Suppression of the Berezinskii-Kosterlitz-Thouless and Quantum Phase Transitions in 2D Superconductors by Finite Size Effects
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l PREPRINT (April 4, 2018)
Suppression of the Berezinskii-Kosterlitz-Thouless and Quantum Phase Transitions in2D Superconductors by Finite Size Effects
T. Schneider ∗ and S. Weyeneth Physik-Institut der Universit¨at Z¨urich, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland
We perform a detailed finite-size scaling analysis of the sheet resistance in Bi-films and theLaAlO /SrTiO interface in the presence and absence of a magnetic field applied perpendicularto the system. Our main aim is to explore the occurrence of Berezinskii-Kosterlitz-Thouless (BKT)and quantum phase transition behavior in the presence of limited size, stemming from the finiteextent of the homogeneous domains or the magnetic field. Moreover we explore the implicationsthereof. Above an extrapolated BKT transition temperature, modulated by the thickness d , gatevoltage V g or magnetic field H , we identify a temperature range where BKT behavior occurs. Itsrange is controlled by the relevant limiting lengths,which are set by the extent of the homogeneousdomains or the magnetic field. The extrapolated BKT transition lines T c ( d, V g , H ) uncover com-patibility with the occurrence of a quantum phase transition where T c ( d c , V gc , H c ) = 0. However,an essential implication of the respective limiting length is that the extrapolated phase transitionlines T c ( d, V g , H ) are unattainable. Consequently, given a finite limiting length, BKT and quantumphase transitions do not occur. Nevertheless, BKT and quantum critical behavior is observable,controlled by the extent of the relevant limiting length. Additional results and implications include:the magnetic field induced finite size effect generates a flattening out of the sheet resistance in the T → T = 0 only. The former prediction is confirmed in both, the Bi-films and the LaAlO /SrTiO interface, as well as in previous studies. The latter is consistent with the LaAlO /SrTiO interfacedata, while the Bi-films exhibit a flattening out. PACS numbers: 74.40.-n, 74.78.-w, 64.60.Ht
I. INTRODUCTION
Over the last two decades, electrical transport mea-surements of thin films near the onset of superconduc-tivity have been studied extensively.
Crucial observa-tions include: the sheet resistance in zero magnetic fieldremains nearly temperature independent at the lowestattained temperature and remains ohmic below theexpected normal state to superconductor transition tem-perature T c ; a magnetic field applied perpendicularto the film generates a flattening out of the sheet re-sistance in the T → the occurrence of asmeared Nelson-Kosterlitz jump in the superfluid den-sity in the absence and presence of a magnetic field. Interpretations of the saturation of the sheet resistancein the T → the occurrence of quantum tunneling ofvortices, and the failure to cool the electrons. On the other hand, more than three decadesago, Beasley, Mooij, and Orlando suggested thatthe Berezinskii-Kosterlitz-Thouless (BKT) transi-tion may be observable in sufficiently large and thinsuperconducting systems. They showed whenever theeffective magnetic penetration depth λ D = λ /d ex-ceeds the sample size [ W s , L s ], where λ is the magneticpenetration depth, d the thickness, W s the width and L s the length of the system, the vortices interact loga-rithmical over the entire sample, a necessary conditionfor a BKT transition to occur. Indeed, as shown byPearl, vortex pairs in thin superconducting systems(charged superfluid) have a logarithmic interaction en- ergy out to the characteristic length λ D = λ /d , beyondwhich the interaction energy falls off as 1 /r . Accord-ingly, as λ D increases the diamagnetism of the super-conductor becomes less important and the vortices in athin superconducting film become progressively like thosein He films. Invoking the Nelson-Kosterlitz relation in the form λ D = λ ( T c ) /d = Φ / (32 π k B T c ), it isreadily seen that for sufficiently low T c ’s, the condition λ D > [ W s , L s ] is in practice accomplishable. Indeed, T c = 1K yields λ D ≃ .
98 cm. Additional limitinglengths include the magnetic length L H ∝ (Φ /H ) / as-sociated with fields applied perpendicular to the film andin the case of ac measurements L f ∝ f − / where f de-notes the frequency. Concentrating on dc measurementsof the sheet resistance one expects that the dimension ofthe homogeneous domains L h sets in zero magnetic fieldthe smallest size so that L = L h = min [ W s , L s , λ D , L h ].As the magnetic field increases this applies as long as L < L H , while for L > L H the magnetic field sets thelimiting length. It controls the density of free vortices n F which determines the sheet resistance ( R ∝ n F ) aswell as the correlation length ( ξ ∝ n − / F ) at and above T c . Accordingly, the correlation length cannot growbeyond L . In this context it is important to recog-nize that the finite size scaling approach adopted hereis compatible with the Harris criterion, stating thatshort-range correlated and uncorrelated disorder is irrel-evant at the BKT critical point, contrary to approacheswhere the smearing of BKT criticality is attributed to aGaussian-like distribution of the bare superfluid-stiffnessaround a given mean value. In this context it shouldbe recognized that irrelevance of this disorder appliesto the universal properties, while the nonuniversal pa-rameters, including T c and the vortex core radius, maychange. The finite size effects stemming from the limitedextent of the homogeneous domains or the applied mag-netic field have a profound influence on the observationof the BKT behavior and have been studied intenselyin recent years. On the other hand, over theyears, consistency with BKT behavior has been reportedin thin films, and in systems exhibiting interfacialsuperconductivity.
Here we extend previous work and ana-lyze the sheet resistance data of Bi-films and theLaAlO /SrTiO interface using the finite size scalingformulas appropriate for the BKT transition, which in-clude multiplicative corrections when present. Thesesystems have been chosen because the data comprise thelow temperature limit, namely
T << T c where T c is theextrapolated BKT transition temperature attained in thelimit of an infinite limiting length L .The paper is organized as follows. In Sec. II wesketch the finite size scaling behavior of the sheet re-sistance adapted to the BKT critical point and presentthe correspondent analysis of the thickness tuned Bi-films and the gate voltage tuned LaAlO /SrTiO inter-face, in the presence and absence of a magnetic field,applied perpendicular to the film or interface. We ob-serve remarkable consistency with the finite size scalingpredictions. In the presence and absence of a magneticfield we identify a temperature range above the extrapo-lated T c where BKT behavior occurs. This temperaturerange is controlled by the relevant limiting length. Inzero magnetic field it is the extent of the homogeneousdomains. It turns out to decrease with the thickness d or gate voltage V g tuned reduction of T c ( d, V g ). Thesurvival of BKT behavior in applied magnetic fields im-plies a smeared sudden drop in the superfluid stiffness at T c ( H ), where it adopts the universal value given by theNelson-Kosterlitz relation. Recently, this behavior hasbeen observed in MoGe and InO x thin films by meansof low frequency measurements of the ac conductivity. Analogously, provided there is a temperature range above T c ( d, V g ) where BKT behavior is present, the smearedjump should also occur in zero field, as observed in var-ious films. An essential implication of the respec-tive limiting length is that the extrapolated phase tran-sition lines T c ( d, V g , H ) are unattainable. As a conse-quence the occurrence of BKT transitions is suppressedand with that the occurrence of quantum phase transi-tions in the limit T c ( d, V g , H ) → the lines T c ( d, V g , H ) exhibit the characteristic quantum criticalproperties. Additional implications of finite size scalingadapted to the BKT transition include: the magneticfield induced finite size effect generates a flattening outof the sheet resistance in the T → T = 0 only. The former prediction is confirmed in both, the Bi-films and the LaAlO /SrTiO interface, as well as in previous studies. The latter isconsistent with the LaAlO /SrTiO interface data, whilethe Bi-films exhibit a flattening out. Finally we explorethe limitations of the quantum scaling approach. II. THEORETICAL BACKGROUND AND DATAANALYSIS
Since only the motion of free vortices dissipate energy,the sheet resistance should be proportional to the freevortex density R ( T ) ∝ n F ( T ) . (1)On the other hand, dynamic scaling predicts therelationship R ( T ) ∝ ξ − z + ( T ) , (2)between the sheet resistance above T c and the corre-sponding correlation length ξ + ( T ) = ξ exp (cid:18) πbt / (cid:19) , t = T /T c − . (3) z is the dynamic critical exponent, the amplitude ξ isrelated to the vortex core radius and b is a nonuniversalparameter related to the vortex core energy. However,approaching T c from above, the aforementioned limitinglengths imply that the correlation length ξ + ( T ) cannotgrow beyond L = L h = min [ W s , L s , λ D , L h ]. Accordingto this a finite size effect becomes visible around T ∗ > T c where ξ + ( T ∗ ) ≃ L. (4)It leads to a characteristic size dependence of the sheetresistance Indeed, Eqs. (2) and (4) imply thatfor z = 2 at T ∗ > T c the sheet resistance adopts the sizedependence σ ( T ∗ ) σ = R R ( T ∗ ) = (cid:18) Lξ (cid:19) (5)To illustrate the experimental situation we consider nextthe sheet resistance data of Yen-Hsiang Lin et al . forBi films of various thickness and the heat conductancedata of Agnolet et al . for a 23 .
42 ˚A thick He film.Both, the sheet resistance in thin superconducting filmsand the heat resistance in He film are supposed to beproportional to the to the free vortex density n F so thataccording to Eq. (2) the respective conductance scales ofa homogeneous film with infinite extent scales for z = 2as σ ( T ) σ = R R ( T ) = exp (cid:16) b R t − / (cid:17) , (6)where b R = 4 π/b. (7)Supposing that the BKT regime is attainable, b R isnearly independent of film thickness, R and T c adoptthe appropriate values, the data plotted as σ ( T ) /σ vs t − / should then fall on the single curve exp (cid:0) b R t − / (cid:1) .In Fig. 1a we depicted this plot for the Bi-films. As t − / increases and with that T c is approached the data nolonger collapse, but run away and flatten out at σ ( T ) /σ values which increase with film thickness d . This be-havior points to a finite size effect where the correlationlength ξ + ( T ) cannot grow beyond the limiting length L so that Eq. (5) applies. As a result the flattening out iscontrolled by the ratio L /ξ which increases with filmthickness and T c . In Fig. 1b we plotted the thicknessdependence of R and of the extrapolated BKT tran-sition line T c ( d ). Apparently the decrease of T c withreduced film thickness points to a quantum phase transi-tion at a critical thickness dc where T c ( d c ) = 0.. Becausethe extrapolated BKT transition temperatures are notattainable due to the limiting length L, it follows thatthese transitions, as well as the possible quantum phasetransition at T c ( d c ) = 0 are suppressed. Nevertheless,slightly above T c , where the data tend to collapse on theBKT line, BKT fluctuations are present. This collapseattests the consistency with the universal and character-istic form of the BKT correlation length (Eq. (6)), whilethe nonuniversal parameters T c and R depend on thefilm thickness d (see Fig. 1b). The reduction of T c and R is attributable to disorder and quantum fluctuations.In particular, the strength of disorder is expected to in-crease with reduced film thickness d . To quantify thisexpectation we consider k F l = (cid:0) h/e (cid:1) /R n , (8)where k F denotes the Fermi wavenumber, l the electronmean free path, and R n the normal state sheet resistance.As disorder increases the mean free path l diminishes, k F l decreases and the strength of disorder increases. In theBi-films considered here k F l varies from 3 . d = 22 . . d = 23 .
42 ˚A. Accordingly, the strengthof the disorder increases substantially with reduced filmthickness or T c . Nevertheless, it does not affect the uni-versal BKT properties but renormalizes the nonuniversalparameters.To classify the relevance of the finite size effect in theBi-films we show in Fig. 2 the corresponding scalingplot of the thermal conductance of a He film. Althoughthe data attain the transition temperature rather closelythere is now sign of a flattening out up to t − / ≃ . . t − / . . T c . In Fig. 3 we de- d (A) 22.24 22.36 22.63 22.89 23.15 23.42 ( d , T ) / ( d ) t -1/2 a) R = / ( k ) d (A) b) T c ( K ) FIG. 1: (color online) (a) Normalized sheet conductance σ ( d, T ) /σ ( d ) of Bi films of thickness d vs t − / = ( T /T c − − / derived from Yen-Hsiang Lin et al . The solid line isthe BKT behavior σ ( d, T ) /σ ( d ) =exp( b R t − / ) for a ho-mogenous and infinite system with b R = 5. (b) Thicknessdependence of the extrapolated T c and R . picted R ( d, T ) /R vs T c ( d ) /T for the Bi films derivedfrom Yen-Hsiang Lin et al . As T approaches T c ( d ) thedata no longer collapse, but run away from the BKT be-havior and flatten out at R ( d, T ) /R ( d ) values whichdecrease with film thickness d . The flattening out ex-tending above T c ( d ) /T > T c ( d ) as well. However, below T c the dy-namic scaling relation (2) is no longer applicable becausethe correlation length is infinite there owing to the diver-gence of the susceptibility. The BKT theory predicts that below T c all vorticesare bound in pairs by the logarithmic vortex interaction,whereupon the linear sheet resistance is zero. Insteadthere is a nonlinear dependence of the voltage on cur-rent since the current can unbind weakly bound pairs. Contrariwise, in a finite sample there will be a popu-lation of free vortices at and below the vortex unbind-ing transition temperature T c . In this tempera-ture regime the linear relationship (1) between sheetresistance and free vortex density still applies, whileEq. (2), relating the sheet resistance to the correla- t h ( T ) / t h t -1/2 FIG. 2: (color online) Thermal conduction σ th ( T ) /σ th of a 23 .
42 ˚A thick He film vs t − / with T c = 1 . et al . The solid line is the BKTbehavior σ th /σ th =exp( b R t − / ) with b R = 1 .
762 and σ th =exp( − . . · − W/K. -4 -3 -2 -1 d (A) 22.24 22.36 22.63 22.89 23.15 23.42 R ( d , T ) / R ( d ) T c (d)/T FIG. 3: (color online) R ( d, T ) /R vs T c ( d ) /T for the Bi filmsderived from Yen-Hsiang Lin et al. The solid line is the BKT-behavior R ( T ) /R = exp( − b R ( T /T c − − / ) with b R = 5. tion length (Eq. (3)), applies at and above T c only.To provide a rough estimate of the free vortex densitywe note that at low temperatures the energy change re-sulting from adding a single vortex in a system of size L is given by ∆ E = ( J ( T ) / R π d Θ R Lξ RdR/R = πJ ( T ) ln ( L/ξ − ), where ξ is the vortex core radiusand J ( T ) = ~ ρ s ( T ) / m = d Φ / (cid:0) π λ ( T ) (cid:1) , (9)denotes the superfluid stiffness at low temperatures( T << T c ). An estimate for the free vortex density fol-lows then from the probability of finding a free vortex from the Boltzmann factor P ( T ) ∝ n F ( T ) ∝ exp( − ∆ E/k B T ) = ( ξ /L ) πJ ( T ) /k B T . (10)Using Eq. (1) we obtain, R ( T ) ∝ n F ( T ) ∝ ( ξ /L ) πJ ( T ) /k B T : T << T c . (11)Invoking the universal Nelson-Kosterlitz relation k B T c = π J (cid:0) T − c (cid:1) , (12)the temperature range of validity is then restricted to T << T c = πJ ( T − c ) / k B . As it should be, for an in-finite system, n F is zero for T ≤ T c . But if the lim-iting length L is finite, the free vortex density vanishesat zero temperature only. This implies an ohmic tail inthe IV characteristic below the extrapolated T c andimpedes a normal state to superconductor transition atfinite temperature in a strict sense. In this context it isimportant to recognize that the standard finite size scal-ing outlined above neglects the multiplicative logarithmiccorrections associated with BKT critical behavior. Arecent renormalization group treatment yields for z = 2and free boundary conditions R ( T ) ∝ ( ( ξ /L ) πJ ( T ) /k B T : L & ξ − ( T )( ξ /L ) / ln (( L lim /ξ ) /b ) : L . ξ + ( T ) , (13)where ξ − ( T ) = ξ exp b | t | / ! , (14)is a diverging length below T c . With Eq. (3) it fol-lows that this thermal length is much smaller than thecorrelation length ξ + ( T ) for the same | t | , because ξ + ( t ) /ξ = ( ξ − ( | t | ) /ξ ) π . (15)The parameter b is fixed by the initial conditions of therenormalization group equations, while the derivationof Eq. (11) identifies ξ as vortex core radius. Further-more, there is the upper bound b < L/ξ − because R ( T ) >
0. Taking the multiplicative logarithmic cor-rection into account Eq.( 5) transforms with Eq. (13)to R ( T c ) R = σ σ ( T c ) = (cid:18) ξ L (cid:19) L/ξ ) /b ) , (16)valid at T ≃ T c .Given R ( T c ) /R and b , estimates for L lim /ξ − arethen readily obtained. Fig. 4a depicts the T c and d dependence of R ( T c ) /R derived from Fig. 3, and theresulting T c dependence of L lim /ξ − is shown in Fig. 4bfor b = 0 . , . b valuessatisfy the lower bound b < L/ξ resulting from the re-quirement, R ( d, T c ) /R ( d ) >
0. Furthermore, b = 0 . b ≈ .
07, derived from large-scale nu-merical simulations. Striking features include the sub-stantial decline of the ratio between limiting length andvortex core radius,
L/ξ , with decreasing T c , and thecomparably low L/ξ <
80 values. Indeed, the run awayis controlled by the magnitude of
L/ξ . The He datashown in Fig. 2 do not exhibit a sign of flattening out upto σ th ( T ) /σ th = 10 , yielding with Eq. (5) the lowerbound L/ξ & . According to this, the run away ob-served in Fig. 1a and Fig. 3 stems from a limiting length L where the ratio L/ξ decreases with film thickness.Nevertheless, there is a temperature range where consis-tency with BKT behavior is observed, but in a strict sensea normal state to superconductor BKT transition is sup-pressed. As a consequence , there is also no film thicknessdriven quantum phase transition where the phase transi-tion line T c ( d ) ends at T c ( d c ) = 0 vanishes at a criticalfilm thickness d c , as could be anticipated from the thick-ness dependence of the extrapolated T c shown in Fig. 1b. -4 -3 -2 R ( d , T c ) / R ( d ) d (A)0.1 0.2 0.3 0.4 a) T c (K)0.1 0.2 0.3 0.4020406080 L / b - 1 0.1 0.05 T c (K) b) FIG. 4: (color online) (a) R ( d, T ) /R ( d ) vs T c and d derivedfrom the data shown in Fig. 3; (b) Estimates for the ratio L/ξ between correlation length and vortex core radius with-out the multiplicative logarithmic correction term ( (cid:13) ) andwith this correction for different b values entering Eq. 16. An essential issue left is the elucidation of the limiting length L min . In principle the magnetic field induced fi-nite size effect offers a direct estimate. A magnetic fieldapplied perpendicular to the film leads to the limitinglength L H = (cid:18) Φ aH (cid:19) / , (17)where a ≈ . T c . In analogy to Eq. (5) the sheetresistance is then expected to scale as R ( H, T c ) = 1 σ ( H, T c ) = fL H = af H Φ , (18)for z = 2 and low fields applied perpendicular to thefilm. In contrast to the zero field scaling form (13),this law holds below T c as well and the additive correc-tion to the leading power law dependence is weak. Themagnetic field induced finite sets then the limiting lengthas long as L H ∝ H − / < L whereby L H increases withdecreasing field and approaches L . Here a runaway fromthe scaling behavior (18) sets in at H ∗ providing for L the estimate L = (cid:18) Φ aH ∗ (cid:19) / . (19)In Fig. 5 we depicted the magnetic field dependence ofthe sheet conductivity of the 23 .
42 ˚A thick Bi film at T = 0 . . T c . Even though the data are rather sparsewe observe in a intermediate magnetic field range consis-tency with the predicted linear and nearly temperatureindependent field dependence of the sheet resistance. No,we focus on the low field behavior of the conductivityshown in Fig. 5. The run away from the 1 /H depen-dence of the sheet conductivity occurs around H = 0 . ≃ H ∗ , yielding with Eq. (19) for the limiting lengththe estimate L ≃ L min /ξ ≃
32, taken from Fig. 4b, we obtain forthe magnitude of the radius of the vortex core radius ξ ≃ . H = H c ≃ . σ vs 1 /H shown in Fig. 5 also reveals a nearlytemperature independent coefficient of proportionality e σ .It implies that the temperature dependence of the sheetresistance at fixed field flattens out, as observed in the23 .
42 ˚A thick Bi film, Analogous behavior was also ob-served in MoGe films, and Ta-films. in a field rangewhere the magnetic field induced finite size scaling ap-proach is no longer applicable. Indeed, in the MoGe filmsthe temperature independent sheet resistance obeys theempirical form σ ( H ) = σ exp ( − aH ) . (22)In the present case it applies according to Fig. 5 at bestabove the critical field only. The unusual empirical formwas attributed to dissipative quantum tunneling of vor-tices from one ”insulating” patch to another. -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 ( - ) H (T) T (K) 0.1 0.2
FIG. 5: (color online) Sheet conductivity σ of the 23 .
42 ˚Athick Bi-film vs magnetic field H at T = 0 . . et al . The solid line is Eq. 18in the from σ = e σ/H where e σ = 1 . · − (Ω − T). Thedashed line is Eq. 21 with σ = 4 . · − Ω − and a = 2 . − . The arrow indicates that this data point marks the zerofield value of the sheet conductivity. As the estimates for L min and ξ stem from rathersparse data a reliability check is inevitable. For this pur-pose we consider the temperature dependence of the cor-relation length ξ + (Eq. (3)) of the 23 .
42 ˚A thick Bi filmin terms of ξ + ( T ) vs t − / with ξ = 6 . ξ + growth with increasing t − / it approachesthe limiting length L = 208 ˚A at t − / ≃ .
38, the rangewhere in this film the run away from BKT behavior oc-curs (see Fig. 1a). Accordingly, we established for the23 .
42 ˚A thick Bi-film reasonable consistency between theestimates for the vortex core radius ξ and the limitinglength L , derived from the magnetic field induced finitesize effect, and the observed zero field behavior of thesheet resistance. Unfortunately, this estimation of ξ and L is restricted to this film because the magnetic field de-pendence of the sheet resistance appears to be missingfor the other films. In any case, the rather small limitinglength L = 208 ˚A points to an inhomogeneous film, withhomogeneous patches of dimension L = L h .In this context it should be kept in mind that there isthe Harris criterion, stating that short-range corre-lated and uncorrelated disorder is irrelevant at the unper- + ( T ) ( A ) t -1/2 FIG. 6: (color online) Correlation length ξ + = ξ exp (cid:16) π ( bt ) − / (cid:17) vs t − / of the 23 .
42 ˚A thick Bi-film with ξ = 6 . π/b = b R / .
5. The dashedline marks L = 208 ˚A. The crossing point at t − / ≃ . T c /T ≃ . turbed critical point, provided that ν > /D , where D isthe dimensionality of the system and ν the critical expo-nent of the finite-temperature correlation length. With D = 2 and ν = ∞ , appropriate for the BKT transition, this disorder should be irrelevant. Given the irrelevanceof disorder, the reduction of the ratio L/ξ with reducedfilm thickness or transition temperature (see Fig.4b) isthen attributable to: (i) increasing vortex core radius ξ with reduced T c combined with a thickness independent L ; (ii) a limiting length L which decreases with film thick-ness combined with a T c independent ξ ; (iii) a thicknessdependence of both, L and ξ , such that the ratio L/ξ decreases with reduced transition temperature. Becausethe vortex core radius is known to increase with reduced T c as ξ ∝ T − /zc with z = 2, we are left with option(i) and (iii). In order to discriminate between these op-tions we estimate ξ ( T c ) from the respective data for the23 .
42 ˚A thick Bi film, namely ξ = 6 . T c = 0 . ξ = gT − / c with g = 4 .
19 ˚AK / . Therough estimates for the thickness and T c dependence of L shown in Fig.7 are then readily obtained from the L/ξ values depicted in Fig. 4b. In spite of the small totalthickness increment of 1 .
18 ˚A there is a strong thick-ness dependence of L , ranging from 50 ˚A to 200 ˚A. Di-rect experimental evidence for superconducting patcheswith an extent of 100 ˚A embedded in an insulating back-ground stems from scanning tunneling spectroscopy in-vestigations on TiNi and InO x films. However, itshould be kept in mind that transport measurements aresensitive to the phase and tunneling experiments to themagnitude of the order parameter. Furthermore, scan-ning SQUID measurements at the interface LaAlO /SrTiO uncovered superconducting regions occupying only asmall fraction of the areas measured. In addition thereare magnetic regions with patches of ferromagnetic re-gions coexisting with a higher density of much smallerscale domains of fluctuating local magnetic moments. L ( A ) d (A)0.1 0.2 0.3 0.4T c (K) FIG. 7: (color online) T c and film thickness dependence ofthe limiting length L of the Bi-films derived from the L/ξ estimates shown in Fig. 4b for b = 0 .
05 and ξ = gT − / c with g = 4 .
19 ˚AK / . To explore the finite size scenario further we turn tothe interface between LaAlO and SrTiO , two excellentband insulators. It was shown that the electric-field ef-fect can be used to map the phase diagram of this inter-face system revealing, depending on the gate voltage, asmeared BKT transition and evidence for quantum crit-ical behavior. Here we revisit the analysis of the tem-perature and gate voltage dependence of the sheet re-sistance data by invoking the approach outlined above.In Fig. 8a we depicted R ( V g , T ) /R vs T c ( V g ) /T andin Fig. 8b the gate voltage dependence of the extrapo-lated transition temperature T c and amplitude R . As T c ( V g ) /T increases Fig. 8a uncovers a flow to and awayfrom the BKT behavior. As T c ( V g ) /T decreases for fixed T c the BKT regime is left, while the rounding of the tran-sition leads with increasing T c ( V g ) /T to a flow away fromcriticality. Nevertheless, in an intermediate T c ( V g ) /T regime the data tend to collapse on the characteristicBKT line. Thus, in analogy to the Bi-films, the collapseattests again consistency with the universal and charac-teristic form of the BKT correlation length (Eq. (6)),while the nonuniversal parameters T c and R depend inthe present case on the gate voltage (see Fig. 8b). Theirreduction points to the occurrence of a gate voltage tunedquantum phase transition around V g ≃ −
100 V wherethe extrapolated transition temperature vanishes. UsingEq. (8) we find that k F l varies from 8 . V g = 80 Vto 13 . V g = +80 V. Accordingly, disorder is present,its strength is comparable to that in the Bi-films butincreases only slightly by approaching the extrapolatedquantum phase transition. In any case, it does not af-fect the universal BKT properties but renormalizes thenonuniversal parameters.To unravel the consistency of the rounded transitionswith a finite size effect, we invoke Eq. (16) to estimate -7 -6 -5 -4 -3 -2 -1 R ( V g , T ) / R ( V g ) T c (V g )/T V g (V) -80 -60 -20 0 +20 +40 +60 +80 a) -120 -100 -80 -60 -40 -20 0 20 40 60 800.000.050.100.150.200.25 T c ( K ) V g (V) b) R ( k ) FIG. 8: (color online) (a) Normalized sheet resistance R ( V g , T ) /R ( V g ) vs T c ( V g ) /T of the LaAlO /SrTiO in-terface at various gate voltages derived from Caviglia et al. The solid line marks the BKT behavior R ( V g , T ) /R ( d ) =exp (cid:16) − b R ( T /T c − − / (cid:17) for a homogenous and infinite sys-tem with with b R = 3 .
43. (b) Gate voltage dependence of theextrapolated transition line T c ( V g ) and R ( V g ). The solidand dashed lines indicate the approach of T c and R to theextrapolated quantum phase transition. the ratio L min /ξ . Fig. 9a shows the T c and d depen-dence of R ( V g , T c ) /R ( V g ) derived from Fig. 8a. Theresulting T c dependence of L/ξ − is shown in Fig. 8bfor b = 0 .
05 and 0 . b = 0 .
05 is comparable to b ≈ .
07, derived from large-scale numerical simulations. In analogy to the Bi-films,important features include the substantial decline
L/ξ with decreasing T c , and the comparably low values of L/ξ , namely
L/ξ <
100 compared to the lower bound
L/ξ & emerging from the He data shown in Fig.2. According to this and in analogy to the Bi-films therun away from BKT behavior as observed in Fig. 9 isattributable to a limiting length L where the ratio L/ξ decreases with reduced T c . Nevertheless, there is a tem-perature range where consistency with BKT behavior isobserved, but in a strict sense a normal state to super-conductor BKT transition is suppressed. -100 -80 -60 -40 -20 0 20 40 60 80 100 -5 -4 -3 -2 -1 a) R ( T c ) / R V g (V)0.00 0.05 0.10 0.15 0.20T c (K)0.00 0.05 0.10 0.15 0.200100200300 b - 0.1 0.05 L / T c (K) b) FIG. 9: (color online) (a) R ( V g , T ) /R ( V g ) vs T c and gatevoltage V g of the LaAlO /SrTiO interface derived from thedata shown in Fig. (8); (b) Estimates for the ratio L/ξ between correlation length and vortex core radius withoutthe multiplicative logarithmic correction term ( (cid:13) ) and withthis correction for different b values entering Eq. 16. An independent confirmation of the finite size scenariodemands the magnitude of L , allowing to determine ξ and with that the temperature dependence of the correla-tion length ξ + ( T ), as well as ξ + ( T ∗ ) = L , where the runaway from BKT behavior should occur. Given the pre-vious estimate derived from the magnetic field inducedfinite size effect L ≃
490 ˚A, (23)for a LaAlO /SrTiO interface with T c ≃ .
21 we obtainwith
L/ξ ≃ ξ ≃ . ξ + ( T ) vs t − / .As the correlation length cannot grow beyond L the runaway from BKT behavior should occur around the cross-ing point between ξ + ( T ) and L at t − / ≃ .
69 corre-sponding to T c /T ≃ .
88. A glance at Fig. 8a reveals that around this value the data of the LaAlO /SrTiO in-terface at gate voltage V g = 80 V ( T c ≃ . /SrTiO in-terface, attributable to a finite size effect stemming froma limiting length L . In the samples with highest T c itsdimension is L ≃
208 ˚A in the Bi-films and L ≃
490 ˚Ain the LaAlO /SrTiO interface. - ( T ) ( A ) t -1/2 FIG. 10: (color online)Correlation length ξ + ( T ) = ξ exp (cid:16) π/ (cid:16) bt / (cid:17)(cid:17) vs t − / of the LaAlO /SrTiO interfacewith T c ≃ .
21 K for ξ = 4 . π/b = b R / . L = 208 ˚A. The crossing point at t − / ≃ .
69 corresponds to T c /T ≃ . Next we turn to the finite size behavior below theextrapolated transition temperature. Here the limitinglength L prevents the thermal length ξ − ( | t | ) to diverge.But compared to ξ + ( | t | ) the thermal length is muchsmaller for the same | t | (Eq. (15)). For this reason L & ξ − ( T ) is expected to hold already slightly below T c . In this regime the sheet resistance is controlled bythe free vortex density where Eq. (13) rewritten in theformln ( R ( T )) = r − s ( T ) T , s ( T ) = πJ ( T ) k B ln Lξ (25)applies. Accordingly, the coefficient s ( T ) controls devi-ations from the 1 /T temperature dependence. At zerotemperature the superfluid stiffness given by Eq. (9)is fixed by the magnetic penetration depth in termsof J ( T = 0) ∝ d/λ ( T = 0), expected to vanish as J ( T = 0) ∝ d/λ ( T = 0) ∝ T c . On the other hand,approaching T c from below, the superfluid stiffness tendsaccording to Eq. (12) to J ( T − c ) = 2 k B T c /π . In additionin both, the Bi-films (Fig. 4b) and the LaAlO /SrTiO interface (Fig. 9b)), ln( L/ξ ) decreases with reduced T c .As a consequence the magnitude of s ( T ) is expected todecrease with reduced T c . In Fig. 11, showing ln( R )vs 1 /T of the LaAlO /SrTiO interface for various gatevoltages, we observe that this supposition is well con-firmed. On the other hand, in the temperature regime ofinterest the data exhibit jitter masking the characteris-tic temperature dependence of the superfluid stiffness in s ( T ). Indeed, the straight lines, corresponding to thenearly temperature independent s ( T ) ≈ T c ln ( L/ξ ),describes the data quite well. To evidence the smearedBKT transition we included in Fig. 11 the characteristicBKT temperature dependence (6) in terms of the dash-dot-dot line. Additional confirmation of this finite sizescenario below T c stems from the observation of an ohmicregime at small currents because it uncovers accordingto Eq. (1) the presence of free vortices. The importantimplication then is: although BKT behavior is observ-able in an intermediate temperature regime above theextrapolated T c , in a strict sense a BKT transition doesnot occur. It is smeared out and the sheet resistance van-ishes at zero temperature only because Eq. (25) reducesin the zero temperature limit to R ( T ) = r exp − (cid:18) πJ ( T = 0) k B T ln L lim ξ (cid:19) = r (cid:18) ξ L min (cid:19) πJ ( T =0) kBT . (26) l n ( R ()) V g (V) -80 -60 -20 0 +20 +40 +60 +80 -1 ) FIG. 11: (color online) ln( R ) vs 1 /T of the LaAlO /SrTiO interface for various gate voltages. The straight lines are Eq.(25): dashed line: V g = −
60 V with r = 5 .
64 and s ( T ) =0 .
044 K; dash-dot-dot line: V g = −
20 V with r = 8 .
78 and s ( T ) = 0 .
87 K; full line: V g = +20 V with r = 9 .
06 and s ( T ) = 1 . V g = +60 V with r = 9 . s ( T ) = 1 . /T c . The dash-dot line marks the BKT behavior (6) at V g = −
20 V with R = 44 kΩ, b R = 3 .
43 and T c = 0 .
119 K.
Contrariwise, the sheet resistance of the Bi-films shownin Fig. 3 does not exhibit a significant temperature de-pendence below T ≈ T c / T ≈ T c /
10. To disen-tangle the scaling regimes below T c more quantitatively,we note that the plot R ( T ) /R vs T c /T should exhibita crossover from a temperature dependent to a tempera-ture independent regime at T ∗ where the diverging length ξ − ( T ) equals the limiting length L min . According to Eqs.(13) and (14) T ∗ follows from Lξ = ξ − ( T ∗ ) ξ = exp b (1 − T ∗ /T c ) / ! . (27)To estimate T ∗ we show in Fig. 12 the temperature de-pendence of ξ − ( T ) in terms of ξ − ( T ) /ξ vs T /T c for theBi-films and the LaAlO /SrTiO interface. Noting thatthe minimum value of L/ξ in the Bi-films is around 3 . /SrTiO interface aroundaround 5 (Fig. 9b) it becomes clear that in both sys-tems T ∗ is close and slightly below T c . As a result, thetemperature regime where ξ − ( T ) > L lim holds is re-stricted to temperatures very close to T c only, while theregime where ξ − ( T ) < L applies sets in slightly below T c . It is the regime where the sheet resistance adoptsthe characteristic temperature dependence given by Eq.(25). A glance at Fig. 11, showing ln( R ) vs 1 /T of theLaAlO /SrTiO interface, uncovers agreement with thistemperature dependence, while the sheet resistance ofthe Bi-films shown in Fig. 3 does not exhibit a signifi-cant temperature dependence below T ≈ T c /
2. Takingthe saturation of the sheet resistance in the BI-films forgranted it implies the breakdown of the BKT behaviorbelow T c , while it applies above T c . The breakdown maythen be a clue that below T c a process is present whichdestroys BKT behavior. On the other hand we have seenthat the LaAlO /SrTiO interface data is at and below T c remarkably consistent with the predicted finite sizeBKT behavior. However, the absence of BKT behav-ior below T c is inconsistent with measurements of thesuperfluid stiffness, uncovering a smeared Nelson-Kosterlitz jump near T c and the presence of superfluid-ity down to the lowest attained temperatures. Given theodd behavior of the Bi-films it should be kept in mindthat a failure to cool the electrons in the low temperaturelimit also implies a flattening of the sheet resistance. Finally, to explore the implications of a magnetic fieldinduced finite size effect below T c we depicted in Fig.13a the temperature dependence of the sheet resistanceof a LaAlO /SrTiO interface with T c ≃ .
19 K at var-ious magnetic fields. Although the data exhibit jitter inthe low field limit the predicted saturation of the sheetresistance in the T → H = 30 mT a crossover to the empiricalform (22) can be surmised as the crossing point of theisotherms around H c = 110 mT is approached. Thiscrossing point is the direct consequence of the fact thatin the covered T range R decreases with decreasing T for H < H c , increases with decreasing T for H > H c ,and is T independent at H c . Noting that the scalingform (18) presumes that density fluctuations are small, which is true for large limiting lengths L H = (Φ /aH ) / ,but not for small, it becomes clear that the applicability0 ( T ) / T c /T LaAlO /SrTiO Bi FIG. 12: (color online) ξ − ( T ) /ξ =exp (cid:16) / (cid:16) b (1 − T /T c ) / (cid:17)(cid:17) vs T /T c for the Bi-films with 1 /b = b R / π ≃ .
398 and theLaAlO /SrTiO interface with 1 /b = b R / π ≃ . L/ξ . L/ξ ≃ . L/ξ ≃ /SrTiO interface(Fig. 9b). of this approach is limited to the low field limit. An-other essential feature emerging from Fig. 13a is theshift of the sheet resistance curves to lower temperatureswith increasing magnetic field. This behavior uncoversthe pair breaking effect of the magnetic field leading ina mean-field treatment to a reduction of T c accordingto T c ( H = 0) − T c ( H ) ∝ H . Adopting the finitesize point of view this behavior relies on the fact that anapplied magnetic field sets an additional limiting length L H = (Φ /aH ) / , giving rise to a smeared BKT tran-sition at a fictitious BKT transition temperature T c ( H )below T c ( H = 0). Contrariwise, in the standard finitesize effect one attains T c in the L → ∞ limit only. Toquantify this option we performed fits to the characteris-tic BKT form (6) of the sheet resistance. A glance at Fig.13a reveals, in analogy to the zero field case (Fig. 8a),agreement in an intermediate temperature range below T c ( H ).Given the consistency with the BKT expression (6)and Fig. (13a) estimates for the fictitious lines T c ( H )and R ( H ) are readily obtained and shown in Fig. 14. T c ( H ) extrapolates to zero around H c = 110 mT wherethe isotherms cross. This behavior suggests a magneticfield induced quantum phase transition where supercon-ducting behavior is lost at zero temperature and the am-plitude R approaches the critical value R ( H c ) ≃ T =0 . T c ( H ) has properties compatiblewith a quantum critical point, where T c = T ( H c − H ) zν applies. z is the dynamic and ν the critical exponent ofthe zero temperature correlation length. The power lawfit included in Fig. 14 yields zν = 1 . ± .
1. It is interest-ing to note that this value is comparable with transportstudies including MoGe, Nb . Si . , , InO x , , andLaAlO /SrTiO interface samples, though these stud- -1 T (K) R () H(mT) 0 10 20 30 40 50 60 80 a) -4 -3 -2 -1 T=0.05 K ( - ) H (mT) b) FIG. 13: (color online) (a) Temperature dependence of thesheet resistance of a LaAlO /SrTiO interface with T c ≃ .
19K at various magnetic fields applied perpendicular to the in-terface taken from Reyren et al . The solid lines are fits tothe BKT form (6) of the sheet resistance with b R = 3 .
43 yield-ing for T c and R the estimates shown in Fig. 14 ; (b) Sheetconductivity vs H at T = 0 .
05 K. The solid line is the empir-ical form (22) with σ = 6 .
79 Ω − and a = 0 .
099 mT − . Thedashed line is Eq. (18) in the from σ = e σ/H where e σ = 8(Ω − mT). ies have limited their analysis to exclude resistance datashowing flattening in the zero temperature limit. In anycase, BKT behavior occurs in an intermediate tempera-ture range only. The extrapolated BKT line T c ( H ) is notattainable because the magnetic field induced finite sizeeffect (Eq. 18)) generates, as observed in Fig. 13a, theflattening out of the sheet resistance in the T → T c ( H ), where the superfluid stiffness adopts theuniversal value given by the Nelson-Kosterlitz relation(12). Recently, this behavior has been observed in MoGeand InO x thin films by means of low frequency measure-ments of the ac conductivity. Although the low fre-quency f = 20 kHz implies an additional limiting length,namely L f ∝ f − / , the magnetic field dependence ofthe blurred Nelson-Kosterlitz jump has been clearly de-1tected and the power law fits to T c ( H ) yielded for zν theestimates 1 . ± .
25 for MoGe and 1 . ± . x . T c ( K ) H (mT) 110100 R ( k ) FIG. 14: (color online) Estimates for T c and R resulting fromthe fits to the BKT form (6) of the sheet resistance included inFig. 13a. The solid line is T c = T ( H c − H ) zν with T = 3 · − (KmT) /zν , H c = 110 mT, zν = 1 . ± . R = R c + R ( H c − H ) ν with R c = 0 .
96 kΩ , R = 0 . / ν , and 2 ν = 2 .
78. These lines indicate the approachto the extrapolated quantum critical point.
Lastly we consider the limitations of the quantum scal-ing form R ( H, T ) = R c G ( x ) , x = H c − HT /zν , (28)applicable close to the quantum critical point. G ( x ) isa scaling function of its argument and G (0) = 1. Itis essentially a finite size scaling function. Indeed at fi-nite temperatures is the divergence of the zero temper-ature correlation length ξ ( T = 0) ∝ ( H c − H ) − ν cut-off by the thermal length L T ∝ T − /z so that x ∝ ( L T /ξ ( T = 0)) /ν ∝ ( H c − H ) /T /zν T . The data for R ( H, T ) plotted vs x = ( H c − H ) /T /zν should thencollapse on a single curve. On the other hand BKT be-havior uncovered in Fig. 13a implies the scaling form (6)rewritten in the form R ( H, T ) = R ( H ) exp (cid:16) − b R / ( T /T c ( H ) − / (cid:17) , (29)where T c ( H )= T ( H c − H ) zν is the transition line shownin Fig. 14. Noting that TT c ( H ) = 1 T x zν , (30)BKT behavior leads with Eqs. (28) and (29) to the ex-plicit scaling form R ( H, T ) = R ( H ) exp (cid:18) − b R / (cid:16)(cid:0) T x zν (cid:1) − − (cid:17) / (cid:19) , (31) valid for any T /T c ( H ) = (cid:0) T x zν (cid:1) − > R ( H, T ) vs z = ( H c − H ) /T /zν obtained from theLaAlO /SrTiO interface sheet resistance data shown inFig. 13a is depicted in Fig. 15a. For comparison we in-cluded the BKT scaling form (31). Apparently, the datado not collapse on a single curve because the amplitude R exhibits a pronounced field dependence (see Fig. 14)and the sheet resistance flattens out for large and smallvalues of the scaling argument z . For fixed H c − H this re-flects the observed flattening out of the sheet resistancein the low and high temperature limits (Fig. 13a). Aglance at Fig. 15b reveals that an improved data col-lapse is achieved by taking the field dependence of theamplitude R into account. Clearly, the flattening outfor small and large z values remains. Noting that forfixed H c − H small z values are attainable at rather hightemperatures only, the respective saturation reflects thefact that in this temperature regime BKT fluctuations nolonger dominate. On the other hand large scaling argu-ments require the incidence of the zero temperature limitwhere the magnetic field induced finite size effect leadsto a flattening out in the temperature dependence andwith that in the z dependence of the sheet resistance inthe z → ∞ limit. Furthermore, the field dependence ofthe amplitude R also implies that the quantum scalingform holds in a unattainable regime close to quantumcriticality only. III. SUMMARY AND DISCUSSION
We analyzed sheet resistance data of thin Bi-films andthe LaAlO /SrTiO interface near the onset of super-conductivity to explore the compatibility with BKT be-havior. On the Bi-films the onset temperature has beentuned by the film thickness, while on the LaAlO /SrTiO interface the gate voltage and the magnetic field, appliedperpendicular to the interface, acted as tuning parame-ter. Noting that BKT behavior involves the transitionfrom a low-temperature state in which only paired vor-tices exist to a high-temperature state in which free vor-tices occur, we demonstrated that finite size induced freevortices below T c prevent the occurrence of a BKT tran-sition in a strict sense. This does not mean, however,that the BKT vortex-unbinding mechanism does not oc-cur and is not observable. Indeed our finite size analysisrevealed that BKT behavior is present in an intermediatetemperature range above the extrapolated BKT transi-tion temperature. This temperature range depends onthe magnitude of the limiting length L while the extrap-olated transition temperature corresponds to the limit L → ∞ . Limiting lengths include he effective magneticpenetration depth λ D = λ /d , the dimension L h ofthe homogeneous domains in the sample, the magneticlength L H ∝ (Φ /H ) / , and in the case of ac mea-surements L f ∝ f − / . L =min[ λ D , L h , L H , L f ] con-trols the density of free vortices n F which determines2 R ( k ) z a) b) H (mT) 50 60 80 90 100 R / R z FIG. 15: (color online) (a) Scaling plot R ( H, T ) vs z =( H c − H ) /T /zν with H c = 110 mT, zν = 1 .
92, and b R = 3 .
43. The solid lines mark the respective BKT scal-ing form (31) with R ( H ) taken from Fig. 14 and T =2 · − (KmT) /zν . (b) Scaling plot R ( H, T ) /R ( H ) vs z = ( H c − H ) /T /zν . The solid line is the BKT scaling form(31). the sheet resistance ( R ∝ n F ) as well as the correlationlength ( ξ ∝ n − / F ) at and above T c . In this temperaturerange the limiting lengths prevent the correlation lengthto diverge. Concentrating on the dc sheet resistance weanalyzed the data using finite size scaling formulas ap-propriate for the BKT transition. The main results for zero magnetic fields include:Above T c we observed in an intermediate temperaturerange consistency with the characteristic BKT behaviorand a thickness or gate voltage dependent BKT tran-sition temperature T c (Figs. 1a and 8a). However, inanalogy to finite systems, the measured sheet resistancedoes not vanish at T c . In this context it should be kept inmind that there is the Harris criterion, stating thatshort-range correlated and uncorrelated disorder is irrel-evant at the unperturbed critical point, provided that ν > /D , where D is the dimensionality of the systemand ν the critical exponent of the finite-temperature cor-relation length. With D = 2 and ν = ∞ , appropriatefor the BKT transition, this disorder should be irrele- vant. Accordingly, the nonvanishing sheet resistance at T c points to a finite size induced smeared BKT transition.Invoking the finite size scaling formula for the sheet resis-tance at T c we obtained estimates for the T c dependenceof the ratio between the limiting length and the vortexcore radius, namely L/ξ (Figs. 3b and 9b). Striking fea-tures included the substantial decline of L/ξ | max ≈ with decreasing T c and in comparison with L/ξ & in He the low value of
L/ξ | max . This difference and the T c dependence of L/ξ imply enhanced smearing of theBKT transition with reduced T c as observed (Figs. 1a,3, and 8a). To disentangle the T c dependence of the lim-iting length L and the vortex core radius ξ we invokedthe magnetic field induced finite size effect allowing toestimate the limiting length directly from magnetic fielddependence of the sheet conductivity at fixed tempera-ture below T c . Unfortunately, in both the Bi-films andthe LaAlO /SrTiO interface, the necessary data is avail-able for the samples with highest T c only. For the 23 . L ≃
208 ˚A, ξ ≃ . /SrTiO inter-face with T c ≃ .
21 K the estimates L ≃
490 ˚A, ξ ≃ . and InO x films, as well as with scanning SQUID mea-surements at the interface LaAlO /SrT iO . To disen-tangle the T c dependence of L and ξ we used the em-pirical relationship ξ ∝ T − /zc with z = 2, revealingthat the extent of the homogenous domains decreasessubstantially with reduced T c (Fig.7). Accordingly theenhanced smearing of the BKT transition with reduced T c was traced back to the reduction of the limiting length L and the increase of the vortex core radius ξ with de-creasing T c .In the low temperature limit and zero magnetic fieldwe observed on the LaAlO /SrTiO interface consistencywith the characteristic finite size scaling form (25) whilethe Bi-films do not exhibit a significant temperature de-pendence below T ≈ T c /
2. Taking the saturation ofthe sheet resistance in the BI-films for granted it im-plies the breakdown of BKT finite size scaling below T c ,while it applies above T c . The breakdown may then bea clue that below T c a process is present which destroysBKT behavior. On the other hand we have seen that theLaAlO /SrTiO interface data is at and below T c remark-ably consistent with the predicted finite size BKT predic-tions. In addition, an absence of BKT-behavior below T c is also incompatible with measurements of the superfluidstiffness, uncovering a smeared Nelson-Kosterlitz jump near T c and the presence of superfluidity down tothe lowest attained temperatures.Subsequently we explored the implications of the mag-netic field induced finite size effect. Considering thetemperature dependence of the sheet resistance at var-ious magnetic fields, applied perpendicular to the inter-face of LaAlO /SrTiO , we observed in an intermedi-3ate temperature range remarkable consistency with thecharacteristic BKT form (6)(Fig. 13a). Fits yielded thefictitious transition line T c ( H ) extrapolating to zero at H c ≃
110 mT where a quantum phase transition is ex-pected to occur (Fig. 14). Indeed, T c ( H ) revealed prop-erties compatible with a quantum critical point, nearwhich T c = T ( H c − H ) zν applies. z is the dynamic and ν the critical exponent of the zero temperature correlationlength. A power law fit yielded zν = 1 . ± .
1. How-ever, this extrapolated line is not attainable because themagnetic field induced finite size effect (Eq. (18)) gen-erates the observed flattening out of the sheet resistancein the T → The survivalof BKT behavior in applied magnetic fields also implies asmeared sudden drop in the superfluid stiffness at T c ( H ),where it adopts the universal value given by the Nelson-Kosterlitz relation (12). Recently, this behavior has beenobserved in MoGe and InO x thin films by means of lowfrequency measurements of the ac conductivity. A key question our analysis raises is whether the ho-mogeneity of 2D superconductors can be improved toreach the quality of He films. Analyzing the sheet resis-tance data of Bi-films and the LaAlO /SrTiO we haveshown that the data are consistent with a finite size ef-fect attributable to the limited homogeneity of the sam-ples. The limited length of the homogenous domainsimpedes the occurrence of a BKT and quantum phasetransitions in the strict sense of a true continuous phasetransition. However, this strict interpretation of the def-inition of a continuous phase transition does not implythat the BKT vortex-unbinding mechanism is not ob-servable and the reduction of the extrapolated T c doesnot reveal properties compatible with a quantum crit-ical point. Indeed, notwithstanding the comparativelysmall dimension of the homogeneous domains, our finitesize analysis revealed reasonable compatibility with BKTand quantum critical point behavior. However, the re-duction of the limiting length with decreasing T c is anessential drawback (Fig. 7). Furthermore, consideringthe expected magnetic field tuned quantum phase tran-sition in the LaAlO /SrTiO interface, it was shown thatthe standard quantum scaling form (28) of the sheet re-sistance applies very close to the unattainable quantumcritical point only (Fig. 15). Indeed, combining the BKTexpression for the sheet resistance with the quantum scal-ing form of the extrapolated transition line T c ( H ), we de-rived the explicit scaling relation (31) uncovering the lim-itations of the standard quantum scaling form. Its main drawback was traced back to the neglect of the magneticfield dependence of the critical amplitude R which variessubstantially by approaching the critical value R c (Fig.14).Finally it should be noted that the finite size scal-ing approach adopted here is compatible with the Harriscriterion, stating that short-range correlated and un-correlated disorder is irrelevant at the BKT critical point,contrary to approaches where the smearing of the BKTtransition is attributed to a Gaussian-like distribution ofthe bare superfluid-stiffness around a given mean value. The irrelevance of this disorder implies, that the univer-sal BKT properties still apply, while the nonuniversalparameters, including T c , the vortex core radius ξ andthe amplitude R , may change. Contrariwise, the rel-evance of disorder at the extrapolated quantum phasetransition, separating the superconducting and metal-lic phase, depends on the universality to which it be-longs. The relevance of disorder is again controlled bythe Harris criterion: if the zero-temperature correla-tion length critical exponent fulfils the Harris inequality ν > /D = 1 the disorder does not affect the quantumcritical behavior. Conversely, if ν < /D = 1 disorder isrelevant and affects the nonuniversal parameters R and T c in the BKT form (2) of the sheet resistance and in par-ticular the reduction of T c . In the magnetic field tunedcase is the field dependence of R and T c attributableto Cooper pair breaking. However, another importantfeature of the of LaAlO /SrTiO interface is the largeRashba spin orbit interaction which originates from thebroken inversion symmetry. It has been shown that itsmagnitude increases with reduced T c , suggesting thatpair breaking occurs in zero magnetic field as well. In-deed,torque magnetometry measurement revealed thatthe LaAlO /SrTiO interface has a magnetic moment,which points in the plane, and has an onset temperaturethat is at least as high as 40 K and persists below theBKT transition temperature. IV. ACKNOWLEDGEMENTS
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