Phase Fluctuations in a Strongly Disordered s-Wave NbN Superconductor Closeto the Metal-Insulator Transition

Mintu Mondal,1 Anand Kamlapure,1,* Madhavi Chand,1,† Garima Saraswat,1 Sanjeev Kumar,1 John Jesudasan,1

L. Benfatto,2 Vikram Tripathi,1 and Pratap Raychaudhuri1,‡

1Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India2ISC-CNR and Department of Physics, Sapienza University, Piazzale Aldo Moro 5, 00185 Rome, Italy

(Received 27 June 2010; published 25 January 2011)

We explore the role of phase fluctuations in a three-dimensional s-wave superconductor, NbN, as we

approach the critical disorder for destruction of the superconducting state. Close to critical disorder,

we observe a finite gap in the electronic spectrum which persists at temperatures well above Tc. The

superfluid density is strongly suppressed at low temperatures and evolves towards a linear-T variation

at higher temperatures. These observations provide strong evidence that phase fluctuations play a central

role in the formation of a pseudogap state in a disordered s-wave superconductor.

DOI: 10.1103/PhysRevLett.106.047001 PACS numbers: 74.55.+v, 74.40.n, 74.62.En, 74.70.Ad

Understanding the role of interaction and disorder is atthe heart of understanding many-body quantum systems.Over the past few decades, it has been realized thatthe ground state in conventional superconductors, oncethought to be resilient to disorder [1], gets destabilized ata critical value of disorder, giving rise to unusual metallicand insulating states [2,3]. However, recent experimentsindicate that even after the global superconductivity isdestroyed, the material continues to possess some of thefundamental properties commonly associated with thesuperconducting state. These include the dramatic obser-vation of magnetic flux quantization in disorder driveninsulating Bi films [4], finite high-frequency superfluidstiffness above Tc in amorphous InOx films [5], and theformation of a ‘‘pseudogapped’’ state [6] above Tc charac-terized by a finite gap in the electronic spectrum. Recenttheoretical investigations [7] also indicate that strongsuperconducting correlation can persist even after globalsuperconductivity is destroyed.

The superconducting state is characterized by a complexorder parameter of the form ¼ jjei where three kindsof excitations can ultimately destroy the superconductingorder: (i) quasiparticle excitations (QE) which primarilyaffect the amplitude jj; (ii) quantum phase fluctuations(QPF) associated with the number-phase uncertaintyrelation; and (iii) classical phase fluctuations (CPF) causedby thermal excitations. In ‘‘clean’’ conventional supercon-ductors, it is sufficient to consider QE alone [8], which arewell described by Bardeen-Cooper-Shrieffer (BCS) andEliashberg mean-field theories [9]. However, strongly dis-ordered superconductors are characterized by poor screen-ing of the Coulomb interactions [10] and small superfluiddensity (ns) due to disorder scattering [9], making themsusceptible to phase fluctuations. Since strong phasefluctuations can destroy the global superconducting orderwell before jj goes to zero, a larger role of phase fluctua-tions could explain the persistence of superconducting

properties above Tc. Numerical simulations [11] alsoshow that the superconducting state can get destroyed bystrong phase fluctuations between domains that emerge inthe presence of strong disorder.In this Letter, we explore the role of phase fluctuations in

strongly disordered superconducting NbN films, using acombination of scanning tunneling spectroscopy (STS) andpenetration depth () measurements. The effective disor-der, characterized by the product of Fermi wave vector(kF) and electronic mean free path (l), ranges from kFl1:24–10:12. In this range, Tc (the temperature where theresistance reaches 1% of its normal state value) decreasesfrom 16.8 K to <300 mK. In films with Tc & 6 K, weobserve that the dip in the tunneling density of states(DOS) associated with the superconducting energy gap(), persists well above Tc; on the other hand ns / 2

is suppressed compared to the BCS estimate, and shows agradual evolution towards linear temperature variation.These observations provide compelling evidence of therole of phase fluctuations in a strongly disordered s-wavesuperconductor.Our samples consist of epitaxial NbN films grown

through reactive magnetron sputtering on single crystallineMgO substrates, with thickness 50 nm, which is muchlarger than the dirty-limit coherence length ( 4–8 nm)[12]. The disorder in these films can be tuned by control-ling the level of Nb vacancies in the NbN crystallinelattice, which is controlled by controlling the Nb=N ratioin the plasma [13,14]. kFl is determined from the resistivity() and Hall coefficient (RH) using the free electron for-

mula kFl ¼ fð32Þ2=3@½RHð285 KÞ1=3g=½ð285 KÞe5=3 (@is Planck’s constant and e is the electronic charge). Theabsolute value of is measured using a low-frequency(60 kHz) two coil mutual inductance technique [15] on8 mm diameter circular films patterned using a shadowmask. The temperature dependence of the tunneling-DOSis probed from the measurement of tunneling conductance

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[GðVÞ ¼ ðdI=dVÞjV] using an in situ, high-vacuum, low-temperature scanning tunneling microscope [16] (STM)operating down to 2.6 K. Tunneling measurements werealso performed on disordered NbN=insulator=Ag planartunnel junctions [17] at 500 mK in a 3He cryostat.

Figure 1(a) shows the conductivity () as a function oftemperature for a series of NbN films. While most of thefilms display ðd=dTÞ> 0 in the normal state [14], even inthe most disordered film [inset (a)], the variation in -T(after superconductivity is suppressed by the applicationof a magnetic field) clearly suggests that ðT ! 0Þ 0.Therefore, all of our samples are on the metallic side ofthe Anderson metal-insulator transition (MIT). Figure 1(b)shows that both the conductivity just before the onset ofthe superconducting transition (0) and Tc asymptoticallygo to zero as kFl ! 1. Therefore, the critical disorderfor destruction of superconductivity coincides with theMIT. While a direct transition from a superconductingto an insulating state with disorder is ubiquitous inultrathin two-dimensional films [2], in 3D, bothsuperconductor-normal metal and superconductor-insulator transitions have been observed and theoreticallypredicted [10,18,19]. At present, we do not know whetherthis coincidence, similar to that reported for 3D boron-doped diamond epitaxial films [20], is merely accidental.

The temperature evolution of the tunneling DOS wasinvestigated through STS measurements performed onseveral samples with different disorder. Figures 2(a)–2(c)show the temperature dependence of the tunneling DOSin the form of an intensity plot [16] of the normalizedconductance GðVÞ=GN [where GN ¼ GðV =eÞ], as afunction of temperature and bias voltage for three differentfilms. For each temperature, the GðVÞ vs V spectrum isspatially averaged over 32 points taken at equal intervalsalong a 150 nm line. The resistance vs temperature for eachsample is shown in the same panel. At the lowest tempera-ture, GðVÞ=GN vs V for the first two samples (shown ininset) show a dip close to V ¼ 0 and two symmetric peaks,as expected from BCS theory. For the sample with

Tc 11:9 K, a flat metallic DOS is restored for T > Tc.However, with increase in disorder a pseudogapped phaseemerges, where the dip in the DOS persists above Tc. In themost disordered sample (Tc 1:65 K), the pseudogap per-sists well beyond the onset of the superconducting transi-tion, up to 6:5 K. We would also like to note that evenabove the pseudogap temperature, in all our samples, a‘‘V’’ shape in theGðVÞ=GN spectra [21] extends up to V =e. This feature, which is not associated with supercon-ductivity, arises from Altshuler-Aronov (AA) type inter-actions in the normal state. To extract the pseudogapfeature associated with superconductivity in the most dis-ordered sample (Tc 1:65 K), we subtract the AA back-ground using the spectra obtained at 8 K [22], fromthe spectra obtained at 2.65 K. The resulting curve

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FIG. 1 (color online). (a) vs T for films with kFl 10:12,8.82, 8.13, 8.01, 5.5, 4.98, 3.65, 3.27, 2.21, 1.68, 1.58, and 1.24.The inset shows an expanded view of ðTÞ for the film withkFl 1:24 in zero field and in magnetic field of 5T. (b) 0 andTc as a function of kFl.

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FIG. 2 (color). (a)–(c) Intensity plots of GðVÞ=GN as a func-tion of temperature and applied bias for three different films(upper panels), and the resistance vs temperature (R-T) in thesame temperature range (lower panels); vertical dotted lines inthe upper panels correspond to Tc. Insets of panels (a),(b) andleft inset of panel (c) show representative tunneling spectra in thesuperconducting state (blue), in the pseudogap state (green) andabove the pseudogap temperature (red). The right inset of panel(c) shows the AA background subtracted spectra at 2.65 K.

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GsubðVÞf¼ ½GðVÞ=GNjT¼2:65 K ½GðVÞ=GNjT¼8 K þ0:76g [right inset of Fig. 2(c)] clearly reveals the presenceof broadened coherence peaks around 2 mV confirming thesuperconducting origin of the pseudogap feature. In addi-tion, the individual line scans [23], reveal that the super-conducting state becomes progressively inhomogeneousas the disorder is increased [11].

To investigate the origin of the ‘‘pseudogap’’ behaviorwe now focus on ðTÞ. 2ðTÞ / ns [ m=ð0e

22Þ,where m is the effective mass of the electron], getsstrongly renormalized [24,25] in the presence of phasefluctuations. Figures 3(a) and 3(b) show the temperaturevariation of 2ðTÞ in a series of NbN films with differentTc. At low disorder the temperature variation agrees wellwith the dirty-limit BCS expression [9], 2ðTÞ=2ð0Þ ¼fðTÞ=ð0Þg tanh½ðTÞ=ð2kBTÞ, where kB is theBoltzman constant. However, as disorder increases twoeffects become noticeable: (i) 2ðT ! 0Þ decreases rap-idly by 2 orders of magnitude and (ii) 2ðTÞ decreasesfaster than the expected BCS temperature variation. Firstwe concentrate on 2ðT ! 0Þ. In the absence of phasefluctuations, 2ð0Þ is reduced by disorder scatteringaccording to the BCS relation [9], 2ð0ÞBCS ¼0ð0Þ0=@. For NbN, we find ð0Þ 2:05kBTc fromtunneling measurements performed at low temperatures(T < 0:2Tc) on planar tunnel junctions fabricated on anumber of samples with different disorder [26] [insetFig. 3(c)]. Figure 3(c) shows that 2ð0Þ 2ð0ÞBCSwithin experimental error for samples with Tc > 6 K.However, as we approach the critical disorder 2ð0Þbecomes gradually smaller than 2ð0ÞBCS, reaching avalue which is 50% of 2ð0ÞBCS for the sample withTc 2:27 K. Also, while in all samples 2ðTÞ saturates

as T ! 0, at higher temperatures, samples with Tc < 6 Kshow a gradual evolution towards a linear-T variation. Thisis most clearly observed in the sample with Tc 2:27 K[Fig. 3(d)].Since the suppression of ð0Þ2 from its BCS value and

linear-T dependence of 2ðTÞ are characteristic featuresassociated with QPF and CPF [27], respectively, we nowtry to quantitatively analyze our data. The importanceof QPF and CPF is determined by two energy scales [8]:the Coulomb energy Ec, and the superfluid stiffnessJ, which can be estimated for a 3D superconductorfrom the relations [25], Ec ¼ ð16e2Þ=ð"1aÞ and J ¼ð@2ansÞ=ð4mÞ, where "1 is the background dielectricconstant and a is the characteristic length scale for phasefluctuations. The suppression of 2ð0Þ from QPF can beestimated using the self-consistent harmonic approxima-

tion [28] which predicts (in 3D), nsðT ¼ 0Þ=ns0ðT ¼ 0Þ ¼e2ðT¼0Þ=6 (ns0 is the bare value in the absence of phase

fluctuations), where 2ðT ¼ 0Þ ¼ ð1=2Þ ffiffiffiffiffiffiffiffiffiffiffi

Ec=Jp

. At thesame time, the energy scale above which phase fluctuationsbecome classical is given by the Josephson plasma

frequency ð@!p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4e2ns=m"1

p ¼ ffiffiffiffiffiffiffiffi

EcJp Þ. We calcu-

late these values for the sample with Tc 2:27 K. ForNbN, we estimate "1 30 from the plasma frequency(12 600 cm1) measured [29] at low temperatures.Taking a 8 nm (Ref. [12]), we obtain Ec 0:3 eV and J 0:14 meV at T ¼ 0, corresponding tonsðT ¼ 0Þ=ns0ðT ¼ 0Þ 0:02. While this value is likelyto have some inaccuracy due to the exponential amplifica-tion of errors in our estimate of Ec or J, the importantpoint is that this suppression is much larger than ourexperimental estimate, 2ð0Þ=2

BCSð0Þ 0:5. On the

other hand "!p 6:5 meV 75 K, so that one should

conclude that CPF cannot be responsible for the observedlinear temperature dependence of 2ðTÞ in this sample.These two apparent contradictions can be resolved by

considering the role of dissipation. In d-wave supercon-ductors, the presence of low energy dissipation has beentheoretically predicted [30] and experimentally observedfrom high-frequency conductivity [31,32] measurements.Recent measurements [33] on amorphous InOx films revealthat low energy dissipation can also be present in stronglydisordered s-wave superconductors [34]. While the originof this dissipation is not clear at present, the presence ofdissipation has several effects on phase fluctuations [25]:(i) QPF are less effective in suppressing ns; (ii) QPF con-tribute to a T2 temperature depletion of ns of the formns=ns0 ¼ 1 BT2 at low temperature where B is directlyproportional to the dissipation, and (iii) the crossover tothe usual linear temperature dependence of ns due to CPF,ns=ns0 ¼ 1 ðT=6JÞ, occurs at a characteristic tempera-ture that is much smaller than @!p=kB, and scales approxi-

mately as Tcl J= , where is a dimensionless measureof the residual conductivity in the SC state. In the samplewith Tc 2:27 K, the T2 variation of ðTÞ2=ð0Þ2 canbe clearly resolved [15] below 650 mK [inset, Fig. 3(d)].

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FIG. 3 (color online). (a),(b) 2ðTÞ for NbN films with differ-ent disorder; the solid lines are the expected temperature varia-tion from dirty-limit BCS theory. (c) 2ð0Þ and 2ð0ÞBCS asfunction of Tc; the inset shows the variation ð0Þ as function ofTc. (d) Temperature variation of ½ðTÞ=ð0Þ2 for the film withTc ¼ 2:27 K; the solid lines are fits to the T2 dependence of2ðTÞ=2ð0Þ at low temperature (T 0:65 K) and the Tdependence at higher temperature; the inset shows an expandedview of the T2 dependence at low temperatures.

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In the same sample, the slope of the linear-T region is 3times larger than the slope estimated from the value of Jcalculated for T ¼ 0. This discrepancy is, however, minorconsidering the approximations involved. In addition, atfinite temperatures ns0 gets renormalized due to QE. Withdecrease in disorder, QE eventually dominates over thephase fluctuations, thereby recovering the usual BCS tem-perature dependence at low disorder. The increased role ofCPF could naturally explain the observation of a pseudo-gapped state in strongly disordered NbN films. We wouldalso like to note that in all the disordered samples 2ðTÞshows an abrupt downturn close to Tc, reminiscent of theKosterlitz-Thouless (KT) transition [15]. However, sincein 3D a KT transition is not expected, we do not know theorigin of this behavior.

To summarize, we observe a progressive increase inphase fluctuations and the formation of a pseudogap statein strongly disordered NbN thin films. These observationssupport the scenario where the superconducting transitionis governed by phase disordering. The remaining questionis on the role of amplitude fluctuations arising from non-equilibrium Cooper pairs, which can also produce a dip inthe tunneling DOS above Tc. While our experiments can-not rule out the possibility of amplitude fluctuations alsoplaying a role [35], within a standard Ginzburg-Landau(G-L) approach this effect is much smaller [36] than thepseudogap feature observed in our samples. However, itwould be worthwhile to critically explore the applicabilityof conventional G-L theory close to the MIT to assess therole of amplitude fluctuations. It would also be interestingto use a comparative analysis between tunneling DOS and to understand the extent to which the pseudogap statein underdoped high-Tc cuprates can be understood withina phase-fluctuations scenario.

We would like to thank T.V. Ramakrishnan, S. Mandal,and M.V. Feigelman for enlightening discussions andV. Bagwe and S. P. Pai for technical help.

*ask@tifr.res.in†chand@tifr.res.in‡pratap@tifr.res.in

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see supplemental material at http://link.aps.org/supplemental/10.1103/PhysRevLett.106.047001 for analy-sis of low temperature 2ðTÞ data, Sec. 4.

[16] See supplemental material at http://link.aps.org/supplemental/10.1103/PhysRevLett.106.047001 for de-tails of the STM, Sec. 1.

[17] S. P. Chockalingam et al., Phys. Rev. B 79, 094509 (2009).[18] A. Kapitulnik and G. Kotliar, Phys. Rev. Lett. 54, 473

(1985); M.Ma and P. A. Lee, Phys. Rev. B 32, 5658 (1985).[19] M. Ma and E. Fradkin, Phys. Rev. Lett. 56, 1416 (1986).[20] T. Klein et al., Phys. Rev. B 75, 165313 (2007).[21] This is also observed in planar tunnel junctions [17].[22] Since the AA background is weakly temperature depen-

dent, we use the spectra taken at the lowest temperaturewhere the low bias feature associated with superconduc-tivity disappears to subtract the background.

[23] See supplemental material at http://link.aps.org/supplemental/10.1103/PhysRevLett.106.047001 for indi-vidual line scans, Sec. 3.

[24] C. Ebner and D. Stroud, Phys. Rev. B 28, 5053 (1983).[25] L. Benfatto et al., Phys. Rev. B 63, 174513 (2001).[26] This is in variance with Ref. [17] where an apparent

increase of 2=kBTc was reported for a much smallerrange of disorder.

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supplemental/10.1103/PhysRevLett.106.047001 for indi-rect evidence of low energy dissipation in our systemobtained from low temperature tunneling DOS, Sec. 2.

[35] J. L. Tallon, J. G. Storey, and J.W. Loram,arXiv:0908.4428.

[36] In the dirty limit, the dip in GðVÞ at V ¼ 0 is expected tobe of the order of ðG=GÞV¼0 ¼

ffiffiffiffiffiffi

Gip

ln½ðT TcÞ=Tc,where Gi ¼ ð1:6=ðkFlÞ3ÞðTc=EFÞ; A. Larkin and A.Varlamov, Theory of Fluctuations in Superconductors(Clarendon Press, Oxford, 2005).

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