Quantitative assessment of pinning forces and the superconducting gap in NbN thin films from complementary magnetic force microscopy and transport measurements
T. Shapoval, H. Stopfel, S. Haindl, J. Engelmann, D. S. Inosov, B. Holzapfel, V. Neu, L. Schultz
QQuantitative assessment of pinning forces and the superconducting gap in NbN thin filmsfrom complementary magnetic force microscopy and transport measurements
T. Shapoval, ∗ H. Stopfel, S. Haindl, J. Engelmann, D. S. Inosov, B. Holzapfel, V. Neu, and L. Schultz IFW Dresden, Institute for Metallic Materials, P. O. Box 270116, D-01171 Dresden, Germany. Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, D-70569 Stuttgart, Germany
Epitaxial niobium-nitride thin films with a critical temperature of T c =
16 K and a thickness of 100 nmwere fabricated on MgO (100) substrates by pulsed laser deposition. Low-temperature magnetic forcemicroscopy (MFM) images of the supercurrent vortices were measured after field cooling in a magnetic fieldof 3 mT at various temperatures. Temperature dependence of the penetration depth has been evaluatedby a two-dimensional fitting of the vortex profiles in the monopole-monopole model. Its subsequent fitto a single s -wave gap function results in the superconducting gap amplitude ∆( ) = ( ± ) meV = ( ± ) k B T c , in perfect agreement with previous reports. The pinning force has been independentlyestimated from local depinning of individual vortices by lateral forces exerted by the MFM tip and fromtransport measurements. A good quantitative agreement between the two techniques shows that for lowfields, B (cid:28) µ H c2 , MFM is a powerful and reliable technique to probe the local variations of the pinninglandscape. We also demonstrate that the monopole model can be successfully applied even for thin filmswith a thickness comparable to the penetration depth. PACS numbers: 74.70.-b 74.78.-w 74.25.Wx 68.37.Rt
I. INTRODUCTION
Vortex pinning is an important characteristic of type II su-perconductors that allows tuning its properties on purposewithout changing the chemical composition. Hence, the in-terpretation of pinning mechanisms, the search for artificialdefects with high pinning potentials, and commensurablepinning effects by ordered arrays of defects remain in thefocus of basic research and application-based engineering.On the one hand, many high-power applications requirematerials with high pinning, i.e. high critical current den-sity. On the other hand, logical applications (i.e. fluxonicsdevices) benefit from low-pinning materials, which can belocally modified by introducing strong pinning sites in orderto tune the dynamics of vortices and rectify their motion. A common way to investigate the pinning strength is tomeasure the sample’s response to an applied magnetic fieldor current, using magnetometry or transport measurements,respectively. These global methods probe the average valueof the pinning force in a material including the collectivedynamics of the elastic vortex lattice. However, the localmodulations of the pinning landscape originating from dif-ferent natural or artificial defects remain inaccessible tothese techniques. This challenge can be addressed by lo-cal imaging methods, such as low-temperature magneticforce microscopy (LT-MFM), which is capable of correlatingthe superconducting (SC) vortex positions with the distri-bution of micro- or nanostructural defects. It effectivelycombines the non-invasive imaging of flux lines with theability to manipulate individual vortices by the stray field ofthe magnetic tip, offering a direct access to the local pinningforce. Nevertheless, reconciling the results of local andglobal measurements often represents a challenge.In this paper, we will estimate the pinning forces in nio-bium nitride (NbN) thin films using two complementarymethods. Local measurements by LT-MFM will be directlycompared to transport measurements. Thin films of NbNhave been chosen due to their high critical temperature, T c ≈
16 – 17 K, that makes them suitable for LT-MFM studiesin a wide temperature range. Moreover, this conventionalsuperconductor has attracted attention due to its recentapplications in sensitive SC bolometers. We will also evaluate the temperature dependence of themagnetic penetration depth, λ ( T ) , by performing a two-dimensional fit of the vortex profiles within the monopole-monopole model. Subsequently, we will use these values toestimate the superconducting energy gap of the NbN film,which we will compare to the direct tunneling measure-ments from the published literature. II. EXPERIMENTAL DETAILS
Epitaxial NbN thin films with a thickness d =
100 nmwere fabricated on single-crystalline MgO (100) substratesby pulsed laser deposition (PLD) using a Nb (99.95 %) tar-get in N atmosphere with a pressure of 5 · − mbar. Thebase pressure in the chamber was 10 − mbar. Prior to thefilm preparation, the Nb deposition rate was measured with (cid:2)(cid:1)(cid:2) (cid:2)(cid:1)(cid:4) (cid:2)(cid:1)(cid:5) (cid:2)(cid:1)(cid:7) (cid:2)(cid:1)(cid:9) (cid:3)(cid:1)(cid:2)(cid:2)(cid:4)(cid:5)(cid:7)(cid:9)(cid:3)(cid:2)(cid:3)(cid:4)(cid:3)(cid:5)(cid:3)(cid:7)(cid:3)(cid:6) (cid:3)(cid:7) (cid:3)(cid:8)(cid:2)(cid:4)(cid:5)(cid:7)(cid:9)(cid:3)(cid:2) (cid:3) (cid:4) (cid:2) (cid:9) (cid:5) (cid:1) (cid:2) (cid:7) (cid:3) (cid:5)(cid:1)(cid:3)(cid:1)(cid:4)(cid:2)(cid:4) (cid:12) (cid:1) D (cid:4) (cid:12) (cid:8) (cid:1) (cid:5)(cid:4)(cid:6)(cid:7) (cid:1) (cid:9) (cid:2)(cid:11)(cid:3)(cid:2)(cid:10)(cid:3) (cid:1) (cid:6) (cid:10) (cid:13) (cid:11) (cid:13) (cid:14) (cid:8)(cid:12) (cid:9) (cid:10) (cid:1) (cid:2) W (cid:3) (cid:4) (cid:1)(cid:2)(cid:9)(cid:3) Fig. 1. (a) Zero-field resistive SC transition with a full width of ∆ T c ≈ µ H c , as a function of thereduced temperature, t = T / T c . Points are the experimental valuesdetermined from transport measurements, whereas the solid linesare empirical fits. a r X i v : . [ c ond - m a t . s up r- c on ] J a n n Inficon XTM / rate monitor. We used a KrF excimer laser( Lambda-Physik ) with a wavelength of 248 nm and a pulseduration of 25 ns. The substrates were heated up to 500 ◦ Cduring deposition. As the on-axis PLD process leads to theformation of droplets on the surface, which pose a severeproblem to the MFM scanning tip, a polishing technique was applied to remove these obstacles, providing a peak-to-valley roughness below 5 nm. X-ray diffraction patternsshowed (00 l ) peaks (simple cubic structure) similar to previ-ous reports. The best samples exhibit a T c (offset) at 16 Kand a sharp resistive SC transition with a width ∆ T c ≈ [ Fig. 1 (a) ] . The temperature dependence of the second crit-ical field, µ H c2 , determined from transport measurements,is shown in panel (b) of the same figure.The LT-MFM measurements have been performed usinga commercial scanning-probe microscope ( Omicron Cryo-genic SFM ). We have used an MFM cantilever (
NanoworldMFMR ) that possesses a force constant k ≈ / m and aresonance frequency f ≈
80 kHz. For the transport mea-surements, a 100 µ m-wide bridge was structured by opticallithography and ion-beam etching. Transport measurementswere performed in the standard four-point configurationusing a 9 T physical property measurement system (PPMS)by Quantum Design . III. MONOPOLE-MONOPOLE MODEL
The magnetic moment of the MFM tip (see sketch inFig. 2) can be well approximated to the first order by amagnetic monopole characterized by a “magnetic charge”˜ m located at a distance δ from the sharp end of the tippyramid. The field distribution from a single flux quantum, φ = × − T m , measured just above the surfaceof the superconductor, is also similar to the magnetic fieldemanated by a magnetic monopole of 2 φ , located at thedepth λ eff = λ below the surface, with λ being themagnetic penetration depth. Hence, the magnetic inductionof the vortex, B ( r , z ) , taken at a distance z above the surface,can be approximated by B ( r , z ) = φ π ( r − r ) + ( z + λ ) e z (cid:0) ( r − r ) + ( z + λ ) (cid:1) / , (1)where e z is the unit vector orthogonal to the film, r is theposition of the vortex core and r is the radial distance fromits center. This leads us to the tip-vortex interaction force F ( r , z ) in the monopole-monopole model, F ( r , z ) = ˜ m B ( r , z ) . (2)Taking into account that the shift of the resonance frequencyof the cantilever measured by MFM, ∆ f , is proportional tothe z -derivative of the normal component of the force thatacts between the tip and the sample, ∂ z F z ( r , z ) , oneobtains the following expression for the measured signal: ∆ f = − f k ˜ m φ π ( r − r ) − ( z + λ + δ ) (( r − r ) + ( z + λ + δ ) ) / . (3)As the magnetic induction of the vortex is maximal at thecenter and decays rapidly with r , the strongest interaction Fig. 2 (color online). A schematic of the MFM imaging procedureand an illustration of the monopole-monopole model. The magne-tized MFM tip, driven by a piezo element, scans above the surfaceof the sample at a given distance z . During measurement, thefeedback loop, which is typically used to stabilize the resonancefrequency of the tip during topographic imaging, is deactivated.One measures a shift of the resonance frequency, ∆ f , inducedby the magnetic field of the vortices, B ( r , z ) . In the monopole-monopole model, described in the text, both the tip and the vortexare approximated by magnetic monopoles at distances z + δ and λ eff from the surface of the sample, respectively. in z direction between the tip and the vortex occurs whenthe tip passes the center of the vortex. Thus, the maximal z -component of this force is reached at r = r :max ( F z ) r = r = ˜ m φ π ( z + λ + δ ) (4)The benefit of the monopole-monopole model is that allspatial parameters of the problem ( z , λ eff and δ ) enter Eqs. 3and 4 additively, hence the sum in the denominator can beredefined as an effective tip-sample distance w = z + λ eff + δ . It represents the distance between imaginary magneticmonopoles within the tip and the vortex, as illustrated bywhite dots in Fig. 2.The ratio between the lateral and vertical forces that actduring scanning typically varies from 0.3 for a tip with aless sharp pyramid to 2 / (cid:112) ≈ This results in the following maximumlateral component of the tip-vortex interaction force:max ( F lat ) ≈ · max ( F z ) . (5)2 µm (a) T = 8.4 K = 50% T c (b) T = 10.0 K = 62% T c (c) T = 11.8 K = 75% T c (d) T = 13.7 K = 87% T c (cid:68) f = . H z r epu l s i v e Fig. 3 (color online). Vortex images measured at a distance z =
30 nm from the surface of a NbN film after field-cooling in − T c , (b) 62% T c , (c) 75% T c , and (d)87% T c . The slow scanning direction is top to bottom. Depinningof vortices by the magnetic tip can be seen in panels (c) and (d). Obviously, non-invasive imaging of vortices by MFM is pos-sible only as long as the vortices are pinned. The tip-vortexinteraction force can be accurately tuned during scanning byvarying the tip-sample separation z . If this force exceedsthe pinning force of an individual vortex at a natural orartificial defect, the vortex can be dragged away from itsinitial position. Likewise, if the tip-sample distance is keptconstant, an increase in temperature can lead to the localdepinning of individual flux lines during scanning due tothe temperature dependence of the pinning force.
IV. LOCAL DEPINNING OF INDIVIDUAL FLUX LINES
We have imaged the temperature dependence of the vor-tex distribution after field-cooling the sample to ∼ µ H z = − z =
30 nm,so that Eq. (4) can be applied. The negative direction ofthe field corresponds to the magnetic repulsion between theMFM tip and the vortex, and therefore the vortices appear asred (dark-grey) objects in the false-color images presentedin Fig. 3. In all panels, the white circles depict the positionsof the vortex cores at base temperature, to emphasize thetip-induced changes in their position as the temperatureis increased. These positions were determined by a two-
Fig. 4 (color online). Temperature dependence of the averagedvortex profile. Blue points are the measured signal, solid lines areleast-squares fits to the monopole-monopole model. dimensional (2D) fitting procedure applied to the regionsindicated by rectangles. The benefit of this method is that,in principle, it allows for a subpixel resolution of the fitting,given that the noise in the measured data is sufficiently low.At temperatures not exceeding ∼ T c [ Fig. 3 (a) ] , non-invasive imaging of vortices takes place, indicating that thepinning exceeds the lateral thrust of the MFM tip. Statisticalimage analysis, such as described by Inosov et al. , revealsthat the vortices form a highly disordered hexagonal latticedue to the pinning by natural defects.At higher temperatures, the decreasing contrast of thevortex profile signifies a natural increase in the penetrationdepth, λ , that characterizes the decay of the magnetic fieldoutside of the vortex core. At 62% T c [ Fig. 3 (b) ] , mostvortices are still in their original positions, implying thatthe tip-vortex interaction force is still lower than the typicalpinning force of a single vortex. Only 2 out of 14 vortices,visualized in the figure, have been irreversibly dragged awayfrom their initial positions (white circles) to the nearestpinning sites with higher pinning potentials. This indicatesthe existence of a slightly modulated pinning landscape inthe NbN film and, hence, a spatial variation of the pinningforce, as expected for natural defects.At 75% T c [ Fig. 3 (c) ] , the movement of nearly every vor-tex by the MFM tip can be seen. Indeed, because mostvortices are irreversibly dragged away by the tip as it passesclose to the core, such vortices appear half-cut in the im-age. Consequently, at this temperature the pinning force forthe majority of the vortices is equal to the maximal lateralforce exerted by the MFM tip onto the vortex, Eq. (5). Onlyone vortex at the bottom-right part of the image remainsTemperature (K): 8.4 9.1 10.0 11.8 w (nm) 305(11) 308(10) 316(10) 383(23) ∆ f ( r = r ) (Hz) 0.182(4) 0.134(3) 0.138(6)max ( F lat ) (pN) 0.74(3) 0.56(2) 0.70(5) Table I. Temperature dependence of the effective tip-sample dis-tance in the monopole-monopole model, w = z + λ + δ ; thefitted peak amplitude from Fig. 4, ∆ f ( r = r ) ; and the correspond-ing lateral tip-vortex interaction, max ( F lat ) . (cid:4) (cid:6) (cid:7) (cid:8) (cid:3)(cid:2) (cid:3)(cid:4) (cid:3)(cid:6) (cid:3)(cid:7) (cid:2)(cid:3)(cid:2)(cid:2)(cid:4)(cid:2)(cid:2)(cid:5)(cid:2)(cid:2)(cid:6)(cid:2)(cid:2) (cid:3) (cid:3)(cid:2) (cid:3)(cid:2)(cid:2)(cid:2)(cid:4)(cid:2)(cid:2)(cid:6)(cid:2)(cid:2)(cid:7)(cid:2)(cid:2) (cid:3)(cid:1) (cid:14) (cid:1) (cid:8)(cid:7)(cid:7)(cid:1)(cid:31)(cid:30)(cid:16)(cid:22)(cid:16)(cid:1)$(cid:26)(cid:27)(cid:31)(cid:1)(cid:25)(cid:27)(cid:29)(cid:30) l (cid:2) (cid:2) (cid:1) (cid:3) (cid:1) (cid:2) (cid:12) (cid:11) (cid:3) l (cid:2) (cid:2) (cid:1) (cid:3) (cid:1) (cid:4) (cid:1) d (cid:6) (cid:8) (cid:5) (cid:9)(cid:10) (cid:1) (cid:2) (cid:12) (cid:11) (cid:3) (cid:17)(cid:24)(cid:30)!(cid:24)"(cid:21)$%"(cid:24)(cid:1)(cid:2)(cid:15)(cid:3) D (cid:2)(cid:7)(cid:3) (cid:14)(cid:1)(cid:9)(cid:5)(cid:13)’(cid:7)(cid:5)(cid:10)(cid:1)(cid:30)(cid:24)(cid:18) (cid:2)(cid:1)(cid:4) (cid:2)(cid:1)(cid:6) (cid:2)(cid:1)(cid:7) (cid:2)(cid:1)(cid:8)(cid:2)(cid:3)(cid:2)(cid:2)(cid:4)(cid:2)(cid:2)(cid:5)(cid:2)(cid:2)(cid:6)(cid:2)(cid:2) $(cid:26)(cid:27) (cid:19)(cid:9)(cid:11)(cid:4)(cid:9)(cid:12)(cid:20) (cid:23) (cid:2) (cid:1) (cid:6) (cid:1) (cid:2) (cid:19)(cid:9)(cid:12)(cid:20)(cid:19)(cid:9)(cid:11)(cid:20) %" (cid:1) (cid:1) l (cid:2) (cid:7) (cid:3) (cid:1) (cid:2) (cid:12) (cid:11) (cid:3) (cid:3) (cid:1) (cid:2)(cid:31)(cid:30)(cid:3) Fig. 5 (color online). Temperature dependence of λ ( T ) + δ/ λ ( ) (left scale).The solid line is a fit to an empirical model with a single s -wavegap. The value of λ ( ) for our film thickness was obtained by aninterpolation of the d -dependent literature data, as shown inthe inset. The solid grey line is an empirical fit. A good agreementof our results with the published value indicates that the δ/ stable, evidencing the locally enhanced pinning force atthis position. On the other hand, three other vortices arefully dragged away as soon as the tip starts crossing theirfield lines. At even higher temperatures [ Fig. 3 (d) ] , vor-tices can no longer be imaged. Here the vortices are beingcontinuously dragged by the tip during scanning.For a quantitative analysis of the vortex profiles, their corepositions r were first determined using 2D fitting. To gathersufficient statistics for the application of the monopole-monopole model, the signal ∆ f from within the neighbor-hood of every vortex (white rectangles in Fig. 3) has beenplotted vs. | r − r | , as shown in Fig. 4. In this figure, everydata point corresponds to a pixel in the original MFM im-age, whereas data points originating from different vorticesare combined in one plot. The resulting clusters of pointscan be fitted to Eq. (3) (solid lines) to obtain the averagevalue of w = z + λ + δ for every temperature with asufficiently small statistical error. The results of these fitsare summarized in Table I. V. LOCAL PINNING FORCES AND THE PENETRATION DEPTH
Now we can proceed to the quantitative estimation of thepinning force. Combining Eq. (3) through (5), we obtainmax ( F lat ) ≈ kwf · ∆ f ( r = r ) . (6)Substituting the fitting results from Fig. 4 and the knownparameters of the tip into this expression, one can calculatethe temperature-dependent lateral tip-vortex forces that arelisted in Table I, with an average value of 0.67 ± As we already know that the local pinning force decreasessufficiently to allow depinning of most vortices at T ≈
12 K = T c , we can use the calculated value as an esti-mate of the mean local pinning force at this temperature, F p ( T =
12 K ) ≈ ( ± ) pN. (7)The results of the same fit also provide a local probefor the temperature-dependent penetration depth, in our case given by λ ( T ) ≈ [ w ( T ) − z − δ ] / z (cid:28) λ is fixedduring measurement and known, so it can be easily sub-tracted (Fig. 5). The second parameter, δ , being a prop-erty of the tip and dependent both on temperature and thestray field of the sample, usually has a large uncertaintyand requires a special calibration of the tip in order to bedetermined. It may lead to a non-negligible constant offsetof the measured penetration depth from its true value. Toquantify the δ/ λ ( ) ≈
205 nm (dark-gray diamond in Fig. 5). The latter hasbeen obtained by interpolating the directly measured λ ( ) values from the literature for different film thicknesses, as shown in the inset of Fig. 5, and taking the intermediatevalue at d =
100 nm. Good agreement between this refer-ence value of λ ( ) and our λ ( T ) + δ/ δ/ et al. , which gives an analytical relationshipbetween λ ( T ) and the SC gap ∆( ) . For a conventional su-perconductor with a single isotropic s -wave gap, it becomes (cid:3) = 1 4 K (cid:3) = 1 2 K (cid:3) = 1 0 K (cid:3) = 8 Kc r i t e r i o n : 1 m V / c m JC (Acm-2) (cid:4) (cid:1) (cid:2) ( T ) Fig. 6 (color online). Dependence of the critical current density, J c , on applied magnetic field for different temperatures: 8 K, 10 K,12 K and 14 K. emperature (K): 8 10 12 14 J c (10 A / cm ) 10 5 3 2 F gp / N (pN) 2.07 1.04 0.62 0.42Table II. Critical current density and average pinning force pervortex at different temperatures evaluated from the transport datain Fig. 6 for B = λ ( T ) = λ ( ) (cid:150) − M (cid:130) ∆( T ) k B T (cid:140)(cid:153) − / , (8)where λ ( ) depends only on the band structure, whereasall the temperature-dependent quasiparticle effects are in-cluded in the approximant function M ( t ) = ( e t / + e − t / ) − (cid:112) π t / + / ( + π t / ) . (9)The temperature dependence of the SC gap in Eq. (8) isapproximated by ∆( T ) = ∆ tanh (cid:18) π (cid:112) T c / T − (cid:19) . (10)The resulting fit yields a value of ∆( ) = ( ± ) meV = ( ± ) k B T c , which perfectly agrees with direct tun-neling measurements and is slightly above the weak-coupling limit of 1.76 k B T c predicted by the Bardeen-Cooper-Schrieffer (BCS) theory. VI. GLOBAL ESTIMATE OF THE PINNING FORCES
While MFM provides access to the pinning force of indi-vidual vortices, the global characterization methods, suchas transport or magnetization measurements, explore thecollective behavior of the flux-line lattice. They evaluate themean pinning force within the whole sample volume, con-sidering also the elastic interaction between individual fluxlines as well as the collective pinning. For small magneticfields, B (cid:28) µ H c , the distance between vortices is largerthan λ . Such vortices can be treated as independent non-interacting objects. In this limit, collective effects can beignored and the pinning force per vortex can be calculatedas F gp / N , where F gp is the average pinning force and N is thenumber of vortices within the sample surface.The pinning force is equal to the maximal sustainableLorentz force that does not move vortices while the currentflows : F gp ( B ) = V J c B = SdJ c B , (11)where J c is the critical current density, S is the surface areaof the sample and d is the sample thickness. The number ofvortices is N = BS /φ , hence the pinning force per vortex is F gp / N = J c d φ . (12) Dependence of J c on the applied magnetic field for tem-peratures between 8 and 14 K is presented in Fig. 6. Theresulting temperature dependence of the pinning force pervortex F gp / N calculated from these J c ( H ) curves is givenin Table II. One can immediately appreciate the agree-ment between the value of F gp / N = T =
12 K with that of ( ± ) pN that we extracted earlier from the MFMdata at a similar temperature. Taking into account that thepinning force varies by nearly a factor of 5 in the studiedtemperature range, such an agreement within 8% betweenlocal and global measurements is indeed remarkable. VII. SUMMARY
To conclude, we found perfect agreement between thevalues of the pinning force per vortex, estimated from localdepinning of individual vortices by the MFM tip and globallyfrom the critical current measurements. We demonstratedthat for low fields, B (cid:28) µ H c2 , MFM is a powerful andreliable method to probe the local space variation of thepinning landscape. The monopole-monopole model, orig-inally derived for d > λ , proved to be successful evenfor thin films with a thickness comparable to the penetra-tion depth. With this knowledge, the quality of such verythin films, that are actually employed for the application inSC bolometers, can be perfectly analyzed using magneticforce microscopy.Finally, we used accurate 2D fitting of the vortex pro-files to extract the London penetration depth of the NbNfilm and the SC energy gap. The statistical errors of thismethod are small enough to ensure that the extracted gap ∆( ) = ( ± ) meV agrees with the directly measuredvalues. Although similar methods of extracting the gapamplitude from the muon-spin rotation ( µ SR), small-angleneutron scattering (SANS), microwave surface-impedance(MSI), or magnetization measurements of the temperature-dependent penetration depth are well developed, theirapplication to the analysis of temperature-dependent LT-MFM images is only becoming a standard practice. Acknowledgments
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