Homogeneous superconducting phase in TiN film : a complex impedance study
aa r X i v : . [ c ond - m a t . s up r- c o n ] D ec Homogeneous superconducting phase in TiN film : a complex impedance study
P. Diener, H. Schellevis, and J.J.A. Baselmans SRON Netherlands Institute for Space Research, 3584 CA Utrecht, The Netherlands. ∗ Delft Institute of Microsystems and Nanoelectronics,Delft University of Technology, 2628 CD Delft, Netherlands. (Dated: July 7, 2018)The low frequency complex impedance of a high resistivity 92 µ Ω cm and 100 nm thick TiNsuperconducting film has been measured via the transmission of several high sensitivity GHz mi-croresonators, down to T C /
50. The temperature dependence of the kinetic inductance follows closelyBCS local electrodynamics, with one well defined superconducting gap. This evidences the recoveryof an homogeneous superconducting phase in TiN far from the disorder and composition driventransitions. Additionally, we observe a linearity between resonator quality factor and frequencytemperature changes, which can be described by a two fluid model.
Thin films of titianium nitride (TiN) have recentlybeen proposed [1] for radiation detection using KineticInductance Detectors [2]. TiN and other strongly dis-ordered superconducting materials have a high surfaceimpedance for radiation with a photon energy hν > µ Ω cm and higher [3], clearlyin violation of the Anderson theorem which states thatdisorder does not affect the properties of the supercon-ducting state [4]. These deviations increase with higherdisorder, leading to systems in which a superconductorinsulator transition (SIT) is observed, typically when theIoffe-Regel parameter k F ℓ ∼
1. SITs have been observedin thin films of several materials by changing the disordervia a film treatment, or by varying the thickness, the filmstoichiometry or the applied magnetic field [5–8]. In TiNa SIT has been evidenced few years ago in ∼ nm thickfilms by magnetic field and thickness changes [9, 10]. Sev-eral studies also report the presence of an inhomogeneoussuperconducting phase close to the SIT [11–15].One expects to recover a classical BCS superconduct-ing phase when going to lower disorder/thicker films. Incontrast, the only one spectroscopic study on a low disor-dered TiN film reports on the presence of a non uniformstate comprising of superconducting and normal areas[16]. These results are discussed in the context of meso-scopic fluctuations close to a superconducting to normaltransition. Indeed, TiN also exhibits a transition withcomposition: as reported recently, superconductivity dis-appear in the N sub stoichiometric range [1]. The pres-ence of an inhomogeneous order parameter in a low dis-ordered TiN film points out the necessity to disentanglethe thickness and composition transitions.In this paper, we report on the complex impedancestudy of several microresonators made of a 100 nm thick, relatively weakly disordered TiN film, having k F ℓ = 12 . µ Ω cm . The film has been charac-terized extensively, and the resonators have internal qual-ity factors up to 10 . This allows an accurate determi-nation of the superconducting gap from the temperatureresonance frequency shift which is directly proportionalto the kinetic inductance or superfluid density changes.The film has been prepared by a DC magnetron sput-tering system. It is sputtered on a nitrogen/argon plasmaat 350 ◦ C on a high resistive HF cleaned silicon substrate(See ref. [17] for more details on the recipe). The thick-ness determined with a scanning electron microscope is d = 100 ± nm . The homogeneity has been checked byX-Ray Photoelectron Spectroscopy (XPS) depth profil-ing : as shown fig. 1, there is no contamination exceptsome oxygen at the surface on few nm, and the T i/N ratio is thickness independent. The stoichiometry deter-mined is 1 : 1 . .
35 ˚ A determinedby X-ray Diffraction (XRD), to be compared to the bulkvalue 4 .
24 ˚ A . XRD spectra also show a favored (200)crystalline orientation. A typical grain size of 10 nm hasbeen determined by transmission electron microscopy inothers TiN films from the same source [17].The film characterizations at low temperature includestandard R(T) and Hall measurements. The resistivityat 10 K , ρ = 92 µ Ω cm is almost constant up to 300 Kand the carrier density is n = 3 .
77 10 cm − . k F ℓ canbe estimated in the free electron model from ρ and n by [18] : k F ℓ = 3 π ¯ h/ ( e ρ √ π n ) giving 12.7 for thisfilm. In addition, fig.2 shows the sharp superconductingtransition observed at T C = 4 . ± . K .Coplanar waveguide resonators have been patternedusing standard contact lithography and dry etching withan SF /Ar gas mixture. One resonator is shown fig.2.The resonators are formed by a central meandered lineof few mm length, 3 µm wide, and slits of 2 µm wide be-tween the central line and the groundplane. They arecapacitively coupled to the feedline by placing one res- (cid:1)(cid:2)(cid:1)(cid:3)(cid:1)(cid:2)(cid:1)(cid:4)(cid:1)(cid:2)(cid:1)(cid:5)(cid:1)(cid:2)(cid:1)(cid:6)(cid:1)(cid:2)(cid:1)(cid:7)(cid:1)(cid:1)(cid:2)(cid:1) (cid:1) (cid:2)(cid:1) (cid:3)(cid:1) (cid:4)(cid:1) (cid:5)(cid:1) (cid:6)(cid:1) (cid:1) (cid:2)(cid:3) (cid:1) (cid:4) (cid:3) (cid:5) (cid:6) (cid:7) (cid:5) (cid:8) (cid:2)(cid:3) (cid:9) (cid:10) (cid:7) (cid:5) (cid:11) (cid:12) (cid:1)(cid:2)(cid:3)(cid:4)(cid:4)(cid:5)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:5)(cid:7)(cid:10)(cid:9)(cid:8)(cid:11)(cid:3)(cid:4)(cid:5)(cid:1)(cid:12) (cid:8)(cid:9)(cid:7)(cid:10)(cid:11)(cid:9)(cid:7)(cid:10)(cid:12)(cid:13)(cid:9)(cid:3)(cid:14)(cid:15)(cid:13)(cid:9)(cid:3)(cid:14) FIG. 1. (Color online) Concentration-depth profiles measuredby XPS depth profiling. The sputter rate calibrated in
SiO at the used setting of the ion gun was 4.4 nm/minute. A smalloxygen contamination is observed at the surface (left) on fewnm, and the Ti and N concentration remain constant in thefilm up to the Si substrate (right). The measured stoichiome-try is 1:1.3, with however an important error due to the 20%uncertainty on the absolute concentration values with thistechnique. onator end alongside it. The feedline is connected tocoaxial cables at both chip sides and its S transmissionis measured using a standard Vector Network Analyzer.A detailed description of the setup can be found in [19].Each resonator gives rise to a dip in S at a fre-quency f − = 2 πxl p ( L k + L g ) C , with l the resonatorlength, L k and L g the kinetic and geometric inductanceper unit length and C the capacitance per unit length. x is a factor depending on the resonator type. Here,half of the resonators are halfwaves (open ended on bothsides) corresponding to x = 2, the others are quarter-waves (short ended on one side) thus x = 4. The in-ductance per unit length is proportional to the surfaceinductance: L s = L k /g with g a factor depending onthe resonator geometry [3]. When the temperature is in-creased, Cooper pairs are broken by thermal excitationsabove the gap, resulting in a change of the kinetic in-ductance. This translate into a resonance frequency shift(See fig.2) and δf ( T ) f = − α δL k ( T ) L k = − α δL s ( T ) L s (1)With α ≡ L k / ( L k + L g ) the kinetic inductance ratio. Forhigh L k ∼ L g films like the TiN film studied here, α islarge enough to be determined precisely from the ratio ofthe experimental resonance frequency at the lowest tem-perature f and the geometrical frequency f g calculatedfrom the CPW dimensions [20].To choose the correct model for L s ( T ), we first esti-mate several characteristic lengths . The London mag-netic penetration depth λ L ≈ nm is determined fromthe measured n and using a quasiparticle mass m ∗ equalto 3 times the electron mass[21]. The BCS coherence (cid:23)(cid:17)(cid:22) (cid:23)(cid:17)(cid:23) (cid:23)(cid:17)(cid:24) (cid:23)(cid:17)(cid:25)(cid:19)(cid:24)(cid:19)(cid:20)(cid:19)(cid:19)(cid:20)(cid:24)(cid:19)(cid:21)(cid:19)(cid:19) (cid:55)(cid:3)(cid:11)(cid:46)(cid:12) (cid:53) (cid:3) (cid:11) Ω (cid:12) FIG. 2. Typical S transmission of one resonator for severaltemperatures between 90 mK and 1 K . When increasing T,the frequency and the quality factor decrease due to quasipar-ticle excitations above the gap. Upper inset: resistive super-conducting transition. Lower inset: photo of one resonatorfrom another chip having the same design. The meanderedresonator line (light grey) is separated to the groundplane allaround (light grey) by slits (dark grey) and is capacitivelycoupled to the feedline (lower straight line). length ξ = ¯ hv F π ∆ ≈ nm is estimated a priori using agap ∆ = 1 . k B T C and a Fermi velocity v F = ¯ h √ π n m ∗ .The mean free path ℓ = τ v F = m ∗ n e ̺ v F ≈ nm iscalculated from the measured ̺ and n . We are nowin position to estimate the effective coherence length ξ eff = (cid:0) ξ − + ℓ − (cid:1) − ≈ . nm and the magnetic pen-etration depth λ eff = λ L (cid:16) ξ ℓ (cid:17) / ≈ nm . Thefilm studied here is in the local ( ξ eff ≪ λ eff ), dirty( ℓ ≪ ξ , λ L ), three dimensional ( ξ eff ≪ d ) and thin filmlimit ( d ≪ λ eff ). In thin films, the current is distributeduniformly and the relation between the square inductanceand λ eff simply reduces to: L s = µ λ eff d (2)In the local dirty limit and at T < T c /
3, the temperaturedependence of λ eff is given by [22]: λ eff ( T ) λ eff (0) = tanh (cid:18) ∆ k B T (cid:19) − / (3)With ∆ the superconducting gap at zero temperature.This is valid in the low frequency limit hf ≪ .It is verified here, since the frequency range used is3 − . GHz corresponding to 12 − µeV which is lessthan 2% of the gap energy. Combining eq.1, 2 and 3 oneobtains: δL s L s = − α δff = 2 tanh (cid:18) ∆ k B T (cid:19) − / − ! (4)Where f , L s and λ eff have been replaced by their valueat the lowest experimental temperature f , L s and λ eff which is correct for δL ( s ) ≪ L s and δλ eff ( T ) ≪ λ eff .This is valid for all temperatures below T ∼ . T c , tem-perature above which the magnetic penetration depth di-verges. As shown by eq. 4, the temperature dependenceof the resonance frequency shift is a direct probe of thesuperconducting gap.The S transmission of 8 resonators of the same chiphas been measured between 90 mK and 1 K . At low tem-perature, the internal quality factors are between 10 and10 . These high values support the conclusions of ref.[23] on the relation between high quality factors and a(200) crystalline orientation in TiN films. The ( f /f g )ratio calculated from the measured f is the same for allresonators, giving α = 1 − ( f /f g ) = 0 . δL s /L s ( T )is shown in fig.3 for all resonators. They exhibit the sametemperature dependence and an excellent fit is obtainedwith eq.4, giving ∆ = 0 . meV . There is however aslightly weaker curvature of the measured δL s /L s ( T )below 0 . K , which is magnified in log scale in the in-set. The curves are between the values expected for∆ = 0 . meV and ∆ = 0 . meV , which may be at-tributed to a small gap decrease close to the film sur-faces due to oxidization or stoichiometry changes, seefig.1. The inflexion point around T ∼ . K , followedbelow by a cut-off of the inductance shift which is notidentical for all resonators/frequencies, is due to the ca-pacitive dielectric variations [24] which becomes non neg-ligible compared to the inductive changes in this temper-ature range. Comparing ∆ and the critical temperature T C = 4 . K , one obtains a ratio ∆ /k B T C = 1 .
50, lowerthan the BCS weak coupling ratio 1.76. This has nothingto do with the SIT : in contrast, as observed by Sacepeet al. in TiN ultra thin films [12], one expect disorder topossibly increase ∆ /k B T C close to the transition. The1.50 ratio may be related to an effect of the grain size: asreported by Bose et al. [25] in niobium thick films, thecritical temperature decreases when decreasing the grainsize and the ∆ /k B T C ratio is slightly reduced, going to1.61 for 18 nm grains. A superconductivity weakeningcan also occur due to interface tunnel exchange at inter-nal and external surfaces[26] as observed in niobium thinfilms[27]. This may also explain why all T C reported forTiN films are below 4.8 K whereas T C = 6 . K in a TiNsingle crystal [28].To go further, we have also compared the temperaturedependence of the frequency and the quality factor Q .Fig. 4 shows the temperature dependence of the normal-ized quality factors δ (1 /Q ) ( T ) = (1 /Q ) ( T ) − (1 /Q ) ( T =0) for all resonators as a function of the frequency shift δ L s / L s T [K] -7 -6 -5 -4 -3 δ L s / L s -1 ] FIG. 3. (Color online) Temperature dependence of the nor-malized surface inductance for the 8 resonators, having f =3 . GHz ( p ), 3 . GHz ( u ), 3 . GHz ( q ), 3 . GHz ( s ),3 . GHz ( v ), 4 . GHz ( r ), 5 . GHz ( t ), 5 . GHz ( u ).The black line is the best fit with eq. 4 giving the gap value∆ = 0 . meV . Inset: same curves in semi-log and inversetemperature scales to magnify the results at the lowest tem-peratures. The dotted line is eq. 4 with ∆ = 0 . meV . − df ( T ). The 8 resonators gives similar results, and ex-hibit a linear behaviour. In the following, we show howthis proportionality between quality factor and frequencyshift can be reproduced analytically in the context of thetwo fluid model. In general, the radio frequency absorp-tion cannot be described by local electrodynamics evenat low temperature and low frequency, due to the nontrivial form of the momentum transition matrix M [29].This holds here however, since we are only interested bythe temperature dependence δ (1 /Q )( T ) and because Mis temperature independent at T < T c / Q is related to the surface impedance Z S = R S + iωL S by Q = ωL total /R total = ωL S /αR S , with R S the surfaceresistance. In the thin film limit, Z S is simply relatedto the complex conductivity σ = σ + iσ by Z − s = σd .At low temperature, σ ≪ σ and we get R S = σ σ d and L S = ωσ d . The complex conductivity in the two fluidmodel is given by [22]: σ = n n e τm ∗ − iωτ + i n s e m ∗ ω (5)With n n the quasiparticle density and τ the momentumrelaxation time. In the limit ωτ ≪ Q = α σ σ = αωτ n n n S (6)The first part of eq.6 is identical to the equation usedby Gao et al. [30] in the context of the Mattis Bardeen δ ( / Q ) - δ f [MHz] FIG. 4. (Color online) Inverse of the quality factor δ (1 /Q ) ( T )as a function of the frequency shift δf ( T ) for the 8 resonators(same symbols as fig.3) between 90 mK and 1 K . The blackline is a linear fit with eq.9 with τ = 1 . ps . theory. Using the properties n n = δn n , δn n /δn s = − Q ( T ) = − αωτ δn S ( T ) n S (7)The quality factor temperature dependence is directly re-lated to the superfluid density changes, as for the kineticinductance: L S ( T ) = m n S ( T ) e d and δn S n S = − δL s L s = 2 α δff (8)By combining eq.7 and eq.8 we obtain the relation be-tween Q and δf : 1 Q = − πτ δf (9)As shown by eq.9, the inverse quality factor is pro-portional to the frequency shift and momentum relax-ation time. This means that losses, which are expectedto be zero at zero temperature, increase proportionallyto the quasiparticle density changes δf ∝ δn n , via thetwo fluid property δn n ∝ δn s . In practice, there arealways residual losses 1 /Q ( T = 0) due to non equilib-rium excess quasiparticles [31, 32], which are subtractedin fig.4. Fitting the results with eq.9 gives the momen-tum relaxation time at T ≪ T c , τ = 1 . − s . Thisis three orders of magnitude longer than in the metallicstate τ m = m ∗ /n e ρ = 3 . − s estimated from theresistivity and the carrier density measured at 10 K . Inthe superconducting state indeed, the momentum relax-ation time strongly increases, typically up to ∼ − s inusual BCS superconductors [22] due to the quasiparticlesvanishing by Cooper pair condensation. The temperature dependence of the resonators fre-quencies and quality factors clearly evidence the pres-ence of one well defined superconducting gap in the den-sity of states. Moreover, the excellent reproducibility be-tween the resonators evidence the good superconduct-ing phase homogeneity in the film. This differs fromprevious STM/S results on a similar 100 nm thick high T C = 4 . K TiN film, which report on the presence ofan inhomogeneous superconducting phase having localnormal areas. As discussed in ref. [16], the detected in-homogeneous gap may be due to mesoscopic fluctuationsat the proximity of a superconductor to metal transition.Indeed, the presence of a composition driven transitionin TiN x has been reported recently[1], with a disappear-ance of the superconducting phase in the low x range,recovered at x = 0 (titanium). However, the film usedin [16] has the characteristic high critical temperatureof overstoichiometric TiN, similar to the one of the filmmeasured here, and our results doesn’t exhibit any sig-nature of the proximity with the composition transition.Additionally, the granular morphology of the two filmsare different. Indeed, the film recipe used here has beenespecially developed to obtain densely packed grains inthe film, corresponding to zone T in the Thornton classifi-cation. As discussed in ref. [17] , a typical low stress TiNfilm is of zone 1, containing many voids between grains.This may lead to important Josephson barriers betweensuperconducting grains, which are expected to play a ma-jor role in the homogeneity of the superconducting phaseof such systems even at relatively low disorder [33].In conclusion, the low frequency complex surfaceimpedance has been studied on microresonators madefrom a thick overstoichiometric TiN film. The resonatorshigh quality factors allow a high sensitivity determina-tion of the inductive and resistive changes with temper-ature. In contrast to the previous spectroscopic studyin a similar film, all our results are in agreement withan homogeneous superconducting phase having a gap∆ = 0 . meV = 1 . k B T C .We thank T. M. Klapwijk, A. Endo, E. F. C. Driessen,P. C. J. J. Coumou, R. R. Tromp and S. J. C. Yates forsupport and fruitful discussions. ∗ [email protected][1] H.G. Leduc, B. Bumble, P. K. Day, B. H. Eom, J. Gao,Sunil Golwala, B. A. Mazin, S. McHugh, A. Merrill, D.C. Moore et al. , Appl. Phys. Lett. , 102509 (2010)[2] Peter K Day, H. G. LeDuc, B. A. Mazin, A. Vayonakisand J. Zmuidzinas, Nature 425, 817 (2003)[3] E. F. C. Driessen, P. C. J. J. Coumou, R. R. Tromp, P.J. de Visser and T. M. Klapwijk, Phys. Rev. Lett. 109,107003 (2012)[4] P.W. Anderson, J. Phys. Chem Solids , 26, (1959)[5] D. Shahar and Z. Ovadyahu, Phys. Rev. B , 10917 (1992)[6] C.A. Marrache − Kikuchi, H. Aubin, A. Pourret, K.Behnia, J. Lesueur, L. Berge and L. Dumoulin, Phys.Rev. B , 144520 (2008)[7] S. Okuma, T. Terashima and N. Kokubo, Solid StateCommun. , 529-533 (1998)[8] E. Bielejec and Wenhao Wu, Phys. Rev. Lett. , 206802(2002)[9] T. I. Baturina, D. R. Islamov, J. Bentner, C. Strunk,M. R. Baklanov, and A. Satta, JETP Lett. , 337-341(2004)[10] N. Hadacek, M. Sanquer, and J.-C. Villegier , Phys. Rev.B , 024505 (2004)[11] T.I. Baturina, C. Strunk, M. R. Baklanov and A. Satta,Phys. Rev. Lett. , 127003 (2007)[12] B. Sacepe, C. Chapelier, T. I. Baturina, V. M. Vinokur,M. R. Baklanov, and M. Sanquer, Phys. Rev. Lett. ,157006 (2008)[13] T.I. Baturina, J. Bentner, C. Strunk, M. Baklanov andA. Satta, Phys. B Cond. Matt. , 500-502 (2005)[14] T. I. Baturina1, A. Yu. Mironov, V. M. Vinokur, M. R.Baklanov and C. Strunk, Phys. Rev. Lett. , 257003(2007)[15] V. M. Vinokur, T. I. Baturina, M. V. Fistul, A. Y.Mironov, M. R. Baklanov and C. Strunk, Nature ,613 (2008)[16] W. Escoffier, C. Chapelier, N. Hadacek, and J.-C. Vil-legier, Phys. Rev. Lett. , 217005 (2004)[17] J.F. Creemer, D. Briand, H.W. Zandbergen, W. van derVlist, C.R. de Boer, N.F. de Rooij and P.M. Sarro, Sen-sors and Actuators A , 416-421 (2008)[18] N. D. Ashcroft and N. D. Mermin, Solid State Physics (Brooks & Cole, 1976).[19] J.J.A. Baselmans, S. Yates, P. Diener and P. de Visser,J. Low Temp. Phys. , 360-366 (2012)[20] J. Gao, J. Zmuidzinas, B.A. Mazin, P.K. Day and H.G.Leduc,Nucl. Instrum. Methods , 585-587 (2006)[21] C. Walker, J. Matthew, C. Anderson and N. Brown, Sur-face Science , 405-414 (1998)[22] M. Tinkham,
Introduction to Superconductivity , (NewYork, McGraw-Hill, 1996)[23] M. R. Vissers, J. Gao, D. S. Wisbey, D. A. Hite, C. C.Tsuei, A. D. Corcoles, M. Steffen and D. P. Pappas, Appl.Phys. Lett. , 232509 (2010)[24] R. Barends, H. L. Hortensius, T. Zijlstra, J. J. A. Basel-mans, S. J. C. Yates, J. R. Gao, and T. M. Klapwijk,Appl. Phys. Lett. , 223502 (2008)[25] S. Bose, P. Raychaudhuri, R. Banerjee, P. Vasa, and P.Ayyub, Phys. Rev. Lett. , 147003 (2005)[26] J. Halbritter, Phys. Rev. B , 14861 (1992)[27] J. Halbritter, Solid State Commun. , 675-678 (1980)[28] W. Splengler, R. Kaiser, A. N. Christensen and G.Muller-Vogt, Phys. Rev. B , 1095-1101 (1978)[29] J.Halbritter, Z. Phys. textbf266,209-217 (1974)[30] J. Gao, J. Zmuidzinas, A. Vayonakis, P. Day, B. Mazinand H. Leduc, J. Low Temp. Phys. , 557-563 (2008)[31] D. E. Oates, A. C. Anderson, C. C. Chin, J. S. Derov,G. Dresselhaus and M. S. Dresselhaus, Phys. Rev. B ,437655 (1991)[32] P. J. de Visser, J. J. A. Baselmans, P. Diener, S. J. C.Yates, A. Endo and T. M. Klapwijk, Phys. Rev. Lett.textbf106, 167004 (2011)[33] B. Spivak, P. Oreto and S. A. Kivelson, Phys. Rev. B77