Tunneling studies in a homogeneously disordered s-wave superconductor: NbN

S. P. Chockalingam,1 Madhavi Chand,1 Anand Kamlapure,1 John Jesudasan,1 Archana Mishra,1,3

Vikram Tripathi,2 and Pratap Raychaudhuri1,*1Department of Condensed Matter Physics and Materials Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba,

Mumbai 400005, India2Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India

3IIC, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667, IndiaReceived 2 February 2009; published 9 March 2009

We report the evolution of superconducting properties as a function of disorder in epitaxial NbN thin filmsgrown on 100 MgO substrates. The effective disorder in these films encompasses a large range, with kFl1.38–8.77. Tunneling measurements reveal that for films with large disorder the superconducting transitiontemperature is not associated with a vanishing of the superconducting energy gap but with a large broadeningof the superconducting density of states. Our results provide strong evidence of the loss of superconductivityvia phase fluctuations in a homogeneously disordered s-wave superconductor.

DOI: 10.1103/PhysRevB.79.094509 PACS numbers: 74.50.r, 74.81.g, 74.78.Db, 74.62.c

I. INTRODUCTION

The effect of disorder on superconductivity1–4 which hasbeen a topic of considerable interest for several decades hasbeen enjoying a serious revival,5–9 which is primarily moti-vated around the phenomenon of the loss of bulk supercon-ductivity not coincident with the loss of pairing. While this isnatural in granular superconductors10 where bulk supercon-ductivity can be destroyed through phase fluctuations be-tween grains, while the pairing amplitude inside individualgrains remains finite, the same is not expected in a homoge-neously disordered system. In fact, it was shown byAnderson1 in the late 1950s that nonmagnetic disorder in as-wave superconductor should not significantly affect the su-perconducting transition temperature Tc as long as the sys-tem remains a metal. However, it has been suggested in re-cent years5–7 that in the presence of strong disorder wherethe mean-free path l is of the order of the inverse Fermiwave vector 1 /kF, the superconductor can spontaneouslybreak up into domains, giving rise to a scenario similar togranular systems. Superconductivity can get destroyed due tophase fluctuations between these domains, giving rise tonovel metallic and insulating phases8,9 with finite pairing am-plitude but no global superconductivity.

Most of the theoretical investigations5,6,9 and experimen-tal studies3,4,7,8,11 on disordered superconductors have con-centrated in the region close to the superconductor-insulatorSI transition where quantum fluctuations are believed todrive the system into an insulating state with a finite gap inthe electronic spectrum. In this paper, we investigate the evo-lution of the superconducting energy gap with tempera-ture in three-dimensional 3D homogeneously disorderedepitaxial NbN films, which are far from the SI transition. Theeffective disorder in these films can be tuned12 from moder-ately clean to strongly disordered limit without destroyingthe crystallographic structure of NbN and the epitaxial natureof these films, making them an ideal test bed for investigat-ing the effect of homogeneous disorder in an s-wave super-conductor. All films reported in the present work have thick-nesses of 50 nm which is much larger than the coherence

length, 05 nm. The effective disorder in these films, char-acterized by the Ioffe-Regel parameter kFl varies in therange of kFl1.38–8.77. Within this range of kFl the Tcdecreases from 16.8 to 7.7 K. However, all our films havecarrier densities typical of a good metal and are therefore farfrom the Anderson superconductor-insulator phase boundary.The central result of this paper is that even far from the SItransition, as kFl approaches unity, the bulk Tc of an s-wavesuperconductor is not associated with vanishing of the super-conducting energy gap but rather a large broadening ofthe quasiparticle density of states DOS.

II. EXPERIMENTAL DETAILS

Epitaxial NbN films were synthesized on 100-orientedsingle crystalline MgO substrates by reactive dc magnetronsputtering by sputtering an Nb target in Ar /N2 gas mixture.The disorder in these films can be controlled by controllingthe deposition parameters during growth. Details of the syn-thesis and characterization of NbN thin films have been re-ported in Ref. 12. Planar tunnel junctions were fabricated byfirst depositing a 300-m-wide NbN strip and oxidizing itssurface at 250 °C in oxygen atmosphere for 2 h. Two tunneljunctions were fabricated on each device by subsequentlyevaporating 300 m cross strips of Ag. Under optimalgrowth conditions,13 this process resulted in highly reproduc-ible tunnel junctions, with high bias V /e tunnel junc-tion resistance varying between 2–10 . Devices fabricatedby oxidizing for longer times had higher junction resistancebut poorer tunneling characteristics in the superconductingstate, possibly due to formation of defect states in the tunnelbarrier. The device was configured in such a way that wecould perform both I-V measurements on the tunnel junctionas well as resistance versus temperature R-T measurementsof the NbN film on which the tunnel junction was fabricated.The conductance GV= dI

dV V versus voltage V character-istics of the tunnel junction were obtained by measuring thecurrent versus voltage I-V characteristics at various tem-peratures and numerically differentiating the I-V data. In allthese measurements the NbN electrode was connected to the

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virtual ground whereas the voltage was applied on the silverelectrode. The Tc and resistivity T of the film on whichthe tunnel junction was fabricated were independently mea-sured using a standard four-probe technique. Hall effect wasmeasured on samples patterned in a Hall bar geometry. Runsof several samples showed that for identical growth condi-tions the Tc of the films are within 10% of each other.

III. RESULTS AND DISCUSSION

Figure 1a shows the variation in Tc with kFl for severalNbN films with different levels of disorder. The values of kFlare determined from the n extracted from Hall effect and thenormal state resistivity n both measured at 18 K usingfree-electron relations.14 Figure 1b shows the variation inn versus kFl for the same films. The inset of Figs. 1a and1b show the variation in Tc and normal state conductivityn=1 /n with n. As a first approximation, both the varia-tions in n and Tc are related to the variation12 in n. While thedisorder tuning in these films is likely to be a result of Nbdeficiencies in highly disordered films, the large change in ncannot be accounted for by chemical considerations alone.12

This points toward an unusual localization of the electroniccarriers with increase in disorder which will be dealt with ina separate paper. The Thomas-Fermi screening length esti-mated from n indicates that the disorder arising from vacan-cies is screened over atomic length scales which is muchsmaller than the coherence length12 5 nm of the super-conductor. These epitaxial films are therefore homogeneous,with no physical source of extrinsic granularity. The kFl val-ues of the NbN films on which tunnel junctions are fabri-cated are estimated from the measured n and Tc using thepolynomial fit to the variations in Tc and n with kFl and

1 /kFl, respectively solid lines in Fig. 1a and 1b.Figures 2a–2c show the GV vs V spectra at different

temperatures down to 2.2 K for three films with differentlevels of disorder corresponding to a Tc=14.9 K, wherekFl6; b Tc=9.5 K, where kFl2.3; and c Tc=7.7 K,where kFl1.4, respectively.15 An asymmetry between posi-tive and negative biases as well as a slope at high bias, simi-lar to the tunneling spectra of high Tc cuprates,16,17 are ob-served for all the spectra. This asymmetry is larger forsamples with smaller kFl. Before theoretically fitting thespectra, the spectra were symmetrized by subtracting a linearbackground passing through the origin. The conductancespectra were fitted with the tunneling equation18

GV = dI

dV

V

=d

dV 1

RN

−

NsENnE − eVfE − fE − eVdE ,

where NsE and NnE=Nn are normalized densities ofstates of the superconducting and normal metal electrodes,respectively, fE is the Fermi-Dirac distribution function,and RN is the resistance of the tunnel junction for V /e.For NsE we used the DOS given by Bardeen-Cooper-Schrieffer BCS theory18 with an additional broadeningparameter19 so that NsE=ReE− i / E− i2−21/2.=h / is formally introduced in this expression to take intoaccount the effect of finite lifetime of the superconduct-ing quasiparticles. However, while fitting a spectrum, phe-nomenologically incorporates all nonthermal sources ofbroadening in the BCS DOS. The symmetrized spectra alongwith the corresponding fits are shown in Figs. 2d–2f. Weobserve an increase in 0 /0 corresponding to their respec-tive values at 2.2 K shown in Figs. 3a–3c from 0.002 inthe most ordered film to 0.054 in the most disordered film,similar to earlier observation in granular Al samples3 anddisordered7 TiN.

The central result of this paper, namely, the temperaturevariation in and obtained from the best fit values of thespectra, is shown in Figs. 3a–3c. In the same graph weshow the T versus T of the NbN films on which the tunneljunctions are fabricated. For the film with least disorderkFl6, and follow the temperature variation expectedfor a strong-coupling superconductor.20 For this film Tc isassociated with a vanishing of the . For the films with largedisorder on the other hand, does not go to zero as T→Tcbut remains finite even for T Tc. This is most clearly seenfor the sample with Tc=7.7 K, kFl1.4 Fig. 3c, where decreases to only 60% of its low-temperature value at 7.3 K.On the other hand increases rapidly as the temperatureapproaches Tc. We can see from the trend in variation in and that Tc phenomenologically corresponds to the pointwhere .

In conventional mean-field theories of s-wave supercon-ductors, such as BCS theory18 or its strong-couplingcounterparts,21 the superconducting Tc is defined as the tem-perature where goes to zero. However, the zero resistancestate in a superconductor is characterized not only by finitepairing amplitude but also by global phase coherence.

0 8 16 240.0

4.0x105

8.0x105

1.2x106

1.6x106

1 2 3 4 5 6 7 8 90

2

4

6

8

10

128

10

12

14

16

0.0 8.0 16.0 24.06

9

12

15

18

σ n(Ω

-1m

-1)

n (X1028 electrons m-3)ρ n(µ

Ωm

)

(b)

Tc(

K)

kFl

(a)

Tc(

K)

n (X1028

electrons m-3)

FIG. 1. Color online a Variation in Tc with kFl in NbN filmswith different levels of disorder; the solid line is a polynomial fit tothe data. The inset shows the variation in Tc with n. b Variation inn with kFl for the same films; the solid line is a polynomial fit inpowers of 1 /kFl. The inset shows the variation in n=1 /n withn and the linear fit to nn.

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Therefore, in principle bulk superconductivity can also belost due to the loss of superfluid stiffness whereby phasefluctuations between different regions of the superconductordestroy the macroscopic zero resistance state even if the am-plitude of the order parameter remains finite. This mecha-nism has been explored10 in the context of Josephson junc-tion JJ networks and granular superconducting films wherethe superconductor can be envisaged as a disordered networkof Josephson junctions. Recent numerical simulations5 ondisordered two-dimensional 2D superconductors as well asscanning tunneling spectroscopy experiments7 on homoge-neously disordered TiN films close to SI transition reveal thatthe situation in a strongly disordered s-wave superconductoris similar to a disordered JJ network. Under strong disorderthe superconductor electronically segregates into supercon-ducting domains separated by insulating regions. In this sce-nario, the bulk Tc corresponds to the temperature where theglobal phase coherence5 is lost between these islands.22 Insuch a situation the amplitude of the order parameter andhence will not go to zero at T Tc but BCS DOS willbroaden due to thermal phase fluctuations. While we cannotmake a detailed comparison with existing numericalsimulations5 which have been performed at T=0 in 2D sys-tems close to the SI transition, the nonvanishing of andthe large increase in as T→Tc in strongly disordered

kFl1 epitaxial NbN films are consistent with this sce-nario.

Extrapolating the T versus T data for the NbN filmwith strong disorder, it seems natural to assume that the gapin the DOS will persist even in the normal state. This natu-rally points to the formation of a state with a finite value ofenergy gap but no global phase coherence for TTc. In sucha situation, the bulk Tc is smaller than the mean-field transi-tion temperature T expected in the absence of phase fluc-tuations. While we cannot obtain spectroscopic informationfor TTc due to low resistance of our tunnel junctions com-pared to the sample resistance, an indirect evidence of thiscomes from the measurement of 20 /kBTc which is a mea-sure of the electron-phonon coupling strength18 within mean-field theories of superconductivity. Figure 4 shows the varia-tion in 20 /kBTc as a function of Tc for six NbN films withdifferent levels of disorder. For the least disordered filmTc14.9 K, 20 /kBTc=4.36 as expected for a strong-coupling superconductor. Since in our films n decreases withincrease in disorder, for films with lower Tc the electron-phonon coupling strength is expected to decrease due to thedecrease in DOS at Fermi level.12 This trend is observed forfilms with Tc9.5 K. However for films with Tc9.5 K,20 /kBTc shows an anomalous increase reaching a value of4.43 for the film with Tc7.7 K. This signals a breakdown

-4 -3 -2 -1 0 1 2 3 40.00

0.06

0.12

0.18

dI/d

V(Ω

-1)

V (m V)

2.17K3.35K4.30K4.90K5.50K6.40K7.00K7.40K

(f)

-4 -2 0 2 40.0

0.1

0.2

0.3

dI/d

V(Ω

-1)

V (mV)

2.23K2.91K4.30K5.05K6.70K7.35K8.0K9.0K

(e)

-6 -4 -2 0 2 4 60.0

0.2

0.4

0.6

0.8

1.0

dI/d

V(Ω

-1)

V (mV)

2.17K3.50K4.50K7.0K10.35K12.35K14.0K14.6K

(d)

-15 -10 -5 0 5 10 150.0

0.2

0.4

0.6

0.8

1.0dI

/dV

(Ω-1)

V (mV)

2.17K3.5K4.5K7.0K10.35K12.35K14.0K14.6K

(a)

-15 -10 -5 0 5 10 150.00

0.05

0.10

0.15

0.20

dI/d

V(Ω

−1)

V (mV)

2.17K3.35K4.3K4.9K5.5K6.4K7.0K7.4K

(c)

-15 -10 -5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

0.30

dI/d

V(Ω

−1)

V (mV)

2.23K2.91K4.30K5.05K6.70K7.35K8.00K9.00K

(b)

FIG. 2. Color online a–cGV-V spectra at different tem-peratures of NbN/oxide/Ag tunneljunction at different temperatureson three NbN films with a Tc

=14.9 K, where kFl6; b Tc

=9.5 K, where kFl2.3; and cTc=7.7 K, where kFl1.4. Sometemperatures have been removedfor clarity. d–f The same set ofspectra corresponding to a–cafter removing the asymmetricbackground points along withthe corresponding theoretical fitslines. The fits are restricted tobias values equal to 2.50 /e.

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of the mean-field scenario where the Tc obtained from theonset of zero resistance is smaller than the mean-field T.Between Tc and T one would expect a state with a gap in thequasiparticle spectrum but no overall phase coherence. Sucha state should in principle be detectable in scanning tunnel-ing spectroscopy measurements at temperatures higher thanTc where the tunneling resistance varies between few100 M to a few G and the contribution of the normalstate resistance of the sample is insignificant.

At this point it is worthwhile to explore whether otherscenarios proposed in disordered superconductors could ex-plain our data. It was suggested by Anderson et al.23 thatclose to kFl1 the loss of effective screening increases theelectron-electron repulsion, leading to a decrease in Tc.Within McMillan theory of strong-coupling superconductorsthis effect leads to an increase in the effective Coulombpseudopotential . While the kFl of our films are in therange where this effect could manifest itself,24 this mecha-nism gives a positive correction to the effective whichdoes not explain the increase in 20 /kBTc or our observation

of finite at T Tc. In another model proposed byBergmann,25 it was suggested that disorder causes an effec-tive increase in the electron-phonon coupling strength due tophonon emission caused by inelastic-scattering processes atimpurity sites. It was qualitatively argued26 that in strong-coupling superconductors, this effect is compensated by“strong-coupling” effects, which in the presence of strongdisorder could cause a decrease in and Tc. While this effectcould in principle account for the increase in 20 /kBTc, it isnot consistent with the nonvanishing of at T Tc.

IV. CONCLUSION

In conclusion, we show that in 3D strongly disorderedNbN epitaxial thin films, the superconducting energy gapdoes not vanish as T→Tc as expected for a conventionals-wave superconductor. Furthermore, 20 /kBTc shows ananomalous increase with increasing disorder in this regime.Our results indicate that the DOS is likely to remain gappedeven in the normal state, resulting in a pseudogap state attemperatures TTc. These results are in qualitative agree-ment with recent theoretical predictions on strongly disor-dered s-wave superconductors close to the SI phase bound-ary. However, the unique feature of our results is that eventhough our films are far from the SI phase boundary, we seethat superconducting Tc is strongly influenced by phase fluc-tuations when kFl approaches 1. These results bear a strongsimilarity with the high Tc superconducting cuprates where afinite energy gap above Tc is well established16 from tunnel-ing measurements. It would therefore be instructive to obtainspectral information on the state at TTc, as well as itspossible spatial variation using techniques such as scanningtunneling spectroscopy. These measurements are currentlyunderway and will be reported in a future paper.

ACKNOWLEDGMENTS

The authors thank T. V. Ramakrishnan for valuable inputat every stage of this work, Nandini Trivedi for illuminatingdiscussions, Vivas Bagwe for technical help, and SangitaBose for critically reading the manuscript.

7 8 9 10 11 12 13 14 15

3.94.04.14.24.34.44.5

8 10 12 14

1.62.02.42.8

2∆0/k

BT

c

Tc (K)

∆ 0(m

eV)

Tc (K)

FIG. 4. Color online Variation in 20 /kBTc as a function of Tc

for NbN films. The inset shows the variation in 0 with Tc.

2 4 6 8 100.0

0.4

0.8

1.2

1.6

0.0

1.0

2.0

3.0

4.0

5.0

ρ(μ

Ωm)

∆,Γ(m

eV)

T(K)

∆∆

ΓΓ

(b)∆

0=1.60 meV

Γ0=0.07 meV

2 3 4 5 6 7 8 90.0

0.4

0.8

1.2

1.6

0.01.02.03.04.05.06.07.0

ρ(μ

Ωm)

∆,Γ(m

eV)

T(K)

∆∆

ΓΓ

(c)∆

0=1.47 meV

Γ0=0.08 meV

3 6 9 12 15 180.0

0.5

1.0

1.5

2.0

2.5

0.0

0.4

0.8

1.2

1.6

ρ(μ

Ωm)

∆,Γ(m

eV)

T(K)

∆

ΓΓ

(a)∆

0=2.80 meV

Γ0=0.006 meV

FIG. 3. Color online Temperature variation of and and for three NbN films with different levels of disorder: a Tc

=14.9 K, where kFl6; b Tc=9.5 K; where kFl2.3; and cTc=7.7 K, where kFl1.4. The values of and obtained fromthe best fits at 2.2 K 0 and 0 are shown in the same figures.

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using the polynomial fit in Fig. 1a. However, we verified thatthe kFl values estimated from n are within 20% of the valuedetermined from Tc.

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