobayashi et al. have proposed that the solubility limit can be bypassed using high-pressure (HP) technique, allowing the entire range from $x = 0$ to $x = 1$ to be available.[14] Superconductivity is found to be strongly robust against In doping, and the end member InTe with a NaCl-type structure also shows superconductivity with $T_{\rm c}$ of 3–3.5 K.[15,16] To get more valuable insight into the potential applications and the possible novel vortex behavior, it is highly desirable to demonstrate the vortex properties of the In-rich compounds of Sn$_{1-x}$In$_{x}$Te systems. In this Letter, we focus on the end member InTe and its Se-codoped analogs synthesized by HP technique to study the critical current density, pinning mechanism, and vortex phase diagram. It is found that the magnetic hysteresis loops (MHLs) exhibit an asymmetric behavior at various degrees, especially for InTe$_{0.9}$Se$_{0.1}$ in which surface barrier is dominant on the vortex dynamic. The second magnetization peak (SMP) in the MHLs is detected in InTe$_{0.9}$Se$_{0.1}$ but only occurs in the field-descending branch. Such a rare phenomenon can be ascribed to the respective roles of the surface barrier and bulk pinning in the field-ascending and field-descending branches. The values of $J_{\rm c}$ exceed $10^{4}$ A/cm$^{2}$ for InTe$_{1-x}$Se$_{x}$, and details on the pinning mechanisms are studied by analyzing the pinning force density in the framework of the Dew-Hughes model. Finally, combining the resistivity and magnetization data, the vortex phase diagrams are also constructed. Experimental Details. The NaCl-type InTe$_{1-x}$Se$_{x}$ ($x=0$, 0.1, 0.2) superconductors were synthesized using a cubic-anvil-type HP apparatus. First, stoichiometric amounts of In, Te, and Se grains were sealed in an evacuated quartz tube and heated at 900 ℃ for 24 h. Then, the resultant sample was ground and pressed into a pellet, which was then placed in a BN crucible and assembled into an HP cell. The cell was heated at 700 ℃ for 2 h under a pressure of 3.5 GPa, followed by rapid cooling to room temperature. The NaCl-type crystal structure was checked by x-ray diffraction (XRD) measurement using a commercial Rigaku diffractometer with Cu $K\alpha$ radiation. Elemental analysis was made by a scanning electron microscope equipped with an energy dispersive x-ray (EDX) spectroscopy probe. The electrical transport measurement was carried out on a Quantum Design physical property measurement system (PPMS) with a standard four-probe technique. Magnetization measurements were performed using the VSM (vibrating sample magnetometer) option of PPMS. It is noted that both resistivity and magnetization measurements were performed on the same sample for each composition. Results and Discussion. Figure 1(a) presents the powder XRD patterns for synthesized InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2) samples. All the diffraction peaks can be well indexed by a NaCl-type structure with space group $Fm\bar{3}m$. The lattice constant of $a$ shows monotonous decrease as 6.163(3) Å, 6.139(2) Å, and 6.108(3) Å for InTe, InTe$_{0.9}$Se$_{0.1}$, and InTe$_{0.8}$Se$_{0.2}$, respectively. The monotonous decrease of lattice constant indicates the successful incorporation of Se atoms with smaller ionic radius into the InTe crystal structure. Our further experiments reveal that the Se solubility limit is at around $x \sim 0.2$ under HP conditions, while extra elemental Se atoms appear as impurity phases in our produced samples with $x > 0.2$ (see Fig. S1 in the Supplementary Materials).
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Fig. 1. (a) Powder XRD patterns of InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2). Inset shows the schematic crystal structure of NaCl-type InTe$_{1- x}$Se$_{x}$. (b) EDX spectrum (main panel) and compositional mappings (right top panel) of In, Te, and Se in the selected rectangular region for a representative sample with $x = 0.2$. (c) Temperature dependence of the normalized resistivity, $\rho /\rho_{_{\scriptstyle 5.5\,{\rm K}}}$, for InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2). (d) Temperature dependence of ZFC and FC magnetization for InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2) under 10 Oe.
The stoichiometry composition of three InTe$_{1- x}$Se$_{x}$ samples was determined by EDX measurements. A representative EDX spectrum for InTe$_{0.8}$Se$_{0.2}$ is shown in the main panel of Fig. 1(b). It is clear that all elements including In, Te, and Se are detected. The right top panel shows the compositional mappings of InTe$_{0.8}$Se$_{0.2}$, signifying that In, Te, and Se are almost homogeneously distributed. For each sample, about 10–15 different spots were selected in the EDX measurements, and the actual stoichiometry composition is found to be very close to the nominal one. Henceforth, we will refer to the samples by the nominal Se content throughout this study. Figure 1(c) shows the temperature dependence of the normalized resistivity $\rho /\rho_{_{\scriptstyle 5.5\,{\rm K}}}$ for the three InTe$_{1- x}$Se$_{x}$ samples at low temperatures. All the samples exhibit sharp superconducting transitions. With Se doping, the onset of transition temperature $T_{\rm c}$ determined by the criteria of 90% of $\rho /\rho_{_{\scriptstyle 5.5\,{\rm K}}}$ is gradually increased from 3.28 K in InTe, 4.17 K in InTe$_{0.9}$Se$_{0.1}$, to 4.65 K in InTe$_{0.8}$Se$_{0.2}$. Such an enhancement of superconductivity is opposite to that tuned by physical pressure, i.e., $T_{\rm c}$ decreases with the application of pressure.[16] Superconductivity is further confirmed by the magnetization measurements. Figure 1(d) shows the temperature dependence of the zero-field-cooled (ZFC) and field-cooled (FC) magnetization under an applied magnetic field of 10 Oe. The obtained $T_{\rm c}$, which is determined by the onset of diamagnetism, is in good agreement with the resistivity data. The calculated superconducting shielding volume fraction after considering the demagnetization effect is around 100%, indicating the bulk nature of superconductivity in InTe$_{1- x}$Se$_{x}$. Figure 2(a) shows the temperature dependence of resistivity under various applied fields for InTe$_{0.9}$Se$_{0.1}$. With increasing field, the superconducting transition gradually shifts to lower temperatures and the width of the transition broadens. We determine the upper critical field $H_{\rm c2}$ by taking the points where the resistivity drops to 90% of $\rho /\rho_{_{\scriptstyle 5\,{\rm K}}}$, as insulated by the dashed blue line. The thus-determined $H_{\rm c2}$ as a function of temperature is plotted in Fig. 2(b), in which the corresponding data of InTe and InTe$_{0.9}$Se$_{0.1}$ are also included. The solid curves represent the fit using the modified Ginzburg–Landau (G-L) model:[17] $H_{\rm c2}(T)=H_{\rm c2}(0)\frac{[1-{(T / T_{\rm c})}^{2}]^{\alpha}}{[1+{(T / T_{\rm c})}^{2}]^{\beta}}$. The fitted $H_{\rm c2}$(0) is estimated to be 4.09 kOe for InTe, 7.03 kOe for InTe$_{0.9}$Se$_{0.1}$, and 9.22 kOe for InTe$_{0.8}$Se$_{0.2}$. All these values are much smaller than the corresponding Pauli limit ($H_{\rm p}(0)=1.84T_{\rm c}$), indicating that the orbital effects are dominant here. The superconducting coherence length $\xi$ of InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2) can be calculated by the relation $H_{\rm c2}$(0) = $\varPhi_{0}$/(2$\pi \xi^{2}$), where $\varPhi_{0}$ is the magnetic flux quantum, as listed in Table 1.
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Fig. 2. (a) Behavior of $\rho /\rho_{_{\scriptstyle 5\,{\rm K}}}$ as a function of temperature under various applied fields ranging from 0 to 6 kOe for InTe$_{0.9}$Se$_{0.1}$. (b) Temperature dependence of the upper critical field $H_{\rm c2}$ for InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2). (c) Initial magnetization curves at different temperatures for a representative sample InTe$_{0.9}$Se$_{0.1}$. The dashed line $(M_{\rm lin}(H)$, Meissner line) is the linear fitting curve at low fields. (d) Temperature dependence of the lower critical field $H_{\rm c1}$ for InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2). The solid lines in (b) and (d) represent the theoretical fitting curves (see text for details).
Table 1. Summary of the superconducting parameters of InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2).
Sample $T_{\rm c}$ (K) $H_{\rm c1}$(0) (Oe) $H_{\rm c2}$(0) (kOe) $\xi$ (nm) $\lambda$ (nm) $\kappa = \lambda /\xi$ $J_{{\rm c}, H =0}(T = 0$ K) (A/cm$^{2}$)
$x = 0$ 3.28 108.5 4.09 28.38 162.87 5.74 $2.77 \times 10^{4}$
$x = 0.1$ 4.17 99.1 7.03 21.65 190.02 8.78 $1.9 \times 10^{4}$
$x = 0.2$ 4.65 110.1 9.22 18.90 184.66 9.77 $3.83 \times 10^{4}$
We also investigated the lower critical field $H_{\rm c1}$ by measuring the initial magnetization curves, as depicted in Fig. 2(c) for a representative InTe$_{0.9}$Se$_{0.1}$ sample. The black dashed line (also called the Meissner line) represents the linear fit at low fields due to the Meissner effect. The penetration field $H_{\rm c1}^*$ (marked by the arrow) at which the vortex starts to enter into the sample is determined from the deviation of the curves from initial Meissner states, details can be found in Fig. S3 in the Supplementary Materials. For a rectangular shaped sample, the actual value of $H_{\rm c1}$ can be deduced from $H_{\rm c1}^*$ when taking the demagnetization effect into account as proposed by Brandt,[18] which is given by $H_{\rm c1}=H_{\rm c1}^{\ast} / {\tanh\sqrt {0.36b/a}}$, where $a$ and $b$ are the width and the thickness of the samples, respectively. Using this relation, the corrected values of $H_{\rm c1}$ as a function of temperature are plotted in Fig. 2(d). The data can be well fitted by the G-L equation $H_{\rm c1}(T)=H_{\rm c1}(0)[1-{(T/T_{\rm c})}^{2}]$, yielding the value of $H_{\rm c1}$(0), i.e., 108.5 Oe for InTe, 99.1 Oe for InTe$_{0.9}$Se$_{0.1}$, and 110.1 Oe for InTe$_{0.8}$Se$_{0.2}$. Using the expression $H_{\rm c1}(0)= \frac{\varPhi_{0}}{4\pi \lambda^{2 }}\ln (\frac{\lambda }{\xi })$ and the coherence length $\xi$ derived above, the magnetic penetration depth $\lambda$ is also calculated and summarized in Table 1. Figures 3(a) and 3(c) show the global isothermal MHLs for InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2), exhibiting typical behavior of type-II superconductors. The shape of MHLs for InTe displays a characteristic anisotropy about the $x$-axis, which is the signature of vortex dynamics contributed by surface barriers.[19] Clearly, such anisotropy behavior is much more significant in InTe$_{0.9}$Se$_{0.1}$ shown in Fig. 3(b), indicating the dominance of surface barrier and relatively low bulk pinning. In the field-ascending branch, the surface barrier inhibits vortex entry and magnetizations at various temperatures are quite different. While in the field-descending branch, the vortex floods out with negligible barriers and magnetization stays a low value until a certain remnant field is reached. With further doping, the MHLs of InTe$_{0.8}$Se$_{0.2}$ become almost symmetric, which means the dominance of bulk pinning and the weakness of surface barrier. Additionally, a pronounced SMP that has been widely observed in many type-II superconductors[20-24] shows up above 2.5 K in InTe$_{0.9}$Se$_{0.1}$, as shown in Fig. 3(d). Thus far, various theoretical models or pictures have been proposed to explain this phenomenon, including a crossover from elastic to plastic ($E$–$P$) vortex creep,[21] vortex order to disorder transition,[22] and vortex lattice structure phase transition,[23] etc. Among them, the picture of $E$–$P$ creep crossover is widely adopted[21] and has been studied in detail in our previous reports.[20,25] As usual, the SMP shifts to lower fields with increasing temperature. Surprisingly, the SMP observed in InTe$_{0.9}$Se$_{0.1}$ only exists in the field-descending branch, distinctly different from the common cases in which the SMP occurs in both the field-ascending and field-descending branches. To the best of our knowledge, such an anomalous SMP behavior has not been observed before in any other superconductors, the origin of which will be discussed in the following. From the MHLs presented in Fig. 3, the critical current density $J_{\rm c}$ can be calculated using the Bean model $J_{\rm c} = 20\Delta M/[a(1-a/3b)]$, where $\Delta M$ is $M_{\rm down}-M_{\rm up}$, $M_{\rm up}$ [emu/cm$^{3}$] and $M_{\rm down}$ [emu/cm$^{3}$] are the magnetization when sweeping fields up and down, respectively, $a$ [cm] and $b$ [cm] are sample widths ($a < b$).[26] Figures 4(a)–4(c) show the field dependence of $J_{\rm c}$ in double-logarithmic scales for InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2). At low temperatures, $J_{\rm c}$ changes slightly at low fields, which is associated with single-vortex region. The self-field $J_{{\rm c}, H =0}$ at $T = 1.9$ K is estimated to be $1.14 \times 10^{4}$ A/cm$^{2}$ for InTe, $1.05 \times 10^{4}$ A/cm$^{2}$ for InTe$_{0.9}$Se$_{0.1}$, and $2.23 \times 10^{4}$ A/cm$^{2}$ for InTe$_{0.8}$Se$_{0.2}$. The values exceeding $10^{4}$ A/cm$^{2}$ are higher than that of Sn$_{0.55}$In$_{0.45}$Te,[13] and comparable to those of iron chalcogenide superconductors FeSe,[27] FeS,[28] and FeTe$_{0.33}$Se$_{0.67}$.[29] Figure 4(d) presents $J_{{\rm c}, H =0}$ as a function of reduced temperature $t=T/T_{\rm c}$. $J_{{\rm c}, H =0}$ at zero temperature can be extracted by linear fitting of the experimental data, as listed in Table 1. $J_{{\rm c}, H =0} (T = 0$ K) first slightly decreases and then increases with Se doping. The lower $J_{{\rm c}, H =0}(T = 0$ K) for InTe$_{0.9}$Se$_{0.1}$ compared with InTe indicates the relatively weak pinning despite its higher $T_{\rm c}$, consistent with the asymmetric MHLs with low bulk pinning.
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Fig. 3. MHLs measured at different temperatures for (a) InTe, (b) InTe$_{0.9}$Se$_{0.1}$, and (c) InTe$_{0.8}$Se$_{0.2}$. (d) Enlarged view of the MHLs of InTe$_{0.9}$Se$_{0.1}$ in the rectangular area shown in (b), where the SMP can be clearly seen in the field-descending branches.
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Fig. 4. Magnetic field dependence of critical current densities for (a) InTe, (b) InTe$_{0.9}$Se$_{0.1}$, and (c) InTe$_{0.8}$Se$_{0.2}$. (d) Critical current densities under zero field $J_{{\rm c}, H =0}$ versus reduced temperature for InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2). The dashed lines are the linear fit to extrapolate the values of $J_{{\rm c}, H =0}$ at zero temperature.
To gain more insight into the pinning mechanisms in InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2), we analyze the scaling behavior of pinning force density $F_{\rm p}=\mu_{0}H\times J_{\rm c}$. Figures 5(a)–5(c) show the normalized pinning force density $f_{\rm p}=F_{\rm p}/F$ peak p as a function of reduced magnetic field $h=H/H_{\rm irr}$ at different temperatures, where the $F$ peak p is the maximum pinning force density and the irreversibility field $H_{\rm irr}$ is defined as the field at which $J_{\rm c}(H)$ extrapolates to zero from the field closure of MHLs. As can be seen, the normalized $f_{\rm p}$–$h$ curves are basically collapsed into a single curve, and two peaks denoted by $h_{\rm p1}$ and $h_{\rm p2}$ are present in InTe$_{0.9}$Se$_{0.1}$ and InTe$_{0.8}$Se$_{0.2}$. It is noted that the $f_{\rm p}$ of InTe$_{0.9}$Se$_{0.1}$ and InTe$_{0.8}$Se$_{0.2}$ are normalized by the high field peak $h_{\rm p2}$ for the convenience of later analysis. As proposed by Dew-Hughes,[30] in the scenario of single vortex pinning mechanism, the scaling of $f_{\rm p}$–$h$ can be well described in terms of the relation $f_{\rm p}\propto h^{p}(1-h)^{q}$, in which the fitted parameters $p$ and $q$ as well as the peak position defined by $h_{\rm p}=p/(p+q)$ can provide information about the nature of pinning. The values of $p$ and $q$ depend on the dimensionality of the pinning defect (point, surface, or bulk), the type of interaction (core pinning or magnetic pinning), and the type of pinning center (normal or superconducting).[30] In general, the nature of pinning for type-II superconductors is classified into two categories, $\delta l$ and $\delta \kappa$ pinning, which arise from the variation in the charge carrier mean free path $l$ and the variation in the superconducting transition temperature $T_{\rm c}$, respectively.[31] According to the Dew-Hughes model,[30] it is known that a low value of $h_{\rm p}$ ($\le 0.33$) indicates a $\delta l$ pinning whereas the higher value of $h_{\rm p}$ ($> 0.5$) corresponds to $\delta \kappa$ pinning.
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Fig. 5. Normalized pinning force density $f_{\rm p}=F_{\rm p}/F$ peak p as a function of reduced magnetic field $H/H_{\rm irr}$ at different temperatures for (a) InTe, (b) InTe$_{0.9}$Se$_{0.1}$, and (c) InTe$_{0.8}$Se$_{0.2}$. The blue dashed lines represent fits to the formula $h^{p}(1-h)^{q}$. The arrows denote the peak and kink positions.
As shown in Fig. 5(a), the data for InTe in the low field regime ($h < 0.43$) can be well described by the scaling function $h^{p}(1-h)^{q}$, which yields $p = 1.36$, $q = 2.95$, and the peak position $h_{\rm p} = 0.32$ as obtained from the experimental data. The extracted parameters are close to the theoretical value that corresponds to the normal point core pinning ($\delta l$ pinning).[30] With increasing fields, the experimental data deviates from the fitting curve and a kink located at $h_{\rm k} \sim 0.6$ appears, similar to the FeSe system,[32] demonstrating that the pinning mechanism at high fields is simultaneously contributed by the $\delta \kappa$ pinning. For InTe$_{0.9}$Se$_{0.1}$, a broad peak also located at $h_{\rm p2} \sim 0.6$ is seen. Fitting the high field data, we obtain $p = 1.6$, $q = 1.25$, and $h_{\rm p} = 0.58$, roughly consistent with the expected value $p = 1.5 q = 1$, and $h_{\rm p} = 0.6$ for $\delta \kappa$ pinning.[30] Moreover, except for the broad peak at $h_{\rm p2} \sim 0.6$, a narrow peak at low fields around $h_{\rm p1} \sim 0.1$ is also detected, as shown in Fig. 5(b). These narrow peaks at low fields have also been seen in our previous work on Sc$_{5}$Ir$_{4}$Si$_{10}$,[33] where it was ascribed to the surface barrier. In InTe$_{0.9}$Se$_{0.1}$, the very asymmetric MHLs shown in Fig. 3(b) have also manifested that surface barriers dominate the MHLs at low fields. Meanwhile, since the bulk pinning is very weak, especially at higher temperatures, the effect of surface barrier becomes more dominant. As a consequence, the low field peaks can be easily seen at higher temperatures, as shown in Fig. 5(b). The absence of such low field peaks in InTe may be related to the strong bulk pinning, which surpasses the surface barrier and leads to the submergence of low field peaks. With further doping in InTe$_{0.8}$Se$_{0.2}$, the low field peaks are still survived but significantly suppressed due to the weakness of surface barrier, as shown in Fig. 5(c). Also, the high field broad peak $h_{\rm p2}$ shifts to lower fields at around 0.32–0.37, signifying the change of vortex pinning. Fitting with scaling function $h^{p}(1-h)^{q}$ gives $p = 0.52$, $q = 0.94$, and $h_{\rm p} = 0.36$, very close to the theoretical $p = 0.5$, $q = 1$, $h_{\rm p} = 0.33$ for magnetic normal volume pinning ($\delta l$ pinning).[30] It should be noted that the poor scaling for certain curves at high fields and high temperatures in InTe$_{0.9}$Se$_{0.1}$ and InTe$_{0.8}$Se$_{0.2}$ should be resulted from the competition between surface barrier and bulk pinning.
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Fig. 6. Vortex phase diagrams of (a) InTe, (b) InTe$_{0.9}$Se$_{0.1}$, and (c) InTe$_{0.8}$Se$_{0.2}$.
Based on the resistivity and magnetization studies, the vortex phase diagrams of InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2) are depicted in Figs. 6(a)–6(c). The irreversibility field $H_{\rm irr}$ is defined by the magnetic field at which $J_{\rm c}(H)$ extrapolates to zero, and the lower critical field $H_{\rm c 1}$ and upper critical field $H_{\rm c 2}$ have been discussed in detail in Fig. 2. $H_{\rm sp}$ in Fig. 6(b) is the magnetic field at which the SMP occurs in the field-descending branch shown in Fig. 3(d). For InTe and InTe$_{0.8}$Se$_{0.2}$, four different regions are clearly seen: (1) Meissner state, a perfect diamagnetic state below $H_{\rm c1}$; (2) vortex solid state, which holds between $H_{\rm c1}$ and $H_{\rm irr}$ and is characterized by a nonzero critical current density; (3) vortex liquid state, the region between the $H_{\rm irr}$ line and $H_{\rm c2}$ line, a state that transforms from the vortex solid state and is dissipative at all currents due to the vortex flow; (4) normal state, the region above $H_{\rm c2}$. For InTe$_{0.9}$Se$_{0.1}$, in addition to the regions present in InTe and InTe$_{0.8}$Se$_{0.2}$, the vortex solid state is further divided into two regions with different vortex dynamics, i.e., elastic and plastic creeps, by the $H_{\rm sp}$ line associated with the $E$–$P$ crossover. It is noted that the SMP in InTe$_{0.9}$Se$_{0.1}$ only exists in the field-descending branch, in contrast to that observed in other superconductors. Here we have a discussion on this phenomenon. For a system with rather asymmetric MHLs, the surface barrier pinning is dominant especially in the field-ascending branch where the vortex entry is inhibited but has less effect on the vortex exit when the field is decreased. Hence, if the bulk pinning exists, it would mainly contribute to the vortex dynamics in the field-descending branch. In view of this, the reason why SMP in InTe$_{0.9}$Se$_{0.1}$ only occurs in the field-descending branch can be well understood. As mentioned above, the occurrence of SMP is intimately related to the $E$–$P$ crossover.[20,21] Meanwhile, in the field-descending branch, the vortex dynamics mainly depends on the nature of bulk pinning in InTe$_{0.9}$Se$_{0.1}$. In such a case, the SMP could also occur as long as the condition for $E$–$P$ crossover is satisfied, despite the bulk pinning is relatively low. While in the field-ascending branch, the vortex dynamic is mainly dominated by surface barrier, and naturally the SMP associated with the $E$–$P$ crossover is difficult to be observed. Moreover, the SMP are smeared out below 2.5 K. As for this point, we would like to emphasize that $E$–$P$ vortex phase transition is a necessary condition but not sufficient for the occurrence of SMP, as we argued before.[20,25] Near the $E$–$P$ phase transition region, as the magnetic field increases the elastic vortex lines soften and vortices can be pinned more easily by additional pinning centers, resulting in the increase of $J_{\rm c}$. On the other hand, the elementary pinning force for individual pinning center also becomes weak, leading to the possible reduction of the sum of elementary pinning force. Therefore, only when the increase of $J_{\rm c}$ contributed by additional pinning centers exceeds the decrease of $J_{\rm c}$ due to the loss of elementary pinning force, the SMP occurs. Otherwise, the SMP is lost even the $E$–$P$ crossover takes place. On the basis of this picture, the absence of SMP at low temperatures in InTe$_{0.9}$Se$_{0.1}$ can be well understood. In summary, we have measured the electrical resistivity and magnetization in InTe$_{1- x}$Se$_{x}$ ($x = 0$, 0.1, 0.2) superconductors. The doping evolutions of the critical current density, pinning mechanism, and vortex phase diagram are systematically studied. The shape of MHLs exhibits an asymmetric behavior at various degrees, especially for $x = 0.1$ in which surface barrier is dominant on the vortex dynamic. Intriguingly, a rare phenomenon that the SMP associated with $E$–$P$ crossover only occurs in the field-descending branch is found in InTe$_{0.9}$Se$_{0.1}$. Such anomalous behavior can be explained by considering the respective roles of the surface barrier and bulk pinning in the field-ascending and field-descending branches. The values of $J_{\rm c}$ are found to be exceeding $10^{4}$ A/cm$^{2}$ for InTe$_{1- x}$Se$_{x}$, signifying good potential for applications. Details on the pinning mechanisms are studied by analyzing the pinning force density versus reduced field in the framework of the Dew-Hughes model. Combining the results of resistivity and magnetization measurements, we finally constructed the vortex phase diagrams. Acknowledgements. This work was partly supported by the National Natural Science Foundation of China (Grant Nos. 12204265 and 12204487), the Natural Science Foundation of Shandong Province (Grant Nos. ZR2022QA040, 2022KJ183, and 2019KJJ020), the National Key R&D Program of China (Grant No. 2018YFA0704300), and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB25000000). 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