Introduction

Fisher’s pivotal paper1 on vortex-glass superconductivity in disordered bulk materials described the state as hosting decaying metastable currents. Prior to this, it was known that in type-II superconductors, metastable currents decay logarithmically over time due to the cumulative dissipation introduced by thermally activated jumps of vortices out of pinning sites (defects). This phenomenon is known as flux creep, and creep measurements can provide experimental access to critical exponents associated with the vortex-glass phase, hence are useful for identifying and characterizing glassiness2. In fact, the primary objective of Fisher’s paper was to show that a sharp equilibrium phase transition exists between the normal state [at high T and fields (H)] and the flux creep phase at low T and H. He argued that a novel thermodynamic phase, the vortex-glass, appears below the phase boundary Tg(H). Subsequently, Nelson and Vinokur3,4 found similarities between the vortex-glass phase and their proposed Bose-glass phase hosted by materials containing correlated disorder (twin and grain boundaries, columnar defects). However, the mechanisms leading to the vortex-glass and Bose-glass phases are distinct. In the former, point disorder encourages wandering and entanglement of flux lines whereas, in the latter, vortices localize on extended, correlated defects3. The two states can be distinguished through measurements in tilted magnetic fields4.

Besides the ability to induce glassiness, interest in columnar defects is further motivated by their strong pinning capacity, associated with large pinning energies and subsequent enhancements in the critical current density (Jc). Pinning from CDs is directional; that is, at high enough fields, pinning is strongest, therefore Jc is highest, when the field is parallel to the CDs5. Despite the strong pinning capacity of CDs, YBCO crystals containing parallel CDs are known to demonstrate extremely high creep rates under certain measurement conditions. At low fields and with increasing temperature, the system evolves from a Bose-glass state hosting half-loop excitations to a non-glassy state in which the half-loops expand, connect with adjacent CDs, and form double-kinks (see Fig. 1). These kinks are unpinned or weakly pinned, therefore can slide relatively unhindered, which allows for rapid transfer of the vortex line between CDs and produces a prominent peak in S(T)6. The peak is quite large —several times higher than the plateau7 in S(T) at ~0.02–0.04 observed in pristine YBCO crystals and associated with a vortex-glass state. Furthermore, when the field is misaligned with the CDs, various staircase structures8 (see Fig. 1a) are known to form; a distinct signature of such structures has not yet been identified in creep measurements.

Figure 1
figure 1

Vortex structures involving columnar defects. Illustration of possible vortex structures (blue lines) in samples with columnar defects (grey tubes). The dotted black lines indicate the direction of the applied field (B) and \({\ell }_{hl}\) labels the half-loop length. The upper panel shows half-loop (left) and double-kink (right) excitations. The lower panel illustrates possible staircase structures that form as a vortex line minimizes its energy by shortening and/or pinning to the ab-plane.

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Many studies have characterized the effects of columnar defects on Jc (θH)5,9,10,11,12,13,14,15,16,17, where θH is the angle of the applied field. Much less is known about the effect of field orientation on the creep rate (S) and, more generally, creep in materials besides YBCO that contain CDs. For example, it is unknown why the peak associated with rapid double-kink expansion in YBCO has not been observed in other materials18,19,20,21,22. Of particular interest is superconductors with low Ginzburg numbers (Gi), such as NbSe2, which can attain significantly lower creep rates23 than superconductors with high Gi, such as YBCO (Gi ~ 10−2). This evokes the question of whether glassy states in low Gi materials manifest as a plateau at such a high S ~ 0.02–0.04 and double-kink expansion creates a peak in S. More generally, it motivates garnering a better understanding of the dynamics of various vortex excitations and glassiness in materials with low Gi.

In this study, we characterize the effect of temperature, magnetic field and field orientation on vortex dynamics in a NbSe2 crystal containing parallel CDs tilted ~30° from the c-axis. First, we observe the expected peak in Jc(θH) when H is parallel to the CDs, and we find that this peak is indeed accompanied by a dip in S(θH). Second, we compare and characterize S(T) and S(H) when the field is parallel to the defects (H || CDs) versus the c-axis (H || c). Last, we find evidence of glassiness in both field orientations.

Sample Fabrication and Measurements

Our experiments are carried out on two undoped 2H-NbSe2 crystals that were grown using iodine vapor transport24 and have dimensions ~0.8 mm × 0.7 mm × 20 μm and ~1.5 mm × 0.3 mm × 8.5 μm (length L × width W × thickness δ). 2H-NbSe2 is a layered transition metal dichalcogenide with an s-wave gap structure that has attracted intense interest25 because it hosts a coexisting incommensurate charge density wave phase and superconductivity below Tc ~ 7 K. Our primary motivation for studying NbSe2 is that it is a clean system (few defects in as-grown crystals) that has a low Ginzburg number (Gi). Scanning tunneling microscopy studies have revealed a low density of Nb and Se vacancies and Nb interstitials in NbSe2 crystals grown by iodine vapor transport26,27,28. One study found a defect density of ~0.4%28. Assuming a coherence length ξab ≈ 7.4 nm, penetration depth29,30,31 λab ≈ 126 ± 3 nm, and upper critical field anisotropy32 of \(\gamma ={H}_{c2}^{ab}/{H}_{c2}^{c}={\xi }_{ab}/{\xi }_{c} \sim 3.2\) (all at T = 0), we estimate \(Gi=({\gamma }^{2}/2){[({\mu }_{0}{k}_{B}{T}_{c})/(4\pi {B}_{c}^{2}(0){\xi }_{ab}^{3}(0))]}^{2}\approx 8\times {10}^{-7}\), where \({B}_{c}={{\rm{\Phi }}}_{0}/[2\sqrt{2}\pi {\lambda }_{ab}{\xi }_{ab}]\) is the thermodynamic critical field.

One crystal (δ = 20 μm) was heavy-ion irradiated with 1.4 GeV208 Pb56+ ions at a dose of 1.45 × 1011 ions/cm2 corresponding to a matching field of 3 T (average distance between CDs ~ 26 nm) at the Argonne Tandem Linear Accelerator System (ATLAS) while mounted with the crystallographic c-axis ~30° from the incident beam. The sample underwent no additional processing steps post-irradiation. We chose to induce tracks at an angle of ~30° from rather than parallel to the c-axis to distinguish the effects of the CDs from those of mass anisotropy and intrinsic correlated defects (e.g., edge and screw dislocations) that are known to produce a peak in Jc(θH) for H || c in YBCO13. Similarly, for tilted CDs, the mere existence of asymmetry between Jc(θH) and Jc(−θH) can provide evidence of correlated pinning.

Transmission Electron Microscopy (TEM) studies were performed on the irradiated crystal. The acquired image shown in Fig. 2a indicates that the columnar amorphous tracks are continuous and almost perfectly parallel to each other, consistent with previous studies33 and with the small splay expected for 1.4 GeV Pb ions. Figure 2b is a higher magnification image showing an angle of ~29° between the radiation direction and the NbSe2 [002] direction. From our TEM work, we measured an average CD diameter of about 4 to 6 nm. In addition to columnar tracks, heavy ion irradiation may induce secondary electrons that act inelastically with the material matrix, producing point defects in between the columnar tracks34,35. There is limited knowledge about the secondary damage produced by heavy ion irradiation. A recent scanning tunneling microscopy study of a heavy ion irradiated Fe(Se, Te) crystal showed that the superconducting order parameter was annihilated inside the columnar tracks and suppressed by the interlaying point defects35.

Figure 2
figure 2

(a) A low magnification TEM image of the irradiated NbSe2 crystal showing continuous and parallel irradiation tracks. (b) A high magnification TEM image showing an angle of ~29° between the radiation direction and the NbSe2 [002] direction. The average size of amorphous tracks is ~4.7 nm.

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Magnetization (M) measurements were collected using a Quantum Design SQUID magnetometer with a rotating sample mount as well as transverse and longitudinal pick-up coils to measure each component of the magnetic moment. By measuring M versus T at 2 Oe, we find that the critical temperature of the irradiated crystal is Tc ≈ 7 K, similar to that in pristine crystals25. We extracted Jc(T) from the magnetization data using the Bean Model36,37, Jc(T) = 20ΔM/W[1 − W/(3L)], for H || c, where ΔM is the difference between the upper and lower branches of the M(H) curve. For the data collected when H || CDs, the tilted field orientation weakens the Lorentz force seen by some of the circulating currents, necessitating a modification of the Bean model38,39: Jc(T) = 20ΔM/W[1 − Wcos(θH)/(3L)]. To measure creep, we record successive measurements of M every 15 s at fixed fields and temperatures, capturing the decay in the magnetization (M ∝ J, where J is the induced current) over time (t). We then calculate the creep rate S[T, H(θH)] = |d ln J/d ln t|. See Methods for more details.

Results and Discussion

Magnetization in different field orientations

Figure 3 compares isothermal magnetic hysteresis loops, M(H), at T = 1.8 K for the pristine crystal for H || c (θH = 0°), and the irradiated sample for both H || c and for the field aligned with the defects (H || CDs, θH = θCD = −31°). The pristine crystal demonstrates dramatically lower magnetization and irreversibility field than the irradiated crystal. This suggests a weak pinning landscape and that the columnar defects in the irradiated crystal are overwhelmingly the predominant source of pinning.

Figure 3
figure 3

Magnetic hysteresis loops when H is aligned with the columnar defects versus the c-axis. Comparison of field dependent magnetization, M(H), at T = 1.8 K for the irradiated NbSe2 crystal for two different field orientations (H parallel to c-axis versus H parallel to columnar defects) and the pristine NbSe2 crystal for H parallel to the c-axis. (inset) Magnification of M(H) loop for the pristine crystal.

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For the irradiated crystal, the magnetization is roughly 5 times higher when the field is aligned with the CDs than with the c-axis. A large enhancement was anticipated and had been observed in previous studies, though the magnitude was less5. This improvement could be attributed to the higher energy used during irradiation (1.4 GeV Pb56+ versus 300 MeV Au26+ in ref.5), which might create straighter, more continuous tracks40.

The dip at low fields μ0H < 0.6 T is caused by the out-of-plane pinning anisotropy. That is, pinning by extended defects along the c-axis (or, in our case, tilted 30° from) should produce a weak dip in M(H) at zero field, while pinning along the crystallographic ab-plane is expected to produce a peak41. At fields below self-field Hsf ≫ H, vortex lines over a large region of the sample peripheries are quite curved. As the applied field is increased (approaching self-field), this region decreases as vortices straighten over a wider portion of the sample center. Columnar defects are more effective at pinning straight vortices. Hence, the initial increase in M with increasing H is caused by a combination of the heightened effectiveness of individual CDs in pinning less curved vortices and growing portions of the sample containing straight vortices. Predicted theoretically41, the peak has been observed in irradiated YBCO42 and Ba(Fe0.93Co0.07)2As2 crystals43.

Additional M(H) loops were collected at T = 4.5 K and at 20 different angles. Select curves are shown in Fig. 4a,b, capturing crossovers into different regimes. Note that the curves converge near zero field. This is because in the very dilute limit and for all field orientations, vortex lines will be oriented normal to the sample surface (aligned with the c-axis) to minimize their energy by shortening5.

Figure 4
figure 4

Magnetic hysteresis loops for different field orientations. (a) Comparison of field dependent magnetization loops, M(H), at T = 4.5 K when the field is applied at several different angles (θH). (b) Select data from (a) highlighting the appearance of weak second magnetization peak when field is tilted far from defects (θH ≥ −2°).

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As the field tilts away from alignment with the CDs (|θH − θCD| > ~6°), the low-field peak progressively shifts to lower fields and eventually disappears. In particular, at θH = −24°, M(H) decreases nearly linearly with decreasing H. Further rotation of the field away from the CDs (θH < −40°, θH > −19°) changes the M(H) behavior. M initially abruptly decays with increasing H, showing similar shape to M when H||c (Fig. 3). As the field is increasingly tilted (θH ≥ −2°), the M(H) curves exhibit a weak second magnetization peak (known as the fishtail effect) between 0.5 T and 1 T. This is most pronounced at θH = 33°, as highlighted in Fig. 4b. The fishtail effect has been observed in a wide variety of materials, including low-temperature superconductors, cuprates, MgB2, and iron-based superconductors44,45 and associated with an equally wide variety of effects, including elastic-to-plastic crossovers, vortex order-disorder phase transitions, and vortex lattice structural transitions44. In fact, a previous study46 reported the appearance of a fishtail in a pristine NbSe2 crystal when the applied field was tilted 30° from the c-axis and attributed it to a vortex order-disorder transition.

Extracted from the M(H) loops, the data is re-plotted as M(θH) at different fields in Fig. 5. The peak at θH = θCD is clear at all fields and M rapidly decays at the slightest field misalignment with the defects, corresponding to a large reduction in Jc. It is important to note that this prominent peak indicates that pinning provided by the CDs is significantly greater than that from any point defects possibly introduced in between the CDs by secondary electrons during the irradiation process. If we compare critical currents when the field is aligned with the CDs versus the c-axis, we find that Jc is ~240 kA/cm2 compared to ~48 kA/cm2, respectively, at 0.6 T. Figure 6 shows such a comparison at 0.3 T over a broad temperature range, displaying an increase in Jc by a factor of ~4 at 4.5 K and ~3 at 1.8 K. Note that the defects are effective even down to the lowest field of 0.2 T, where Jc is only ~10% lower than at the maximum. This is consistent with all data in Fig. 5 being well above Hsf ~ Jcδ ≤ 550 Oe at this temperature. At most angles, lower fields produce higher M. However, for θH > 0°, some low field curves cross, resulting in non-monotonic M(H) that is consistent with the regime in which the fishtail is observed (Fig. 4b).

Figure 5
figure 5

Large peak in magnetization when applied field is aligned with columnar defects. Magnetization (M) versus magnetic field orientation (θH) in NbSe2 crystal containing columnar defects tilted ~−30° from the crystallographic c-axis. Data is shown for T = 4.5 K and multiple values of the magnetic field.

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Figure 6
figure 6

Temperature dependence of the critical current. Critical current (Jc) versus temperature for the irradiated crystal for both H || c and H || CDs in a field of 0.3 T.

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Vortex creep when field is aligned with CDs versus c-axis

To analyze vortex excitations and the potential for glassy dynamics, we measured the dependence of the creep rate on temperature and field orientation. First, we consider two creep models: the Anderson-Kim model and collective creep theory. A defect (or collection of defects) can immobilize a vortex segment (or a bundle of vortex lines) by reducing the vortex line energy by the pinning energy UP(T, H), which is the energy barrier that must be overcome for vortices to move. The Lorentz force induced by the persistent current J then reduces UP to an activation barrier Uact(T, H, J) and the vortex hopping rate is ~\({e}^{-{U}_{act}/{k}_{B}T}\). The Anderson-Kim model2, which neglects vortex elasticity and therefore does not predict glassy behavior, often accurately describes creep at low temperatures T ≪ Tc. It assumes Uact(J) ∝ UP|1 − J/Jc| for J/Jc. As UP is nearly temperature-independent at low T, S is expected to increase linearly with increasing T, resulting in2 S(T) ≈ kBT/UP. At high temperatures, S(T) steepens as UP(T) decreases.

Collective creep theory2 predicts that the temperature dependence of the creep rate is

$$S=\frac{{k}_{B}T}{{U}_{P}+\mu {k}_{B}T\,\mathrm{ln}\,(t/{t}_{0})},$$
(1)

where t0 is the effective hopping attempt time and C ≡ ln(t/t0) ~ 25–30. Here μ > 0 is the glassy exponent indicating the creep regime: μ = 1/7, 3/2 or 5/2, and 7/9 are predicted for creep of single vortices, small bundles (size less than the penetration depth λab) and large bundles (size greater than λab) of flux, respectively. At low temperatures T ≪ Tc, UP ≫ μkBTln(t/t0) such that S(T) ≈ kBT/UP, coinciding with the Anderson-Kim prediction.

We now compare creep data for the irradiated crystal in two different field orientations: H || CDs and H || c. Note that our measurements are restricted to low fields because at high temperatures and fields, the magnetic signal is quite small when H ||c. Figure 7a shows the measured creep rate versus field orientation at 4.5 K and 0.5 T. Creep is clearly minimized when the field is aligned with the defects; S is an order of magnitude smaller for H || CDs than for H || c. In fact, aligning the field with the defects suppresses creep at all fields and temperatures measured in our study, e.g., the comparison of S(H) in both field orientations at 1.8 K shown in Fig. 7b.

Figure 7
figure 7

Creep in irradiated NbSe2 crystal. (a) Creep rate (S) versus magnetic field orientation (θH) in the irradiated NbSe2 crystal and (inset) the extracted effective pinning energy U*. (b) Comparison of field-dependent S and (c,d) temperature-dependent S when the field is aligned with the columnar defects versus the c-axis. The solid black lines are linear fits to the low temperature data. The dotted grey curve in c is data for the pristine sample when the field is aligned with the c-axis.

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Comparing creep data for the irradiated sample to the pristine crystal can only be performed at very low fields because the measurement signal produced by the pristine crystal at higher fields is near the lower limit of our measurement sensitivity. The temperature dependence of the creep rate in the pristine crystal and the irradiated crystal at 0.02 T is shown in Fig. 7c. For both field orientations, S increases linearly with T up to 5.5 K, qualitatively adhering to the Anderson-Kim description. Despite the very low applied field, the CDs are effective at lowering creep when H || CDs, but not when H || c, seen from a comparison to the data from the pristine sample.

Considering collective creep theory, if UP ≪ CμkBT, S(T) should plateau at S ~ 1/Cμ. Such a plateau is predicted in the case of glassiness, such that S ~ 0.02−0.04, equivalent to typical observations of plateaus in YBCO single crystals7 and iron-based superconductors39,47,48,49,50,51,52. For our NbSe2 crystal, Fig. 7d shows S(T) at μ0H = 0.3–0.5 T for the two field orientations. In all cases, in Fig. 7c,d the creep rates are well below the usual collective creep plateau. The simplest interpretation is that UP is not negligible compared to CμkBT (see eq. 1), which is in agreement with the pinning energy estimates described below. Although, consistent with this scenario, most of the S(T) curves in Figs 7c,d are monotonically increasing, Fig. 7d also shows a broad temperature insensitive region in the 0.5 T data for H || c (S ~ 0.003) and a narrower one in the 0.3 T data for H || CDs (S ~ 0.002). However, interpretation of these data as indicative of a plateau at much lower than usual values would imply Cμ ~ 300–500, producing unphysically large values of either μ (10–17) or C (120–200); note that typical C and μ values7,12 give Cμ < 75. Finally, quantum creep may be a significant component of our measured creep rates at these low temperatures, in which case, adding a temperature independent (and unfortunately unknown) contribution would imply an even smaller thermal creep contribution.

A plateau in S(T) is the most apparent manifestation of glassy vortex dynamics. In its absence, we need a different approach to assess the nature of the vortex depinning excitations. Analysis of the current dependence of the effective activation energy U * ≡ T/S can provide direct experimental access to μ without the need for assumptions regarding UP. According to collective creep theory2, the activation barrier depends on the current as

$${U}_{act}(J)=\frac{{U}_{P}}{\mu }[{(\frac{{J}_{c0}}{J})}^{\mu }-1],$$
(2)

where Jc0 is the temperature-dependent critical current in the absence of flux creep. Considering the Arrhenius hopping rate ~\({t}_{0}^{-1}{e}^{-{U}_{act}(J)/{k}_{B}T}\) and equations (1) and (2), the effective pinning energy is

$${U}^{\ast }\equiv \frac{T}{S}={U}_{P}\times {({J}_{c0}/J)}^{\mu },$$
(3)

where μ > 0 for glassy creep and μ → p < 0 for plastic creep53. Consequently, the exponent can easily be extracted from the slopes of U* vs 1/J on log-log plot. From Fig. 8, we see distinct elastic-to-plastic crossovers for all sets of data. At low T the dynamics is clearly glassy at both field orientations, with μ ~ 1. This is one of the main experimental findings of this study. As T increases the dynamics turns plastic, with p in agreement with the expectation for the motion of dislocations in the vortex lattice (p = −0.5)54.

Figure 8
figure 8

Elastic to plastic crossovers. Dependence of the effective energy barrier U* on the inverse current density when the field is aligned with the columnar defects (upper panel) versus the c-axis (lower panel). The extracted exponents μ and p are displayed in the plot, where μ = 1 is expected for glassy behavior and p = −0.5 for plastic flow.

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For H || CDs, glassy dynamics with μ ~ 1 is expected for a Bose-glass state characterized by half-loop formation. However, glassiness was unforeseen for H || c. In this configuration, we expected to see evidence of staircase structures (see Fig. 1), which form when the field is tilted away from the CDs by an amount greater than the lock-in angle (θL), but less than the trapping angle (θt). Yet in the simplest scenario staircases should be non-glassy, as finite length kinks easily slide along CDs. So, several possibilities should now be considered: θH = 0° is within the lock-in angle and half-loop excitations are responsible for μ ~ 1, the dynamics of the staircase vortices is glassy, or this orientation is beyond θt and the CDs do not produce correlated pinning (so glassiness arises from standard random collective pinning).

A Bose-glass state formed when the field is aligned with CDs (and vortices are localized on these defects) will be robust to small changes in field orientation. That is, when the field tilted away from the CDs by an angle less than θL, vortices will remain completely pinned by the CDs. This results in a plateau in M(θH) for |θH − θCD| < θL that has been observed in cuprates55,56,57,58 and Co-doped BaFe2As214. Though our data is too coarse to determine if there is a lock-in effect and identify θL, we see from Fig. 5 that the magnetization is greatly reduced at θH = 0° versus θH = θCD. So, θH = 0° is clearly well beyond the lock-in angle. Consistently, θL is expected to be very small in our NbSe2 crystal (see estimate below). On the other hand, the asymmetry of M(θH) around θH = 0°, which can only arise from the tilted CDs, suggests that staircases are present at this orientation55.

Having eliminated half-loops and random collective pinning as the cause of μ ~ 1 at H || c, we consider the possibility of a vortex-glass state or an anisotropic glass involving both columnar and point disorder, as predicted in ref.4. Segments of a single vortex line could be alternatingly pinned by adjacent CDs and interstitial point defects. As the current and thermal energy act on the vortex, the segments pinned by point defects might wander/entangle (instead of sliding like kinks). Alternatively, interactions among weakly pinned kinks may create “kink bundles” that, by analogy with the 3D vortex bundles, should exhibit glassy collective creep with μ ~ 1. In either case, if the phase for H || CDs is indeed a Bose-glass then the system experiences a field-orientation-driven transition from a Bose-glass (H || CDs) to an anisotropic glass (H || c). As the expected exponent μ ~ 1 is identical for a vortex glass, Bose glass, and anisotropic glass, measurements of the exponent alone cannot distinguish between vortex configurations that lead to glassy dynamics. The real fingerprint of the Bose glass is the presence of a lock-in effect.

In light of this, we find it important to mention an alternate possible scenario: even for H nominally parallel to the CDs, a slight field misalignment θH − θCD > θL could lead to staircase formation. Such a misalignment is challenging to avoid when \({\theta }_{L}\approx \frac{4\pi \sqrt{2{\varepsilon }_{l}{\varepsilon }_{r}}}{{{\rm{\Phi }}}_{0}H}\) is very small. Here \({\varepsilon }_{\ell }=({\varepsilon }_{0}/{\gamma }^{2})\,\mathrm{ln}\,({\lambda }_{ab}/{\xi }_{ab})\) is the line tension (in the nondispersive limit and disregarding anisotropy factors \(\varepsilon ({\theta }_{CD})={[{co}{{s}}^{2}({\theta }_{CD})+{si}{{n}}^{2}({\theta }_{CD})/{\gamma }^{2}]}^{1/2}\sim 1\)); εr is the pinning energy per unit length; and ε0 = (Φ0/4πλab)2 is the line energy4. From the relation Jc/J0 ≈ 0.6εr/ε0, where \({J}_{0}={{\rm{\Phi }}}_{0}/({3}^{3/2}\,\pi {\mu }_{0}{\lambda }_{ab}^{2}{\xi }_{ab})\) is the depairing current density2, using2,29,30,31,32 ξab(1.8 K) = ξab(0)[1 − T/Tc)]−1/2 ~ 8.6 nm, λab(1.8 K) = λab(0)[1 − (T/Tc)4]−1/2 ~ 126 nm, we estimate J0(1.8 K) ≈ 75 MA/cm2 in our crystal, thus εr/ε0 ≈ 0.01 and \({\theta }_{L}(T=0)\approx \frac{8\,Oe}{H}\). This corresponds to θL(1.8 K) ≈ 0.2° for μ0H = 0.2 T, decreasing with both T and H. It is thus possible that we are observing staircases in both configurations and the differences in Jc and S arise from the much larger number of kinks for H || c. Additional studies with an angular resolution finer than θL would be needed to elucidate this point.

Pinning energies

The effectiveness of CDs is typically assessed by evaluating the measured pinning energies, which can be calculated from the creep data. The scale of the pinning energy in a superconductor59 is approximately the condensation energy \({U}_{P1} \sim ({B}_{c}^{2}/2{\mu }_{0})V\) within a coherence volume \(V \sim {V}_{c}=\,(4\pi /3){\xi }_{ab}^{3}/\gamma \). For NbSe2, we calculate that UP1 ~ 160–300 K within our measurement T range. From the Fig. 7a inset, we see that the effective activation energies U* extracted from our creep measurements plummets from being considerably greater than to comparable to UP1 as the field rotates from H || CDs to H ||c. This is because pinning energies larger than UP1 can be achieved through individual strong pinning by defects larger than Vc, as is the case for our CDs.

Columnar defects are most effective at pinning vortices of smaller core size \(\sqrt{2}{\xi }_{ab}\le R\) (where R is the CD radius)2,3,4,12,60. This is not easily achieved in low-Tc superconductors, which tend to have large coherence lengths. When \(R < \sqrt{2}{\xi }_{ab}\) (as is the case for our sample), under ideal pinning conditions εr ≈ ε0(R/2ξab)2. Considering an average R ~ 2.5 nm for the CDs in our crystal, at T = 1.8 K we obtain εr/ε0 ≈ 0.02, about twice our aforementioned experimental value determined simply from Jc/J0. This demonstrates that the CDs in our crystal indeed behave as strong correlated disorder, producing about half of the ideal pinning. For comparison, analogous calculations predict that CDs in YBCO should ideally produce Jc ~ J0, while experimental Jc values fall short of that by a factor of ~3 to 4.

A vortex pinned to an isolated CD may depin when the half-loop length is \({\ell }_{hl} \sim {\xi }_{ab}{[{\varepsilon }_{r}{\varepsilon }_{\ell }/{\varepsilon }_{0}^{2}]}^{1/2}({J}_{0}/{J}_{c})\) (the half-loop nucleus reaches a critical radius). In this case, the associated pinning energy2 is \({U}_{h\ell } \sim {\varepsilon }_{r}{\ell }_{hl}\). Note that the transverse size of the half-loop depends on competition between the elastic energy \({\varepsilon }_{\ell }{U}_{h\ell }/{\ell }_{hl}\) and pinning energy \({U}_{h\ell }\), and that the critical size occurs when the Lorentz force \(J{\Phi }_{0}{\ell }_{hl}{U}_{h\ell }/c\) matches the elastic energy2,4. A system containing half-loops therefore exhibits a glassy response because the half-loop energy barrier increases with decreasing current. For our NbSe2 sample when H || CDs, using2,29,30,31,32 ξab(4.5 K) ~ 12.4 nm, λab(4.5 K) ~ 138 nm, (thus J0(4.5 K) ≈ 42.5 MA/cm2) and our measured Jc(4.5 K, 0.5T) ~ 180 kA/cm2, we calculate the following: ε0(4.5 K) ~ 1000 K/nm, εr(4.5 K) ~ 0.007ε0 ~ 7 K/nm, \({\ell }_{hl}(4.5K) \sim 130\,\,nm\) and \({U}_{h\ell }(4.5\,K) \sim 1000\,{\rm{K}}\). This is somewhat smaller than our experimental U *(4.5 K, 0.5T) = T/S~3500 K, but consistent given the simplicity of the estimates. First, we note that the calculation of εr(4.5 K) based on Jc/J0 is likely an underestimate, as Jc may be reduced due to CDs discontinuities, vortex bending, and the possibility that some vortices may be occupying interstitial positions outside the CDs. Alternatively, if we use the estimate εr ≈ ε0(R/2ξab)2, for R = 2.5 nm we obtain εr(4.5 K) ~ 10 K/nm; and \({U}_{h\ell }(4.5\,K) \sim 1300\,{\rm{K}}\). We note that the calculation is highly sensitive to slight changes in the parameters, e.g., R ~ 2–3 nm yields \({U}_{h\ell }\, \sim 670\,\mbox{--}\,2300\,{\rm{K}}\). In fact, the effective CD size may be larger because the irradiation induced tracks may depress the superconducting order parameter over a farther distance than the diameter measured by TEM due to, e.g. lattice strain. Second and perhaps more importantly, the above analysis neglects vortex-vortex interactions, which should be considered because the lateral dimension of the half-loops4 \({u}_{h\ell }(4.5\,K) \sim {[{\varepsilon }_{r}/{\varepsilon }_{l}]}^{1/2}{\ell }_{hl} \sim 20\,\mbox{--}\,30\,{\rm{nm}}\) is not negligible compared to the vortex lattice parameter a0 ~ 70 nm for μ0H = 0.5 T. Hence, repulsion of neighboring vortices produces a caging effect that increases the effective pinning energy, stiffening the lattice and reducing S.

Conclusions

In conclusion, we have studied the dependence of vortex dynamics on the orientation and magnitude of the applied magnetic field in a NbSe2 crystal containing tilted columnar defects. As most studies of creep in samples containing columnar defects have been limited to heavy ion irradiated YBCO, studying NbSe2 has allowed us to probe effects applicable to materials with lower Ginzburg numbers and larger vortex size to columnar track ratios. Specifically, we demonstrated that the critical current is maximized and creep is concomitantly minimized when the field is aligned with the defects (T = 4.5 K, μ0H = 0.5 T). This result was not necessarily intuitive, as the rapid expansion of double-kinks can promote fast creep when H || CDs in YBCO (at low temperatures and fields below the matching field). We also found that H || CDs preferentially produced lower creep rates than H || c over our entire measurement range, and that both field orientations resulted in glassy behavior. A Bose glass state is indeed expected when the field is aligned with the CDs. Yet the existence of glassiness when the field is misaligned is quite fascinating and suggestive that staircase structures might be able to entangle or localize in a way that leads to glassy behavior.

Many open questions remain. First, it is unclear why a distinct, large peak in S(T) resulting from double-kink expansion has only been observed in YBCO. Second, do other materials containing CDs show glassiness when the field is oriented in a way that is favorable for staircase formation? In addition to testing this in other low Gi materials, it would be interesting to test in highly anisotropic samples in which pinning to the ab-plane is highly favorable over the c-axis. Third, is the potential anisotropic glass state enabled by secondary damage that appears in between the columnar tracks? These results motivate further studies of creep rates at various field orientations in other heavy ion irradiated materials.

Methods

TEM images

The TEM specimen of the irradiated NbSe2 crystal was fabricated in a focused ion beam and the microstructure was characterized by using FEI Tecnai F30 Transmission electron microscopy (TEM, 300 kV).

Magnetization Measurements

Magnetization measurements were collected using a Quantum Design SQUID magnetometer with a rotating sample mount, and transverse and longitudinal pick-up coils to measure each component of the magnetic moment, mt and ml, respectively. The angle of the field was verified by calculating \({\theta }_{H}={\tan }^{-1}({m}_{t}/{m}_{l})\), the total moment \(m={m}_{l}/\,\cos \,{\theta }_{H}\), and the magnetization M = m/δLW (where δ μm is the thickness, W is the width, and L mm is the length). Creep data were taken using standard methods7. Firstly, the field was swept high enough (ΔH > 4H*) that the sample was fully penetrated with magnetic flux and in the critical state. Then, successive measurements of M were recorded every 15 s, capturing the decay in the magnetization (M ∝ J) over time (t). Last, the time was adjusted to account for the difference between the initial application of the field and the first measurement and S = |d ln M/d ln t| is calculated from the slope of a linear fit to ln M-ln t. Tc was determined from the temperature-dependent magnetization at H = 2 Oe.