I-V characteristics of the vortex state in MgB2 thin films

TL;DR

This study investigates the I-V characteristics of MgB2 thin films under various magnetic fields, revealing vortex liquid-glass transitions and scaling behaviors consistent with quasi-2D vortex glass theory, with deviations at higher fields suggesting complex vortex dynamics.

Contribution

It demonstrates the applicability of quasi-2D vortex glass theory to MgB2 thin films and identifies deviations at high magnetic fields indicating mixed vortex states.

Findings

Vortex liquid-glass transitions observed at fields 0-5T.

Scaling behavior consistent with quasi-2D vortex glass theory at low fields.

Deviations from scaling at fields >= 5T, suggesting complex vortex dynamics.

Abstract

The current-voltage (I-V) characteristics of various MgB2 films have been studied at different magnetic fields parallel to c-axis. At fields \mu0H between 0 and 5T, vortex liquid-glass transitions were found in the I-V isotherms. Consistently, the I-V curves measured at different temperatures show a scaling behavior in the framework of quasi-two-dimension (quasi-2D) vortex glass theory. However, at \mu0 H >= 5T, a finite dissipation was observed down to the lowest temperature here, T=1.7K, and the I-V isotherms did not scale in terms of any known scaling law, of any dimensionality. We suggest that this may be caused by a mixture of \sigma band vortices and \pi band quasiparticles. Interestingly, the I-V curves at zero magnetic field can still be scaled according to the quasi-2D vortex glass formalism, indicating an equivalent effect of self-field due to persistent current and applied magnetic field.

Figures

Tables

Table 1. Table 1: Quasi-2D VG scaling parameters at different fields.

${\mu }_{0}H$ (T)	$T_g\left(K\right)$	$\nu$	$z$
0.0	39.94	1.12	6.61
0.1	39.28	1.30	6.08
0.5	35.95	1.37	6.42
1.0	31.4	1.32	6.12
3.0	15.4	1.17	6.58

Equations

$$\frac{E}{j{\left(T-T_{\mathrm{g}}\right)}^{\nu (z+2-D)}}=f_{\pm }\left(\frac{j}{{\left|T-T_{\mathrm{g}}\right|}^{\nu (D-1)}}\right).$$

$${\rho }_{\mathrm{lin}}={\mathrm{d}E/\mathrm{d}j|}_{j\to 0}\propto {\left(T-T_{\mathrm{g}}\right)}^{\nu (z+2-D)}.$$

$$E(j){|}_{T=T_{\mathrm{g}}}\approx j^{(z+1)/(D-1)}.$$

$$\frac{E}{j}\exp \left[{\left(\frac{T_0}{T}\right)}^p\right]=g\left(\frac{j}{T^{1+{\nu }_{2\mathrm{D}}}}\right),$$

$${\rho }_{\mathrm{lin}}\propto \exp [-(T_0/T{)}^p].$$

Full text

I Introduction

Since the discovery of the two-gap superconductor MgB2 in 2001,MgB2found the mechanism of its superconductivity and vortex dynamics has attracted considerable interests. The two three-dimension (3D) $\pi$ bands and two quasi-two-dimension (quasi-2D) $\sigma$ bands in this simple binary compound seem to play an important role in the superconductivity,coherence_length as well as the normal state properties.MR_normal ; LiQ The two sets of bands have different energy gaps, i.e., about 7 meV for the $\sigma$ bands, and about 2 meV for the $\pi$ bands.gaps1 ; gaps2 And the coherent length of the $\pi$ bands is much larger than that of the $\sigma$ bands coherence_length . Many experiments have demonstrated that the $\pi$-band superconductivity is induced from the $\sigma$-band and there is a rich evidence for both the interband and intraband scattering. Owing to the complicated nature of superconductivity in this system, its vortex dynamics may exhibit some interesting or novel features. Among various experimental methods, measuring the current-voltage ($I$-$V$) characteristics at different temperatures and magnetic fields can provide important information for understanding the physics of the vortex state. Up to now, the transport properties of MgB2 have been studied on both polycrystalline bulk samplesMgB2IVpoly and thin filmsMgB2IVfilm . In both cases, the $I$-$V$ characteristics demonstrated good agreement with the 3D vortex glass (VG) theory. This was partially due to the limited magnetic fields in the experiment. In addition, it has been shown that the properties of MgB2 are very sensitive to the impurities and defects introduced in the process of sample preparation, and the vortex dynamics must be influenced, too. Therefore, it is necessary to investigate the vortex dynamics in high quality MgB2 epitaxial thin films and to reveal the intrinsic properties of the vortex matter in this interesting multiband system. In this paper, we present the $I$-$V$ characteristics of high-quality MgB2 thin films measured at various temperatures and magnetic fields. The vortex dynamics in this system is then investigated in detail.

II Experiment

The high-quality MgB2 thin films studied in this work were prepared by the hybrid physical-chemical vapor deposition techniquefilm on (0001) 6$H$-SiC substrates. All the films had $c$-axis orientation with the thickness of about $100\;$nm. Fig. 1 (a) shows the $\theta$-$2\theta$ scan of the MgB2 film, and the sharp $(000l)$ peaks indicate the pure phase of the $c$-axis orientation of MgB2. In order to show the good crystallinity of the film, we present in Fig. 1(b) the same data in a semilogarithmic scale which enlarges the data in the region of small magnitude. It is clear that, besides the background noise, we can only observe the diffraction peaks from MgB2 and the SiC substrate, i.e., there is no trace of the second phase in the film. The $c$-axis lattice constant calculated from the MgB2 peak positions was about 3.517 Å(bulk valueMgB2found : 3.524 Å). The $\phi$ scan (azimuthal scan) shown elsewherefilm indicated well the sixfold hexagonal symmetry of the MgB2 film matching the substrate. The full width at half maximum (FWHM) of the 0002 peak taken on the film in $\theta$-$2\theta$ scan [MgB2 0002 peak in Fig. 1(a)] and $\omega$ scan [rocking curve, shown in Fig. 1(c)] is 0.15∘ and 0.39∘, respectively. The scanning electron microscopy (SEM) image in Fig. 1(d) gave a rather smooth top surface view without any observable grain boundaries, which suggested that the film had a homogeneous quality. Ion etching was used to pattern a four-lead bridge with the effective size of $380\times 20\;\mu\mathrm{m}^{2}$. The resistance measurements were made in an Oxford cryogenic system Maglab-Exa-12 with magnetic field up to 12 T. Magnetic field was applied along the $c$ axis of the film for all the measurements. The temperature stabilization was better than $0.1\%$ and the resolution of the voltmeter was about 10 nV. We have done all the measurements on several MgB2 films, and the experimental data and scaling behaviors are similar; so, in this paper, we present the data from one film.

In Fig. 2, we present the resistive transitions ($R$-$T$ relations) of a MgB2 thin film measured at various magnetic fields in a semilogarithmic scale. The current density in the measurement was about $500\;\mathrm{A/cm}^{2}$, much smaller than the critical value for low temperatures, 10^{6}\;\mathrm{A/cm}^{2}\WenHHPRB2001 . It can be determined from Fig. 2 that the sample had a superconducting transition temperature of $T_{\mathrm{c}}=40.05\;$K, with a transition width of about $0.5\;$K. Its normal state resistivity was about $2.45\;\mu\Omega\,$cm and the residual resistance ratio [$\equiv\rho(300\;\mathrm{K})/\rho(42\;\mathrm{K})$] was about $6.4$. The $I$-$V$ curves were measured at various temperatures for each field, and then we got the electric field ($E$) and the current density ($j$) according to the sample dimension. The current density was swept from 5 to $10^{5}\;\mathrm{A/cm}^{2}$ during the $I$-$V$ measurements.

III Theoretical models

In the mixed state of high-$T_{\mathrm{c}}$ superconductors with randomly distributed pointlike pinning centers, a second-order phase transition is predicted between VG state and vortex-liquid state.Fisher_theory The $I$-$V$ curves at different temperatures near the VG transition temperature $T_{\mathrm{g}}$ can be scaled onto two different branchesFisher_experiment by the scaling law

[TABLE]

The scaling parameter $z$ has the value of 4–7, and $\nu\approx$ 1–2; $D$ denotes the dimension of the system with the value 3 for 3D and 2 for quasi-2D quasi2D ; $f_{+}$ and $f_{-}$ represent the functions for two sets of the branches above and below $T_{\mathrm{g}}$. Above $T_{\mathrm{g}}$, the linear resistivity is given by

[TABLE]

At $T_{\mathrm{g}}$, the electric field versus the current density curve satisfies the relationship

[TABLE]

In 2D superconductors at $\mu_{0}H=0\;$T, a Berezinskii-Kosterlitz-Thouless (BKT) transition was found at a specific temperature $T_{\mathrm{BKT}}$.BKT At $T_{\mathrm{BKT}}$, $E\propto j^{3}$, which is a sign of the BKT transition. A continuous change from the BKT transition at zero field to a quasi-2D VG transition, and then to a true 2D VG transition with $T_{\mathrm{g}}=0\;$K was found in TlBaCaCuO film,Wen_TBCCO which shows a field-induced crossover of criticalities.

A 2D VG transition may exist in a true 2D system with $T_{\mathrm{g}}=0\;$K, i.e., there is no zero-resistance state at any finite temperatures. The $E$-$j$ curves can be scaled by2D_VG

[TABLE]

where $T_{0}$ is a characteristic temperature, $\nu_{2\mathrm{D}}\approx 2$, and $p\geq 1$, while $g$ is a scaling function for all temperatures at a given magnetic field. The linear resistance is given by

[TABLE]

This 2D scaling law can be achieved in the very thin films2D_YBCO or in highly anisotropic systems at high magnetic fields.WenHHPRL97 ; WenHHPRL98

IV Experimental results and discussions

IV.1 Quasi-two-dimension vortex-glass scaling in the low-field region ($\mu_{0}H<$ 5 T)

The $E$-$j$ characteristics have been measured at various magnetic fields up to 12 T. In Fig. 3 we show the typical example at $\mu_{0}H$ = 1 T for (a) $E$-$j$ curves and (b) the corresponding $\rho$-$j$ curves in double-logarithmic scales. It is obvious that when the temperature goes below some particular value (this is actually the vortex-glass transition temperature $T_{\mathrm{g}}$ according to following discussions), the resistivity falls rapidly with decreasing current density and finally reaches the zero-resistance state which is the characteristic of the so-called VG state. At the temperatures above $T_{\mathrm{g}}$, the resistivity remains constant in small current limit. The current density of $500\;\mathrm{A/cm}^{2}$ used in $\rho$-$T$ measurement shown in Fig. 2 lies in this linear resistivity regime from about $10^{-3}$ to $1\;\mu\Omega\,$cm. Consequently, these data sets provide the basic information on scaling if the data are describable by the VG theory.

The inset in Fig. 4 shows the data of the $\rho_{\mathrm{lin}}$ versus $(T-T_{\mathrm{g}})$ and the fit to Eq. (2). The data are the same as those shown in Fig. 2 for $\mu_{0}H=1\;$T, and the attempt $T_{\mathrm{g}}$ value is $31.4\;$K. In this double-logarithmic plot, the slope of the linear fitting gives just the exponent of $\nu(z+2-D)$, and the determined value is $8.08\pm 0.05$. In order to have reasonable values for $\nu$ and $z$, the dimension parameter $D$ needs to be chosen as 2, i.e., the investigated system has the property of quasi-2D, which is similar to the situation found in BiSrCaCuO.quasi2D ; zhang This is further supported by the VG scaling of the data at 1 T. As shown in the main frame of Fig. 4, the scaling experimental $E$-$j$ curves form two universal branches corresponding to the data above and below $T_{\mathrm{g}}$ ($31.4\;$K) with $\nu=1.32$ and $z=6.12$. At very large current density or a temperature near the onset of superconducting transition, the free flux flow regime dominates and, hence, the data do not scale. The symbols in the figure denote the range of the data well described by the scaling law.

The situation at $\mu_{0}H=3\;$T is similar to that at $\mu_{0}H=1\;$T. As shown in Fig. 5(a), the determined parameters are $T_{\mathrm{g}}=15.4\;$K, $\nu=1.17$, and $z=6.58$. Interestingly, the previous work on MgB2 film MgB2IVfilm indicated that the 3D VG scaling theory ($D=3$) is a better choice in describing the $I$-$V$ characteristics in this system, though this experiment was done at magnetic fields lower than 1 T. Moreover, the $I$-$V$ curves were demonstrated to scale well by using the argument of $j/(T\left|T-T_{\mathrm{g}}\right|^{2\nu})$. The same conclusions were also drawn on the polycrystalline MgB2 samples MgB2IVpoly . In order to clarify this issue, we also analyzed our data using the form suggested in Ref. 8. As shown in Fig. 5(b), such a scaling with $j/(T\left|T-T_{\mathrm{g}}\right|^{\nu(D-1)})$ as the scaling variable is worse than that with $j/\left|T-T_{\mathrm{g}}\right|^{\nu(D-1)}$. Most importantly, the dimension parameter $D$ is still required to be 2 instead of 3 as proposed in Refs. 7 and 8. This confusion can be easily understood in terms of the two-band superconductivity of MgB2. As we know, there are two types of bands contributing to the superconductivity of MgB2, namely, the 3D $\pi$ bands and the 2D $\sigma$ bands. Therefore, the structure of the vortex matter must be affected by both of them. Although the superconductivity of $\pi$ bands, induced possibly by that of $\sigma$ bands, is much weaker, it provides a large coherence length with 3D characteristics in the low-field region. Therefore, the vortices in this system may be quasi-2D like and, at the same time, they can possess large cores characterized by the coherence length of the $\pi$ band superfluid. In this sense, the quasi-2D scaling should be more appropriate than the 3D one. However, when a higher disorder is induced in the system, especially in the boron sites, the interband scattering gets stronger and the anisotropy decreases, which may lead to a 3D vortex scaling. In this case, a more rigid vortex line can be observed, especially at low fields.JinH The good quasi-2D scaling at 1 and 3 T demonstrated here suggests that the phase transition from VG to the vortex liquid in MgB2 resembles that in the high-$T_{\mathrm{c}}$ superconductors. Together with the data shown below, we can safely conclude that a vortex glass state with zero linear resistivity can be achieved in the low field region due to the presence of the finite superfluid density from the $\pi$ bands. Regarding the VG scalingLobb a principal requirement is a proper determination of $T_{\mathrm{g}}$, namely the temperature with a straight $\log E$-$\log j$ curve in the low dissipation part. The tolerance for $T_{\mathrm{g}}$ variation is very small (about $\pm 0.3\;$K). With an inappropriately chosen $T_{\mathrm{g}}$, the scaling quality dramatically deteriorates and, simultaneously, the values of $\nu$ and $z$ quickly deviate from those reported above and those proposed by theory. This validates our analysis here.

IV.2 Anomalous vortex properties in high field region

As shown in Fig. 2, when the magnetic field reaches 6 T, no zero-resistance state can be observed down to the lowest temperature, here 1.7 K. Consequently, no VG transition exists above 1.7 K at this field, as shown in Fig. 6. The shape of the curve at $T=1.7\;$K suggests that the resistivity goes to a finite value as the current density approaches zero.JiaY As shown in Fig. 7, the $\rho_{\mathrm{lin}}$ versus $T-T_{\mathrm{g}}$ seriously deviates from linearity for any possible $T_{\mathrm{g}}$ value, indicating the inapplicability of Eq. (2) in the present case. Correspondingly, the quasi-2D scaling law fails here. A natural explanation is that, with increasing field, the 3D supercurrent from $\pi$ bands is seriously suppressed gaps2 ; tunnelling and the quasi-2D vortex structure transforms into a 2D-like one dominated by the $\sigma$ band superfluid. In Fig. 8, we show our attempt to apply 2D VG scaling on the data. Surprisingly, this attempt also failed, even though this model has been successfully applied to the layered superconductors with large anisotropy (or 2D property) such as Tl- and Bi-based high-$T_{\mathrm{c}}$ thin films at high magnetic fields.WenHHPRL97 ; WenHHPRL98

The most reasonable explanation for this anomaly is that the supercurrent contribution from the $\pi$ bands is much easier to suppress by the magnetic field than that from the $\sigma$ bands, since the gap in the $\pi$ bands is several times smaller than that in the $\sigma$ band. We suggest that at high magnetic fields (above 5 T), a different vortex matter state is formed, composed of quasi-particles from the $\pi$ bands and vortices formed mainly by the residual superfluid from the $\sigma$ band. The $\pi$-band quasiparticles diminish the long range phase coherence of the superconducting phase, which leads to a finite dissipation. Once the long range superconducting phase coherence is destroyed by the proliferation of a large amount of these $\pi$-band quasiparticles, neither 3D nor quasi-2D VG scaling is applicable. Such a mixed state is obviously difficult to be simply described by any known scaling theory. Recently, scanning tunneling microscopy studies showed that the quasiparticles of the $\pi$ bands disperse over all of the superconductor, both within and outside the vortex coresSTMvortex , which strongly supports our arguments. This is the basis for the explanation of the nonvanishing vortex dissipation at high magnetic fields in a zero temperature limit found recently on MgB2 thin films.JiaY

IV.3 Self-field effect at $\mu_{0}$H = 0 T

For a 2D layered superconductor in zerofield, the above mentioned BKT transition may exist and be reflected in the $I$-$V$ characteristics Wen_TBCCO . In the present MgB2 samples, we have not found any evidence of this transition in low magnetic fields which would be consistent with the quasi-2D (instead of 2D) configuration of the vortex matter. Moreover, both the $E$-$j$ curves and the $\rho$-$j$ curves (as presented in Fig. 9) are similar to the situation of $\mu_{0}H=1\;$T. Considering the narrow transition width at zero-field, we did the measurement carefully with an increment of 0.05 K. Obviously, there is no $E(j)$ curve which satisfies the $E\propto j^{3}$ dependence, as expected by the BKT theory. Since the current can induce self-generated vortices, it might be interesting to look at whether the quasi-2D VG model applies here.

Similar in Sec. IV A, we present $\rho_{\mathrm{lin}}$ versus $(T-T_{\mathrm{g}})$ in a double logarithmic plot. From this graph, we determined the exponent in Eq. 2 (as shown by the inset of Fig. 10(a)). A good quasi-2D scaling was obtained with parameters $T_{\mathrm{g}}=39.94\;$K, $\nu=1.12$, and $z=6.61$, as presented in Fig. 10(a). Using the parameters determined here, one finds a self-consistency with the value of $\nu(z+2-D)$, as determined in fitting the linear resistivity [Eq. (2)]. Both the temperature dependence of $\rho_{\mathrm{lin}}$ and the scaling curves at $\mu_{0}H=0\;$T are similar to the situation at small field $\mu_{0}H=0.1\;$T [shown in Fig. 10(b)] and $\mu_{0}H=0.5\;$T (not shown in this paper), except for the slight differences of the scaling parameters. The scaling parameters including the ones at $\mu_{0}H=0.5\;$T are listed in Table 1. It was proven that current and magnetic field exhibit analogous effects in suppressing superconductivity and generating quasiparticles in conventional superconductors.HIDOS Similarly, the current-induced self-field may lead to a similar effect in the vortex state as an applied magnetic field. Nonetheless, the good agreement of this simple scaling law with the zero-field data is interesting and worth studying in detail. Moreover, the values of $\nu$ and $z$ for zero field are very close to those for $\mu_{0}H=1$ and $3\;$T, indicating a similar vortex dynamics in the whole low-field region.

V Summary

We have measured $I$-$V$ curves on high-quality MgB2 films at various magnetic fields and temperatures. At magnetic fields below $5\;$T including the zero field, the curves scaled well according to the quasi-2D VG theory instead of the 3D model, in good agreement with the multiband superconductivity of MgB2 contributed from the strong 2D $\sigma$ bands and weak 3D $\pi$ bands. At the fields above $5\;$T, the curves did not scale according to any known VG scaling laws, accompanied by the disappearance of a zero-resistance state. Based on our result combined with recent tunneling experiments, a different vortex state was suggested, namely, a state where the vortices composed of the superfluid from the $\sigma$ bands move through the space filled with numerous quasiparticles from $\pi$ bands.

VI Acknowledgments

This work is supported by the National Science Foundation of China, the Ministry of Science and Technology of China (973 project: 2006CB601000 and 2006CB921802), and the Knowledge Innovation Project of the Chinese Academy of Sciences (ITSNEM). The work at Penn State is supported by NSF under Grants Nos. DMR-0306746 (X.X.X.), DMR-0405502 (Q.L.), and DMR-0514592 (Z.K.L. and X.X.X.), and by ONR under grant No. N00014-00-1-0294 (X.X.X.).