Charge-4 e and Charge-6 e Flux Quantization and Higher Charge Superconductivity in Kagome Superconductor Ring Devices

The flux quantization is a key indication of electron pairing in superconductors. For example, the well-known h= 2 e flux quantization is considered strong evidence for the existence of charge-2 e , two-electron Cooper pairs. Here we report evidence for multicharge flux quantization in mesoscopic ring devices fabricated using the transition-metal kagome superconductor CsV 3 Sb 5 . We perform systematic magneto-transport measurements and observe unprecedented quantization of magnetic flux in units of h= 4 e and h= 6 e in magnetoresistance oscillations. Specifically, at low temperatures, magnetoresistance oscillations with period h= 2 e are detected, as expected from the flux quantization for charge-2 e superconductivity. We find that the h= 2 e oscillations are suppressed and replaced by resistance oscillations with h= 4 e periodicity when the temperature is increased. Increasing the temperature further suppresses the h= 4 e oscillations, and robust resistance oscillations with h= 6 e periodicity emerge as evidence for charge-6 e flux quantization. Our observations provide the first experimental evidence for the existence of multicharge flux quanta and emergent quantum matter exhibiting higher-charge superconductivity in the strongly fluctuating region above the charge-2 e Cooper pair condensate, revealing new insights into the intertwined and vestigial electronic order in kagome superconductors.


I. INTRODUCTION
Superconductivity was discovered more than one century ago and described by the Bardeen-Cooper-Schrieffer (BCS) theory in terms of the condensation of charge-2e Cooper pairs [1,2].Flux quantization in units of the charge-2e flux quantum h/2e, where h is the Planck constant and e is the elementary charge, is a key signature of the electron pairing in BCS superconductors.The observations of the h/2e flux quantization served as key experimental evidence of the BCS theory of conventional superconductors (SCs) [3][4][5].Superconductivity via the condensation of higher charges, such as bound states of electron quartets or sextets (i.e.charge-4e or charge-6e), has been proposed theoretically [6][7][8][9][10][11][12][13][14][15][16].Despite intensive theoretical studies, whether higher-charge superconductivity exists and the many of its basic properties remain mysterious and illusive due to the lack of experimental evidence.For charge-Q superconductivity with Q=2ne to arise, there must exist experimental evidence for flux quantization in units of the charge-Q flux quantum Φ ℎ/2.
Here, we report the observation of flux quantization in units of  (h/4e) and  (h/6e) in ring devices of the newly emerged kagome superconductor CsV 3 Sb 5 .We fabricate superconducting CsV 3 Sb 5 ring devices with various sizes and perform systematic magneto-transport measurements.At low temperatures, magnetoresistance oscillations with period of the charge-2e flux quantum h/2e are detected, as expected from the flux quantization for charge-2e superconductivity.The h/2e oscillations are suppressed and replaced by resistance oscillations with h/4e periodicity as the temperature is increased.When the temperature is further increased toward the onset of the superconductivity, the h/4e oscillations vanish and novel resistance oscillations with a period equaling to 1/3 of the period of h/2e oscillations emerge.Careful measurements and analysis on many ring devices indicate consistently that these resistance oscillations with the period equaling to 1/2 and 1/3 of the h/2e oscillations reveal the existence of flux quantization in units of cha observ in units point t quantu  [17,18 s [17,18].K before effects of g electron-l and topo ity and pai  We then carry out systematic magneto-transport measurements on the CsV 3 Sb 5 ring devices.When the applied perpendicular magnetic field drives the sample into a sufficiently resistive state, noticeable oscillations in a magnetic field range from 2000 Oe to 7000 Oe are observed on the measured resistance (R)-magnetic field (H) curves shown in Fig. 7(a) at temperatures from 0.1 K to 1.0 K.By subtracting the smoothly rising (in H) resistance backgrounds in the R-H curves, the oscillations become more pronounced as shown in Fig. 2(b) at various temperatures, where dashed and solid lines guide the dips and peaks, respectively.We index the dips and peaks of the oscillations by an integer or half-integer n and the corresponding magnetic field by H n .The index n is obtained by H n /ΔH, here, ΔH is the period of the oscillation.The positions of the dashed or solid line associated with an index n are chosen to make sure that most of the corresponding dips and peaks on different curves can be properly located.Plotting H n versus n reveals a clear linear dependence [Fig.2(h)], indicating that these oscillations are periodic in H, as expected from the quantization of the magnetic field flux through the ring   , where n is an integer, in unit of the charge-2e superconducting flux quantum Φ 0 .Using the measured period of the oscillations in Fig. 2(b), i.e.ΔH 2e ~ 750 Oe, and setting Φ 0 =h/2e=SꞏΔH 2e , we obtain the effective ring area S ≈ 0.0276 μm 2 , which is marked by the red rectangle in the inset of Fig. 2(a).Note that if the effective area were larger, the condition for flux quantization would imply Cooper pairs of fractional charged quasiparticles, which is unlikely for this system.We thus consider the oscillations with period of ~ 750 Oe as the charge-2e (h/2e) oscillations with the effective area close to inner hole area.We will return to the physical implications in the discussion section.Surprisingly, upon increasing the temperature to 1.65 K and above into the broad superconducting transition region, new oscillations in a magnetic field range from 2000 Oe to 5000 Oe are observed [Fig.2(c)] with a different period in magnetic field ΔH ~ 350 Oe as confirmed by the linear relation between n and H n [Fig.2(h)].These emergent oscillations correspond to a magnetic flux through the ring Φ=SꞏΔH=0.97×10 -1 Wb, using the same effective area S ≈ 0.0276 μm 2 identified above and marked in the inset of Fig. 2(a).Intriguingly, this value of the flux is very close to one-half of the charge-2e flux quantum, i.e. the charge-4e flux quantum  =h/4e =1.03 ×10 -15 Wb.
More strikingly, we find that the oscillations with period of ~350 Oe vanish when the temperature is further increased.Above 2.3 K, brand new oscillations in magnetic field emerge with the period ΔH ~ 250 Oe [Figs.2(d)-2(g)].This value is remarkably 1/3 of that in the charge-2e oscillations (~ 750 Oe) observed at low temperatures and corresponds to resistance oscillations due to flux quantization in units of an emergent charge-6e flux quantum  =h/6e =0.69×10 -15 Wb, under the same effective area (S ≈ 0.0276 μm 2 ).Indeed, the charge-6e resistance oscillations in Figs.2(d)-2(g) are strong and robust over a wide field range [Figs.8,9], exhibiting magnetic flux quantization   down to zero magnetic field with a resistance minimum at n=0 [Figs.2(d), 2(e)].In Fig. 2(i) and Fig. 11(a), the temperature evolution of the fast Fourier transform (FFT) of the resistance oscillations is presented in a waterfall plot.The positions of the FFT peaks show that the periodicity of the flux quantization in unit of the magnetic flux quantum changes from Φ 0 =h/2e at low temperatures to  =h/4e at higher temperatures, and then to  =h/6e at still higher temperatures on approaching the onset of the superconducting transition.Similar results have also been observed consistently in other CsV 3 Sb 5 ring devices of similar dimensions.The results obtained on another device are summarized in Fig.With increasing temperature, the changes in periodicity from h/2e to h/4e and then to h/6e can be clearly observed.Note that the weak peaks appearing around 0.5×(2e/h) on FFT curves at T < 0.6 K correspond to the secondary peaks, which may originate from the slight amplitude fluctuation of the h/2e oscillation signals.
It is important to note that these set of thick-rimmed ring devices leave room for the possibility that the oscillations with period of ~350 Oe and ~250 Oe were also h/2e oscillations, but with larger effective areas.In the following, we will show that this possibility can be excluded and the effective area remains a constant for h/2e, h/4e and h/6e oscillations in a given CsV 3 Sb 5 ring device.First, as shown in Fig. 2(i) and Fig. 11(a), the FFT peak positions for h/2e, h/4e, and h/6e oscillations respectively do not change with the temperature, indicating that the corresponding effective area maintains a constant with increasing temperatures.
Second, there are no intermediate periodicities between the h/2e (h/4e) and h/4e (h/6e) oscillations, which further demonstrates the constant effective area for the observed oscillations.
To further scrutinize the evidence for the extraordinary charge-6e flux quantization and directly exclude the possibility of assigning the h/6e oscillations as those of h/2e under a different effective area, we fabricated micron-sized CsV 3 Sb 5 ring devices with much larger hole area and ratio of hole size to wall width.The experimental data on device s2 with a hole area ~ 0.96 μm 2 are presented in Fig. 3.At low temperatures, resistance oscillations with period ~ 13.6 Oe in magnetic field are observed.Using the measured period of the oscillations in Fig. 3(b), i.e.ΔH 2e ~ 13.6 Oe, and setting Φ 0 =h/2e=SꞏΔH 2e , we obtain the effective area S ≈ 1.52 μm 2 , which is marked by the red rectangle in the inset of Fig. 3(a).If a larger effective area, e.g. with boundaries in the middle of the rim (S middle ~2.01 μm 2 ) were considered as the effective area, then the flux Φ= S middle ꞏΔH=2.734×10 -1 Wb≈h/0.77e,would lead to fractional charges, which is unlikely in this system.Therefore, the oscillations at low temperatures in the large ring device s2 are h/2e oscillations.
When the temperature is increased to 2.5 K and above, novel emergent oscillations are again observed [Fig.3(e)], as in the smaller, thick-rimmed devices.The periodicity of these new oscillations is confirmed by the linear relation between n and H n [Fig.3(f)], which reveals a new period in magnetic field ΔH ~ 4.6 Oe, equaling to 1/3 of the period of the h/2e oscillations (~13.6 Oe) observed at low temperatures.These small-period oscillations (~4.6 Oe) cannot be mistaken for h/2e oscillations with a larger effective area in the micron-sized device s2, because if they were, the corresponding effective area (green rectangle in the inset of Fig. 3(a)) would be much larger than even the outer area of the ring device, which is physically impossible.Therefore, the oscillations with period of ~ 4.6 Oe are unambiguous evidence for flux quantization in units of novel higher-charge magnetic flux quantum.Considering the extremities of device s2, i.e. an outer area S outer ~2.97 μm 2 and an inner area S inner ~0.96 μm 2 , the oscillations period ~4.6 Oe imply that the flux quantum must satisfy: Φ inner ≤ Φ ≤ Φ outer , where Φ inner = S inner ꞏΔH=0.44×10 -1 Wb≈h/9e and Φ outer = S outer ꞏΔH=1.366×10 -1 Wb≈h/3e.This means that the oscillations with period of ~4.6 Oe do not correspond to the 2e but the multi-charge of the flux quantum in the range of 3e to 9e. after subtracting smooth backgrounds.The dashed and solid lines label the oscillation dips and peaks, respectively.For clarity, data curves are shifted.The period of the oscillations (4.6 Oe) is nearly 1/3 of that of the h/2e oscillations (13.6 Oe).If the oscillations with period of ~4.6 Oe were h/2e oscillations, the effective area (~4.5 μm 2 , shown by the green rectangle in the inset of (a)) would be obviously much larger than the outer area of the ring device (~2.97 μm 2 ), which is physically impossible.If we consider the effective area of the oscillations with period of ~4.6 Oe is the same as that of the h/2e oscillations with period of ~13.6 Oe, then the period of ~4.6 Oe will correspond to a periodicity of h/6e.(f) n-H n index plot of the h/6e oscillations.Here, n is an integer or half integer, representing the oscillation dip and peak, respectively.The linear relation between n and H n confirms that the magnetoresistance oscillations are periodic.(g) FFT results of the h/6e oscillations.Note that the oscillation range for FFT is not exactly the same as the range shown in (e).
The evidence for a temperature-independent effective area discussed above encourages the use of the same effective area (S≈1.52 μm 2 ) determined for the low-temperature charge-2e oscillations with period ~13.6 Oe and the oscillations with period ~4.6 Oe at higher temperatures, which corresponds to a magnetic flux through the ring Φ=SꞏΔH=0.699×10 -1 Wb.This value is remarkably close to the periodicity in unit of the charge-6e superconducting flux quantum  =h/6e =0.69×10 -15 Wb.Similar results have also been obtained in two additional large ring devices s4 [Fig.13] and s5 [Fig.14].These results on devices with much larger hole areas provide arresting evidence for the discovery of robust h/6e oscillations in CsV 3 Sb 5 ring devices.In Fig. 4, we present the temperature evolution of the magnetoresistance oscillations as color intensity plots in unit of the magnetic flux Φ/Φ 0 over the broad superconducting transition region in the CsV 3 Sb 5 ring devices.With the increase of the temperature, the periodicity of the oscillations changes from h/2e to h/4e and then to h/6e in small thick-rimmed ring devices represented by s1, and from h/2e to h/6e in micron sized ring devices represented by s2, recapitulating our finding of multi-charge flux quantization in unit of charge-4e and charge-6e superconducting flux quanta.superconductors.For ordinary BCS superconductors, this transition region is narrow (the SC transition is sharp) and the melting of coherent charge-2e Cooper pairs is all that happens wherein, resulting in the canonical behavior of h/2e oscillations [33].In contrast, the transition region in our kagome CsV 3 Sb 5 ring devices is rather broad, as can be seen in the R-T curves.It separates the charge-2e SC ground state below the zero-resistance temperature and the normal state above the SC onset temperature.The multi-charge flux quanta emerge in the broad transition region with strong SC fluctuations, as the temperature is increased in the melting of the charge-2e condensate ground state.To further substantiate this fundamental difference and develop new insights, we studied ring devices made of conventional superconductor Nb [Figs.15,16] using identical fabrication methods.Two Nb ring devices are fabricated using the same etching technique, under the same etching parameters, and protected by the PMMA layer of the same thickness.The measured R-T curve reveals a sharp superconducting transition in the Nb ring device n1 [onset temperature ~7.6 K and zero resistance temperature ~7.0 K, Fig. 15(a)] and n2 [onset temperature ~7.6 K and zero resistance temperature ~7.2 K, Fig. 16(a)].Indeed magnetoresistance measurements show only h/2e oscillations in Nb ring devices in a narrow temperature regime [Fig.15(c), Fig. 16(d)] above the zero-resistance transition.The corresponding effective area has a radius lying in the middle of the wall [inset of Fig. 15(a) for device n1 and Fig. 16(a) for device n2], which is in accord with the theoretical predictions for the Little-Parks oscillations in conventional superconductor ring devices [34].These experiments reassure that our device fabrication, measurements, and analysis have been tested to reproduce the expected physics in ordinary BCS superconductors, and further demonstrate that the remarkable observation of the multi-charge flux quantization is due to the never before encountered higher-charge superconducting quantum states in extended fluctuation region of CsV 3 Sb 5 ring devices.
Another intriguing property, distinct from ordinary superconductors having sharp superconducting transitions, is that the observed resistance oscillations with different flux quantization in the strongly fluctuating superconductivity region all have the same effective area close to the inner hole area in the CsV 3 Sb 5 ring devices [inset of Figs.2(a), 3(a)].In a recent insightful paper, Han and Lee [35] provided a possible theoretical explanation for our experimental observation.They studied the effective area for transporting charge-4e and charge-6e bound states across our thick-rimmed geometry devices (CsV 3 Sb 5 ring device s1, s3) using the space-time formulation of time-dependent Ginzburg-Landau theory.Because the superconductivity is strongly fluctuating in the broad transition regime in the CsV 3 Sb 5 ring devices, the optimal path is found to stick to the edge of the inner hole in order to reduce the fluctuations by proximity to the open area of the hole, leading to the effective area close to the inner area for h/4e and h/6e flux quantization.The result also applies to the strongly fluctuating charge-2e state [35].The combined experimental and theoretical findings land convincing support for the observation of resistance oscillations with h/4e and h/6e flux quantization.We now turn to the possible origin of the observed charge-4e and charge-6e flux quantization.There are theoretical proposals for fractional flux quantum in spin-triplet   superconductors [36,37], and in vortices trapped at domain walls and twin boundaries in time-reversal symmetry breaking superconductors [38].However, these do not describe our observation because the fractional flux quantization under these settings only leads to nontrivial phase shift in the quantum oscillations, while the periodicity of the oscillations would remain at Φ 0 as the pairing is still charge-2e in nature.More theoretical discussions are available in Appendix B. While we cannot rule out the possibility where a fractional flux quantum results from a multicomponent superconductor where the phases of different charge-2e condensates wind differently in the magnetic field [39], the observed sequential changes in the flux quantization under thermal melting are unnatural to be accounted for in this scenario.
The distinct charge-4e and charge-6e flux quanta observed in the flux quantization with increasing temperatures naturally suggest a sequential destruction of the phase coherent charge-2e superconductivity and the emergence of phase coherent bound states of 4-electrons (or two Cooper pairs) and 6-electrons (or three Cooper pairs) in the strongly fluctuating transition region of the ring structures.This scenario is consistent with the theoretical proposal of the putative charge-4e and charge-6e superconductivity as vestigial ordered states following the melting of a novel charge-2e superconducting state that breaks crystalline symmetry [13] as evidenced by the PDW order observed in the kagome superconductor CsV 3 Sb 5 [23].In addition to the broken global U(1) symmetry, such a PDW superconductor has a multicomponent charge-2e order parameter that simultaneously breaks translation and time-reversal symmetry, where the relative chiral phases of the different components creates an emergent hexagonal vortex-antivortex lattice carrying the momentum of the PDW.The destruction of the PDW order by thermal fluctuations is thus described by the well-known melting theory of a vortex-antivortex lattice [40][41][42].With increasing temperatures, the topological defects of positional order, i.e. dislocations, unbind and proliferate, which destroy the PDW order and charge-2e superconductivity.The composite uniform charge-4e and charge-6e order parameters [13], decoupled from the proliferating dislocations, can therefore become the primary quasi-long-range vestigial ordered states.The predicted charge-4e state has chiral d+id symmetry and requires residual orientation order, whereas the charge-6e state is isotropic with s-wave symmetry [13].
Intriguingly, the melting of the vortex-antivortex lattice into an isotropic liquid phase can indeed be staged and exhibit an intermediate liquid crystal phase, termed as a hexatic, with orientation order [40][41][42].This happens when the topological defects associated with rotation, i.e. the disclinations, remain bound succeeding the proliferation of dislocations that restores translation symmetry.Our observation of charge-4e flux quantization in the intermediate temperature range on the small ring device [Fig.2(c)] is thus consistent with such a hextic charge-4e state with the superconducting correlation length exceeding the circumference of the inner hole.Further increasing the temperature causes the unbinding and proliferation of the disclinations, which destroying the orientation order and the charge-4e state, as the melting into the isotropic liquid is completed, giving rise to the isotropic s-wave charge-6e superconductor.The sharp FFT peaks located at 6e/h for charge-6e flux quantization in Fig. 2(i) with the nearly perfect periodicity down to zero magnetic field are indeed consistent with the most robust s-wave charge-6e state in the isotropic phase proposed in this scenario.The chiral phase in the vestigial hexatic charge-4e state, on the other hand, couples strongly to the strain fields, supercurrent fluctuations, and disclination defects [13], which can limit the correlation length and hinders the ability of the charge-4e bound states to move through the ring structure phase coherently.As a result, in contrast to the robust isotropic charge-6e state, the charge-4e state is less robust, which is consistent with our observation that the h/4e oscillations are relatively weak in small ring devices and difficult to discern in the micron-sized ring devices with much larger hole areas (Table 1).Moreover, a direct melting transition from the vortex-antivortex lattice to the isotropic liquid phase is also possible in the micron-sized ring devices, resulting in the single higher-charge vestigial ordered charge-6e superconductivity.Although the 3Q chiral PDW offers a physically intuitive picture for our observations, the origin of the observed charge-6e flux quantization is wide open for future investigations, as is the mechanism for superconductivity and PDW order in the kagome superconductors.
In summary, we discovered quantum oscillations with the periodicity of multi-charge flux quantization in mesoscopic CsV 3 Sb 5 superconducting ring devices, which suggests the possibility of higher-charge superconductivity.Our experimental findings bring new insights into the rich and fascinating quantum states in the kagome superconductors and provide ground work for exploring the physical properties of unprecedented phases of matter formed by multi-particle bound states.

ACKNOWLEDGMENTS
A portion of this work was carried out at the Synergetic Extreme Condition User Facility (SECUF).We acknowledge discussions with Li Lu, Yanzhao Liu, Yi Liu, Yanan Li, Haoran Ji and Jingchao Fang, and technical assistance from Jiawei Luo, Pengfei Zhan, Chunsheng Gong, Zhijun Tu, Yinbo Ma, and Gaoxing Ma.

APPENDIX A: MATERIALS AND METHODS
Crystal growth.Single crystals of CsV 3 Sb 5 were grown by using the self-flux method [28].In a typical growth, the mixture of Cs ingot (purity 99.75%), V powder (purity 99.9%) and Sb grains (purity 99.999%) was put into an alumina crucible and sealed in a quartz ampoule under argon atmosphere.The quartz ampoule was heated up to 1273 K for 12 h and held for 24 h.Then it was rapidly cooled down to 1173 K in two hours and slowly cooled down to 923 K. Finally, the CsV 3 Sb 5 single crystals were separated from the flux by using a centrifuge.In order to prevent the reaction of Cs with air and water, all the preparation processes except the sealing and heat treatment procedures were carried out in an argon-filled glove box.
Devices fabrication.The thin CsV 3 Sb 5 flakes were firstly exfoliated from bulk single crystals using the scotch tape and then transferred onto the 300-nm-thick SiO 2 /Si substrates, which were pre-cleaned in oxygen plasma for 5 min at ~70 mTorr pressure.By a standard electron beam lithography process in a FEI Helios NanoLab 600i Dual Beam System with PMMA 495A11 as a resist, electrodes were patterned, and metal electrodes (Ti/Au, 6.5/180 nm) were deposited in a LJUHVE-400 L E-Beam Evaporator.Note that the PMMA 495A11 resist was not baked.Finally, the PMMA 495A11 layers were removed by standard lift-off process and CsV 3 Sb 5 flake devices with four metal electrodes were obtained.All the device fabrication processes were carried out in an argon-filled glove box with the O 2 and H 2 O levels below 0.1 ppm.
Then, PMMA 495A11 electron-beam resist (800 nm thick, without baking) was spin coated on the CsV 3 Sb 5 flake devices as the protection layers for etching the CsV The AFM DimensionI

ONAL ORS
can host different bands/Fermi surfaces, spins, or even coupled with different momenta as in the case of pair density wave superconductors.The basic idea is that in a magnetic field that causes the phase winding of the multicomponent order parameters, if one or more components do not wind, or wind differently, then the single-valuedness of the wave function results in a fractionally quantized flux quantum.This seemingly simple proposal has not been realized over the years, possibly because that in a real superconductor, these multicomponent condensates are usually phase-locked together by the Josephson coupling, and the phases of different order parameter components wind together to give the expected flux quantization.
Consider the p+ip superconductors as an example.This is a spin-triplet pairing state with equal-spin pairing.The order parameter is a mixing of |1,1> and |2,2>, with 1 and 2 denoting the two spin components.The single-valuedness of the wave function requires the flux quantization Φ= Φ 0 (n 1 +n 2 )/2, where Φ 0 =h/2e is the usual flux quantum, and n 1 and n 2 are the integer number of times that each spin-component winds around 2π. Consider the case where n 1 and n 2 are both nonzero, but differ by an odd integer due to the different Berry phases of electrons with opposite spins, such that n 1 + n 2 = 2n + 1 and Φ=Φ 0 (n+1/2), giving rise to the half-flux-quantum (HFQ).Note that this HFQ only causes a shift in Φ, leading to a π phase shift of the quantum oscillations, while the period of oscillations would remain to be Φ 0 .This is, however, very different from our observations of the quantum oscillations with h/6e periodicity corresponding to .Furthermore, the existing experimental evidence does not support spin-triplet pairing in kagome superconductors [27,32,[43][44][45].
Next consider the case where one component doesn't wind, i.e. either n 1 or n 2 equals zero.Then, Φ= times an integer.This can in principle lead to quantum oscillations with a flux quantization h/4e.This picture can be generalized to superconductors with N number of degenerate components.Depending on the various hypothesized inter-component Josephson couplings, oscillations with fractional flux quantization is possible and mostly favored by Φ 0 /N.Our observations do not rule out such a scenario, but the observation of the temperature evolution of the flux quantization from h/2e to h/6e makes it unnatural even at a phenomenological level.We note that in all the scenarios discussed above, the pairing is still charge-2e in nature, i.e. a Cooper pair, whereas our observations suggest more naturally that the successive transitions in the periodicity of the quantum oscillations is connected to charge-6e pairing.Furthermore, to our knowledge, oscillations with h/4e periodicity have never been experimentally reported in p+ip system.
Hexagonal pair density waves (PDWs) turn out to offer a more physically intuitive account of our observations, in addition to having been observed in bulk CsV 3 Sb 5 FIG. 1 CsV 3 Sb yellow of the dimens FIG. 3 (a) Res with o observ sample represe The in false-c oscillat dashed effectiv the ins represe observ results exactly .h/2e and sistance as onset tem ved.Inset s e protecte ented by g nner area, olored im tions with d lines lab ve area of set of (a).enting the ved, indica of the h y the same d h/6e osc s a functio mperature shows the d by PM gray.The , middle mage is ~0.h period o bel the os f the h/2e o (c) n-H n i oscillatio ating that h/2e oscill e as the ran illations in on of temp ∼ 4.1 K false-colo MMA layer scale bar area and 96 μm 2 , õf ~13.6 O scillation d oscillation index plot n dip.A l the magne lations.N nge shown n the supe perature f and zero ored imag r is repre in the fal outer ar ~2.01 μm 2 Oe after s dips.For ns (~1.52 μ t of the h/ linear rela etoresistan ote that t n in (b).(e erconducti from 5 K t o-resistanc ge of the C esented by lse-colored rea of the and ~2.97 ubtracting clarity, d μm 2 ) is ma /2e oscilla ation betwe nce oscilla the oscilla e) Oscillati ing CsV 3 S to 0.5 K. ce temper CsV 3 Sb 5 ri y blue and d image re e ring es 7 μm 2 , resp g smooth ata curves arked by th ations.Her een n and ations are ation rang ions with p Sb 5 ring de Supercond rature ∼1 ing device d the sub epresents stimated f pectively.backgroun s are shif he red rect re, n is an H n can b periodic.ge for FFT period of ~ evice s2.ductivity 1.4 K is e s2.The bstrate is 500 nm.from the (b) h/2e nds.The fted.The tangle in n integer, e clearly (d) FFT T is not ~ 4.6 Oe FIG. 4 ring de CsV 3 Sb (b) sh respect curve differe (a) and region 4e and magne maps ( the per white r interme regime oscillat temper ring de h/6e. This work was financially supported by the National Natural Science Foundation of China (Grant No. 12488201), the National Key Research and Development Program of China (Grant No. 2018YFA0305604, No. 2018YFE0202600), the National Natural Science Foundation of China (Grant No. 11974430), Beijing Natural Science Foundation (Z180010, Z200005), the Innovation Program for Quantum Science and Technology (2021ZD0302403) and the China Postdoctoral Science Foundation (2022M720270).Z.Q.W. acknowledges support from the U.S. Department of Energy, Basic Energy Sciences Grant No. DE-FG02-99ER45747 and the Cottrell SEED Award No. 27856 from Research Corporation for Science Advancement.
The dashed and solid lines label the resistance dips and peaks, respectively.The period of the h/4e and h/6e oscillations is ~ 119 Oe and ~80 Oe, respectively.For clarity, data curves in (c) and (d) are shifted.(e) n-H n index plot of the h/2e, h/4e and h/6e oscillations.(f), (g) FFT results of the oscillations at various temperatures.Note that the oscillation range for FFT is not exactly the same as the range shown in (b)-(d).When the temperature is increased, the periodicity changes from h/2e to h/4e and then to h/6e.

ENT RESU NG DEVI V 3 Sb 5
7.25 to 7.3 K) were observed in the Nb ring device.The effective area of the h/2e oscillations is ~0.268 μm 2 , which is marked by the red rectangle in the inset of (a).(d) h/2e oscillations in the Nb ring device.