Disorder and magnetic field induced Bose-metal state in two-dimensional Ta x (SiO 2 ) 1−x granular films
Zhi-Hao He, Hua-Yao Tu, Kuang-Hong Gao, Guo-Lin Yu, Zhi-Qing Li
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p Disorder and magnetic field induced Bose-metal state in two-dimensional Ta x (SiO ) − x granular films Zhi-Hao He, Hua-Yao Tu, Kuang-Hong Gao, Guo-Lin Yu, and Zhi-Qing Li ∗ Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology,Department of Physics, Tianjin University, Tianjin 300354, China National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics,Chinese Academy of Science, Shanghai 200083, China (Dated: September 9, 2020)The origin of the intermediate anomalous metallic state in two-dimensional superconductor ma-terials remains enigmatic. In the present paper, we observe such a state in a series of ∼ x (SiO ) − x ( x being the volume fraction of Ta) nanogranular films. At zero field, the x & x . x ≃ x & R (cid:3) ( T, H ) ( T and H being the temperature and the magnitude of magnetic field) datanear the crossover from the anomalous metal to superconductor and in the vicinity of the anomalousmetal to insulator transition, respectively, obey unique scaling laws deduced from the Bose-metalmodel. Our results strongly suggest that the anomalous metallic state in the Ta x (SiO ) − x granu-lar films is bosonic and dynamical gauge field fluctuation resulting from superconducting quantumfluctuations plays a key role in its formation. I. INTRODUCTION
Recently, the anomalous metallic state in two-dimensional (2D) superconductors has attracted greatattention [1–20]. The main characteristic of the anoma-lous metal state is that when the sample is cooled downto a certain temperature, the resistance will drop dra-matically as if the system were approaching a supercon-ducting ground state, but then saturates to a value thatcan be orders of magnitude smaller than that of normalstate with further decreasing temperature. The anoma-lous metallic state has been observed in disordered 2Dsuperconductor films [1–6], mechanical exfoliated crys-talline 2D superconductors [9–11], the electric-double-layer transistor [7, 8], and Josephson junction array [14–16], and can be tuned by disorder, magnetic field, andgate voltage. Several theoretical frameworks, includingBose-metal model [21–24], vortex tunneling model [25],binary composite structure model with superconductinggrains embedded in a normal-metal film [20, 26, 27], andnew two-fluid model [28], have been proposed to disclosethe nature of the intermediate metallic state. However,its origin is still intensely debate. In the Bose-metal andvortex tunneling models, the systems are both assumedto be composed of superconducting grains and insulat-ing matrix. Thus the superconductor-insulator granularcomposites are important model system for studying the ∗ Corresponding author, e-mail: [email protected] anomalous metallic state. In fact, the disorder drivenanomalous metallic state has appeared in the earlier lit-eratures of 2D granular superconductors and the did notreceived much attention at that time [29, 30]. Thus it isnecessary to explore the nature of the anomalous metallicstate in superconductor-insulator granular system.In this paper, we systematically investigate the low-temperature electrical transport properties of a seriesof Ta x (SiO ) − x (with x being the volume fraction ofTa) granular films with thickness ∼ x (SiO ) − x films and can not only be tuned by disor-der but also by magnetic field. The scaling relation inresistance with magnetic field and temperature can bedescribed by the prediction of Bose-metal scenario. II. EXPERIMENTAL METHOD
Our Ta x (SiO ) − x films were deposited on Al O sin-gle crystal substrates by co-sputtering Ta and SiO tar-gets in Ar atmosphere. Al O single crystal possessesrelative higher thermal conductivity, which is favorablefor the thermal equilibrium between the film and sampleholder in liquid helium temperatures for electrical trans-port measurement. The base pressure of the chamber FIG. 1. Bright-field TEM images for Ta x (SiO ) − x films with x values of (a) 0.45, (b) 0.66, (c) 0.75, and (d) 0.8. The insetsin (a) and (c) are the grain-size distribution histograms forthe x ≃ .
45 and 0.75 films, respectively, and in (b) and(d) are the selected-area electron-diffraction patterns of thecorresponding films. is less than 8 × − Pa, and the deposition was carriedout in an argon (99.999%) atmosphere of 0.5 Pa. Dur-ing deposition, the substrate temperature was kept at200 ◦ C and the Ta volume fraction x was regulated bythe sputtering powers applied in the two targets. Thefilms were also simultaneously deposited on the polyimide(Kapton) and copper grids coated with ultrathin carbonfilms for composition and microstructure measurements.The narrow rectangle shape films (2 mm ×
10 mm), de-fined by mechanical masks, were used for transport mea-surement. To obtain good contact, four Ti/Au electrodeswere deposited on the films. The distance between thetwo voltage Ti/Au electrodes is 2 mm.The thickness of the films was controlled by deposit-ing time and the exact thickness of the films was deter-mined by the atomic force microscopy (AFM Multimode-8, Brucker). It is indicated that the thicknesses of thefilms are 9 . ± . x ineach film was obtained from energy-dispersive x-ray spec-troscopy analysis (EDS; EDAX, model Apollo X). Themicrostructure of the films was characterized by trans-mission electron microscopy (TEM, Tecnai G2 F20). Theresistance versus temperature and magnetic field wasmeasured using standard four probe ac technology, inwhich a Keithley 6221 and a SR 830 lock-in amplifierwere used as the ac current source and voltmeter, respec-tively. During measurements, the frequency of the cur-rent is set as 13.33 Hz and the applied bias varies from 50to 100 nA for different films. For the isothermal current-voltage curves measurement, the current was supplied x=1 x=0.66 x=0.80 x=0.60 x=0.78 x=0.52 x=0.75 x=0.50 x=0.73 x=0.45 x=0.71 T (K) () R FIG. 2. Sheet resistance as a function of temperature from 2down to 0.04 K for the Ta x (SiO ) − x films (0 . . x ≤ by the Keithley 6221 and the voltage was monitored bya Keithley 2182A. The low temperature and magneticfield environments were provided by dilution refrigerator(Triton 200, Oxford). III. RESULTS AND DISCUSSIONS
Figure 1 shows the bright-field TEM images and se-lected area electron diffraction (SAED) patterns for filmswith x ≃ .
45, 0.66, 0.75 and 0.80. The dark regionsare Ta granules and the bright regions are SiO insu-lating matrix. Thus these films reveal typical granularcomposite characteristics. The mean-size of Ta granules(d) for the x ≃ .
45, 0.66, 0.75 and 0.80 films is ∼ ∼ ∼ ∼ matrix are amorphous.Figure 2 shows temperature T dependence of the sheetresistance R (cid:3) for films with 0 . . x . .
80 from 2 Kdown to 40 mK. Clearly, the normal-state sheet resistance R N decreases with increasing x at a certain temperature.For the x & .
75 films, the low-temperature sheet resis-tance changes little with decreasing temperature in thenormal state, and then rapidly drops to zero below a tran-sition temperature T c with further decreasing tempera-ture, where the superconducting transition temperature T c is defined as the temperature at which the sheet re-sistance drops to 0 . R N . The value of T c decreases withreducing x , and is listed in Table I. For the x ≃ . R N to ∼ R N asthe temperature decreases from 0.25 to 0.1 K, and thentends to be a constant with further decreasing tempera-ture. Thus the the anomalous metallic state is achievedin the Ta x (SiO ) − x granular films via tuning the volumefraction of Ta, i.e., tuning the disorder of the system. Forthe 0 . . x . .
71 films, a very slow increase in R (cid:3) isvisible upon cooling, e.g., the variation of R (cid:3) is less than I ( A) T’ BKT = 0.25 K V ( m V ) (b) T (K)(a) () R x = 0.75 T (K)
FIG. 3. (a) The sheet resistance R (cid:3) versus temperature T forthe x ≃ .
75 film. The solid curve is the least-squares fits tothe Halperin-Nelson formula. (b) Voltage versus current (indouble logarithmic scales) measured at constant temperatureranging from 0.22 to 0.35 K for zero magnetic field, the stepsare 0.01 K from 0.22 to 0.28 K. Inset: α as a function of T .The dashed line represents the α = 3 line.
10% when the films is cooled from 2 K down to 40 mK.For the x ≃ .
45 film, the sheet resistance increases a fac-tor of 4 in this temperature range, indicating the film hastransformed into insulator. We will focus on the x & . A. Berezinskii-Kosterlitz-Thouless transition
A remarkable feature for 2D superconducting films,no matter homogeneous or inhomogeneous, is the oc-currence of Berezinskii-Kosterlitz-Thouless (BKT) tran-sition [31–34], which in turn becomes a criterion to judgewhether a superconductor is 2D. The basic picture ofthe BKT transition is the existence of thermally excitedvortices which are bound in vortex-antivortex pairs be-low the phase transition temperature T BKT and dissoci-ated above. In Fig. 3(a), we present R (cid:3) variation with T for the x ≃ .
75 film from 0.5 down to 0.2 K. Itis found that the temperature dependence of the sheetresistance in the superconducting transition region canbe well described by the Halperin-Nelson formula [35] R (cid:3) = R exp[ − b ( T /T
BKT − − / ], where R is a prefac-tor, b ∼ T BKT is the BKT transitiontemperature, which can be determined via extrapolat-ing the linear part of [d ln R (cid:3) / d T ] − / versus T curve to[d ln R (cid:3) / d T ] − / = 0. For the x ≃ .
75 film, the valueof T BKT is ∼ R = 7215 Ω and b = 0 .
92. On the other hand, in theBKT transition region the current dependence of voltageobeys V ∼ I α ( T ) law in small current limit, and the valueof α ( T ) is α ( T ) = 3 at T BKT . These relations valid forboth homogeneous and inhomogeneous (granular films or U / k B ( K ) (b) H (T) -1 ) (a) () R FIG. 4. (a) Logarithm of resistivity as a function of T − at different magnetic field for the x ≃ R (cid:3) = R ( H ) exp( − U ( H ) /k B T ). (b)Activation energy U ( H ) /k B , obtained from the slopes of thesolid lines in Fig. 4(a), as a function magnetic field for the x ≃ proximity-coupled arrays) 2D superconductors [31–34].Figure 3(b) shows the voltage versus current at some se-lected temperatures in double logarithmic scales for the x ≃ .
75 film. Clearly, log V varies linearly with log I insmall current limit, indicating the relation V ∼ I α is sat-isfied. The inset of Fig. 3(b) shows α as a function of T obtained via linear fitting of the log V -log I data. Inspec-tion of the figure indicates that α ( T ) decreases from ∼ T BKT ≃ .
25 K, which isidentical to that determined by the Halperin-Nelson for-mula within the experimental uncertainty. Similar phe-nomena have also been observed in the x > .
75 films,and the transition temperature T BKT , together with thefitting parameters R and b , is listed in Table I. The ex-istence of BKT transition in the x & .
75 films suggeststhat these films are 2D with respect to superconductivity.
B. Anomalous metallic state
For the x & .
75 films, the superconductivity wouldgradually disappear once a moderate magnetic field per-pendicular to the film plane was applied. Consideringthe temperature dependence behaviors of the R (cid:3) undermagnetic field are similar for the x & .
75 films, we onlypresent and discuss the results obtained in the x ≃ . R (cid:3) (in logarithmic scale) as afunction of 1 /T at different magnetic fields, as indicated.The low-temperature state of the film remains supercon-ductivity (zero resistance) at low field. When a moder-ate field with magnitude H c . H . H c ( H c is uppercritical magnetic field, while H c is the critical magneticfield for superconductor to anomalous metal transition TABLE I. Relevant parameters for the Ta x (SiO ) − x films with x & .
73. Here x is volume fraction of Tb, T c is the super-conducing transition temperature, T BKT is the BKT transition temperature, R and b are the parameters in Halperin-Nelsonformula, U and H are the parameters in Eq. (1), H c (0) is upper critical field at 0 K, d is the the mean-size of Ta granules, ξ (0) is the Ginzburg-Landau coherence length at zero-temperature, and H c is the critical field at which the isotherms of the R (cid:3) versus µ H cross at a point. R (cid:3) (2 K) T c T BKT d R b U /k B H H c (0) ξ (0) H c x (Ω) (K) (K) (nm) (Ω) (K) (T) (T) (nm) (T)1.0 227 0.91 0.82 −
667 1.04 7.22 1.06 1.51 14.5 1.800.80 345 0.66 0.62 5.6 41534 1.18 3.67 0.69 0.96 18.2 1.220.78 434 0.39 0.34 5.3 4520 0.78 1.26 0.39 0.52 24.7 0.710.75 493 0.32 0.25 5.1 7215 0.92 1.75 0.22 0.45 26.6 0.650.73 585 − − − − − − which will be defined later) is applied, the sheet resis-tance starts to drop rapidly with decreasing temperaturenear T c , and then tends to saturate with further decreas-ing temperature. The saturation value increases with theenhancement of the field until the field reaches H c . Thevalues of µ H c (with µ being the permeability of freespace) for the x & .
75 film are summarized in Table I.The features of R (cid:3) suggest that an anomalous metallicground state presents in these x & .
75 films when amoderate field is applied.From Fig. 4(a), one can see that the log R (cid:3) (ln R (cid:3) )decreases linearly with increasing 1 /T in the transi-tion region near T c . Thus R (cid:3) ( T ) satisfies R (cid:3) = R ( H ) exp( − U ( H ) /k B T ) in the transition region, where R ( H ) is a prefactor, k B is the Boltzmann constant, and U ( H ) is the activation energy under field H . This formof R (cid:3) ( T ) means that the vortex-antivortex pairs are un-bound with increasing temperature and the resistance inthe transition region is governed by the motion of ther-mal activated individual vortices. Figure 4(b) shows theactivation energy U ( H ) variation with H extracted fromleast-squares fits to the linear part of the log R (cid:3) versus1 /T data in Fig. 4(a). Clearly, the activation energy U ( H ) variation with H is consistent with the thermallyassisted collective vortex-creep model in 2D system [36], U ( H ) = U ln( H /H ) (1)where U is the vortex-antivortex binding energy and H ∼ H c (0) is the field above which all the vortex-antivortex pairs are almost broken down. The fittedvalues of U and H are listed in Table I. The valueof H c (0) can be estimated by linear extrapolating thelow temperature H c ( T ) data to 0 K. Thus the in-planeGinzburg-Landau coherence length at zero temperature ξ (0) is obtained via µ H c (0) = Φ / πξ (0) with Φ be-ing flux quantum. The values of H c (0) and ξ (0) arealso summarized in Table I. Insepectation of Table I in-dicates the value of H is less than the upper criticalfield H c (0) for each film. Since the thermally assistedcollective vortex-creep model is heuristic [36], the devia-tion is acceptable. The coherence length ξ (0) is greaterthan the thickness t for each film, which confirms the 2D H=0.25 T V ( m V ) (a) -1 ) () R (b) I ( A) H c ( T ) T (K)
FIG. 5. (a) Logarithm of the sheet resistance as a functionof T − from 1 down to 0.04 K at a field of 0.25 T for the x ≃ . , . , .
78 and 0.80 films. (b) I - V curves for the x ≃ .
75 film measured at different fields. Inset: H c as a functionof temperature for the x ≃ .
75 film. superconductor characteristics of the films.Figure 5(a) shows the R (cid:3) as a function of 1 /T at 0.25 Tfor the 0 . . x . .
80 films. The x ≥ . . . x < .
80 films exhibitmetallic behavior below ∼ . x on the films is the enhancement of disor-der. Thus the anomalous metallic state in Ta x (SiO ) − x granular films can also be driven by disorder besides mag-netic field. The reduction of x decreases the mean-sizeof Ta granules and increases the separation between Tagranules, thus increases the tunneling resistance, whichresults in the enhancement of the superconducting quan-tum phase fluctuations at low temperatures [37]. In ad-dition, the T c of superconducting particles with inter-mediate electron-phonon coupling strength (i.e., the T c and superconducting energy gap ∆(0) of the bulk super-conductor satisfies 2∆(0) ∼ . k B T c ) decreases with de-creasing particle size due to quantum size effect [38, 39].The value of 2∆ /k B T c of bulk Ta is ∼ ∼ x=0.75(b)
280 mK 250 mK 200 mK 150 mK 100 mK 60 mK (H-H c0 ) (T)x=0.78
350 mK 300 mK 250 mK 200 mK 150 mK 90 mK 50 mK (H-H c0 ) (T)(a) () R () R T (K) T (K)
FIG. 6. Sheet resistance as a function of H - H c in doublelogarithmic scales at different temperatures for (a) x ≃ . x ≃ .
75 films. The solid lines are least-squares fitsto Eq. (2). Inset: The exponent 2 ν in Eq. (2) as a functionof T for the corresponding films. ∆(0) (or T c ) of Ta particles will decrease with decreasingparticle size (or x in granular films), which is just whatwe observe in the Ta x (SiO ) − x granular films. For theTa x (SiO ) − x granular films with a certain fixed x , thenumbers of grains of Ta with a certain size versus thegrain size tends to follow a normal distribution (see in-sets in Fig. 1). Thus the values of ∆(0) for Ta grains alsoobey the normal distribution law, which could enhancethe fluctuations of the superconductor order parameter.On the other hand, the external magnetic field not onlyreduces the superconductor energy gap but also destroythe coherence between the superconducting grains, whichalso increases the superconducting quantum phase fluc-tuation. Thus the quantum fluctuations of the supercon-ductor order parameter plays a key role in the supercon-ductor to anomalous metal transition.Figure 5(b) shows the voltage V as a function of cur-rent I at 60 mK at different field for the x ≃ .
75 film. Inthe larger current part, the I - V curve exhibits nonlinearbehavior, and the curve measured in the increasing cur-rent process does not overlap with that in the decreasingprocess at a moderate field. As the field increases, thehysteresis in I - V curve gradually disappears. The min-imum field at which the hysteresis disappears is definedas the critical field H c for superconductor to anomalousmetal transition [3, 4, 9]. H c is temperature dependentand shown in the inset of Fig. 5(b). For the x ≃ . H c decreases from 0.11 to 0.04 T as T increasesfrom 0.04 to 0.2 K. The hysteretic in I - V curves was alsoobserved in homogeneous Ta film [3, 4], 2D crystallineNbSe films [9] and granular Bi film [41] under a mag-netic field. The characteristics of I - V curves in our filmsare similar to that in the Josephson junction in the under-damped region [43], which is consistent with the granularstructure of our films. C. The scaling relation for the resistance near thephase transitions
Das and Doniach have proposed a Bose-metal modelto explain the anomalous metallic state in 2D supercon-ductors [22, 23]. They investigated the quantum phasefluctuations in granular superconductors in the absenceof disorder. In their picture, the anomalous metal phaseis dominated by dynamical gauge field fluctuations whichresults from superconducting quantum fluctuations. Vor-tices and antivortices moving in a dynamically fluctu-ating gauge field tend to form a quantum liquid whengauge field fluctuations overcome the quantum zero pointmotion of the vortices. They argue that the groundstate of the system are determined by three energy: theJosephson-coupling energy J , the onsite repulsion energy V , and repulsion energy among the nearest neighbors V ( V and V are related to the inverse of the capacitancematrix of the grains). When V /J + 8 V /J is less than˜ b (˜ b ≃ .
78 in Ref. 22), the system is superconduct-ing; when V /J + 8 V /J is greater than ∼ p V /J ,the system is insulaing; and in the intermediate region,the system is in Bose-metal state. In the granular super-conductor films, J = ( R Q / R N )∆ ( R Q = h/ e with h being the Planck’s constant and e the electron charge)decreases with decreasing x , and a crossover from super-conducing to Bose-metal states would occur as V and V are of comparable order of magnitude. This is just whatwe see in the x ≃ .
73 film at zero field.When a small magnetic field is applied, the coher-ence of the superconducting region will be graduallysuppressed and the coupling energy J will be reduced.The uncondensed bosons (vortex and antivortex) startto emerge at H > H c , which drives the system intothe Bose-metal phase. Thus the superconductor to Bose-metal transition is associated with the unbinding of(quantum) dislocation-antidislocation (or vortex and an-tivortex) pairs and finite resistance is induced by the freedislocations (vortices). Near the superconductor to Bose-metal transition and on the metallic side, the resistancescales with [23] R (cid:3) ∼ ( H − H c ) ν , (2)where H c is the critical field for superconductor to Bose-metal transition and ν is the scaling parameter. As thefield increases further, quantum zero point motion of thevortices increases, and it gradually overtakes the dynam-ical gauge field fluctuations. When the field is greaterthan a critical value H c , quantum zero point motion ofthe vortices dominates the vortex dynamics. The vorticesform a superfluid phase and the film is insulating. Con-sidering the insulator to Bose-metal transition is a phasetransition from a vortex superfluid to a gapless nonsuper-fluid phase, Das and Doniach proposed a two parameterscaling relation for resistance across the transition [23] R (cid:3) (cid:20) T /νz δ (cid:21) ν ( z +2) = f ( δ/T /νz ) (3) -3 -2 -1 0 1 2 310 -5 -3 -1 x=0.75(b)
60 mK 100 mK 150 mK 200 mK 250 mK 300 mK /T
1/ z (T/K
1/ z ) ( T / z / ) ( z + ) ( K ( z + ) / z / T ( z + ) ) R
300 mK60 mK H C =0.67 T H (T) R ( ) (a)
50 mK120 mK150 mK200 mK250 mK300 mK x=0.78
FIG. 7. (a) R (cid:3) ( T /νz /δ ) ν ( z +2) as a function of δ/T /νz atdifferent temperatures and fields near the anomalous metalto insulator transition for (a) x ≃ x ≃ . νz is taken as 4/3 in both (a) and (b). Themagnetic field data are taken from 0.4 to 0.9 T for both filmsat a certain fixed temperature. The inset in (b) is R (cid:3) versus µ H measured at different temperatures (300, 200, 100, and60 mK from up to down on the left side of the cross point) forthe corresponding film. where δ = µ ( H − H c ) with H c being the critical fieldfor metal-insulator transition, ν and z are scaling expo-nents. Similar to the scaling relation R (cid:3) ∼ F ( δ/T /νz )near the superconductor to insulator transition in dis-ordered 2D superconductors [43], Eq. (3) was derivedbasing a second order quantum phase transition. Nearthe transition there is a diverging correlation length ξ ∼ | δ | − ν and a vanishing frequency characteristic fre-quency Ω ∼ ξ − z . Das and Doniach compare the exper-imental data of MoGe films with Eq. (3) and find thedata collapse onto two different branches with ν = 4 / z = 1 for H > H c and H < H c .Figure 6 (a) and 6(b) show R (cid:3) as a function of µ ( H − H c ) in double-logarithmic scales for the x ≃ . R (cid:3) varies lin-early with log µ ( H − H c ) in the vicinity of H c , indi-cating the scaling relation in Eq. (2) is present in thefilms. The linear parts of the curves are least-squaresfitted to Eq. (2), and the results are shown in Fig. 6 bysolid lines. The insets in Fig. 6 present temperature de-pendence of the exponent 2 ν . For the x ≃ .
78 (0.75)film, the value of 2 ν gradually decreases from ∼ .
04 K to higher temperature. 2 ν ≃ x = 0.78 H (T) () R () R T = 50 mK(a) x = 0.75(b) T = 60 mK H (T)
FIG. 8. R (cid:3) as a function of magnetic field for (a) x ≃ .
78 filmat 60 mK, and (b) x ≃ .
75 film at 50 mK. The solid curves arethe least-squares fits to Eq. (4).
When the field is much larger than H c ( T ), the scalinglaw in Eq. (2) is not valid. According to the theoret-ical prediction of Bose-metal model, Eq. (3) should besatisfied in the vicinity of Bose-metal to insulator tran-sition. Figure 7 shows R (cid:3) ( T /νz /δ ) ν ( z +2) as a functionof δ/T /νz near the transition with νz = 4 /
3. The insetof Fig. 7(b) presents R (cid:3) as a function of µ H measuredat different temperatures for the x ≃ .
75 film. Theisotherms of R (cid:3) versus curves cross at a critical mag-netic field H c , at which the critical sheet resistance R c (cid:3) is ∼
495 Ω. The value of R c (cid:3) decreases from ∼
495 to ∼
231 Ωas x increases from ∼ H c foreach film is listed in Table I. Figure 7 clearly indicatesthat the R (cid:3) ( H, T ) data at
H < H c and H > H c col-lapse onto two different branches. The value of z is setas 1 due to the long range Coulomb interaction betweenthe Bosons in 2D system. We also compare our R (cid:3) ( H, T )data with the scaling relation R (cid:3) ∼ F ( δ/T /νz ) (see Sup-plemental Materials Fig.S1). The value of ν , determinedby the method in Ref [44], is 1 .
30 for x ≃ .
78 film and1 .
22 for x ≃ .
75 film. It is found that the formula worksat high temperature regime, but fails to describe the low-temperature R (cid:3) ( H, T ) data near H c . Thus the validationof both Eqs. (2) and (3) in Ta x (SiO ) − x ( x & .
75) filmsstrongly suggests the intermediate metallic state in this2D granular composite is the Bose-metal phase.
D. Comparison with other models
Shimshoni et al ascribe the dissipation in 2D disor-dered superconductor to quantum tunneling of vorticesthrough thin superconducting constrictions [25]. In thisscenario, the system is treated as a percolation networkcoupling with a fermionic bath and the resistance in theanomalous metallic state is caused by the vortices tun-neling across the narrow superconducting channel. Thenthe low-temperature resistance at a field H can be writ-ten as [7, 25] R (cid:3) ∼ h e κ − κ (4)with κ ∼ exp (cid:20) C h e R N (cid:18) H − H c H (cid:19)(cid:21) , where C is a dimensionless constant of order unity, and R N is the normal state resistance. It has been foundthe experimental data on amorphous MoGe films [1, 2]and ZrNCl [7] electric-double-layer transistor can be welldescribed by Eq. (4). Figure 8 presents the low tempera-ture isothermal R (cid:3) - µ H curves for the x ≃ .
78 and 0.75films, as indicated. The solid curve are the best fit (usingleast-squares method) to Eq. (4), in which the adjust pa-rameter C is 0.467 for the x ≃ .
78 film and 0.326 for the x ≃ .
75 film. Clearly, Eq. (4) can only describe the R (cid:3) - µ H data at low field regime, e.g., µ H . .
30 T for the x ≃ .
78 film and µ H . .
20 T for the x ≃ .
75 film.The experimental R (cid:3) - µ H data at relative larger fieldregion deviates to the prediction of Eq. (4). This sug-gests that the R (cid:3) ( T ) data in Ta x (SiO ) − x films cannotbe fully explained by the quantum tunneling of vortexmodel.To explain the field-induced intermediate metallicphase in amorphous superconducting film, Galitski et al introduce a two-fluid formulation consisting of fermion-ized field-induced vortices and electrically neutralizedBogoliubov quasiparticles (spinons) interacting via along-ranged statistical interaction [28]. The “vortexmetal” phase can be obtained in their theory, and thetheory also predicates a large peak in low-temperaturemagnetoresistance. Form Fig. 8, one can see that the resistance increases monotonically with increasing mag-netic field up to H c (0) and no magnetoresistance peakappear over the whole magnetic field range. Thus the in-termediate metallic phase in Ta x (SiO ) − x granular filmscannot be explained by the “vortex metal” theory either. IV. CONCLUSION
The low temperature transport properties of a seriesof 2D Ta x (SiO ) − x granular films are systematically in-vestigated. At zero field, the low temperature statesof the films undergo transitions from superconductor toanomalous metal and then to insulator with decreasing x . BKT transition is observed in those films with su-perconducting ground state ( x & . x & .
75 films. It is found that the experimental R (cid:3) ( H, T ) data near the transitions from superconduct-ing to anomalous metallic states and anomalous metallicto insulating states are consistent with the theoreticalscaling relations of the Bose-metal model, and cannot bedescribed by the predictions of other related models. Ourresults suggest that the intermediate anomalous metallicstate in the 2D granular films originates from strong dy-namical gauge field fluctuations caused by the disorderand magnetic field.
ACKNOWLEDGMENTS
The authors are grateful to Professor J. J. Lin (Na-tional Chiao Tung University) for valuable discussion.This work is supported by the National Natural Sci-ence Foundation of China through Grant No. 11774253(Z.Q.L.) and Grant No.11774367 (G.L.Y.). [1] A. Yazdani and A. Kapitulnik,
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