Nonlocal versus local vortex dynamics in the transversal flux transformer effect
Florian Otto,1,
*
Ante Bilušić,1,2 Dinko Babić,3Denis Yu. Vodolazov,4Christoph Sürgers,5and Christoph Strunk11Institute for Experimental and Applied Physics, University of Regensburg, D-93025 Regensburg, Germany
2Faculty of Natural Sciences, University of Split, N. Tesle 12, HR-21000 Split, Croatia
3Department of Physics, Faculty of Science, University of Zagreb, Bijenička 32, HR-10000 Zagreb, Croatia
4Institute for Physics of Microstructures, Russian Academy of Sciences, 603950, Nizhny Novgorod GSP-105, Russia
5Karlsruhe Institute of Technology (KIT), Physikalisches Institut and DFG-Center for Functional Nanostructures, P.O. Box 6980, D-76049 Karlsruhe, Germany
共Received 24 March 2010; published 20 May 2010兲
In this follow up to our recent letter关F. Ottoet al., Phys. Rev. Lett. 104, 027005共2010兲兴, we present a more detailed account of the superconducting transversal flux transformer effect共TFTE兲 in amorphous共a-兲NbGe nanostructures in the regime of strong nonequilibrium in local vortex motion. Emphasis is put on the relation between the TFTE and local vortex dynamics, as the former turns out to be a reliable tool for determining the microscopic mechanisms behind the latter. By this method, a progression from electron heating at low tem- peratures T to the Larkin-Ovchinnikov effect close to the transition temperature Tc is traced over a range 0.26ⱕT/Tcⱕ0.95. This is represented by a number of relevant parameters such as the vortex transport entropy related to the Nernst-type effect at lowT and a nonequilibrium magnetization enhancement close toTc. At intermediateT, the Larkin-Ovchinnikov effect is at high currents modified by electron heating, which is clearly observed only in the TFTE.
DOI:10.1103/PhysRevB.81.174521 PACS number共s兲: 74.25.Uv, 74.25.F⫺, 74.78.Na
I. INTRODUCTION
Applying a transport currentIto a type-II superconductor in the mixed state may result in vortex motion and power dissipation if the driving forcefdron vortices共per unit vortex length d兲 exceeds the pinning force. For a homogeneous mixed state, fdris given by the Lorentz force fL=j0, where j is the transport current density and 0 the magnetic flux quantum. When effects related to jleave thermodynamics of the mixed state unchanged, which happens at low j, any nonlinearity in the voltage 共V兲 vs I curves is caused by a competition between fL and the pinning force. Further in- crease in j not only enhances fL but can also change the thermodynamic properties ifjbecomes large enough.1,2Such a strong nonequilibrium 共SNEQ兲 corresponds to a mixed state that is distinct from its low-j counterpart. This difference—and not the pinning force—then leads to nonlin- ear, or even hysteretic, V共I兲 in measurements over a wide range ofI.1–4
The SNEQ mixed state has different backgrounds at low T and at high T. At low T, as modeled by Kunchur,2 the electron-phonon collisions are too infrequent to prevent elec- tron heating共EH兲to a temperatureTⴱabove the phonon tem- perature T0, which leads to a thermal quasiparticle distribu- tion function that is set byTⴱrather thanT0. This causes an expansion of vortex cores. Close toTc, the dominant effect is the time variation in the superconducting order parameter⌬ while the heating is negligible and the distribution function acquires a nonthermal form as calculated by Larkin and Ovchinnikov 共LO兲.1 In consequence, vortex cores shrink. A detailed consideration ofV共I兲in the two regimes3,4supported that: EH was identified at low T and the LO effect close to Tc. However, this conclusion relied on a somewhat intricate numerical analysis, which called for a more obvious proof of viability in order to rule out other possible scenarios.5
Recently, an alternative experiment provided a stronger support to the picture outlined above. This evidence came from dc measurements of the transversal flux transformer effect共TFTE兲—the latter was introduced by Grigorievaet al.
in Ref.6—in a sample ofa-NbGe.7,8The TFTE is a nonlocal phenomenon where the voltage response Vnl, representative of vortex velocity, to a localI in a mesoscopic film is mea- sured in a remote region where I= 0. In the TFTE, the flux coupling is transversal to the magnetic inductionB共perpen- dicular to the film plane兲and is caused by the in-plane re- pulsive intervortex interaction, which complements the lon- gitudinal flux transformer effect of Giaever9where the flux is coupled alongBover an insulating layer. First reports on the TFTE referred to lowIboth in low-frequency ac共Ref.6兲and dc共Ref.10兲measurements, where it was found thatVnlwas odd inI, i.e.,Vnl共−I兲= −Vnl共I兲. This was a consequence of the local driving force fL⬀Iacting as a pushing or pulling loco- motive for a train of vortices in the region of I= 0.
In Ref. 8, this behavior—found again at lowI—changed dramatically at highI, whereVnlreversed sign to eventually become symmetric, exhibiting Vnl共−I兲=Vnl共I兲. Remarkably, the sign of this evenVnl共I兲was opposite at lowTand close to Tc. This implied that the local SNEQ mixed states were com- pletely different, which turned out to be consistent with EH 共TⰆTc兲 and the LO effect 共TⱗTc兲 in the I⫽0 region.
Hence, the TFTE has offered a new possibility for distin- guishing between EH and the LO effect in a manner that is free of numerical ambiguities mentioned before since only the sign ofVnlhas to be measured. The cause ofVnlwith EH or the LO effect in the I⫽0 region can be described by generalizing the magnetic-pressure model of Ref. 10 to fdr which is different from fL and depends on the type of the local SNEQ.8At lowT, the origin offdris aTgradient at the interface of the I⫽0 and I= 0 regions so Vnl is the conse- quence of a Nernst-type effect.11 Close to Tc, vortices are driven by a Lorentz-type force induced at the interface and
1098-0121/2010/81共17兲/174521共11兲 174521-1 ©2010 The American Physical Society
stemming from a novel enhancement of diamagnetism in the LO state relative to that in equilibrium.
In this paper, we give a timely account of other results of the experiment of Ref. 8. These refer to eight temperatures from 0.75 KⱕTⱕ2.80 K 共i.e., 0.26ⱕtⱕ0.95, where t
=T/Tc兲 and the whole range of applied magnetic fieldBext where the TFTE could be observed at a givenT.7EH persists up to 2 K 共t= 0.68兲 above which the LO effect takes place.
The T evolution of the SNEQ vortex dynamics is presented through changes in a characteristic high-IvoltageVnlⴱ. In or- der to account for the phenomenon quantitatively,Vnlⴱ is com- bined with the nonlocal resistance Rnl=Vnl/I which is de- fined for the low-I linear response regime and contains information on the pinning efficiency. Quantities characteris- tic of the TFTE with a given local SNEQ are traced in T ranges of their relevance. These are the vortex transport en- tropy S below 2 K and the nonequilibrium magnetization 共M兲 enhancement␦M in the LO state above 2 K. A special attention is paid to results at 2 K, where the LO effect is modified by EH above a certainI, which leaves a clear sig- nature only inVnl共I兲.
II. EXPERIMENT
The sample of Ref.8—a nanostructureda-Nb0.7Ge0.3thin film—was produced by combining electron-beam lithogra- phy and magnetron sputtering onto an oxydized Si substrate.7 The layout of the sample is presented schematically in Fig.
1共a兲. The film thickness is d= 40 nm, the width is W
= 250 nm共in and around the channel兲and the channel length isL= 2 m. The relevant coordinate system共with unit vec- tors xˆ,yˆ,zˆ兲 is indicated. Bext=Bextzˆ is perpendicular to the film plane. In measurements of Vnl共I兲, one applies ⫾兩I兩 be- tween the contacts 1 and 2共local lead兲. The corresponding兩j兩 decays exponentially away from the local lead, over a char- acteristic length ⬃W/ⰆL.6,7 Vortices in the channel are pressurized by the locally driven ones8,10and move along the channel at nonlocal velocity unl=⫾兩unl兩xˆ. This induces an electric field E=B⫻unl that is measured as ⫾Vnl between the contacts 3 and 4共nonlocal lead兲. The direction ofunl, and consequently the sign of Vnl, depends on the type of fdr, which will be addressed in Sec.III B.
The same sample is used to measure the local dissipation.
In this case,Iis passed between 1 and 3, and the local volt- age drop Vl is measured between 2 and 4. Since W is the sample width for all current paths共apart from a weak modu- lation of j in the local-lead area adjacent to the channel兲, j
⬇I/Wdis effectively the same both for measurements ofVnl andVl, which permits to useVl共I兲 as a representative of the local vortex dynamics forVnl共I兲at the sameTandBext. Mea- surements of Vl also provide important parameters of the sample,12which are:Tc= 2.94 K, the normal-state resistivity
n= 1.82 ⍀m, the diffusion constantD= 4.8⫻10−5 m2/s,
−共dBc2/dT兲T=Tc= 2.3 T/K, whereBc2 is the equilibrium up- per critical magnetic field, and the Ginzburg-Landau param- eters = 72, 共0兲= 7.0 nm, and 共0兲= 825 nm.7,8 The low pinning, characteristic of a-NbGe, allowed for dc measure- ments of Vnl⬃10– 200 nV, which was at the level of Rnl
⬃0.1 ⍀in the low-I linear regime. All measurements were carried out in a standard3He cryostat.
III. LOCAL AND NONLOCAL DISSIPATION VS NONEQUILIBRIUM VORTEX DYNAMICS
In this section, we give a brief overview of the SNEQ vortex-motion phenomena ina-NbGe films. Due to the sim- plicity of vortex matter and weak pinning in these systems,4 the discussed topics are related to fundamental issues of vor- tex dynamics rather than to sample-dependent pinning or pe- culiar vortex structure in exotic superconductors. We discuss limitations in the reliability of information that can be ex- tracted from Vl共I兲 only and the potential ofVnl共I兲in identi- fying the microscopic processes behind an SNEQ mixed state.
A. Types of SNEQ in vortex motion
In Fig.2, we plot exemplary共nonhysteretic兲 E共j兲curves extracted from Vl共I兲 of the sample under discussion. The correspondingIis shown on the top axis, the simple conver- sion beingI关A兴↔j关100 MA/m2兴. We choose two charac- teristic temperatures where the SNEQ is well defined, these are: 共a兲 for EH, T= 0.75 K 共t= 0.26, Bc2= 4.69 T兲, and 共b兲 for the LO effect, T= 2.5 K共t= 0.85,Bc2= 1 T兲. The values of b=Bext/Bc2are selected to demonstrate the cases of rela- tively strong共b⬃0.43– 0.45兲and weak共b⬃0.65– 0.68兲non- linearities inVl共I兲at both temperatures.
W W
L >> W x y Bext
1 2
3 4
I V
nl(a) geometry of the TFTE
LOCALLEAD
T(x)
x T*
T0 LT
(b) local electron heating at lowT
x Bext
B(x) µ0M(x)
(c) local magnetization enhancement close toLM Tc
CHANNEL
NONLOCALLEAD
unl
z
unl
unl jM
FIG. 1. 共a兲Schematic representation共not to scale兲of the TFTE geometry, as used in Ref.8.Bextis applied perpendicularly to the 共x-y兲film plane.Iis passed between the contacts 1 and 2, andVnlis measured between the contacts 3 and 4. 共b兲 Temperature profile along the sample in the regime of EH in the local lead.共c兲Profiles of Bext,B, and0M 共all in thez direction兲along the sample, and consequentjM, in the regime of the LO effect in the local lead. In 共b兲and共c兲, the direction ofunldoes not depend on the polarity ofI.
At first sight, there is no obvious difference between the curves in Figs. 2共a兲 and2共b兲 but a closer look reveals that those in Fig.2共a兲exhibit slightly sharper changes of curva- ture than their high-T counterparts. A difference can also be noted at high dissipation where EⱗEn=nj. In Fig. 2共a兲, there is an electric field Ec, appearing at moderate j and indicated by the solid circles, above which E=En within 0.1%. In contrast, the curves in Fig.2共b兲slowly creep toward Enbut stay below by more than 1% over the whole range of j. Thus, there are some features which point to different ori- gins of the two types ofE共j兲but these are barely visible and therefore difficult to spot.
Another way of determining the physics behind such a nonlinear E共j兲 is to analyze the set of curves at a same T numerically.2–4,7 At low T, one can concentrate on steep jumps of E共j兲 at low b by the method of Ref. 2, or can address the high-Epart in the spirit of Ref. 3for allb, both approaches being based on the assumption of a change Bc2共T0兲→Bc2共Tⴱ兲due to EH. The latter method results in a determination of Ec which, according to a model based on the b dependence of the Gibbs free energy density close to Bc2,3,4 should be well approximated by Ec=Ec0共1 −b兲. The result of this procedure for Vl共I兲 at T= 0.75 K is shown in the inset to Fig. 2共a兲.7The extracted Ecis displayed by the solid circles and the solid line is a linear fit with Ec0
= 900 V/m. This analysis also clarifies the meaning ofEc: at E=Ec, the heating destroys superconductivity, i.e., Tⴱ
=Tc共Bext兲, or, equivalently,Bext=Bc2共Tⴱ兲.
The framework for analyzing E共j兲 close to Tc is different.3,4,7 In this case, one uses the LO expression for E共j兲, which describes a dynamic reduction in the vortex- motion viscosity coefficient.1The main quantity to be de- termined from E共j兲 is the characteristic LO electric field ELO=uLOB, whereuLOis the LO vortex velocity. TheseE共j兲 can exhibit a steep jump accompanied by a hysteresis 共for low ELO兲or can be smooth 共for highELO兲,3,4which follows the prediction of the LO formula forE共j兲and permits extrac- tion of ELO irrespective of the presence or absence of the jump. The positions of ELO are in Fig. 2共b兲 shown by the open circles and the same symbols are used for plottingELO against Bext in the corresponding inset. The approximation B⬇Bextis justified by兩M兩ⰆBextfor a high-superconductor in the mixed state. The solid line is a linear fit with uLO
= 205 m/s.
The extracted Ec and ELO follow the predicted depen- dences reasonably well but still not as good as in Ref.
3—where measurements were carried out on a 5 m wide microbridge—which also holds for the overall agreement of the shape of the experimentalE共j兲 with the models outlined above.7We believe that the main reason for this discrepancy lies in the characteristic times involved in establishing an SNEQ in such narrow strips. This can be demonstrated by the following consideration. The time required for a creation/
destruction of the LO state is the relaxation time of nonequi- librium quasiparticle excitations, which is close toTcgiven by ⬃e,phkBTc/兩⌬兩 with e,ph being the electron-phonon scattering time and kB the Boltzmann constant.13 For the given uLO⬇205 m/s and other sample parameters, is around 1.5 ns.7 On the other hand, the time of vortex tra- versal across our sample in the LO regime is on the order of
W⬃W/uLO⬇1.2 ns, i.e., about the same as. This was not the case in Ref. 3 where the LO state fully developed be- cause ofⰆW. A similar analysis, leading to the same con- clusions, can be done for EH as well.
There are several messages of the above overview. First, the shape ofE共j兲can be almost the same for distinct SNEQ mixed states with hardly detectable differences. Second, nu- merical analyses can also be of limited reliability if the samples are very small. Moreover, any combination of these qualitative and quantitative approaches could fail to give a proper answer on the nature of an SNEQ when T is neither low nor close toTc, i.e., when a competition between EH and the LO effect may occur. The latter point will be addressed more closely in Sec.IV C.
B. TFTE vs local vortex dynamics
LocalV共I兲curves in the mixed state are generally mono- tonic and odd inI, apart from their possible weakly hyster- etic behavior at low b.4 In contrast, Vnl共I兲 measured over a wide range ofIis nonmonotonic and at first glance lacks any even or odd symmetry.7,8This is a consequence of different contributions to fdr, which do not have the same I depen- dence. At low j, the driving force fdr=fL is purely electro- magnetic, as the mixed-state thermodynamics in the local lead remains essentially intact. For that reason,fLis odd inj, and the resultingVnl共I兲is odd too. On the other hand, SNEQ FIG. 2. Local E共j兲 curves共solid lines兲 withEn=njshown by
the dashed lines. The values ofBextandbare given in the legends.
共a兲 T= 0.75 K, whereBc2= 4.69 T. The solid circles representEc which is in the inset plotted against共1 −b兲together with a linear fit 共solid line兲 given by Ec0= 900 V/m. 共b兲 T= 2.5 K, where Bc2
= 1 T. The open circles displayELO, in the inset plotted againstBext together with a linear fit 共solid line兲 corresponding to uLO
= 205 m/s.
at high jin the local lead is a thermodynamic state different from that in the channel and it is this difference which pro- duces the SNEQ part offdr. This part does not depend on the sign of j because the creation of a local SNEQ is set by 兩j兩 and the resultingVnl共I兲cannot be odd. Consequently, a wide- range sweep from −I to +I results in Vnl共I兲 of a rich structure,7,8 which is advantageous in determining the phys- ics behind an SNEQ mixed state.
A generalization of the model of Ref. 10 for Vnl as a response to fdrcan reasonably well account for the complex- ity of Vnl共I兲 in Ref. 8. This approach relies on a plausible assumption that vortices in the local lead push or pull those in the channel due to intervortex repulsion, and that the vor- tex matter is incompressible against this uniaxial magnetic pressure. The pressurizing occurs at theW-wide interface of the local lead and the channel, see Fig. 1共a兲. The pushing/
pulling force is produced bynWXvortices under the direct influence of fdr, whereX is the distance over which fdrex- tends in the xdirection and n=B/0 is the vortex density.
The number of vortices in the channel isnWLand the mo- tion of each of these vortices is damped by a viscous drag 共per unit vortex length兲unl. The driving and damping forces are balanced, i.e., fdr⫻共nWX兲=共unl兲⫻共nWL兲 hence unl
=fdrX/L determines Vnl=BunlW. As before, we can ap- proximateB⬇Bextfor a high-superconductor to obtain the nonlocal current-voltage characteristics
Vnl共I兲=WBextX
L fdr共I兲. 共1兲 This expression does not apply below a certain magnetic fieldBd共T兲that originates in the pinning in the channel and also in the vicinity of the phase transition at Bc2共T兲. More precisely, Vnl= 0 belowBd and close to Bc2 so the TFTE is always restricted to a range ofBext.6–8,10
Whenfdr=⫾兩fL兩xˆ=⫾共0兩I兩/Wd兲xˆ for the sample orienta- tion in Fig. 1共a兲, vortices in the local lead contribute to fdr over the whole width, andX=W. This results in
Vnl共I兲=WBext0
Ld I=RnlI. 共2兲 The above expression satisfies Vnl共−I兲= −Vnl共I兲 and as well introduces Rnl as a measure of the TFTE efficiency.Rnlde- pends entirely on the channel properties, in particular, on for vortices out of SNEQ. In Ref.10, the use of a theoretical
=f of pining-free flux flow reproduced the experimental values ofRnlwhen the pinning was negligible共close toTc兲.
When the pinning became stronger, at low temperatures,Rnl was lower than that calculated for pure flux flow but re- mained constant, i.e., Vnl共I兲 was still linear. This property was assigned to the motion of a depinned fraction of vortices in the channel, which was affected by a shear with the pinned 共or slower兲vortices but responded linearly to I.10 These ef- fects can be parametrized by introducing an effective ˜
⬎f which does not depend onI.
We now turn to the TFTE at low T, where EH underlies the local SNEQ. The correspondingT共x兲 is sketched in Fig.
1共b兲. In the local wire,T=Tⴱwhich over a lengthLTdrops to T=T0 in the channel. The driving force is a thermal force
produced by the T gradient11 and this behavior belongs to the class of Nernst-type effects. More precisely, fdr=fT
= −S共T/x兲xˆ⬇S关共Tⴱ−T0兲/LT兴xˆ is always in the positivex direction because S⬎0, i.e., it drives vortices away from the local lead. WithX⬇LT, one obtains
Vnl共I兲=SRnld
0
␦T共I兲, 共3兲
where ␦T共I兲=Tⴱ共I兲−T0 and Rnl is the same as in Eq. 共2兲.
Here,Vnl共−I兲=Vnl共I兲becausefTstems from the difference of thermodynamic potentials in the local and nonlocal regions.
Notably, LTdoes not appear in Eq.共3兲but it is still an im- portant parameter in context of the magnitude of fTand the applicability of the model—which requires LTⰆL. For the sample of Ref. 8, this condition is fulfilled because the esti- matedLTin the relevantTrange of measurements共0.75–1.5 K兲is between 125 nm共at 1.5 K兲and 295 nm共at 0.75 K兲.7
As explained before, the SNEQ close toTccorresponds to the LO effect. It follows from a calculation in Ref.8, which is presented in more detail in Appendix A, that the nonequi- librium diamagnetic兩M兩=兩Mneq兩in the LO state is larger than 兩M兩=兩Meq兩 in equilibrium. This results in spatially nonuni- form profiles of0M andB, where 0= 4⫻10−7 Vs/Am, as depicted in Fig.1共c兲. The energy for producing the spatial inhomogeneity ofMandBis supplied by the applied current which maintains the LO state in the local lead. The non- uniformity of M creates a current densityjM=共ⵜ⫻M兲yyˆ at the interface that stretches over X=LM. Therefore, jM
= −共M/x兲⬇共Mneq−Meq兲/LM⬍0, i.e., jM is always in the negative y direction. This leads to a Lorentz-type force fdr
=fM= −兩jM兩0xˆ that drives vortices toward the local lead.
Hence
Vnl共I兲=关Rnld兴␦M共I兲, 共4兲 where␦M共I兲=兩Mneq共I兲−Meq兩 andRnlis again the same is in Eq. 共2兲. SinceM also determines thermodynamic potentials, Vnl共−I兲=Vnl共I兲but of the sign which is opposite to that in Eq.
共3兲. As before,LM drops out from the expression for Vnl共I兲 but should be addressed because it is an important parameter in both the magnitude and the extent of fM. The issue ofLM
is, however, less straightforward than that ofLT.
In Ref. 8, it was shown that the reason for ␦M was a nonequilibrium gap enhancement near the vortex cores in the LO state. The net effect is an increase in the magnetic mo- ment of a single-vortex Wigner-Seitz cell. In the equatorial plane, the dipole magnetic field of an individual cell opposes Bextin other cells and in this way reducesB. Therefore, the larger the gap enhancement, the larger the diamagnetic re- sponse. The gap enhancement occurs at the expense of qua- siparticles within the cores, which have energies below the maximum兩⌬兩maxof 兩⌬兩 in the intervortex space. These qua- siparticles can penetrate into the surrounding superfluid by Andreev reflection only, i.e., up to a distance of about the coherence length—which is the first candidate forLM. On the other hand, this process is a single-vortex property whereas␦Mrequires a many-vortex system. The second can- didate isL=
冑
Dbut this length is more specific of quasi- particles with energies above 兩⌬兩max. There is, however, athird candidate as well. This is the intervortex distance a0
⬃共0/B兲1/2 which plays a crucial role in the screening of Bext as explained above. Thus, we believe that the proper estimate for LM is a0 although this matter is certainly still open to debate. In any case,LMⰆLholds.
IV. RESULTS AND DISCUSSION
Henceforth, we turn to experimental results which support the concepts presented above. General trends in Vnl共I兲 are demonstrated using experimental curves at 共T,Bext兲 points where the TFTE is maximal for the two local SNEQ regimes.
These are shown in Fig. 3: 共a兲 for EH, at T= 0.75 K and Bext= 3.2 T, and 共b兲 for the LO effect, at T= 2.5 K and Bext= 0.45 T. TheBc2values are 4.69 and 1 T, respectively, thus t= 0.26 and b= 0.68 in 共a兲, and t= 0.85 andb= 0.45 in 共b兲. Note that the corresponding local dissipation curves are displayed in Fig.2.
We first return to Fig.1共a兲to explain the signs in Vnl共I兲 plots. Bext is always directed as shown, I⬎0 represents j downwards, andVnl⬎0 meansunlleftwards, i.e., toward the local lead. TheVnl共I兲saturates at highIboth in Figs.3共a兲and 3共b兲but the sign of the saturation voltage is opposite in the two regimes. The saturation occurs for most of measured Vnl共I兲, except when there is a physical reason共see Sec.IV C兲 for the saturation to be shifted beyond the maximum usedI of 4 – 5 A. Without introducing a significant error, instead of characterizing Vnlstrictly by the saturation value, we use Vnlⴱ=Vnl共兩I兩= 4 A兲, indicated by the arrows, to represent the strength of the TFTE at a local SNEQ. Another measure of the 共overall兲TFTE efficiency isRnlwhich can be extracted from the antisymmetric part Vnl=RnlI corresponding to fdr
=fLat low I, as indicated by the dashed lines.
The difference between the curves in Figs.3共a兲 and3共b兲 becomes striking at high I, in contrast to that between the curves in Figs. 2共a兲 and 2共b兲. This implies availability of information from Vnl共I兲 without any in-depth analysis. For example, at兩I兩⬇1.5 A, whereE=Ecin Fig.2共a兲andVnl共I兲 in Fig.3共a兲either changes sign共for I⬎0兲or starts to be flat when I⬍0 strengthens further. The asymmetry originates in fT andfL acting in the same direction for I⬍0, and in the opposite directions when I⬎0. The same Vnlⴱ for I⬍0 and I⬎0 is a consequence of fL= 0 for E⬎Ec. Besides being completely different, theVnl共I兲in Fig.3共b兲exhibits no sharp features. This is consistent with the LO effect not leading to a destruction of superconductivity in the range ofI used, as already pointed out in Sec.III.
We shall consider these and other issues in more detail later but it is worthwhile to begin by a simple plot of Vnlⴱ againstbfor allTwhere our TFTE data were collected.7This is done in Fig.4. It is seen thatVnlⴱ⬍0 forT= 0.75, 1, and 1.5 K, which implies the local EH, andVnlⴱ⬎0 atT= 2.5, 2.6, 2.7, and 2.8 K, suggesting the LO effect in the local lead. There is, however, an intermediate behavior at T= 2 K, where Vnl共I兲 does not show a proper saturation and Vnlⴱ does not clearly belong to either of the two regimes. These three cases are addressed separately below.
A. TFTE well belowTc
In order to understand different contributions to Vnl共I兲, it is appropriate do decompose it into Vnl⫾共I兲=关Vnl共I兲
⫾Vnl共−I兲兴/2. The symmetric part Vnl+ is representative of the thermodynamic forces fT and fM whereas the antisym- metric partVnl− accounts for the electromagnetic forcefL. The result of this approach for theVnl共I兲in Fig.3共a兲is displayed in Fig. 5共a兲 and is typical of the low-T regime. Vnl−⬀I is found at low I, withVnl+ at the same time being very small, and this suggests fdr⬇fL. As I increases,Vnl− at some point starts to decrease andVnl+⬍0 simultaneously to grow, which implies a transition toward fdr⬇fT. Eventually, around FIG. 3.Vnl共I兲in the presence of共a兲EH, and共b兲the LO effect in
the local lead, for measurements where the overall TFTE strength is maximal in the two regimes. Slopes of the linear dashed lines de- termineRnl. The arrows point toVnlⴱ and, in共a兲, also toVnlatI共Ec兲, see the text. The values of important parameters are given in the legends and the corresponding local dissipation is presented in Fig.
2.
FIG. 4. Plot ofVnlⴱ vsbfor all T where TFTE data were col- lected, as indicated in the legend. For the local EH共open symbols兲, Vnlⴱ⬍0, and for the local LO effect共solid symbols兲,Vnlⴱ⬎0. At T
= 2 K共gray diamonds兲, there is no proper saturation ofVnl共I兲, and Vnlⴱ does not exhibit a well-defined behavior.
I共Ec兲⬇1.5 A,Vnl− drops to zero andVnl+ approaches a con- stant value.
TheTⴱ共I兲characteristics exemplified in Ref.8indicates a one-to-one correspondence of EH in the local lead and Vnl⫾共I兲. Analysis of the Vl共I兲 in the superconducting state 关Tⴱ⬍Tc共Bext兲兴 by the method of Ref. 3 connotes that Tⴱ共I兲 first increases slowly and then jumps very steeply in the I window where the above-discussed steep changes in Vnl⫾共I兲 occur.7,8The high-I part, whereVnl−= 0 andVnl+⬇const., cor- responds to the normal state in the local lead. Noise measurements7,14in this regime indicate a marginal increase in Tⴱ with increasing I hence one can assume Tⴱ⬇Tc共Bext兲 regardless ofI. Therefore, there is a relatively abrupt transi- tion from fdr⬇fLto fdr⬇fTwhenTⴱis close toTc共Bext兲 关the experimental results for which are shown in the inset to Fig.
5共a兲兴.
The strongest effect of fT occurs at Tc共Bext兲ⱗTⴱ, i.e., when the local lead is in the normal state. In this regime, vortices nucleate somewhere within the lengthLTaway from the local lead, move toward the channel due to theT gradi- ent, and push vortices in the channel. This situation is differ- ent from that in conventional measurements of the Nernst effect,15,16 because here T gradients are very strong 共
⬃1 K/ m兲, the number of vortices under the direct influ- ence offTis small, and the voltage corresponds to the motion of vortices which are in an isothermal environment 共the channel remains at T=T0兲. Strong lateral temperature varia- tions over a-NbGe microbridge films共also on oxidized Si兲 due to EH at low T were also observed in a noise experiment.17 This gives an additional support to the reality of spatially dependent separation of the electron temperature Tⴱ and the phonon temperature T0 at least for the given substrate-film interface properties.18
In Fig.5共b兲, we showRnl共b兲atT= 0.75, 1, and 1.5 K, i.e., for temperatures where the local SNEQ corresponds to EH 关the overall magnitude ofRnl共T兲will be discussed later兴. In Fig.5共c兲, we use the same symbols to plotS共b兲obtained by inserting Rnl, Vnl=兩Vnlⴱ兩, and ␦T=关Tc共Bext兲−T0兴 into Eq. 共3兲. The intricacy of the experimental situation has been outlined above so it is not straightforward to analyze Sin terms of the Maki formula19,20 S=0兩Meq兩/T 关where Meq⬇共Bext
−Bc2兲/2.32 02 for Bext not much below Bc2兴 which ap- plies to a weak T gradient over the whole sample and no local destruction of superconductivity by heating. On the other hand, if fTis really the relevant fdr, then the extracted S should still be reasonable in terms on the order of mag- nitude. This is indeed the case, since ourSdoes not depart significantly neither from the estimate by the Maki formula withT=T0, givingS⬃0.1– 0.2⫻10−12 Jm−1K−1, nor from the values in experiments of Ref.15共Nb films兲and Ref.16 共Pb-In films兲, where it was found S⬃0.05– 1.5
⫻10−12 Jm−1K−1andS⬃0.2– 5⫻10−12 Jm−1K−1, respec- tively. Thus, we conclude that our results for the TFTE at low T are consistent with the picture of local EH and the consequent Nernst-type effect.
B. TFTE close toTc
The method of analyzingVnl⫾共I兲can also be applied to the TFTE at TⱗTc. For the Vnl共I兲 in Fig. 3共b兲, this results in Vnl+共I兲 andVnl−共I兲 displayed in Fig.6共a兲. Let us first discuss FIG. 5. 共a兲Vnl+共I兲andVnl−共I兲, as indicated, for theVnl共I兲 in Fig.
3共a兲. Inset: experimental Tc共Bext兲. 共b兲 Rnl vs b for measurements where the SNEQ in the local lead is caused by EH.共c兲Sagainstb, plotted with the same symbols as in共b兲and calculated as explained
in the text. For共b兲and共c兲,T is indicated in the legend to共b兲. FIG. 6. 共a兲Vnl+共I兲andVnl−共I兲 for theVnl共I兲in Fig.3共b兲, as indi- cated.共b兲Rnlvsbfor measurements where the SNEQ in the local lead is caused by the LO effect. 共c兲 Magnetization enhancement
␦M共b兲, calculated using Eq. 共4兲. 共d兲 Interface current jM共b兲, ex- tracted from␦M共b兲. For共b兲–共d兲,Tis indicated in the legend to共b兲.
Vnl−共I兲. As before, Vnl−⬀I at low I, but—in contrast to the low-Tbehavior—this is followed by a slow decay ofVnl− asI increases, not by a sharp drop to zero. The linear part of Vnl−共I兲is again a consequence offLdominating infdrat lowI whereas the decrease inVnl−共I兲at high Ican be explained by a reduction of fL in the high-dissipation regime of vortex motion. Namely, whenEⱗEn, which can be a consequence either of an SNEQ or ofbⱗ1 in a close-to-equilibrium situ- ation, a significant fraction of jis carried by quasiparticles.1 This normal current does not lead to asymmetry in the profile of⌬around the vortex core, which is set by the supercurrent density js, and it therefore does not contribute to fL.21 The observed progressive reduction in Vnl−共I兲 as b grows8 is in support to this picture.
The main information about the SNEQ is contained in Vnl+共I兲 which increases monotonically with increasingIuntil it saturates. As explained before, Vnl+ represents fM that is given by ␦M at T=T0. As I increases, ␦M grows until the core shrinking reaches its limit1 at 共t兲共1 −t兲1/4, when the increase in␦M must saturate.8This simple consideration ex- plains the shape ofVnl+共I兲qualitatively. Quantitatively, we can use Eq. 共4兲 and Rnl共b兲, shown in Fig. 6共b兲, to calculate
␦M共b兲. The result of this procedure is shown in Fig.6共c兲. It can be seen that␦Mis around 50 A/m, which is a very small value corresponding to⬃60 T. However,␦M is not small on the scale of 兩Meq兩 which is of the same order. Moreover, the gradient of M occurs over a small distance of the inter- vortex spacing a0⬃共0/Bext兲1/2 which—for the given Bext range—takes values between 60 and 140 nm. The calculated interface current jM=␦M/a0is plotted againstb in Fig.6共d兲, where it can be seen that it is comparable to a typicaljin our experiment.
There are also other issues of relevance for the TFTE at TⱗTc. In our measurements, SNEQ develops in the local- lead area adjacent to the channel, as well as in the W-wide parts of the local lead along theydirection, see Fig.1共a兲. The local lead widens up further away and j is smaller there, which introduces additional interfaces of the SNEQ and close-to-equilibrium mixed states. In the presence of an SNEQ in the local lead, vortices do not simply traverse the SNEQ area 共as they do when fdr=fL兲: they all move either away共TⰆTc兲or toward共TⱗTc兲it. This must modify vortex trajectories in order to maintain n=B/0 via complex vor- tex entry/exit paths in and around the SNEQ area. At T ⰆTc, the problem is less troublesome because the strongest effects occur when EH has destroyed superconductivity and there are no vortices in the SNEQ area. Close toTc, on the other hand, there are vortices everywhere, their sizes and velocities being spatially dependent. Obviously, their trajec- tories must be such that a local growth ofnis prevented, as this would cost much energy due to the stiffness of a vortex system against compression. Moreover, while there is experi- mental evidence for a triangular vortex lattice in the channel,22this cannot be claimed for the SNEQ area where the above effects could cause a breakdown of the triangular symmetry. This may be complicated further by sample- dependent pinning landscape, edge roughness, etc., but our simple model can nonetheless still account for the main physics of the phenomenon. Another subject related to effect of the sample geometry on the magnitude of␦Mis discussed in Appendix B.
Last but not the least, our results may have implications for other topics as well. We have shown that there are two thermodynamic forces that can incite vortex motion and set its direction. Gradients of TandM can be created and con- trolled by external heaters and magnets, and it therefore seems that a combination of these two approaches can be useful in elucidating the presence of vortices or vortexlike excitations in different situations. For instance, current de- bate on the origin of the Nernts effect in high-Tc
compounds23could benefit from supplements obtained in ex- periments based on applying a gradient of M in an isother- mal setup.
C. TFTE at intermediateT
We have shown in previous sections that the SNEQ mixed states at TⰆTc and TⱗTc have different physical back- grounds. However, the situation is less clear at intermediate T. For instance, analysis of local V共I兲 at T= 2 K in Ref.3 was not conclusive, and these data were used only later in a qualitative consideration of another phenomenon.24 The same applies to Vl共I兲 at T= 2 K of this work and this is where the TFTE is crucial in determining the nature of the corresponding SNEQ mixed state.
In Fig. 7共a兲, we present Vnl共I兲 at T= 2 K 共t= 0.68兲 and Bext= 1.2 T共b= 0.53兲, the shape of which is markedly differ- ent from those in Fig. 3. There are pronounced minima and maxima for both polarities ofI, there are only indications of a saturation of Vnl共I兲 at the maximum current used, etc. A better understanding of the underlying physics can again be obtained from the correspondingVnl−共I兲andVnl+共I兲, which are FIG. 7. 共a兲Vnl共I兲atT= 2 K共t= 0.68,Bext= 1.2 T, andb= 0.53兲 where neither EH nor the LO effect can give a conclusive descrip- tion ofVl共I兲.共b兲Vnl−共I兲andVnl+共I兲, as indicated. The latter exhibits a change of the sign, which is suggestive of the appearance of EH on top of the LO effect which dominates at lowerI. Inset to共a兲:I1and I2, in the main panels indicated by the arrows, againstBext.
shown in Fig. 7共b兲. At I⬍I1, there is a usual behavior Vnl−
⬀I, characteristic of the linear action of fL. Looking back at Fig.7共a兲, one can see thatI=I1corresponds to the minimum of Vnl共I兲 on the I⬍0 side. Vnl+共I兲 for I⬍I1 is positive and grows with increasing I as well, which is suggestive of the LO effect gradually taking place. WhenIis increased further, Vnl−共I兲 begins to decay in a way similar to that in Fig. 6共a兲, whereas Vnl+共I兲⬎0 continues to grow until I=I2 is reached, which is a current just after the maximum of Vnl共I兲 on the I⬎0 side. Characteristic currentsI1andI2are in the inset to Fig. 7共a兲 plotted vs Bext. The decrease of Vnl+共I兲 after I has exceededI2implies a reduction offMby fTthat appears due to EH at high I. Eventually, fT prevails and Vnl+ becomes negative but not constant as in Fig.3共a兲, which suggests that the superconductivity has survived in the form of a heated LO state. Coexistence of the LO effect and EH was actually predicted theoretically1,18 but experimental confirmations have been facing difficulties related to weak sensitivity of local V共I兲 to such subtle effects. At T= 2 K, conditions for this coexistence are just right:Tis still close enough toTcfor the quasiparticle distribution function to assume the LO form but the number of phonons is too small for taking away all the heat if the energy input is large. Finally, now it becomes clear why analyses of localV共I兲at intermediateTdo not give a proper answer on the microscopic mechanisms behind these curves: the SNEQ changes its nature along theV共I兲.
D. SNEQ regimes in theT-Bextplane
We complete our discussion by mapping the TFTE results for the appearance of different SNEQ regimes, which is shown in Fig.8. The TFTE occurs in a restricted area of the T-Bext plane. The lower boundary of its appearance is af- fected mainly by the pinning in the channel, which impedes vortex motion therein and consequently leads toVnl= 0 when it becomes strong enough at low T and Bext. The upper boundary is at the present time less understood. It may reflect a smearing out of superconducting properties as most of the sample volume becomes normal so that signatures of some phenomena become immeasurably small. However, one can
also not rule out that it may be associated with high-b fluc- tuations which ina-NbGe films seem to appear in an appre- ciableB region blow Bc2.25 While a full mapping of SNEQ mixed states requires a combination of Vnl共I兲 andVl共I兲 re- sults, there are situations whereVl共I兲is of little use andVnl共I兲 is decisive, for instance, in showing that EH and the LO effect can coexist at intermediateT.
SinceRnlis also required for understanding and quantify- ing the TFTE in different regimes, in the inset to Fig.8 we show theTdependence of its representativeRpwhich is the maximum ofRnl共b兲extracted from allVnl共I兲at a givenT, see Figs. 5共b兲and 6共b兲. Actually, Rp is a good estimate for the peak value ofRnl共Bext兲curve obtained by sweepingBextiso- thermally at a low I, which was the method of Refs. 6 and 10. Fora-NbGe samples in Ref.10,Rp共T兲was several times higher than here because these samples had such a low pin- ning that⬇f applied close toTc. However, the shapes of the two Rp共T兲 curves are very similar. Rp is high at low T because it occurs at highBext, andRp⬀Bext. There is also an upturn ofRp before the TFTE disappears atTc, because the pinning close to Tc weakens, this reduces ˜ and enhances Rp⬀1/˜. This similarity implies that the TFTE does not suf- fer much from pinning as long as the main effect is in f
→˜ due to the shear between vortices moving at differentunl 共which may also include unl= 0 for some of them兲.
V. SUMMARY AND CONCLUSIONS
In this follow up to Ref.8, we present a broader perspec- tive on the TFTE at different local vortex dynamics. At least in weak pinning materials—where fundamental phenomena in vortex motion dominate over sample-dependent pinning—
the TFTE is a powerful diagnostic tool for vortex dynamics in the local lead. The TFTE is particularly helpful at high applied currents I, where the local mixed-state thermody- namics is altered. In this case, while the local dissipation curves offer only meager evidence for the microscopic pro- cesses being different at low and high temperatures T, the TFTE leaves no doubt: the sign of the nonlocal voltage is opposite in the two cases. This is a consequence of the non- equilibrium quasiparticle distribution function being funda- mentally different at low and highT, which results in differ- ent thermodynamic properties.
At lowT, the entire quasiparticle system is heated locally.
This leads to an expansion of vortex cores, and the corre- sponding TFTE stems from a T gradient at the interface of the local and nonlocal regions. This Nernst-type effects pushes vortices away from the local region. Close toTc, the isothermal Larkin-Ovchinnikov effect takes place in the local region, resulting in a shrinkage of vortex cores and an en- hanced diamagnetic response. The magnetization gradient at the interface drives vortices toward the local region by a Lorentz-type force. The TFTE at intermediate T shows that the Larkin-Ovchinnikov effect appears at moderateIbut it is modified by electron heating at higher I, which cannot be concluded from the local current-voltage curves.
Remarkably, these effects—including the TFTE with vor- tices being locally driven by the Lorentz force—can all be accounted for by a simple model of the magnetic pressure FIG. 8. Regions of different SNEQ mixed states, as extracted
from the TFTE data, plotted in the T-Bextplane. Inset: T depen- dence of the maximum nonlocal resistanceRp.
