Magnetoresistance hysteresis in the superconduting state of Kagome CsV₃Sb₅

Hysteresis is a phenomenon whereby the response of a system to an external disturbance, such as magnetic field ($H$), electric field, temperature ($T$) or mechanical stress, depends on the history of applied perturbation. Understanding of hysteresis is beneficial for designing materials of improved quality and predicting novel properties [1, 2, 3, 4, 5, 6]. In superconductors, the presence of hysteretic magnetoresistance has attracted considerable attention, because of its relation to several intriguing dynamics, including magnetic vortices, ferromagnetism, chiral superconducting domains and spin-triplet paring symmetry [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Relevant investigation will provide valuable insights for the development of novel superconducting materials and various quantum technologies, ranging from energy transmission to quantum computation.

As a member of Kagome superconductors, CsV₃Sb₅ (CVS) has garnered significant interest due to its unique electronic properties and diverse quantum orders of matter [17, 18]. In addition to the fully gapped superconductivity [19, 20, 21, 22], a pair density wave was detected by scanning tunneling microscopy, showing remarkable resemblance to that in cuprates [23]. Moreover, multicharge flux quantization was observed in mesoscopic CVS ring devices, suggesting the possibility of higher charge superconductivity [24]. Numerous theoretical studies have predicted chiral superconductivity in CVS, given the Kagome lattice formed by vanadium atoms and breaking of time-reversal symmetry (TRS) in the normal state [25, 26, 27, 28, 29, 30, 31, 32]. Recently, a technique combining the zero-field superconducting diode effect and interference patterns was employed to demonstrate the presence of TRS-breaking superconducting order with dynamic domains in CVS [33].

In this study, we performed electrical transport measurements on CVS nanoflakes under various DC bias current ($I_{\mathrm{DC}}$), AC excitation current ($I_{\mathrm{AC}}$) and $T$. In the superconducting state, an intriguing hysteresis of magnetoresistance within a field range of $\pm$ 20 mT was observed when sweeping the field in opposite directions. In this context, the potential roles of vortex pinning effects and chiral superconducting domains were discussed in elucidating the origin of the observed phenomena.

Single crystals of CsV₃Sb₅ were grown through flux methods by using Cs (purity 99.8%) bulk, V (purity 99.999%) pieces and Sb (purity 99.9999%) shot as the precursors and Cs_0.4Sb_0.6 as the flux agent. The starting elements were placed in an alumina crucible and sealed in a quartz ampoule in an argon-filled glove box. The ampoule was then gradually heated up to 1000 ℃  in 200 h and held at that temperature for 24 h in an oven. It was subsequently cooled down to 200 ℃ at a rate of 3.5 ℃/h. The resulting product was immersed in deionized water to remove the flux. Finally, shiny CsV₃Sb₅ crystals with hexagonal shape were obtained. CsV₃Sb₅ nanoflakes were mechanically exfoliated from the bulk crystals with a thickness exceeding 40 nm, using Nitto blue tape and transferred onto silicon substrates capped with 300 nm SiO₂. The contacts were deposited with Ti (5 nm)/Au (80 nm) via electron beam evaporation. In order to improve the quality of the contacts, the flakes were cleaned by Ar plasma prior to deposition.

The transport measurements were performed in Quantum Design physical property measurement system (PPMS). The resistance ($R$) and differential resistance (d$V$/d$I$) for three devices (D1-D3) were measured by a standard four-terminal method. d$V$/d$I$ in the positive ($I_{\textrm{+}}$) and negative ($I_{\textrm{-}}$) bias current regimes was collected by sweeping ($I_{\mathrm{DC}}$) from 0 to $I_{\textrm{max+}}$ and $I_{\textrm{max-}}$, respectively. $I_{\mathrm{DC}}$ was supplied by a current source meter (Keithley 2450). A lock-in amplifier (Stanford Research, SR830) combined with a 100 k$\Omega$ buffer resistor was used to offer small $I_{\mathrm{AC}}$ ($11~{}\mathrm{Hz}-173$ Hz, $1~{}\mu\mathrm{A}-50~{}\mu$A) to detect the differential resistance (d$V$/d$I$ = $V_{\textrm{AC}}$/$I_{\textrm{AC}}$).

Figure. 1(a) illustrates $R$ versus $T$ for D1 in a full temperature range, which indicates high sample quality with a residual-resistance-ratio (RRR) of 60. The transitions of charge density wave (CDW) and superconductivity occur at $T_{\rm CDW}$ = 82 K and $T_{\rm c}$ = 4.3 K, respectively. Fig. 1(b) shows the magnetoresistance measured at $T=1.8$ K and $I_{\rm AC}$ = 3 $\mu$A by sweeping $H$ in a large field range, i.e. from 0 T to 2 T (red) and back from 2 T to 0 T (blue) respectively. It clear that two curves collapse without detectable hysteresis. As $H$ increases, the fast enhancement of finite resistance could be attributed to vortex depinning effect. The absence of hysteresis observed in Fig. 1(b) suggests that this pinning effect is not prominent [34, 7, 8, 9, 10].

In Fig. 1(c), we present d$V$/d$I$ as a function of $I_{\rm DC}$ at 1.8 K, where the superconducting state sustains up to 420 $\mu$A. The discrepancy between positive ($I_{\rm+}$) and negative ($I_{\rm-}$) sweeping curves arises from the superconducting diode effect, as reported in our previous work[33]. The presence of multiple transition peaks implies signatures of superconducting domains, likely chiral domains [33]. Figure 1(d) present d$V$/d$I$ versus $H$ for $I_{\mathrm{DC}}=100-400~{}\mu$A, corresponding to the vertical lines in Fig. 1(c). At $I_{\rm DC}=200-400~{}\mu$A, these curves exhibit a notably bow-tie-shaped hysteresis loop within the superconducting state. This resembles the behavior in ferromagnetic superconductors [11, 12], where the positions of dips (peaks) in the d$V$/d$I$ curves define the coercive field $H_{\mathrm{co}}$. The hysteresis merges at slightly higher field ($\pm 20$ mT), corresponding to the full polarization of domains, denoted as $H_{\rm hys}$. However, it is well-established that there is no ferromagnetism in CVS [17, 18]. On the other hand, in the superconducting mixed state, vortex depinning occurs once its Lorentz force exceeds the pinning force caused by defects, with the former proportional to the product of the external field and current density ($J$) [35]. Therefore, the higher the current, the lower the depinning field. This is in stark contrast with the observation in Fig. 1(d), wherein even as $I_{\mathrm{DC}}$ increases, both $H_{\mathrm{co}}$ and $H_{\rm hys}$ remain roughly unchanged.

In a chiral superconductor, it is energetically favorable to form domain structures between degenerate superconducting phases of opposite chiralities. Similar to ferromagnetic domains, these superconducting domains are pinned by disorder or defects. The field strength, exceeding the pinning force, can switch the chirality of domains, resulting in hysteresis during field cycling [36]. This field is independent of the applied current, consistent with the observation in Fig. 1(d). At $I_{\rm DC}$ = 100 $\mu$A, the hysteresis is negligible since the resistance is zero within $\pm H_{\rm hys}$. Subsequent data will extend the discussion in the frame of chiral domains.

In Fig. 2(a), we examine the field-dependent resistance at $I_{\mathrm{DC}}=0$ for device D1 by varying $T$. At low $T$, the resistance is nearly zero, displaying no detectable hysteresis when the field cycles between $\pm$50 mT. As $T$ approaches $T_{\mathrm{c}}$ and above, $R$ becomes finite and an observable bow-tie-shaped hysteresis loop emerges, similar to that in Fig. 1(d). However, the width of the hysteresis, for instance indicated by $2H_{\mathrm{co}}$ $\sim$ 1.4 mT, becomes much smaller than the value of $\sim$ 3.2 mT in Fig. 1(d). This weaker hysteresis can be ascribed to the reduction of the domain pinning force due to strengthened thermal fluctuation as $T$ rises, which aligns with the chiral superconducting domain scenario. At even higher $T$ (in the normal state), this hysteresis disappears implying that it is inherent to the superconducting state. It is worth noting that vortices have been depinned at $T>T_{\rm c}$, making them irrelevant here.

We note that artificial hysteresis may occur such that the field sweep rate exceeds the data collection rate of the computer in the measurements of $R$-$H$ curves. To check this, we measured $\textrm{d}V/\textrm{d}I$ versus $H$ for D1 by varying the field sweep rates from 0.05 mT/s to 0.3 mT/s, as shown in Fig. 2(b). All curves exhibit hysteresis and collapse with each other. It seems insensitive to the sweep rate, ruling out this possibility. Moreover, hysteresis can also be induced by heating effects from the field sweep around 0 mT. In this case, the characteristics of hysteresis would depend on the heating power related to the sweep rate. This is clearly not the case here. Therefore, the hysteresis observed in our devices is unlikely to be artificial.

Similar hysteresis can be reproduced in other CVS devices. In Fig. 3(a), we inspect the $R$-$H$ curves with different $I_{\mathrm{AC}}$ at 1.8 K and $I_{\rm DC}=0~{}\mu$A for D2. As $I_{\rm AC}$ increases, the depinning and subsequent motion of vortices generate finite resistance in a magnetic field. At $I_{\rm AC}$ = 20 $\mu$A, the resistance remains small with marginal hysteresis. As $I_{\rm AC}$ increases from 30 to 50 $\mu$A, the hysteresis of magnetoresistance becomes increasingly remarkable, which resembles the data collected by varying $I_{\mathrm{DC}}$ or $T$ in Fig. 1 and 2.

In the following, let’s delve into the switching characteristics of hysteresis by varying the maximum sweeping field ($H_{\mathrm{max}}$). The measurements were performed on D3 with $I_{\rm DC}=0~{}\mu$A and $I_{\rm AC}=50~{}\mu$A at $T=1.8$ K. During the measurements, we initially ramped $H$ from 0 to a positive field $H_{\mathrm{max}}^{\mathrm{+}}$, then returned to $H_{\mathrm{max}}^{\mathrm{-}}$ and finally back to $H_{\mathrm{max}}^{\mathrm{+}}$ again. As seen in Fig. 3(b), the hysteresis in the $R$-$H$ curve is indetectable at $\left|{H_{\rm max}}\right|$ = 1 mT. As $\left|{H_{\rm max}}\right|$ increases to 2 mT, a discernible hysteresis emerges. Further increasing $\left|{H_{\rm max}}\right|$ results in a pronounced hysteresis, as observed for $\left|{H_{\rm max}}\right|=3$ and $5$ mT. This hysteresis showcases a retarded manner, i.e. the $R$-$H$ curves shifts to negative (positive) as the field sweep from the $H_{\mathrm{max}}^{\mathrm{+}}$ ($H_{\mathrm{max}}^{\mathrm{-}}$). This phenomenon is a typical feature of domain switching with $H_{\mathrm{co}}$ marked by the vertical dashed lines in Fig. 3(b). At $H_{\mathrm{max}}\approx H_{\mathrm{co}}$, the chirality of domains is minimally switched during the field cycling, resulting in the absence of hysteresis. As $H_{\mathrm{max}}$ rises, an increasing number of domains are switched in both sweeping protocols towards $H_{\mathrm{max}}^{\mathrm{\pm}}$ and leads to a more conspicuous bow-tie-shaped hysteresis, as seen in the curves for $\left|{H_{\rm max}}\right|=5$ mT.

Note that in several candidates of chiral superconductors, such as Sr₂RuO₄ [15] and Bi/Ni bilayer [14], an ”advanced” form of hysteresis has been observed in the transport properties (e.g. superconducting critical current) of their Josephson junction or SQUID devices. This feature is thought to arise from the response of chiral domains to the interaction between the Meissner screening supercurrent and the change in the applied field. This phenomenon requires the system to be in the Meissner state [36, 14]. In contrast, in our case, detecting the hysteresis is challenging in a Meissner state, since it will be hidden by zero resistance. A large DC or AC current is required to stimulate the vortex motion and generate finite resistance in the mixed state.

The aforementioned hysteresis in CVS is properly interpreted by the model involving the pinning of TRS-breaking domains in the superconducting state. This aligns with recent literature demonstrating a TRS-breaking order parameter with dynamic superconducting domains in CVS [33]. Previous theoretical studies have also proposed a chiral superconducting phase based on the Kagome lattice of CVS [37, 38, 39]. Therefore, the observed hysteretic magnetoresistance may serve as an indirect evidence for the possible existence of chiral superconductivity.