Evidence of fractional matching states in nanoperforated Nb thin film grown on porous silicon
aa r X i v : . [ c ond - m a t . s up r- c on ] D ec epl draft Evidence of fractional matching states in nanoperforated Nb thinfilm grown on porous silicon
M. Trezza , C. Cirillo , S. L. Prischepa and C.Attanasio , Laboratorio Regionale SuperMat, CNR-INFM Salerno and Dipartimento di Fisica “E. R. Caianiello”, Universit`adegli Studi di Salerno, Baronissi (Sa) I-84081, Italy State University of Informatics and RadioElectronics, P. Brovka street 6, Minsk 220013, Belarus NANO MATES, Research Centre for NANOMAterials and nanoTEchnology at Salerno University, Universit`a degliStudi di Salerno, Fisciano, (Sa) I-84084, Italy
PACS – Transport properties of superconductors
PACS – Vortex lattices, flux pinning, flux creep
PACS – Porous materials
Abstract. - Resistive transitions have been measured on a perforated Nb thin film with a latticeof holes with the period of the order of ten nanometers. Bumps in the dR / dH versus H curveshave been observed at the first matching field and its fractional values, 1/4, 1/9 and 1/16. Thiseffect has been related to different vortex lattice configurations made available by the underlyinglattice of holes. Introduction. –
The nucleation of Abrikosov vor-tices [1] in the mixed state of type-II superconductors withperiodic artificial pinning centers attracted a great atten-tion since 1970’s. Recent progress in the fabrication ofnanostructures provides the possibility to realize super-conducting thin films containing artificial defects as pin-ning sites with well-defined size, geometry and spatial ar-rangement [2, 3]. Vortex pinning was extensively exploredby many groups to develop a fundamental understandingof flux dynamics and for its relevance in applications whichrequire enhancements of the critical current density. Thus,several types of artificial pinning centers, such as square,rectangular or triangular arrays, have been introduced ina controlled way in the superconducting films. In particu-lar, the use of regular array of pinning centers such as an-tidots [2,4–6] or magnetic dots [3,7–9] brings to new com-mensurability effects, which give additional insight intothe pinning properties of vortices. The most notable phe-nomenon for these studies is the so-called matching ef-fect which occurs when the vortex lattice is commensuratewith the periodic pinning array. This situation occurs, inparticular, at fractional or integer values of the so-calledfirst matching field H = Φ /a , i.e., when the appliedfield H corresponds to one flux quantum, Φ = h/ e , perunit cell area, a , of the pinning array. Here a is the lat-tice constant of the pinning arrangement. As a result, at the matching field, the critical current density, J c , is dras-tically enhanced [3, 4, 10] and moreover, as a consequenceof the Little-Parks effect [11], the upper critical magneticfield is increased at the matching values. Recently anti-dot arrangements with a big variety of symmetries havebeen investigated. Matching effects have been reported inperforated Nb thin films for antidots lattices with shortrange order [12], or quasiperiodic fivefold Penrose struc-tures [13]. Moreover asymmetric pinning arrays have beensuggested as superconducting rectifiers [14].If the artificial structure of defects is created by litho-graphic technique, the matching fields are usually in therange of a few oersteds. For this reason, matching ef-fects are observed in a very narrow temperature region,close to the critical temperature T c , for a reduced value t = T /T c ≥ .
95. In order to both increase the match-ing field and decrease the temperature where the effectis present, the period of the pinning structure should bereduced to less than 100 nm. This gives, in fact, the possi-bility to increase H up to 1 tesla or even higher. A reason-able method to achieve this goal is to use self-assembledsubstrates, such as, for example, Al O templates withcharacteristic features in the nanometric scale [15]. Thepore diameter in Al O substrates could easily be variedin the range 25-200 nm with porosity (i.e. interpore spac-ing) around 50%, and this gives the possibility to achievep-1. Trezza et al. matching fields of thousands of oersteds [15]. To prepare Al O substrates bulk Al [16], Al foils [17], and depositedthick Al films were used [15].Very recently, another very promising material for self-assembled substrates and an optimum candidate for theNb growth was proposed, namely, porous silicon (PS) [18].PS is constituted by a network of pores immersed in ananocrystalline matrix [19] and it is a material which of-fers a considerable technological interest in different fields,as for instance micro and optoelectronics [20] and gas sens-ing [21, 22]. The diameter of pores, Ø, in PS can easilybe varied from 200 nm down to 5 nm by using substrateswith appropriate doping (n or p) and different regimes ofanodization. The porosity, in fact, can be varied in therange 30-90% by adjusting parameters such as the acidsolution, the anodizing current density and the illumina-tion of the substrate during the anodization. The regu-larity of the pores arrangement, however, is of the orderof 10% lower than the one observed in Al O templatesobtained by electrochemical oxidation [23]. It has beendemonstrated [18] that thin Nb films deposited on PS sub-strates can inherit their structure. The resulting samplesthen consist of porous Nb thin films with in plane geo-metrical dimensions, a and Ø, comparable with the su-perconducting coherence length, ξ ( T ). In these samples,matching fields of the order of 1 Tesla were experimentallyobserved [18].Aim of this work is to deepen the study of the matchingeffect in superconducting Nb thin films deposited on PS.Superconducting properties were investigated by trans-port measurements in the presence of magnetic fields ap-plied perpendicularly to the samples surface, down to t =0.52. As a consequence of the high density of the porenetwork, the (H,T) phase diagram presents a deviationfrom the classic linear dependence. This effect appearsat the matching field H ≈ R ( H ) curves, dR / dH , which can be observed in cor-respondence of the first matching field and its fractionalvalues. Fabrication. –
Porous layers were fabricated byelectrochemical anodic etching of n-type, 0.01 Ωcm,monocrystalline silicon wafers. The electrochemical dis-solution was performed in 48% water solution of HF, ap-plying a current density of 20 mA/cm . The anodizationtime was chosen in the range of 0.5 - 4 min in order toget porous layers with a thickness ranging from 0.5 to4 µ m. The pores extend on a surface of about 1 cm .The integral porosity was estimated by gravimetry to beof about 50% [24]. The resulting porous substrates haveØ=10 nm and a = 40 nm. For this lattice, if the formula H = Φ /a for the square lattice is used, the expectedfirst matching field is H = 1.3 Tesla.Nb thin films were grown on top of the porous Si sub-strates in a UHV dc diode magnetron sputtering systemwith a base pressure in the low 10 − mbar regime andsputtering Argon pressure of 3 . × − mbar. In orderto reduce the possible contamination of the porous tem-plates, the substrates were heated at 120 ◦ C for one hourin the UHV chamber. The deposition was then realizedat room temperature after the cool off of the substrates.Films were deposited at typical rates of 0.33 nm/s, con-trolled by a quartz crystal monitor calibrated by low-anglereflectivity measurements. Since the effect of the periodictemplate would be reduced when the film thickness, d Nb ,exceeds the pore diameter, Ø, [18] the Nb thickness waschosen to be 8.5 nm for the sample analyzed in this paper.A reference Nb thin film of the same thickness was grownon a non-porous Si substrate in the same deposition run. Experimental results and discussion. –
The su-perconducting properties were resistively measured in a He cryostat using a standard dc four-probe technique onunstructured samples. The critical temperature was de-fined at the midpoint of the R ( T ) transition curves. Thevalue of the transition temperatures of the film grown onthe porous substrate and of the reference sample in the ab-sence of the magnetic field were T c = 3.83 K and T c = 4.53K, respectively. The critical temperature depression in thecase of the porous sample is consistent with what alreadyreported in literature for films grown both on Al O [15]and on PS [18]. The first step for the characterizationof the behavior of the porous Nb sample in the presenceof perpendicular magnetic field is the determination of its( H , T ) phase diagram. The temperature dependence ofthe perpendicular upper critical field, H c ⊥ , was obtainedperforming resistance vs. field, R ( H ), measurements atfixed values of the temperature with a temperature stabil-ity of 1 mK. H c ⊥ was defined at the midpoint of each ofthe R ( H ) curves.In Fig. 1 the ( H , T ) phase diagrams of the Nb thin filmsare shown. In general, the perpendicular upper criticalfield of superconducting films of thickness d obeys a lin-ear temperature dependence, H c ⊥ ( T ) = (Φ /2 πξ k )(1- T / T c ) [25]. ξ k is the Ginzburg-Landau coherence lengthparallel to the sample surface at T = 0. The temperaturedependence of ξ k is ξ k ( T ) = ξ k / p − T /T c . Another su-perconducting parameter to be taken into account is themagnetic field penetration depth, λ , whose temperaturedependence is λ ( T ) = λ / p − T /T c , where λ is the pen-etration depth at T =0.The H c ⊥ ( T ) curve obtained for the Nb film depositedon porous Si template, reported in Fig. 1(a), presentssome peculiarities, which indicate that the superconduct-ing properties are influenced by the introduction of theporous array. In fact, if the H c ⊥ second derivative versusthe temperature is plotted we can see that it changes itssign from positive to negative at H ≈ Fig. 1: Left scale: Perpendicular upper critical field H c ⊥ vs.temperature of the Nb thin film with d Nb = 8.5 nm grown on(a) porous template and (b) non-porous reference substrate.The linear fits to the data close to T c are also shown. Rightscale: dH c ⊥ / dT versus temperature. The inset shows thecomparison between the second derivatives as functions of thereduced temperature of two samples, grown on the porous tem-plate (full circles) and on the non-porous template (open cir-cles). (Color online). value is very close to the nominal first matching field thatwe expect for the porous Si template, H ≈ H it follows that the period of the porous tem-plate is a = 42 nm. In the following we will identify a ≡
42 nm. In Fig. 1(b) is reported the H c ⊥ ( T ) curve for theNb reference film of the same thickness deposited on thenon-porous template. As expected the H c ⊥ ( T ) behavioris linear over the all temperature range and the H c ⊥ sec-ond derivative versus temperature does not present anypeculiarity except for a shallow peak near T c . In the insetof Fig. 1(b), for sake of comparison, the dH c ⊥ / dT ver-sus the reduced temperature is reported for both the Nbfilms, in order to point out the difference in their magni-tude. A fit to the data close to T c with the expression for H c ⊥ ( T ) reported above, yields a value of the Ginzburg-Landau coherence length at T = 0, ξ k = 9.1 nm and ξ k = 9.5 nm, resulting in a superconducting coherence length ξ S = 5.8 nm and ξ S = 6.0 nm, for the Nb porous sampleand the Nb reference film, respectively. The values of ξ k are significantly smaller than the BCS coherence length ofNb, ξ = 39 nm [26], indicating that our films are in dirtylimit regime with an electron mean free path of l = 1.38 ξ k / ξ ≈ xy plane are larger than ξ k ( T ), the expression for H c ⊥ ( T ),reported above, is verified in the whole temperature range.The Ginzburg-Landau parameter, κ = λ (0)/ ξ k , can be es-timated using the expression κ = 0.72 λ L / l = 9.6, where λ L = 39 nm is the London penetration depth of Nb [26].Ratios of ξ k / a ≈ λ (0)/ a ≈ a =42 nm, are larger than in previous works [17, 28] on perfo-rated Nb samples, and indicate that we are in presence ofindividual vortex pinning [29]. Moreover, the pore diame-ter, Ø, in our PS template is comparable with the vortexcore dimension at T =0, Ø ≈ ξ k . This means that thesaturation number, n S = Ø2 ξ S ( T ) , defined as the maximumnumber of vortices that fits into a pore with diameter Ø,is less or equal to 1, so that each pore can trap only onefluxon [30]. Subsequently multiquanta vortex lattice [2]cannot be observed in our system.Now we move to a more careful inspection of the R ( H )curves of the Nb porous film. This will lead to the obser-vation of a peculiar behavior of these transitions, whoseanalysis represents the main subject of this work. In Figs.2(a) and 2(b) R ( H ) curves obtained for two different val-ues of the temperature, T = 3.490 K and T = 3.531 K,respectively, are presented.At first glance both the curves are rather smooth anddo not present any structures or enlargements due, forexample, to sample inhomogeneities. However if the de-pendence of the first derivative dR / dH versus the appliedmagnetic field is analyzed, some distinct features can beobserved. In particular in both the curves a small localmaximum is present at specific values of the magnetic field.Let’s focus on the position where the bumps, as indicatedby an arrow in Fig. 2, start to develop. The bumps in thefirst derivative reflects the presence of a small dip in thecorresponding magnetic field dependence of the resistance R ( H ) at the same value of H . This effect was ascribed to apinning enhancement when the period of the vortex struc-ture is commensurate with the period of the antidots [31].The bumps in the dR / dH appear indeed in our curves atvalues of the magnetic fields H n when the magnetic fluxthreading each unit cell is equal to the flux quantum, Φ ,or to fractional values of Φ . In Fig. 2(a), where the R ( H ) measurement at T = 3.490 K is shown, the pecu-liarity in dR / dH is, in fact, observed at H bump ≈ a ′ = 128 nm, i.e. about three times the interpore spac-ing of this analyzed sample, a = 42 nm. Consequentlythis field value corresponds to one-ninth of the matchingp-3. Trezza et al. Fig. 2: Left scale: R ( H ) measurement at (a) T = 3.490 Kand (b) T = 3.531 K. Right scale: dR / dH versus the appliedmagnetic field. In both panels the arrow indicates the fieldwhere the bump is present. (Color online) field H /9 ≈ R ( H ) measurement at T = 3.531 K is shown, thebump in dR / dH develops at H bump ≈ a ′′ = 178nm, which is about four times the interpore spacing ofthis sample. Consequently this field value corresponds toone-sixteenth of the matching field H /16 ≈ H ≈ R ( H ) measurements at differ-ent temperatures have been performed and the behaviorof all the corresponding dR / dH curves has been analyzed.A selection of these curves is reported in Fig. 3. Someof them have been obtained by sweeping the field upwardand downward and no hysteresis has been detected.For instance, the curves at T = 2.551 K and T = 3.304K present a bump at H bump = H and H bump = H /4,respectively. By comparison a curve with no bump, mea-sured at temperature T = 2.805 K, is also shown. In allcurves the fields at which the bumps are observed are re- H H H H T = 2.551 K T = 2.805 K T = 3.304 K T = 3.490 K T = 3.531 K H (Tesla) / d R / d H ( / T e s l a ) Fig. 3: First derivatives, dR / dH , as a function of the appliedmagnetic field at different temperatures. The arrows indi-cate the field where the bump is present for each temperature.(Color online) lated to the first matching field through the relation: H = H / n with n = 1,...,4. The temperatures at which bumpsare observed, the corresponding fields and their values nor-malized to H , the ξ S values, the vortex-vortex distances, a , and their values normalized to a , are summarized inTable 1.We argue that the presence of the observed bumps inthe dR / dH curves can be related to different vortex latticearrangements made possible by the lattice of holes. Thespecific vortex lattice configurations occurring at the firstmatching field and at its fractional values are shown inFig. 4.In the case of H bump / H = 1 a commensurate square vor-tex configuration is formed, where each pore is occupiedby a fluxon and the side of this square array is just a = 42 nm. Increasing the temperature the vortices diam-eter ( ≈ ξ S ) and their reciprocal distance increase, as re-ported in Table 1. When H bump / H = 1/4, 1/9 and 1/16a square vortex lattice is again obtained with a = 87 nm,128 nm and 178 nm, respectively. This means that thepores act as an ordered template of strong pinning cen-ters, which is able to preserve the long range positionalorder of the flux lattice also at low fields value, i.e. athigher vortex spacing. As already pointed out the opti-mization of the vortex structures leads to the formation oflarger square flux lattices with respect to the underlyingartificial pinning array with the lattice constant a exactlyequal to na . The vortices tend to be placed as far fromeach other as possible due to the repulsive interaction be-tween them and at the same time they want to follow theimposed square potential induced by the antidots. Thisconstraint gives a = a √ l + k , where l and k are inte-ger numbers. Therefore, we should expect the fractionalmatching fields at H = H k/l = Φ /a = Φ / [ a ( l + k )]p-4vidence of fractional matching states in Nb thin film grown on porous silicon Table 1: Temperatures at which the bumps are observed, cor-responding fields and their values normalized to H , ξ S valuesat that temperature, vortex-vortex distances, a k/l , and theirvalues normalized to a = 42 nm. T (K) H bump (T) H bump H ξ S (nm) a k/l (nm) a k/l a H / ( l + k ) [32]. We observed bumps at fractionalmatching fields H / , H / and H / . The other bumpsexpected from the equation above at fractional fields H k/l with k = 0 have not been observed. All the fields valuesat which the bumps in the dR / dH appear are shown aspoints of coordinates ( H bump , T ) in Fig. 5.In this figure the solid lines correspond to the matchingfields of different order, as calculated assuming an inter-pore spacing a = 42 nm, through the formula H =Φ /a . The dotted lines are obtained considering a de-viation from the corresponding mean interpore distanceof the order of 10% [18]. It is worth noticing that all thedata fall into the range theoretically estimated, suggestingthat the observed peculiarities in the R ( H ) curves can beindeed ascribed to commensurability effect between theporous structure of the Nb film and the vortex lattice.The distribution of the experimental points is consistentwith the observation that a certain temperature depen-dence of the matching effect can be found for the case ofshort-range ordered templates [12]. We would also pointout that the effect is observable in our sample only up to H = H , due to the very high value of the first matchingfield. The second matching field in fact is H = 2 H =2.32 Tesla. From a linear extrapolation of the H c ⊥ curve,it follows that in order to see at this field a bump in the dR / dH we should measure a R ( H ) curve at T = 1.73 K,temperature which cannot be reached in our experimentalsetup. All the field values reported above have been calcu-lated assuming a square lattice. The measured field valuesdo not match with the ones calculated if a triangular ar-ray for the pores is considered. In fact, at T = 3.490 K(see Fig. 2(a)) the structure in the dR / dH curve for a tri-angular lattice would have been observed at a field 2 / √ H / = 0.126 Tesla, where no peculiarfeature has been detected. This supports our assumptionof considering a square lattice of holes in our system. Conclusions. –
Matching effects have been reportedfor Nb thin film grown on porous silicon. Due to the ex-tremely reduced values of the interpore distance the effectis present at fields values higher than 1 Tesla and down toreduced temperatures as low as t ≃ H , T ) phase diagram and Fig. 4: Vortex lattice configurations occurring at the firstmatching field and its fractional values. Increasing the tem-perature the vortices diameter and their reciprocal distanceincrease, as reported in Table 1. Blue circles represent theholes, pink ones represent the vortices. (Color online) H a = 4a H a = 3a H a = 2a H =1.16Ta =42nm H bu m p ( T e s l a ) T (K)
Fig. 5: The points of coordinates ( H bump , T ) identify the valuesof the fields and temperatures at which bumps have been ob-served in the dR / dH curves at fixed temperatures. The solidlines correspond to the different matching field orders achievedwith the interpore spacing a = 42 nm, while the dotted linesare obtained forasmuch as the regularity of the pore distance isachieved within the 10 percents of the average distance. (Coloronline) p-5. Trezza et al. in the R ( H ) transitions. The latter in particular revealthe formation of fractional matching states. As it was ar-gued in many works the vortex configuration at fractionalmatching fields are characterized by striking domain struc-ture and associated grain boundaries [33, 34]. The pres-ence of multiple degenerate states with domain formationat the fractional field, directly observed with scanning Hallprobe microscopy [33], seems to be high probable in ourfilms. The reduced regularity of our templates, in fact,could be compensated by the formation of domain wallsof different complexity. The particular domain configura-tion is of course a matter of energy balance between thecost in energy for the wall formation and the energy gaindue to the vortex pinning. REFERENCES[1]
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