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Abstract

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Physica 404(2004)176-18l C rlsevier.com/locate/physc

PHYSICA G

EI-SEVIER

Traits of the insulatorformed under the superconductor-insulator transition
* V.F. Gantmakher
Institute of Solid State Physics RAS, Chernogolouka 142432, Russia

Various experimental observations the magnetic-field-induced of superconductor-insulator transition are described and comparedwith differenttheoreticalmodels:one basedon boson-vortex duality, next exploring the propertiesof granular superconductors the third analyzingeffectofthe superconducting and fluctuationsin the magneticfield at low temperature. the modelspoint to the existence pairwiseelectroncorrelationsat the Fermi-levelof the insulator. All of Theseso-calledlocalizedpairs should vanish in high magneticfields. The localizedpairs apparentlycome from the parity effect in ultrasmall quasigrains-local minima of the random potential which can admit only small limited numberof electrons. O 2004Elsevier B.V. All riehts reserved. PACS:'14.40.+k: 74.78.-w
Keywords: Superconducting fluctuations; Localization; Superconductor-insulator transition; Vortices

When superconductivity is destroyed by changingsome parameters,either intrinsic (carriersdensity, (magnetic levelof disorder) extrinsic or field) ones,the material can turn not only into normal metal but into insulator as well. We'll discusshere magnetic-field-induced superconductor-insulator transitions(SIT). A systemof delocalizedelectrons characterizedbythe product of is the Fermi-wavevector and mean free path tpl. If the carrier density in the material is low and the level of disorderis high, so that kpl x I and the material would be on the insulatingsideof the metal-insulatingtransition if it were not for the
-' 1.1, .'7-96- 5223 6; f ax:+'t-9 6-52497| . t9 0 (V.F. Gantmakher). E-mail address: gantm@issp.ac.ru

superconductivity,then the magnetic field may turn the superconductor into insulator. The main sign of SIT is the fan-like set of the resistance curvesR(7): they go down with decreasing the of temperature at fields below the critical, B 4 8., and go up at fields B ) 8". The list of materialswhich display this kind of behaviorcontainsamorphousMo.,Ge1-, [] and Mo.,Si1-, [2] films, amorphousInO, films [3,4], ultrathin films of Be [5], crystalline films of Nd2-.,Ce,CuO4+y16,7l of TiN" [8]. Two typiand cal examplesof such sets of curves relevant to differentlimits are presented Figs. I and 2. In in (Fig. l), the growth of the Nd2-,Ce.,CuOa*n resistance the non-superconducting on side of the field-inducedSIT with decreasing temperatureis below l0%; this reminds rather a metal with

0921-45341$seefront matter @ 2004ElsevierB.V. All rishts reserved doi:I0.I 0l 6/j.physc.2003. I0.034


V.F. Gantmakher PhysicaC 404 (2004) 176-181 I

177

q

r(K) Fig. l. Experimentalset of curves for crystalline Ndz-,Ce,CuOa*, films [7]. At fields aboveB" the resistivitychanges are (see below lfflo. Two upper curvescross at low temperatures enlargedpart of this plot below, on Fig. 4(b)).
+

+
I

+ oi
or

4e+. t t \ '-r
Ar
*

lr

)'XX_-s___:_:
?-

g
a<

ovv--[--

-

=*
- -+- -o- -a-x- -v- -tr- 3.m - 2.5 - 2.3 2.1 - 1.9 - t.7 0.8

v+,
I
I

tr
l l -0

0.2

0.4 7(K)

0.6

Fig. 2. Temperature dependence the resistivity of highof resistiveamorphousInO, films at differentmagneticfields [4].

quantum conductivity correctionsthan an insulator. Amorphous InO, representsthe opposite limit. Dependingon non-stoichiometryparameter 3 - x, uprise of the low-temperature resistance may be of 50o/o, may be almost l0-fold as on or Fig.2, or may evenreachmore than five ordersof magnitudein the temperature interval 0.07-l K [9] with typical for an insulator exponentialtemperature dependence the resistance. all cases, of In thereexistsa critical value of the magneticfield B" at which the resistance doesnot dependor almost doesnot dependon temperature, R(?) = const: R".

All the transport data in the above-listed experiments were interpreted as SIT in twodimensional (2D) electron systems.For the systems with comparatively trifling growth of the resistance(Mo1-*Ge1-", Mo"Si1-", Ndz-"Ce,CuOa1"), this interpretation is based on scaling hypothesis [0] which asserts that the metallic ground stateis prohibited for 2D-systems and that ground states only insulatingand superconducting are possible.Hence the film stateswhich do not display tendencyto becomesuperconducting, i.e. thosewith low-temperature derivative 0R/07 < 0, should be accepted insulating. as The theoretical model of 2D-SIT which was suggested [1,12] appealsto similarity of two in kinds of bosons:Cooper pairs and vortices,called the boson*vortexduality. The model considers the insulating phaseas a condensate vortices with of localizedCooper pairs just as the superconducting phaseis a condensate Cooper pairs with localof ized vortices. The fanlike set of the resistance yield of this theory. The curyesR(Z) is an essential insulating phase which appears as the result of such SIT is rather specific; it contains pairwise correlationsbetweenthe localizedelectronsas the remnant of the superconductingpairing. It is called the Bose-insulator[3] and the correlated electronsare referred as localized electron pairs. Of course, existenceof such specific insulator should be confirmed experimentally.Though the theory [1,12] describes only vicinity of the SIT and studiestransport only insidethe critical region on the (I, B)-plane, the confirmation came from high field region, well abovethe critical field 8". In most cases,the fan-like shapeof the set of R(7, B : const) curves is accompaniedby the (NMr) in higher fields negativemagnetoresistance insulatingsideof the SIT, far awayfrom the on the criticalregionlB-B"l (.B". In Fig. l, the NMr looks as crossing of two upper curves at low temperatures. is demonstratedin detail in Fig. It 3(a) and (b) for InO" films with higher resistance. Noteworthy that total reduction of the resistance in high fields is alwaysof the sameorder of magnitude as its initial uprise aboveR". The NMr has been studiedand explainedboth experimentallytl5l and theoretically tl6l for granular superconductors. gap Superconducting in


178
II I= 6OmK 70 80 (a) I lo 150

V.F. Gantmakher PhvsicaC 404 (2004) 176-181 I
rl 32mK 42 48 60 79 il4 195
I I

III

100 80 60y

I I I

I I I

{
,10

fluctuation corrections 6o to the conductivity of 2D superconductors the low temperature7 ( to field .B) B"z(0).The 4(0) and the high magnetic correction 6o includesAslamazov-Larkin, Maki(DOS) terms.In Thompsonand density-of-states the dirty limit it can be written in the form
6o:

20 0

(r)#rl- ^; *+ tt) +4(r{t' t)],
(l)

o

3

6

e

t'r,f,

3

6

e

t2

Fig. 3. Set of isotherms R(I: const,B)for amorphous lnO, films [4]. In the fields region I the material remains superconducting,label III marks the NMr region.(a) Magneticfield normal to the film. The theory[1,12] relates the vicinity of to the boundary betweenthe regionsI and II. (b) Magnetic field parallel to the film (seediscussion the end of the paper). at

the spectrumof grainsleadsto exponentiallysmall number of excitationsat low temperatures inside the grains, and henceto exponentiallysmall oneparticle intergrain tunneling current. When this tunneling process controls the total resistance, the granular material behaves insulator. The magas netic field destroysthe gap, increases one-particle intragrain density of statesat the Fermilevel and reduces the tunneling resistanceand hence the resistance the granular material as a whole, of There is intrinsic similarity betweenthe basic conceptsof the scaling theory for Bose-insulator [11,12]and of the theory for granular superconductor [6]: pairs localized in the disordered materials and Cooper pairs quasilocalizedinside grains;the strong magneticfield destroyspairwise correlations the Bose-insulator in and the superconductivity inside the grains in the granular insulator. But neither theory can be applied directly to NMr in homogeneously disordered materials.The theory [11,12]relatesonly to the vicinity of the SIT, the NMr happens beyondthe range of its action. And expansionof the theory developedfor granular materials to the homogeneouslydisordered onesis far from obvious. The progresscame from the recent paper by Galitski and Larkin [7]. They succeededin extending calculations of the superconducting

y where ry' is digamma-function, : (ll2y')(blt), : et : 1.781is the exponential Euler'sconof l' stant, and t: TlT"o( I and b: (B - B"z(f))l B"z(0)< I are reducedtemperatureand magnetic field. The particular feature of this expression are the negative terms. They originate from the depression DOS g near the Fermi-levelby fluctof uative pairing of carriers.This becomes important if disorder and magneticfield make ineffective the transport by Cooper pairs. The remaining transport dependson the DOS. The magneticfield oppresses pair correlationsand increases DOS the the at the FermiJevel; this leads to growth of the conductance, to the NMr. i.e. (l) To comparethe expression with experiment we shall revisitFig. I and presentin Fig. 4(b) the part of the set of R(B) curvesfor low-temperature Nd2-,Ce.-CuOa-;r. The curves R(B) measuredat 2.85and 3.18T both showat leasttendency the to superconducting behavior at the lowest available temperature.The curvesat 3.5 T and even at 4 T may be a maximum at lower temperaturebut the curve at 5 T certainly not. So we have the sequence: the superconducting state (B < 3.5 - 4 T)-the non-superconducting state (B :5 T)-the state with NMr (B :7 T). Uncertainty in the critical field B. is not of vital importance. To make the theoreticalexpression suitablefor the comparison,Aronov-Altshuler quantum correction 644o: -("'lDln(TlT.) from the diffusive channeltypical for normal metalswas added [7] to the formula (l):

o: oo*6o(8, - *lhelT.) rl
n

(2)

(herea and Z- are adjustableparameters; allowing for a * I we make flexiblethe relation between the


V.F. Gantmakher PhysicaC 404 (2004) 176-181 I

t79

0.24

crossover from logarithmic to exponential tem2D perature dependence non-superconducting in gas, with only diffusive channel being electron wasestitemperature7oo important. The crossover x (e2lh)ln Zs6 as mated[8] from the relation os T* - 3!-"-zt*rtt,
kel

o.20

(3)

3.5
q

3.18

2.85

o

',r*r,

4

s

Fig. 4. Comparisonof (a) the set of curvesderived from the theory [7] and (b) the experimental set for crystalline part of the plot Ndz-,Ce"CuO+" films. Panel(b) is an enlarged from Fig. I (from [7]).

*h"r. ," was the Fermi energy [8]. Acting in the samemannerand equatingthe right part of the of relation (2) to zero,one getstemperature?"0 the function to crossover bosonicinsulator stateas the of the magneticfield. The valuesof ft near maxima turn to be much larger then foo Gig. 5). Similar results were obtained by numerical simulations [9,20]. It was demonstratedthat the attractive interaction stimulated localization by combining single particles into pairs. This effect happenedin 3D as well [20]. This was the second indication that SIT could existin 3D materials;the first one followed from the model of superconducting grains.The possibledimensionalityof the Bose-insulatoris the next request to the experimenters. All experiments[-8] were done with films. It is usually difficult to say whether a film is thick enough to be consideredas 3D. For instance,the

contributions from Cooper and difrrsive channels; One to the latter is assumed be field independent). in 4 remarkableresemblance qualcan seein Fig. featuresof panels(a) and itatively distinguishable (b)-there is separationbetweenlow field curves "bend which "bend down", and high field which up"; there is also high field NMr at low temperature. So, the SIT experimentsare approachedfrom differentsidesby threetheoreticalmodelsbasedon the boson-vortex duality |l,l2l, on granularity p6l, and on superconducting fluctuations[7]. A1l three lead to similar conclusion:the specificinsulator with pairwise correlations near the Fermileveldoesexist. fluctuationsmay stimulatethe Superconducting electron localization. This can be demonstrated by the same trick as that used by Larkin and Khmel'nitskii [8] to determinethe position of the

ro-2
/
10
.U Lo l0-4

krl=7

h

kol=9

l0loo lo-' ro-' lo-t loo lot

B lBc2-1

for temprature7no severalreducedvaluesof Fig. 5. Crossover the mean free path / calculatedby equating to zero the right part of the Eq. (2) for the fieldsvaluesup to B:l.2B"z(0). the Dotted lines qualitativelydesignate asymptoticparts of the decurves.Horizontal lines approximatelymark levelsof ?"oo rived for the samevaluesoftp/ [7].


180

V.F. Gantmakher PhysicaC 404 (2004) 176*181 I

thicknessof 200 A of InO" films in experiments [4,14] was much larger than the mean free path I x l0 A, but apparentlyslightly lessthan the superconducting coherentlength (. The model based on the boson-vortexduality can be appliedonly to 2D-systems. But just necessity this duality can of be checkedin experiment.Revisiting Fig. 3 one can see that the behavior of isotherms R(B) is qualitatively the samewith the field normal and parallel to the film. But in parallelfield the vortices cannot be treated as particles and the duality principle doesnot work. HenceNMr and pairwise correlationsof localizedelectronshave more general origin. Qualitativemodel of the 3D localizedpairs can be proposedwhen exploring the low-sizelimit of the granular material. A grain of size a has level spacing 6er (ga3)-t' it retains superconductivity until 6eremainslessthen the superconducting gap A*. In the opposite limit of ultrasmall grains 6e) /.", the superconductivity replacedby the is parity effect.For an isolated grain this effectwas calculatedin [21].A level in the grain occupiedby two electronshas energylessby lo x 6elln(6e//..) (6e) /..), (4) than when it is occupiedonly by one electron.This would mean a Eaplp in one-particlespectrumfor the multitude of suchgrains:when hopping from a double-occupied level, the electronhas to provide the left one with energy /0. Sure, this is only a rough notion but it can be usedas a starting point in describinghomogeneously disorderedinsulator with pairwiseelectroncorrelations. To summarize,field and temperature dependence of superconductingfluctuations describe negativemagletoresistance 2D-materials with of moderateresistivityat fieldsabovethe critical field .8";thesefluctuationsmay stimulatelocalizationin the magneticfield. This supportsthe model of pair localization in the process of superconductorinsulator transition in high-resistive materials which follows from the idea of 2e-boson-vortex duality. Granularity of the superconductors leads to the sametransport phenomena:to the superconductor-insulatortransition and to the negative magnetoresistance; hence granular and homogeneously disordered high-resistive materials are

hardly distinguishable.Pair localization apparently can be derived from the parity effect in ultrasmallquasigrains a 3D disordered of material. This would result in the gap at the Fermi-levelin the spectrumof one-particleexcitations.

Acknowledgements Author appreciates discussions cooperation and with all his coworkers in the field of superconductor-insulator transitions, in particular, with late V.M. Teplinskii,with M.V. Golubkov, V.T. Dolgopolov,A.A. ShashkinG.E. Tsydynzhapov, V.N. Zverev,and T.A. Baturina. This work was supportedby grantsRFBR 02-02-16782 NShand 2t70.2003.2.

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