Superconducting film magnetic flux transformer with micro- and nanosized branchesLevan Ichkitidze

Citation: AIP Advances 3, 062125 (2013); doi: 10.1063/1.4812700 View online: http://dx.doi.org/10.1063/1.4812700 View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/3/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Perpendicular applied magnetic field dependence of Josephson current and measurement of trapped magneticflux in Nb superconducting thin film by vibrating sample magnetometer J. Appl. Phys. 105, 07E312 (2009); 10.1063/1.3072446 Flux transformers made of commercial high critical temperature superconducting wires Rev. Sci. Instrum. 79, 025107 (2008); 10.1063/1.2885610 Variable transformer for controllable flux coupling Appl. Phys. Lett. 86, 152504 (2005); 10.1063/1.1899746 Superconducting multiturn flux transformers for radio frequency superconducting quantum interference devices J. Appl. Phys. 88, 5966 (2000); 10.1063/1.1322382 Simulation of the superconducting multiturn flux transformer integrated with a coplanar resonator Appl. Phys. Lett. 76, 3606 (2000); 10.1063/1.126721

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AIP ADVANCES 3, 062125 (2013)

Superconducting film magnetic flux transformer withmicro- and nanosized branches

Levan Ichkitidzea

Department of Biomedical Systems, National Research University of Electronic Technology,MIET, Zelenograd, Moscow, 124498 Russia

(Received 28 April 2013; accepted 10 June 2013; published online 25 June 2013)

The object of the study is a superconducting film magnetic flux transformer compris-ing two square shaped loops with the tapering active strips and a magnetosensitivefilm element between them. It is shown that splitting of the active strips into par-allel micro- and nanosized superconducting branches and slits increases the gainfactor of the transformer, i. e., the concentration of an external magnetic field onthe magnetosensitive element, by a factor of more than four. C© 2013 Author(s). Allarticle content, except where otherwise noted, is licensed under a Creative CommonsAttribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4812700]

I. INTRODUCTION

Magnetic fields B of nanosized objects and biological objects are close to the level of themagnetic vacuum (< 0,01 f T) and can be measured only by modern highly sensitive magnetometersand magnetic systems at an obtainable distance from these objects.b

Weak magnetic fields (B ≤ 10 pT) are currently measured by different magnetometers:1–4

SQUIDs (superconducting quantum interference devices), atomic, optical magnetometers, etc. Themost sensitive of them are SQUIDs based on the effect of superconducting electrons tunnelingthrough a weak link (Josephson junction or transition), but they do not measure the absolute valueof a magnetic field and only detect changes in it. For SQUIDs, the magnetic field resolution δB,i. e., the minimum detectable magnetic field, is ∼1 f T.

As a magnetosensitive element (MSE), any materials with sufficient nonlinearity of their mag-netic characteristic can be used, for example, Hall sensors, materials and structures based on theeffect of giant magnetoresistance (GMR), and granular traditional or ceramic high-temperature su-perconducting (HTS) materials. However, in order to improve the important parameters of a magneticfield sensor, in particular, to reduce δB, it is necessary to use concentrators of a measured (external)magnetic field that are called the magnetic flux transformers (MFTs). For this purpose, the propertyof superconductors to preserve the magnetic flux in a closed circuit without loss is often used.

The MFT elements based on HTS film materials are used in many magnetometers, where MSEsare Josephson junctions (for SQUIDs),5 Hall sensors,6 sensors based on the GMR effect,7 sensorsbased on the magnetoresistive effect in ceramic HTS materials,8–10 etc.

As was shown in studies,11–13 the efficiency of the MFT can be increased by optimal fragmen-tation of its active strip into numerous parallel micro-, submicro-, and nanosized branches and slits.In this case, the MFT is separated from the MSE by an insulator film and concentrates an externalmagnetic field in the direction parallel to the substrate surface. In this work, we present the resultsof the calculations of the MFTs based on HTS film materials in a magnetic field sensor (MFS).Superconducting film loops serve as an MFT; as an MSE, different magnetoresistive elements canbe used. The MFT and MSE lie in one plane and are separated from one another by gaps; a magnetic

ae-mail: leo852@inbox.rubThese are magnetic nanoparticles, superparamagnetic nanoparticles, carbon nanotubes with catalytic nanoparticles or

encapsulated with magnetic nanoparticles, nanosized elements of very-large-scale integrated circuits, etc. In addition, themagnetic field produced by activation of the human brain neurons is meant.

2158-3226/2013/3(6)/062125/8 C© Author(s) 20133, 062125-1

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062125-2 Levan Ichkitidze AIP Advances 3, 062125 (2013)

FIG. 1. Layout of the MFT and MSE: (a) substrate 1, MSE 2, and superconducting MFT loops 3; (b) MFT active strip 4consisting of numerous branches (enlarged). The shaded and unshaded areas show superconducting branches 5 and slits 6,respectively.

field to be measured is concentrated in the direction perpendicular to the substrate surface. We inves-tigate the possibility of improving the important parameters of the MFT by local fragmentation of itsactive strips into numerous parallel superconducting branches and slits at a technological linewidthresolution of 20 ÷ 10000 nm.

II. MATERIAL AND METHODS

The object of the study is the factor F of the effective concentration of a magnetic field on theMSE for the case when the HTS-film-based MFT and the MSE lie in one plane and do not intersect.To increase F , the MFT active strips were split into several parallel branches in the areas adjacentwith the MSE. The MFT was calculated with regard to the size effect when the current distributionin superconducting films significantly depends on their width.

The MFT comprises two active strips with the MSE symmetrically positioned between them(Figs. 1(a) and 1(b)).

The value of the factor F of the effective concentration of a magnetic field on the MSE isestimated as follows. In an external magnetic field B the magnetic flux shielded by the MFT loop3 is calculated as: φ = A · B, where A – the MFT loop area. The screening current value IS iscalculated as: IS = φ

/(L + M), where L – the MFT loop inductance, M – the sum of the mutual

inductances between the right and left loop, and also between the loops and MSE. It is known thatthe value of L is one (or more) order of magnitude larger than the total mutual inductance M .

The MFT active strip width ws is one (or more) order of magnitude smaller than the width ofthe other MFT parts. This results in a substantial growth of the critical current density and thus theincrease of the external magnetic field concentration in the neighborhood of the MFT active stripand on MSE.

The MFT loop inductance is mainly determined by the inductance L of the MFT active strip. Inthe case when the latter is split into several branches, each with the inductance Li (i = 1, 2, ..., j ,where j, n = j − 1 – the number of superconducting branches and slits in the MFT active strip,

respectively),c the total inductance L j =j∑

i=1(L−1

i )−1 grows insignificantly in relation to L; the

cThe case of j = 1, n = 0 corresponds to the unsplit MFT active strip.

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062125-3 Levan Ichkitidze AIP Advances 3, 062125 (2013)

corresponding factor KL is calculated as:

KL ( j) = L j

L≈ ws

j∑i=1

wbi

, (1)

where wbi − the width of the i-th branch.The estimates show that the increase of number of branches j in the MFT active strip results in

the change of the total mutual inductance relative to the case of j = 1 and the change of KL , bothchanges being of the same order of growth. Therefore, the total mutual inductance M j , correspondingto the case of the MFT active strip splitting into j branches, is one (or more) order of magnitudesmaller than L j .

In an external magnetic field in the MFT active strip branches induced are the screening currents

ISi (j∑

i=1ISi = IS), flowing in the vicinity of MSE and influencing it by the magnetic field Bi⊥ in the

point (x0,y0) inside MSE. The reference point (0,0) is located in the centre of the i-th branch uppersurface. The value of Bi⊥ is calculated by formula:

Bi⊥ = μ0 · ISi

8π · λ · h· [

0∫

−l

0∫

−2h

e− x+lλ · (x0 − x)

(y0 − y)2 + (x0 − x)2dydx +

+l∫

0

0∫

−2h

e− l−xλ · (x0 − x)

(y0 − y)2 + (x0 − x)2dydx],

(2)where l = wbi/2 and h are the i-th branch half-width and half-thickness, respectively, μ0 = 4π ·10−7

H/m − the magnetic constant, ISi/4λh ≤ Jc, Jc − the critical current density for the HTS of MFT,λ − the London penetration depth for the HTS. In all cases it is supposed that the length of thebranches significantly exceeds wbi and the technological linewidth resolution wa . In the calculations

considered were: B⊥( j) = 2j∑

i=1Bi⊥, where Bi⊥ was calculated by formula (2); 〈B⊥( j)〉 − the

value of B⊥( j) averaged along the MSE width w0; the rectangular quadrature method was used fornumerical integration.

The relative gain factor FB , which is calculated with no regard to the change in the resultinginductance of branches, is calculated as:

FB = 〈B⊥( j)〉〈B⊥( j = 1)〉 , (3)

where 〈B⊥( j = 1)〉 − the magnetic field averaged along the MSE width w0 for the case of j = 1.Then the final gain factor F calculated with the use of formulas (1) and (3) is approximated as:

F ≈ FB

KL. (4)

In all the calculations with the use of formulas (1)–(3), it was assumed that slit width wp and thewidths of the gaps between the nearest MFT and MSE edges coincide with technological linewidthresolution wa . Width ws of the MFT active strip and widths wbi of its branches were assumed tobe multiple of wa . For the specified values of wa , we determined the optimal splitting of the MFTactive strip into branches for attaining the maximum value of F(Fmax).

III. RESULTS AND DISCUSSION

The calculations were made for the two variants: the active MFT strip widths are ws = 30 μm(variant 1) and ws = 1 μm (variant 2).

Variant 1. Figure 2 illustrates the variation in KL and F as a function of number of branches jin the MFT active strip for differentwp. The case j = 1 corresponds to the continuous MFT activestrip without splitting into branches. It can be seen that the dependences of KL on j monotonicallygrow within (1 ÷ 2) (Fig. 2(a)), while the dependences of factors F on j have maxima Fmax

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062125-4 Levan Ichkitidze AIP Advances 3, 062125 (2013)

FIG. 2. Dependences of (a) the relative inductance variation and (b) gain factor on the number of branches at ws = 30 μmand wa = wp , μm: 1 (♦), 2 (�), and 5 μm (�).

TABLE I. Parameters of the MFT with the optimal splitting of its active strip into superconducting branches and slits(variant 1).

wa = wp , μm Optimal splitting, wbi , μm Fmax

1 1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-1-11 4.322 2-2-2-2-2-2-2-2-2-14 2.855 5-5-5-5-5-10 1.6410 10-10-10-10 1.17

(Fig. 2(b)). The smaller the slit widthswp, the higher are the maxima Fmax and the larger are thenumbers of branches j∗corresponding toFmax.

Table I gives the values of Fmax (Fig. 2(b)) for the optimal splitting of the MFT active strip intobranches at wp = 1, 2, 5, and 10 μm. Slits (wp) and branches (wbi ) start alternating from the gapbetween the MFT and the MSE. The initial values given in large bold italics show the gap widths.The slit widths are given with normal bold italics and the superconducting branch widths – with theregular type. All the values are given in μm.

In the calculations, the following values were used: λ = 200 nm, Jc = 1010 A/m2, h = 10 nm,ws = 30 μm, and w0 = 10 μm. Table I gives the variants of the optimal splitting of the MFT activestrip for technological linewidth resolution wa ; i. e., wp = 1, 2, 5, and 10 μm.

As the value of wp is decreased, the number of branches and slits corresponding to the optimalsplitting grows and the maximum values of the MFT gain factor increase. It can be seen that theefficiency of the concentration of a magnetic field on the MSE grows with the number of branchesin the MFT active strip: for 2 branches, the concentration grew by 17% relative to the unsplit MFTstrip and for 10 branches, by 332%. It should be noted that the optimal splitting can be with differentbranch widths; at a large number of branches, their widths will strongly differ; e. g., at wp = 1 μm,the first branch width is wb1 = 1 μm and the tenth branch width is wb10 = 11 μm. It is importantthat at the optimal splitting the branch width can grow or remain invariable but cannot decrease withincreasing distance from the MSE.

The results presented here were computed by enumerating possible variants of the splitting ofthe MFT active strip into branches (in particular, ≈8 · 105 variants for wa = wp = 1 μm), whichrequired serious computational burden. Further decrease in wp at a fixed value of the MFT activestrip width ws will undoubtedly lead to even larger computer resource consumption.

According to the calculation procedure, in order to optimize the computation time, first averagedconcentrations of a magnetic field on the MSE were calculated by formula (2) for all possiblelocations and widths of branches at specified technological parameters (λ, ws , w0, wa , etc.). Then,the variants of the splitting of the MFT active strip were enumerated to choose the optimal variantcorresponding to the maximum concentration of a magnetic field on the MSE with regard to the

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062125-5 Levan Ichkitidze AIP Advances 3, 062125 (2013)

FIG. 3. Dependences of (a) the relative inductance variation and (b) gain factor on the number of branches at ws = 1 μmand wa = wp , nm: 20 (♦), 50 (�), and 100 (�).

TABLE II. Parameters of the MFT with the optimal splitting of its active strip into superconducting branches and slits(variant 2).

wa = wp , nm Optimal splitting, wbi , nm Fmax

20 20-140-20-160-20-180-20-200-20-240 1.3450 50-200-50-300-50-400 1.27100 100-400-100-500 1.19

change in the resulting inductance of branches. The combinatorial growth of the number of splittingvariants does not allow us to consider this algorithm to be universal. To optimize the computationalalgorithm, further investigations are needed.

Variant 2. In the second variant of the calculation, in order to improve the important parametersof the sensor (in particular, to reduce δB), we studied the possibility of the transition to the nanosizedtechnological linewidth resolution. Figure 3 shows the variation in KL and F as a function of numberof branches j in the MFT active strip at different nanosized values of wp. Here, the case j = 1corresponds to the unsplit active strip. In the calculation, we used the following values: λ = 200 nm,Jc = 1010 A/m2, h = 10 nm, ws = 1 μm, wp = wa = 20, 50, and 100 nm. In contrast to variant 1,the widths ws = 1 μm and w0 = wp were used. Similar to variant 1, the dependences of KL onj monotonically grow within (1 ÷ 2) (Fig. 3(a)) and the dependences of F on j have maxima(Fig. 3(b)).

Table II gives the values of Fmaxin accordance to Fig. 3(b) for the optimal splitting of the MFTactive strip into branches at wp = 20, 50, and 100 nm. The gap and slit widths (nm) are givenwith bold italics and the superconducting branch widths – with the regular type.d As before, withdecreasing wp the number of branches and slits corresponding to the optimal splitting grows and themaximum values of the resulting MFT gain factor increase. Table II gives the variants of the optimalsplitting of the MFT active strip for wa = wp = 20, 50, and 100 nm.

It can be seen that the efficiency of the concentration of a magnetic field on the MSE grows withthe number of branches in the MFT active strip: for 2 branches, Fmax grew by 19% relative to thecase of the unsplit MFT strip and at 5 branches, by 34%. Thus, we observe the noticeably smallerbenefit in Fmax as compared to variant 1. In variant 2, after attaining its maximum value, F rapidlydrops as the number of branches is increased; at j∼ 2 j∗, we have F< 1 (Fig. 3). Hence, the splittinginto numerous branches leads not the growth but, just the opposite, to the reduction of F relative tothe case of the unsplit MFT active strip.

dWe used the same designations in Tables I and II.

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062125-6 Levan Ichkitidze AIP Advances 3, 062125 (2013)

FIG. 4. Optimal sizes (nm) and location of MFT branches and MSE strip at the initial parameters λ = 200 nm, Jc = 1010

A/m2, h = 10 nm, ws = 1 μm, w0 = 20 nm, wp = wa = 20 nm. (1) Slits, (2) MSE strip, and (3) MFT active strip branches.Figure scale is not respected.

In variant 1, the splitting into j∼2 j∗ branches did not lead to the sharp drop of F ; theinequalityF> 1 was always valid (Fig. 2); in other words, the split MFT active strip was stillmore effective than the unsplit one (F = 1). This is explained by manifestation of the size effectonly in those superconducting films whose widths are much larger than λ: e. g., in variant 1, λ =200 nm and ws = 30 μm. In variant 2, the values λ = 200 nm and ws = 1 μm are of the same orderof magnitude; therefore, here the size effect is not as pronounced as in variant 1. Consequently, thesplitting of the MFT active strip is more effective in variant 1 than in variant 2, which is confirmedby our calculations (see Tables I and II): Fmax≈ 4.32 (variant 1) and Fmax≈1.34 (variant 2).

Figure 4 shows the structure of the MFT branches and MSE strip for j = 4 in the relative scaleaccording to their optimal sizes and location calculated at the following initial parameters: λ = 200nm, Jc = 1010 A/m2, h = 10 nm, ws = 1 μm, w0 = 20 nm, and wp = wa = 20 nm.

According to Fig. 3(b), the maximum gain factor is attained at the optimal number of branchesj = 5 in the MFT active strip. However, at j = 4 the value of F is smaller than Fmax at j = 5 by anegligible value (∼0.05%).

The results obtained can be used for estimating possible enhancement of the efficiency of theweak magnetic field sensor considered in,7 in which a material with the GMR effect was used asthe MSE and an HTS Y-Ba-Cu-O film was used as the MFT. In this sensor, the active MFT stripwidth is 1 μm and, according to the data given in Table II, at the optimal splitting of its MFT activestrip, the gain factor of the sensor will grow by ∼34%. One may expect that the magnetic fieldresolutionδBwill correspondingly decrease and the dynamic range of the MFS will broaden.

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062125-7 Levan Ichkitidze AIP Advances 3, 062125 (2013)

In the calculations, we took into account the high density of the critical current (Jc = 1010

A/m2) typical of single-crystal or highly textured films of the HTS Y-Ba-Cu-O materials used as theMFT. However, the additional growth of the gain factor due to the size effect will allow using HTSfilms with the low densities of the critical current Jc∼108-109 A/m2 as the MFT (e. g., Bi-2212 andBi-2223 materials), which are relatively easy to fabricate.

IV. SUMMARY

The analysis of the obtained results shows that fragmentation of the MFT active strips intomicro- and nanosized superconducting branches and slits (the slit width lies within ∼20-10000 nm)makes it possible to increase the MFT gain factor and, correspondingly, the concentration of anexternal magnetic field on the MSE, by a factor of more than 4 in the case of a wide (∼30 μm) MSEand by more than 30% in the case of a narrow (∼1 μm) one.

The value of Fmax can be further increased by decreasing the gap between the MFT activestrips and the MSE or by using the materials with high Jc and low λ, e. g., niobium films as theMFT. Indeed, in heteroepitaxial (highly textured, nearly single-crystal) niobium layers on sapphiresubstrates at the temperature T ∼ 4K the values Jc∼ 107 A/cm2 and λ∼ 50 nm are attained.14–16

The choice of the heteroepitaxial niobium layers as the MFT will apparently lead to the growth ofFmax by more than an order of magnitude in variant 1 and in several times in variant 2 relative to thevalues Jc∼106 A/cm2 and λ ∼ 200 nm for the HTS materials considered here. The niobium filmshave already demonstrated their higher efficiency (higher values of F) as compared to the films inthe Y-123 system when used as a material for the MFT with continuous active strips.7 Certainly,the growth of F and Fmax will ensure the reduction of the magnetic field resolution of the MFS (δB∼ 1/F)11–13 for detecting weaker magnetic fields.

The structure considered above is planar: the MFT and MSE lie in one plane and do not intersect.Consequently, such a single-layer film sensor of weak magnetic fields is much easier to fabricate ascompared to the multilayer structures often used in SQUIDs.

At present, there are no high-temperature superconductors that would allow connecting theirends such as to the closed ring had no superconductivity (magnetic flux) loss and could serve as anMFT element. Therefore, novel HTS-film-based magnetic flux transformers have been intensivelydeveloped and the existing ones have been improved.17, 18 We believe that the magnetic flux trans-former based on HTS film materials with the nanostructured active strips will facilitate solving theabove-mentioned problem.

ACKNOWLEDGMENT

This study was partially supported by the Ministry of Education of the Russian Federation (statecontract No 16.740.11.0765, agreement 14.B37.21.0567) and by the Foundation for Promotion ofSmall Enterprises in Science and Technology (contract 10678r/19537).

The author thanks Professor S.V. Selishchev for the support of this study and A.N. Mironyukfor the help in calculations.

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062125-8 Levan Ichkitidze AIP Advances 3, 062125 (2013)

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