Upper critical field and superconductor-metal transition in ultrathin niobium films

Recent studies suggest that in disordered ultrathin films superconducting (SC) state may be intrinsically inhomogeneous. Here we investigate the nature of SC state in ultrathin Nb films, of thickness d ranging from 1.2 to 20 nm, which undergo a transition from amorphous to polycrystalline structure at the thickness \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \simeq 3.3$$\end{document}d≃3.3 nm. We show that the properties of SC state are very different in polycrystalline and amorphous films. The upper critical field (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{c2}$$\end{document}Hc2) is orbitally limited in the first case, and paramagnetically limited in the latter. The magnetic field induced superconductor-metal transition is observed, with the critical field approximately constant or decreasing as a power-law with the film conductance in polycrystalline or amorphous films, respectively. The scaling analysis indicates distinct scaling exponents in these two types of films. Negative contribution of the SC fluctuations to conductivity exists above \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{c2}$$\end{document}Hc2, particularly pronounced in amorphous films, signaling the presence of fluctuating Cooper pairs. These observations suggest the development of local inhomogeneities in the amorphous films, in the form of proximity-coupled SC islands. An usual evolution of SC correlations on cooling is observed in amorphous films, likely related to the effect of quantum fluctuations on the proximity-induced phase coherence.


Arrhenius law
In Fig. S1(b-d) we show the effect of the magnetic field on the dependence of R sq /R N on 1/T (on logarithmic scale) for two polycrystalline films, d = 9.5 nm (b), and d = 5.3 nm (c), and for amorphous film d = 1.4 nm (d). The polycrystalline films exhibit clear Arrhenius behavior over the large portions of the data, what indicates flux activation regime. Following the previous analysis of such regime in niobium films by others 3 and us 4 , from the slope of linear portions of the Arrhenius plots we may extract the activation energy for the vortex pinning, ln (R sq /R N ) = −U (H)/k B T + K(H). Here U (H) is the zerotemperature activation energy and K(H) is the coefficient in the linear T correction. In polycrystalline films the U shows ln H-dependence at high fields (µ 0 H 1 T), and power-law dependence at lower fields; as d decreases and polycrystalline/amorphous boundary is approached the ln H region expands towards lower fields, down to about 0.1 T. Similar ln H-dependence has been reported previously for thin Nb films 3,4 . In case of amorphous film with d = 1.4 nm [ Fig. S1(d)] the linear portion of the Arrhenius plots is limited to high temperature range, T 0.27 K; at lower T saturation of resistance gradually sets in. The saturation does not shift to higher T as the magnetic field is increased, therefore, we do not believe that magnetic field causes the heating. However, we cannot exclude the possibility that the saturation may be caused by sensitivity of the film to external noise, this issue will be studied in a future investigation. The Arrhenius portions of the plots may be fitted as in the case of polycrystalline films, revealing that the magnetic field dependence of the activation energy in all films is given by U (H) = U 0 ln (H 0 /H), with the parameters H 0 and U 0 shown in Fig.S1(e). The H 0 is the magnetic field at which U extrapolates to zero; it is very close to H c2 in polycrystalline films with d=9.5 and 5.3 nm, but drops substantially below H c2 on the approach to polycrystalline/amorphous boundary, indicating that the disorder destroys vortex pinning at substantially lower magnetic field. The prefactor U 0 depends on the film thickness as a power law, U 0 ∼ d 3 . Both the logarithmic dependence on the magnetic field, and the dependence of the prefactor on the film thickness are expected in case of collective vortex pinning in thin films 5 . In particular, the prefactor should be proportional to d/Λ 2 e f f , where Λ e f f is the effective penetration depth in thin film,

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inversely proportional to d, what results in the observed film thickness dependence. More details on this behavior will be published elsewhere.
In polycrystalline films the flux may remain pinned at the lowest T below melting line, H m (T ), creating zero-resistance vortex glass phase as has been verified in case of thick Nb films 6 . We use standard method to determine this line 7,8 , from the fit of linear relation to the dependence of (d(lnR)/dT ) −1 versus T , as shown in the inset to Fig.S1(f) for the film with d = 9.5 nm. The main figure shows that the melting line extrapolates to the H 0 for T = 0. While this method results in considerable error bars, it is clear that the melting line is situated substantially below the H c2 (T ) line, which is expected in case of ultrathin films. This dependence is further confirmed by the measurement of I-V curves at T = 2 K, as indicated by green star in the figure. Similar procedure may be used in all polycrystalline films, and in the thicker amorphous films, down to d = 2.2 nm. On the other hand, in case of thinnest amorphous films (d = 1.3 or 1.4 nm) we were not able to determine unambiguously such melting line.
Figs.S2(a-b) show the results of I-V measurements for two films, d = 9.5 nm measured at T = 2 K (a), and d = 1.3 nm measured at T = 50 mK (b). In (a) we see a behavior typical for superconductor, with zero voltage at low current, and abrupt increase of the voltage once the critical current is reached. In (b), on the other hand, a gradual increase of the voltage starts as soon as the current starts to grow, and this is the case even in the absence of the magnetic field. This suggests that in this film true zero resistive state may be absent. Interestingly, in the vicinity of the critical current a clear evidence of hysteresis is seen, i.e. the abrupt transition to normal state occurs at higher increasing current than the transition to quasi-superconducting state on decreasing current. This hysteretic behavior suggests the possible involvement of the vortex pinning and depinning. Such interpretation is consistent with the fact that on increasing magnetic field hysteresis disappears, since at higher field activation energy decreases, what reduces pinning. We note that we cannot exclude the possibility that hysteresis may arise because of local heating of some areas of the film. However, it may also be related to flux avalanche phenomenon, which prevents vortex pinning on decreasing current at exactly the same current value at which depinning occurs on increasing current. Similar behavior has been observed in thin tantalum films, and interpreted as arising from non-thermal origin 9 .  Figure S2. I-V characteristics measured at various magnetic fields for films with d = 9.5 nm at T = 2 K (a), and d = 1.3 nm at T = 50 mK (b). In (b) the arrows indicate hysteresis observed for increasing and decreasing current.

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Isotherm crossing Fig. S3 shows the dependence of R sq on B for series of temperatures below 1K for a film with d = 1.4 nm. To extract B c and R c we determine the crossing points of consecutive isotherms. Starting from high temperatures, when T is reduced from 1 K towards 0.4 K, as shown by grey curved arrow, the crossing points, marked by crosses at the intersections of dashed lines, shift towards lower B. However, when T is reduced below 0.36 K, as shown by black curved arrow, the crossings points (marked by stars at intersections of continuous lines) reach some minimum value of B, and possibly even start to increase slightly (this increase is within experimental error of the measurement).
The inset to Fig. S3 shows the T -dependence of B c and R c values determined by this procedure. Within experimental accuracy the B c and R c are seen to be constant below the temperature T 0 marked in the figure. The value of B c , averaged for T < T 0 , is equal to 1.46 ± 0.01 T.

Upper critical field
According to WHH theory the upper critical field in the dirty limit can be calculated using the following equation 10 : When α = 0 and λ so = 0, in the absence of the spin-paramagnetic effect and the weak spin-orbit interaction upper critical field is described by: In BCS weak-coupling superconductors with 2∆/kT c = 3.52, A is equal to 0.69. Since the Nb is an intermediate-coupling superconductor, 2∆/kT c is larger, so that A is larger as well. It has been shown by Park and Geballe 11 that the value of the 2∆/kT c increases with decrease of the film thickness for amorphous Nb-films. Therefore, in our fits we have used the estimates of 2∆/kT c reported in Ref. 11 . We have verified that in order to achieve satisfactory fit in the thinnest films, we have to increase the value of 2∆/kT c from 3.7 (as in bulk Nb) to 4.5.
In the fitting procedure, we first determine the slope dH c2 /dT | T c from linear fits to H c2 (T ) line in the vicinity of the T c . Subsequently, the slope is treated as fixed parameter, while Maki parameters α and λ so are treated as adjustable parameters. The values of dH c2 /dT | T c and 2∆/kT c used in the fits, together with Maki parameters estimated from the fits, are shown in Table S1.

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Scaling analysis Fig. S4(a) shows the plot of (∂ R sq /∂ B) B c versus 1/T for several amorphous films. It is evident that the data do not follow straight lines with the same slope across whole investigated temperature range. Instead, we observe two different T -ranges with distinctly different slopes, marked at the top of the figure by arrows: high-T range (T 0.5 K) with larger slope, and low-T range (0.12 K T 0.35 K) with smaller slope. In addition, at the lowest T (T 0.12 K) the R sq is saturated (labeled "sat"), so that the slope is zero. When calculating the derivative, we took care to determine it at actual crossings of consecutive isotherms (actual B c ), which shifts with T in high-T range; this shift, however, does not change the slope. Note that the crossover between high-T range and low-T range is not the same for all films, but increases with increasing d; in fact, it occurs approximately at the temperature T 0 , below which B c is constant. The T 0 is marked by small vertical arrows in Fig.S4(a). On the other hand, the B c remains constant at the crossover between low-T range and saturation region; there is no any apparent signature of this crossover in the B c value. Interestingly, in both high-T , and low-T ranges the slopes for many different films are similar. To emphasize this point, we plot in Figs.S4(b) and S4(c) the data for low-T and high-T ranges, respectively, 6/11 after shifting the data for different films vertically, so all of them overlap the data for the film with d = 1.4 nm (this amounts to multiplying data for each film by a different constant, C 1 or C 2 for high-T or low-T range, respectively). In the high-T range we include, in addition, the data for polycrystalline films -it is seen that they show exactly the same slope as the data for amorphous films. From the slope of straight line, fitted to all data for different amorphous films in low-T range, average critical exponent is determined: νz = 2.2 ± 0.2. Similar procedure applied to high-T range gives the average inverse of the slope equal to 0.6 ± 0.1. While in polycrystalline films this quantity may be identified as critical exponent, in case of amorphous films this is less likely, because the B c and the R c are not constant across the high-T range, so that one parameter scaling may be invalid -therefore, we call it "apparent critical exponent".
We summarize the scaling analysis results in Fig.S4(d), where points show the d-dependence of νz and "apparent νz", extracted separately for each film using method (1) and (2)

Superconducting fluctuations
We first compare our experimental data to formulas provided by Galitski and Larkin 12 , and we conclude that reasonably good fit may be achieved. It is restricted, however, to low-T , and to the vicinity of H c2 , i.e. h 2.5. The data are fitted by the following equations: where r = h/3.562t, ψ is digamma function, and G n is the normal state resistance. In these equations ∆G is in units of G 0 = e 2 /2π 2 ℏ, t = T /T c0 , and h = H/H c2 (0) − 1. We use H c2 (0) determined from the WHH expressions. Parameters for the fit are α, β and G n (G n is close to G value at maximum h). In addition, in order to fit all the data with minimal variations on the other parameters it has been necessary to allow T to vary slightly, by about ±0.015 K. The fitting process goes as follows. First, a set of acceptable parameters is obtained for a single ∆G(t, h) versus h curve. Using these parameters as a starting point, individual fits for ∆G(t, h) versus h curves were done at different temperatures, yielding α, β and G n for each value of t. After fitting all the curves, α(t), β (t) and G n (t) were kept constant and T c0 was varied in order to improve the fit of all ∆G(t, h) versus h curves simultaneously. Deviations were then addressed by adjusting each curve's T .
The results of the fits, together with fitting parameters, are displayed in Fig. S5. The fitting becomes worse as t is increased, and diverges from the data at h > 2.5. We note that, while these formula do not provide the ideal fits, the essential result of negative ∆G is well reproduced.   Figure S5. Comparison of experimental data (points) and theory of Ref. 12 (lines); parameters of the fit are listed in the table.
Next, we compare experimental data to theory of Glatz et al. 13 . In order to avoid tedious calculations, we use simplified approach, fitting to the data asymptotic functions provided by theory for several distinct domains in the phase diagram, as defined by Glatz et al. 13 . These are the following domains: domain IV (region of quantum fluctuations), domain V (quantum to classical region), domain VI-VII (classical, strong fields), and domain IX (high magnetic fields). We disregard as insignificant all residual parts of the order of (t/h) 2 . Since in our experiment the temperature is not zero, but finite, even at the lowest T the increase of the field drives the system first through domain V, and subsequently, at higher field, through domain IV. Accordingly, we break the range of the data for the fittings into three different regions: (1) low-T , low-h (2) low-T , high h (3) high-T . The first two regions cover the same T -range (T ≤ T 0 ), but are distinguished by the h value. In region (1) we use a combination of asymptotic expressions from domains V and IV to fit the data (the first one gives positive, and the second negative contribution), while in region (2) it is enough to use expressions from domain IV. Finally, the data in the region (3) are well fitted by combination of expressions from domains VI-VII and IX. Note that when expressions from two domains are included, it is necessary to include additional, field-independent parameter, in order to offset the excess asymptotic value of ∆G. Accordingly, we use the following functions to fit the data: Here γ E = 1.78107 is Euler's constant, ∆G is in units of G 0 = e 2 /2π 2 ℏ, t = T /T c0 , and h = H/H c2 (0)− 1. The factor 0.88 comes from the definition of dimensionless magnetic field (0.88H/H c2 (0)), and in case of thin Nb films replaces the usual factor of 0.69 (valid for weak-coupling superconductor), as discussed in case of WHH equations. Finally, the constant H R is given by H R = H c2 (T )/H c2 (0); it deviates slightly from 1 for temperatures close to T c0 , but it approaches 1 on cooling down to t = 0.5. We use T c0 = 1 K, and H c2 (0) determined from the WHH expressions. A, B and C are fitting parameters. Parameter C is a h-independent constant, added in order to offset the excessive background resulting from the inclusion of asymptotic expressions from two domains, or/and in order to account for any deviation of the G 8.5 from the real normal-state conductivity.   Figure S6. Table: parameters of the fit of asymptotical expressions from Glatz et al. 13 The Eq.S7 describes low-t data with different set of parameters for low-h (region (1)) and high-h (region (2)), while Eq.S8 is used to fit high-t data (region (3)). At the top of Fig.S6 we show the table with all sets of parameters for different regions; the T -dependence of these parameters is plotted in Fig.S6 (1) and (2), and for high-t region (3), respectively.
In the region (2) (low-t, high-h) the system remains exclusively in the domain IV. At the lowest T the A parameter is negative, so that first part of Eq.S7 gives negative contribution to fluctuation conductivity; the contribution of the second part is positive, while the constant C is small and negative. This results in the total negative contribution, which approaches zero at large h. On the other hand, in the region (1) (low-t, low-h), around the minimum on the ∆G(h) curve) the two terms in Eq.S4 result from two domains: domain V (quantum to classical region, it gives positive contribution to first term) and domain IV (quantum region, it gives negative contributions to both terms) -this is because in our experiment we keep t constant, and increase h, therefore, the system is driven first through domain V, and subsequently through domain IV. Because of contributions from two domains the offset C is sizeable. The resulting total ∆G(h) curve describes very well the minimum, which appears in region (1). Note that the boundary between regions (1) and (2) is t-dependent, and shifts from h=1.24 up to h=2.2 when t is increased from 9/11 0.1 up to 0.4.
In Eq.S8 the first term (t-and h-dependent) comes from formula for domain VI-VII. It is positive for small h, describing well small-h behavior of ∆G. However, to obtain the correct dependence for large h we have to include, in addition, second term, which is h-dependent part of the formula from domain IX (t-and h-independent terms from domain IX are included in the constant C). Note that parameter A extrapolates smoothly from region (3) into region (1) at t = 0.4. In fact, the experimental data for T = 0.4 K are reasonably well fitted both by Eq.S7 and by Eq.S8, confirming the fact at that this temperature a crossover occurs from quantum to classical fluctuations.
The final curves for ∆G for the three different regions of the fits are compared with experimental data for d = 1.4 film in Fig. 3(b) in the main paper. Interestingly, the negative contribution exists also in case of polycrystalline film with d = 9.5 nm, although it is far less spectacular. Fig.3(c) (main paper) shows the ∆G = G − G 9 (G 9 is the conductivity at 9 tesla) for this film. Since the H c2 is larger in this film, the data are limited to smaller h-range, so we can test the behavior at two lowest measured temperatures only (2 and 3 K); in addition, the data are more noisy. Nevertheless, it is clear that the minimum appears; moreover, the data may be described by the expressions from region (1) (fits of expressions from region (3) are unsuccessful).