Introduction

The existence of vortices in the mixed state of a superconductor is one of the most fascinating research topics in superconducting physics1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16. The vortices can form liquid, crystalline and glassy phases. The discovery of high critical temperature (TC) superconductors (HTS) has made the detailed study of vortex phases possible because of the large anisotropy and thermal fluctuations in HTS systems1,2,3,4,5,6,7. In the presence of strong disorder, the zero resistance state corresponds to a vortex glass (VG) phase and with increasing temperature the transition to the vortex liquid phase induces a dissipative state2,17. This transition has been demonstrated by transport measurement in HTS systems like YBa2Cu3O7 (YBCO)1,3, Pr2-xCexCuO4-y (PCCO)7 and Bi2Sr2-xLaxCuO6 + δ (BSLCO)4,6. However, VG transition-like behavior in conventional low critical temperature superconductors (LTS) has only been observed in very few systems, like Nb8,9,10, In11 and Mo3Si13 films.

In the last few decades, a technique – focused ion beam induced deposition (IBID) has been developed and successfully used to fabricate low dimensional superconducting materials including W18,19,20,21,22,23. The reported TC of amorphous W-based nanostructures fabricated by IBID is around 5.2 K18, which is much higher than that of single-crystal bulk W (12 mK)24. This synthesis technique and the associated increase in TC makes it possible to observe different vortex states in mesoscopic W system. In two dimensional W systems fabricated by IBID, the direct observation of vortex melting utilizing scanning tunneling microscopy/spectroscopy (STM/STS) has been reported25. We report here a systematic study of V-I properties of W nanostructures.

In this paper, W strips with different dimensions are fabricated using a focused ion beam-chemical vapor deposition (FIB-CVD) system. Experimental studies of electronic transport properties, especially the V-I curve measurement of individual W strips at various temperatures and magnetic fields are reported. Under zero magnetic field, the strip jumps from superconducting to normal state directly as the current is increased to a temperature dependent critical value. Under a finite magnetic field at temperatures below a specific value Tg, the jump to the high resistance state is preceded by a state of low residual resistance. The size of the jump in the V-I curve decreases with increasing temperature and vanishes as T approaches Tg. For T > Tg, the V-I curves are continuous resembling a sigmoid function with a maximum curvature that decreases with increasing temperature. When plotted in log-log scale, the V-I curve at T = Tg shows a power law dependence. Curves at T > Tg show a concave curvature with increasing I and curves measured at T < Tg prior to the jump show a convex curvature. This evolution in the shape of the V-I curves is similar to that found in HTS and type II superconducting films mentioned above, which has been explained by VG transition theory.

Results

A commercial FIB etching and deposition system (FIB/SEM FEI Quanta 200 3D) utilizing a beam of 30 keV singly-charged Ga+ ions was used for sample synthesis. The substrate used was a lightly p-doped Si wafer with a 1 μm thick Si3N4 insulating layer on the surface. Tungsten hexacarbonyl [W(CO)6] gas was injected on the sample surface near the focused beam scanned area through a needle nozzle. The ion beam was then scanned over the sample surface in the designed pattern, depositing the non-volatile component of the precursor on the surface. The volatile component was removed and the base pressure before introducing the W(CO)6 gas was lower than 1 × 10−5 Torr.

During deposition the system pressure was in the range of ~ 10−5 Torr. Lateral nanostrips with various lengths, widths and thicknesses were deposited by a 30 pA ion-beam current onto the substrate. Changing the ion-beam current or the volume per dose can change the superconducting properties of the resulting W strips. The sample W strips were connected to large Pt contact pads, which serve as electrodes for four-terminal transport measurements, with four W strips whose deposition parameters were the same as that of the sample. The normal Pt electrodes were also synthesized using the IBID technique. A scanning electron micrograph (SEM) of a W nanostrip with 4 connecting W strips can be seen in the inset of Figure 1(a). The volume per dose used in deposition is 6.8 × 10−1 μm3/nC for samples 1 and 3 and 3.4 × 10−1 μm3/nC for sample 2. The length, width and thickness of the samples were measured by a scanning electron microscope. Transport measurements were carried out in a physical property measurement system (PPMS-Quantum Design). The data from three samples are shown in this paper. The sample dimensions are 1.43 μm long, 250 nm wide and 50 nm thick (sample 1); 3.2 μm long, 250 nm wide and 50 nm thick (sample 2) and 3.5 μm long, 200 nm wide and 30 nm thick (sample 3). The thickness of a similarly deposited W strip was measured by first cutting the strip with ion beam and then examined in a SEM chamber at a tilted angle. We estimate the uncertainty in the thickness to be no more than 5 nm. The elemental concentrations of our nanostrips detected by energy dispersive spectrum (EDS) are about 40%, 40% and 20%, for W, C and Ga, respectively, which is consistent with the value detected by electron probe microanalysis (EPMA)18.

Figure 1
figure 1

Transport measurements of the W strip.

(a) Resistance vs. temperature of W strip (sample 1). Inset is the SEM image of the sample with 4 connecting W electrodes; (b) Resistance vs. temperature of the W strip in different magnetic fields applied perpendicular to the plane of the sample; (c) Magnetoresistance of the sample at different temperatures.

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An SEM image of sample 1 is shown in the inset of Figure 1(a). The length of the sample is defined as the distance L between the inner edges of the two voltage electrodes. Figure 1(a) shows resistance as a function of temperature (R-T) for sample 1 at an excitation current of 50 nA in the temperature range of 2–300 K at zero magnetic field (H). This sample exhibits a superconducting transition and zero residual resistance below TC ~ 4.8 K. The sample does not show metallic behavior in the normal state and the resistance actually shows a 10% increase when it is cooled from 300 K to 8 K. This non-metallic behavior is likely a consequence of the presence of impurities such as carbon, Ga and oxides and also high density of defects in the amorphous W strips formed during the FIB-CVD fabrication process18,23. In order to study the superconductivity near or below TC, the R-T curves in the low temperature regime (2 K ~ 6 K) are measured. Figure 1(b) shows R-T curves in the low-temperature regime measured at different H. In this paper, the applied magnetic fields are always perpendicular to the plane of the sample and substrate. The R-T curve of the W strip at zero field exhibits a superconducting transition at TC = 4.9 K (onset TC) and the width of the transition is about 0.2 K. On increasing H, the superconducting transition broadens and shifts to lower temperatures. Figure 1(c) shows the resistance of the W strip as a function of H at different temperatures. A superconducting transition with an onset critical field (HC2) larger than 8 T at 2 K is seen. The HC2 of the strip decreases with increasing temperature. At a temperature of 6.0 K, the superconducting transition is barely noticeable.

Discussion

Although bulk W is a type I superconductor, the W strips measured in these experiments can be thought of as a type II conventional superconductor due to its finite thickness and amorphous structure with high density of defects as seen in transmission electron microscope (TEM) studies23. The disorders and defects act as pinning centers for vortices. For type II conventional superconductor Nb8,9,10, the electric field – current density (E-J) isotherms measured at various temperatures at a given field value show qualitatively different behavior demarcated by a VG transition temperature, Tg. Near Tg, the curves show a power law relationship. For T > Tg, the E-J curves exhibit ohmic behavior and the resistivity remains constant at low currents. For T < Tg, E ~ exp (-J/Jμ) resulting in a convex curvature in the E-J curves which is a signature of the vortex glass phase6.

According to the VG transition theory6, the V-I curves at different temperatures near Tg can be scaled into two different branches by the scaling law,

where is the exponent of the coherence length , and is the dynamical exponent of the coherent time respectively, D is the dimension of the system and are the scaling functions above and below Tg. In the vortex liquid phase for the temperature above Tg, the linear resistivity at low current behaves as,

The V-I isotherm at Tg should satisfy the power law,

with . Previous theoretical and experimental studies show that and are reasonable values for a 3D VG phase transition1,6,7,9. In recent years, quasi-2D VG transitions have been reported and the ranges of and are similar to those for 3D12.

The V-I curves of the W strips are obtained at various magnetic fields and different temperatures. In this paper, we have measured the V-I curves by ramping the current from low to high and from high to low values. Identical results are found and we report only the curves collected when the current is being scanned downward. Figures 2(a)–2(c) show V-I curves measured between 2 and 6 K at applied fields of 0 Oe, 10 kOe and 20 kOe respectively. At zero field, the measured voltage jumps abruptly from zero to the normal state value at well-defined temperature dependent critical current. However, at a field of 10 kOe (Figure 2(b)), at temperature below Tg, V first increases slowly with I. This is followed by an abrupt jump and then a more gradual increase with further increase in I towards ohmic behavior. V-I curves measured at T > Tg do not show an abrupt jump and the curves are continuous resembling a sigmoid function. The maximum curvature of these curves decreases with increasing temperature. Similar evolution of V as a function of the current is seen at a field of 20 kOe (Figure 2(c)). The value of Tg decreases with magnetic field. The V-I curves shown in Figures 2(b) and 2(c) show that Tg at 10 kOe is near 4.0 K and at 20 kOe is near 3.4 K. We will show below that Tg can be more precisely identified by a more quantitative procedure. We define the critical current IC0, where the voltage changes from zero to non-zero and critical current IC1, where the voltage jumps rapidly.

Figure 2
figure 2

V-I measurement of the W strip at different temperature and under magnetic field.

(a–c) V-I characteristics measured at temperatures ranging from 2 to 6 K for H = 0 kOe, 10 kOe and 20 kOe, respectively. The red arrow points out that the measurement temperature changes from 2.0 K to 6.0 K. (d–f) V-I characteristics in log-log scales The dashed lines in panels (e) and (f) highlight power law V-I dependence at T = Tg (4.05 K for measurements taken at 10 kOe and 3.4 K at 20 kOe).

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The V vs. I dependence at low current is revealed in more details in log-log plots. Figure 2(d) clearly shows that at zero field the V-I curves show an abrupt change from the superconducting to the normal ohmic behavior at well defined ICs for all T < TC. For measurements at fields of 10 and 20 kOe, the log V - log I curves (Figures 2(e) and 2(f)) at currents smaller than IC1 show a smooth evolution in the curvature from being convex to concave with increasing temperature. The curves with different curvature are separated by a curve showing power law dependence measured at or near T = Tg. This behavior is reminiscent of the vortex-glass state with the crossover temperature TC-O corresponding to the VG transition temperature - Tg. When the temperature is below Tg and the sample is in the vortex glass state, the resistance falls rapidly with decreasing current and is zero below IC0. At temperatures above Tg, the sample is in the vortex liquid state, the V-I curves exhibit ohmic behavior and the resistance remains constant even at small excitation currents. Tg can be more precisely identified by plotting the slope (dV/dI) as a function of I for each T numerically computed from the data of Figures 2(b) and 2(c). For T < Tg, the curves show a sharp and theoretically infinite dV/dI peak at some current, for T > Tg, the dV/dI shows a low and wide peak (shown in Figures 3(a) and 3(b)). After plotting the inverse of the maximum dV/dI values shown in insets of Figures 3(a) and 3(b) for each temperature (black line), the changing rate of the slope can be calculated as a function of temperature shown in blue line in the inset of Figures 3(a) and 3(b). The Tg of sample is the point which has the maximum changing rate of inversed maximum dV/dI value vs. T (4.05 K for the curves under a field of 10 kOe and 3.4 K for 20 kOe). A red dashed line in Figures 2(e) and 2(f) represents the V-I behavior at Tg. The fitted line satisfies the relation— where is 3.15 and 3.16 for Figures 2(e and 2(f)) respectively. The values at different fields are in close agreement.

Figure 3
figure 3

dV/dI-I plots at temperatures ranging from 2 to 6 K for 10 kOe and 20 kOe, respectively.

The insets show the inverse of the maximum values of dV/dI (black line) and the changing rate of the slope of the black line. The blue line exhibits that the max changing rate is at 4.05 K and 3.4 K for 10 kOe and 20 kOe, respectively.

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The inset of Figure 4 shows the log R-T curve of sample 1 at 10 kOe with fit based on Eq. (2). The Tg value is 4.05 K. The slope of the fitting curve is . In order to get reasonable values of and within the limits mentioned above ( and ), the dimension D is taken to be 3 and the fitting parameter values are found to be and . This means our W strip system shows quasi-3D behavior. This is reasonable since the coherence length of W strip is about 6 nm17, which is much smaller than the thickness and width of the sample. Figure 4(a) shows the scaling collapse of the V-I isothermals according to Eq. (1) based on the experimental V-I curves at 10 kOe and the obtained parameter values ( and ).The experimental V-I curves formed two universal branches corresponding to the data above and below Tg (4.05 K). Similar universal scaling behavior can be obtained for the V-I curves at a field of 20 kOe (Figure 4(b)) with parameter values of , and .

Figure 4
figure 4

Quasi-3D VG scaling of the V-I curves measured at 10 kOe and 20 kOe respectively.

The insets show a logarithmic plot of the temperature dependence of the linear resistivity along with the fit used to obtain the scaling parameters.

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The V-I isothermals under different magnetic fields at 2 K (Figure 5(a)) are qualitatively similar to the V-I curves at different temperatures at fixed magnetic fields. The difference is that the V-I curves are separated into two parts by a particular field - Hg shown by a red dashed line in Figure 5(a). The isotherms are concave at the fields higher than Hg, while they are convex at fields lower than Hg. Hg plays the same role for these curves as Tg did in the discussion above. When the applied field is smaller than Hg, the voltage drops rapidly with decreasing current. At fields larger than Hg, the V-I curves exhibit ohmic behavior and the resistance remains constant in small current. This behavior exhibits the characteristics of a vortex glass-liquid transition. Hg can be considered to be the critical field of the VG phase transition. On the basis of the VG transition theory, the Eq. (1) can be rewritten in the following form:

Utilizing Eq. (4) and an approach similar to the one described above (also see Supplementary Information), the related parameter values can be obtained (, and ), the V-I isothermals collapse onto two different branches as shown in Figure 5(b). The values of z and ν are a little deviation from the reasonable ranges mentioned above, which is possibly due to the limited thickness and width of the nanostrip.

Figure 5
figure 5

V-I curves and Quasi-3D VG of the W strip at 2 K.

(a) V-I characteristics measured at 2 K for magnetic fields ranging from 5 to 80 kOe; (b) Quasi-3D VG scaling of the V-I curves measured at 2 K.

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Figure 6(a) is the H-T phase diagram for sample 1 (W strip). The values of the upper critical field HC2, are extracted from the R-H curves at different temperatures shown in Figure 1(c). The fitting line in Figure 6(a) is based on the empirical temperature dependence of HC (Eq. (5)).

In Figure 6(a), the black dots are Tg at 1 T and 2 T obtained from the V-I curves in Figure 3. The dashed fitting line is obtained using empirical formula (6)17,

is the field corresponding to the vortex glass melting temperature (Tg). From the V-I data at different temperatures and under different magnetic fields (see Figures 2(e) & 2(d), Figure 5(a)), the value of Tg at 10 kOe and 20 kOe, Hg at 2 K can be derived. With two values of parameter ν obtained by fitting Eq. (2) earlier (0.99 and 0.89 under 10 kOe and 20 kOe field respectively), the fitting lines with two ν values are almost the same. We found that the data point - Hg at 2 K (the star mark) in Figure 6(a) lies on the red dashed fitting line which is the VG melting line. According to the phase diagram, each Tg corresponds a unique field along the transition curve. Thus, the Hg should be equivalent to Tg on the melting line.

Figure 6
figure 6

The phase diagram and JC varies with T of the W strip.

(a) The phase diagram of magnetic field dependence of the melting temperature. (b) JC calculated from IC at various temperatures and perpendicular magnetic fields. JC0 is plotted by solid symbols and JC1 is drawn by open symbols. For 0 Oe data, the values of JC0 and JC1 are the same.

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According to Figure 2, the W strip is in totally superconducting state (zero voltage) below IC0, then the voltage increases slowly till IC1 and jumps to the normal state. Figure 6(b) shows the critical current density JC0 and JC1, calculated from IC0 and IC1 respectively, changing with temperature at 0, 10 and 20 kOe. The JC0 vs T curves obey the empirically parabolic law (Eq. (7)). For 0 Oe, the JC0 and JC1 are the same. The JC0 is at 2 K under 0 Oe field (The value in β - W film, of which TC is 3.35 K, is about 26). With the temperature and magnetic field increasing, the values of JC decreases.

The results discussed above were verified on the other two W strip samples by measuring their V-I characteristics at 2 K at different applied H. The crossover behavior (the curvature changes from convex to concave) discussed above was also observed in these samples at a particular field (Hg). In Figures 7(a) and 7(b), the V-I isothermals for samples 2 and 3 are scaling collapsed onto two branches obtained by using parameters: , and for sample 2, , and for sample 3. According to Figure 7, sample 2 obeys the VG scaling better than sample 3. This may be because that the thickness and width of sample 3 are distinctly smaller than those of sample 1 and sample 2. The small thickness and width limit the applicability of quasi-3D and quasi-2D VG transition theory to sample 3 and quasi-1D size effects may need to be taken into account in this situation.

Figure 7
figure 7

Quasi-3D VG scaling of the V-I curves measured at 2 K for sample 2 and 3.

(a) Quasi-3D VG scaling of the V-I curves measured at 2 K for sample 2; (b) Quasi-3D VG scaling of the V-I curves measured at 2 K for sample 3.

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In summary, the V-I curves of the W strips fabricated by focused ion beam & chemical vapor deposition are measured at different magnetic fields and temperatures. The scaling analysis of the V-I isothermals of W nanostrip (sample 1 and sample 2) follows the quasi-3D VG transition theory well. Every VG transition temperature Tg has a corresponding Hg along the VG melting line i.e. Hg appears to be equivalent to Tg in VG transition. However, the thinnest W strip (sample 3) shows a larger deviation from the conventional quasi-3D VG fitting. This may be because quasi-1D effects are not negligible in this situation27 and have to be considered in future theoretical studies.