Analysis of the Ghost and Mirror Fields in the Nernst Signal Induced by Superconducting Fluctuations
AAnalysis of the Ghost and Mirror Fields in the Nernst Signal Induced bySuperconducting Fluctuations
A. Glatz,
1, 2
A. Pourret, and A.A. Varlamov Materials Science Division, Argonne National Laboratory,9700 S. Cass Avenue, Argonne, Illinois 60639, USA Department of Physics, Northern Illinois University, DeKalb, Illinois 60115, USA Universit Grenoble Alpes, CEA, IRIG, PHELIQS, F-38000 Grenoble, France CNR-SPIN, c/o DICII-Universit di Roma Tor Vergata, Via del Politechnico, 1, 00133 Roma, Italia (Dated: August 3, 2020)We present a complete analysis of the Nernst signal due to superconducting fluctuations in a largevariety of superconductors from conventional to unconventional ones. A closed analytical expressionof the fluctuation contribution to the Nernst signal is obtained in a large range of temperatureand magnetic field. We apply this expression directly to experimental measurements of the Nernstsignal in Nb x Si − x thin films and a URu Si superconductors. Both magnetic field and temperaturedependence of the available data are fitted with very good accuracy using only two fitting parameters,the superconducting temperature T c0 and the upper critical field H c2 (0). The obtained values agreevery well with experimentally obtained values. We also extract the ghost lines (maximum of theNernst signal for constant temperature or magnetic field) from the complete expression and alsocompare it to several experimentally obtained curves. Our approach predicts a linear temperaturedependence for the ghost critical field well above T c0 . Within the errors of the experimental data,this linearity is indeed observed in many superconductors far from T c0 . I. INTRODUCTION
In a superconductor above its critical temperature, T c0 ,global superconducting coherence vanishes, leaving be-hind droplets of short lived Cooper pairs. Supercon-ducting fluctuations, discovered in the late 1960s, haveconstituted an important research area in superconduc-tivity as they are manifested in a variety of phenom-ena. Today their investigation has emerged as a power-ful tool for quantifying material parameters of new su-perconductors. In this regard, the observation of a gi-ant Nernst signal (three orders of magnitude larger thanthe value of the corresponding coefficient in typical met-als) over a wide range of temperatures and magneticfields attracted great attention of the superconductivitycommunity and caused lively theoretical discussions .Important milestones were its discovery in underdopedphases of high-temperature superconductors , later inthe normal phase of conventional superconductors , innormal phases of overdoped, optimally doped, and inthe underdoped superconductors La . − x Eu . S r x CuO ,Pr − x Ce CuO , and, finally, the observation of thecolossal thermo-magnetic response in the exotic heavyFermion superconductor URu Si . Today, it is com-monly agreed that this effect is related to superconduct-ing fluctuations, and its profound relationship to the fluc-tuation magnetization is well established .One of the characteristic features of the fluctuationsinduced Nersnt signal is its non-monotonous dependenceon applied magnetic fields. The latter follows from avery generic heuristic arguments: the fluctuations in-duced Nersnt signal is the response to an applied crossedmagnetic field and temperature gradient, N (fl) = β (fl) xy R (cid:3) ,where β (fl) xy is the off-diagonal component of the fluctua- tion induced contribution to the thermoelectric tensor and R (cid:3) is the film sheet resistance. Hence, it is zeroat H = 0 (where the thermoelectric tensor is diagonal)and it should vanish in very strong fields, which sup-press fluctuations . Indeed, a maximum of the Nernstsignal as function of the magnetic field has been widelyobserved . The study of the temperature dependenceof the field at which the Nernst signal is maximum, the ghost (critical) field H ∗ ( T ), acquired special significancefor HTS compounds, since the authors of Refs. 10 and11 have proposed to use it for the precise determinationof the second critical field H c2 (0), often inaccessible fordirect measurements because of its huge value. II. THE ISSUE OF THE GHOST FIELDTEMPERATURE DEPENDENCE
The analysis of the experimental data obtained fromthe HTS compound Pr − x Ce CuO led the authors ofRef. to the hypothesis that the temperature depen-dence of the “ghost critical field” is described by theexpression: H ∗ exp ( T ) = H c2 (0) ln TT c0 . (1)The prefactor in front of the logarithm with H c2 (0) wasdetermined by observation that H c2 (0) is the only em-pirical parameter that characterizes the strength of su-perconductivity. The authors stated that “the character-istic field scale encoded in superconducting fluctuationsabove T c ”, is equal to the field needed to kill supercon-ductivity at T = 0 K and we share this motivation. Theargument, justifying the logarithmic dependence of H ∗ on temperature was based on the statement, that the a r X i v : . [ c ond - m a t . s up r- c on ] J u l FIG. 1. The magnetic field and temperature dependence of the fluctuation part of the Nernst coefficient. top-left:
A viewon the t = 0 plane with the ghost temperature line in blue (light gray) indicating the maximum of the Nernst coefficient forconstant h . top-right: A view on the h = 0 plane with ghost field line in green, indicating the maximum of the Nernst signal forconstant fields. bottom: Zoom on to the quantum fluctuations (QF) region at t = 0 close to h = h c . The red (dark gray) lineindicates the contour, where the Nernst signal is zero. In a very small area of the QF region, the Nernst coefficient becomesnegative for t (cid:46) .
02 and ˜ h (cid:46) .
15 (see text). maximum of the Nernst signal should correspond to thefield, where the magnetic length of a fluctuation Cooperpair, L H , becomes equal to its “size”. We agree withthe latter: in terms of the qualitative picture of super-conducting fluctuations, one can see how moving alongthe H c2 ( T ) line the Ginzburg-Landau long wave-lengthscenario gradually transforms into the precursor of anAbrikosov vortex lattice: a set of clusters of rotatingfluctuation Cooper pairs (FCP) in magnetic field, whichare relatively small (of size ∼ ξ BCS ) . Yet, in or-der to practically apply this correct ideological idea, theauthors of Refs. 8–11, and 16 extrapolate the Ginzburg-Landau (GL) expression for the FCP coherence length ξ FCP ( T ) = ξ GL ( T ) ∼ ξ BCS / (cid:113) ln TT c0 , obtained with theassumption of closeness to the critical temperature , tothe region of high temperatures T (cid:29) T c0 . Indeed, thisprocedure leads to Eq. (1).However, at this point we need to stress that thereis no theoretical justification for such an extrapolationprocedure. Moreover, it leads to the obviously incor-rect conclusion that at high temperatures, the size of FCPs becomes much less than ξ BCS . The correlationlength ξ FCP ( T ), identified with the fluctuation Cooperpair “size”, should be determined from the pole of thetwo-particle Green function, or, idem , of the fluctuationpropagator . For arbitrary temperatures and magneticfields in impure superconductor, the general form of thelatter is: L ( R ) − n ( − iω, q z ) = (2) − ρ e (cid:104) ln TT c0 + ψ (cid:16) + − iω +Ω H ( n + )+ Dq z πT (cid:17) − ψ (cid:0) (cid:1)(cid:105) . Close to the critical temperature, where ln TT c0 ≈ T − T c0 T c0 = (cid:15) (cid:28)
1, and in zero magnetic field it takes the standardform of the diffusive mode, after expansion of the ψ -function: L (0 , q ) = − ρ e (cid:18) (cid:15) + πDq T (cid:19) − . (3)Analyzing the pole of this expression, L − (0 , q ) = 0,one indeed obtains ξ FCP ( T → T c0 ) ∼ q − ∼ ξ GL ( (cid:15) ) ∼ ξ BCS / √ (cid:15) . Yet, far from the critical temperature, withthe assumption that ln TT c0 (cid:29)
1, the ψ -function in Eq. (3)with large argument, should be replaced by its asymp-totic logarithmic expression and one obtains L (0 , q ) = − ρ e ln − (cid:18) Dq πT c0 (cid:19) . (4)The pole of this expression is given by q − ∼ (cid:113) πT c0 D resulting in ξ FCP ( T (cid:29) T c0 ) ∼ q − ∼ ξ BCS . Hence,the qualitative argumentation justifying Eq. (1) is un-founded.In Ref. the authors looked for an analytical expres-sion for the ghost field by proposing a scaling argumentsbased on the general expression for fluctuations inducedNersnt signal (see Refs. ), valid in a wide range oftemperatures and magnetic fields. It was noticed thatthe magnetic field enters only normalized by tempera-ture, while the latter also appears in the theory as pa-rameter ln ( T /T c0 ). This observation allowed them toobtain the following expression for the ghost field, whichis very different from Eq. (1): H ∗ KV ( T ) = H c2 (0) (cid:18) TT c0 (cid:19) ϕ (cid:18) ln TT c0 (cid:19) , (5)where ϕ ( x ) is some smooth function which satisfies thecondition ϕ (0) = 0. It is easy to see that Eq. (5) coincideswith Eq. (1) only in the very particular case of ϕ ( x ) = x exp( − x ).Due to the extremely cumbersome nature of the gen-eral expression for the fluctuations induced Nersnt signal,none of the authors of Refs. 2, 3, and 17 succeeded obtain-ing an analytical expression for the temperature depen-dence of the ghost field valid far from the critical temper-ature. Yet, simple equating of the asymptotic expressionsvalid at low fields and high temperatures ln t (cid:38) , h (cid:28) VIII ) N (fl) ( T, H ) ∼ (cid:18) ξ L H c2 (cid:19) (cid:18) HH c2 (0) (cid:19)(cid:18) T c0 T (cid:19) ln − TT c0 (6)(here L H c2 = c eH c2 (0) ∼ ξ ) and that one valid forhigh fields h (cid:29) max { , t } (see Table I, domain IX ) N (fl) ( T, H ) ∼ (cid:18) L H c ξ (cid:19)(cid:18) TT c0 (cid:19) (cid:18) H c2 (0) H (cid:19) ln − HH c2 (0) (7)leads to the conclusion that at sufficiently high temper-atures ( T (cid:38) T c0 ) the ghost critical field should grow asfunction of temperature almost linearly (with logarithmicaccuracy): H ∗ ( T ) ∼ H c2 (0) (cid:18) TT c0 (cid:19) . (8) In Ref. 13, a general computational approach to thedescription of fluctuation phenomena in superconduc-tors, valid in the whole phase diagram, numerical fluc-tuoscopy , was presented. In the following we will applythis method for the determination of the true tempera-ture dependence of the ghost field in the Nernst signaland its comparison with experimental data. III. CONSISTENT DERIVATION OF THEGHOST FIELDA. Theoretical Foundation: fluctuations inducedNersnt Signal β xy h 1.021.051.101.201.301.502.003.004.00 β max -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 2.5 3 3.5 4 β xy t 0.700.720.750.800.851.01.52.03.0 β max FIG. 2. fluctuations induced Nersnt signal as function of(a) magnetic field at constant temperatures (above T c0 , i.e., t > h c (0)), indicated in the legend, with dashed maximacurve as function of field. The general expression for the fluctuation contributionto the Nernst signal of 2D superconductors, valid beyondthe line H c2 ( t ), can be presented in the form suitable forthe numerical analysis as : domain t and h range description π N (fl) / N I h = 0, (cid:15) (cid:28) T c0 2 eHξ ( T )3 c = h (cid:15) II (cid:15) (cid:28) h (cid:28) T c0 , above the mirror reflected H c2 -line 1 − (ln 2) / III h − H c2 ( t ) (cid:28) (cid:15) (cid:28) H c2 -line (cid:15) + h IV t (cid:28) (cid:101) h region of quantum fluctuations − γ E t ˜ h V t / ln(1 /t ) (cid:28) ˜ h (cid:28) t (cid:28) t ˜ h VI ˜ h (cid:28) t / ln(1 /t ) (cid:28) H c2 ( t (cid:28) γ t ˜ h ( t ) VII t / ln(1 /t ) (cid:46) (cid:101) h ( t ) (cid:28) h ( t ) (cid:20) H c2 ( t ) π t ψ (cid:48)(cid:48) ( + H c2( t ) π t ) ψ (cid:48) ( + H c2( t ) π t ) (cid:21) VIII ln t (cid:38) , h (cid:28) t high temperatures π ht ln t IX h (cid:29) max { , t } high magnetic fields π th ln h TABLE I. Asymptotic expressions (obtained in Ref. 2) for fluctuation corrections to the Nernst signal in different domains ofthe phase diagram (see Fig. 3). Where N ≡ π (cid:126) k B R (cid:3) , ˜ h = h − h c2 ( t ) h c ( t ) (cid:28)
1, and h c (0) = H c2 (0) (cid:101) H c2 (0) = π γ E = 0 .
69 (see text). N (fl) = N π (cid:34) M t (cid:88) m =0 ( m +1) ∞ (cid:88) k = −∞ (cid:26)(cid:18) η (2 m +3)+ | k |E m + η (2 m +1)+ | k |E m +1 (cid:19) (cid:0) E (cid:48) m −E (cid:48) m +1 (cid:1) +2 η [ η (2 m +1)+ | k | ] E (cid:48)(cid:48) m E m +2 η [ η (2 m +3)+ | k | ] E (cid:48)(cid:48) m +1 E m +1 (cid:27) +4 π M t (cid:88) m =0 ( m + 1) (cid:90) ∞−∞ dx sinh πx (cid:40) η Im E m Im ( E m + E m +1 ) + (cid:2) η ( m + 1 /
2) Im E m + x Re E m (cid:3) Im ( E m +1 + 2 η E (cid:48) m − E m ) |E m | + η Im E m +1 Im ( E m + E m +1 ) + (cid:2) η ( m + 3 /
2) Im E m +1 + x Re E m +1 (cid:3) Im ( E m +1 + 2 η E (cid:48) m +1 − E m ) |E m +1 | + 2 x Im ln E m E m +1 (9) − E m + E m +1 ) (Im E m Im E m +1 +Re E m Re E m +1 ) |E m +1 | |E m | (cid:104) η (cid:18) m + 32 (cid:19) Im E m +1 − η (cid:18) m + 12 (cid:19) Im E m + x E m +1 −E m ) (cid:105)(cid:27)(cid:21) with N = ek B R (cid:3) (cid:126) . Here the function E m ≡ E m ( t, h, | k | ) = ln t + ψ (cid:20) | k | + 12 + η (cid:18) m + 12 (cid:19)(cid:21) − ψ (cid:18) (cid:19) (10)is the denominator of the above mentioned fluctuationpropagator. Its derivatives with respect to the argument x are related to polygamma functions: E ( n ) m ( t, h, x ) ≡ ∂ n ∂x n E m ( t, h, x )= 2 − n ψ ( n ) (cid:20) x η (cid:18) m + 12 (cid:19)(cid:21) . (11)We use here the convenient combination η = hπ t of thedimensionless temperature t = TT c0 and magnetic field h = H (cid:101) H c2 (0) . The latter is normalized by the value of thesecond critical field obtained by linear extrapolation of itstemperature dependence near T c0 : (cid:101) H c2 (0) = Φ / (cid:0) πξ (cid:1) ,where Φ = πc/e is the magnetic flux quantum. Thevalue of the magnetic field (cid:101) H c2 (0) is 8 γ E /π times largerthan the Abrikosov’s value for the second critical field H c2 (0): h = H (cid:101) H c2 (0) = π γ E HH c2 (0) = 0 . HH c2 (0) . (12)In analogy to (cid:15) , which measures the closeness to T c0 inzero field, we introduce ˜ h ( t ) = h − h c ( t ) h c ( t ) , where ˜ h (0) mea-sures the closeness to the true critical field at zero tem-perature. Despite the apparent divergence of Eq. (9) (weintroduced the natural upper limit of the summation overLandau levels M t ∼ ( T c τ ) − , with τ being the electronelastic scattering time) it in fact converges due to intri-cate cancellations in two divergent orders of the transport(Kubo) and magnetization current fluctuation contribu-tions (see Refs. 2, 17, and 19). This can be verified byexpanding all functions E ( n ) m ( t, h, x ) and their derivativesin Eq. (10) over Landau level differences in the limit oflarge numbers. Hence, the result of summations doesnot depends on the cut-off parameter. This fact is alsoconfirmed by the direct numerical evaluation. B. Numerical Analysis of the fluctuations inducedNersnt Signal
The fluctuation contribution to the Nernst signal inthe whole phase diagram beyond the line H c2 ( t ) as sur-face plot in accordance to Eq. (9) is presented in Fig. 1.Fig. 2 shows selected isomagnetic and isothermal cutsof this surface plot. The asymptotic expressions for theNernst signal in different domains of the phase diagramare summarized in table I and the corresponding domainsare indicated in Fig. 3.Close to the critical temperature T c0 , (domains I - III )where fluctuations have Ginzburg-Landau thermal char-acter, the Nernst signal is positive and grows in magni-tude approaching the transition line ( h − H c2 ( t ) (cid:28) h ( m ) ( t ) = t −
1, separates the linearand non-linear regimes in the magnetic field dependenceof the Nernst signal. The isothermal Nernst signal graphs in Fig. 2a, showthe well known non-monotonous behavior of the Nernstsignal for temperatures above T c0 . The dashed line con-nects the maxima for various fixed values of temper-ature, indicating the “ghost field temperature depen-dence”, which at sufficiently high temperatures is welldescribed by the expression h ∗ ( t ) ≈ .
12 ( t − . , (13)fit shown in Fig. 3. One can see that its linear dependenceon temperature corresponds to our qualitative pictureabove and is quite different from the logarithmic law usedin Refs. 10 and 11.Of special interest is the study of the low-temperatureregime of fluctuations, close to the upper critical field H c2 (0) (domains IV – VI ). Here a crossover line, t (qt) ( h ) = (cid:101) h , exists, which separates the purely quantum regime atvanishing temperatures (domain IV ) and the region oflow temperatures, but where fluctuations already acquirethermal character (domain VI ). It is interesting, that inthe quantum regime the fluctuation contribution to theNernst signal is negative in a very small t - h area, whereit depends linear on temperature and diverges as ˜ h − ap-proaching the transition point (see the insert in Fig. 1).This change of the sign in the fluctuation thermoelectricresponse is similar to the negative fluctuation conductiv-ity close to the quantum phase transition in the vicin-ity of H c2 (0), found in Ref. 21. These negative valuescomes from the diffusion coefficient renormalization con-tribution, which exceeds the positive, but fading away ALterm in this region. In the quantum-to-classical crossoverregion (domain V ), the Nernst signal becomes positiveand less singular ( ∼ ln t ˜ h ). Increasing the temperature,one goes over into the region of thermal fluctuations (do-main VI ) and sees that the Nernst signal continue to growas ∼ t / ˜ h .In the isomagnetic Nernst signal plots above the sec-ond critical field, shown in Fig. 2b, one sees, similarlyto the situation above T c0 , that the Nernst signal tem-perature dependence at fixed fields is non-monotonous and has maximum. The line connecting these maximacan be called the ”ghost temperature line” and it is welldescribed by the linear dependence t ∗ ( h ) ≈ .
65 ( h − . , (14)for h > . t + 0 .
13 fit).In the following we will use these insights and completeexpression, Eq. (9), for the Nernst signal to fit experi-mental data allowing to perform a characterization of thesuperconducting material. In particular, we can extractthe values of T c and H c2 (0), without using any ‘artificial’convenience criteria (like half width of the transition re-gion, 90% of the resistance decay, the temperature wherethe derivative of resistance is maximal or has an inflectionpoint, etc). In a ‘simplified’ version one can just use theghost field and ghost temperature lines (Eqs. (13)-(14))for fitting instead of the non-trivial fluctuoscopy , the fullfitting procedure of the Nernst signal. IV. Nb x Si − x EXPERIMENTS
In order to verify our theoretical studies, measure-ments on two stoichiometrically identical samples ofNb x Si − x were performed, labelled samples 1 & 2 inthe following. The Nb concentration, x , was fixed at x = 0 .
15. These amorphous film samples were preparedunder ultrahigh vacuum by e-beam coevaporation of Nband Si with precise control over concentrations and de-posited on sapphire substrates. Such films typically un-dergo a metal-insulator transition when x decreases.The two samples have different thicknesses, whichmostly controls their critical parameters, since the nom-inal concentration is the same: Sample 1 (2) was 12.5(35) nm thick, its experimental midpoint T (exp)c0 was0.165 (0.380) K (resistively measured in zero field) andits upper critical field H (exp)c2 (0) was 0.36 (0.91) T. Thezero temperature coherence lengths for both samples are19.7nm and 13nm, respectivelyThe Nernst coefficient is obtained by measuring thethermoelectric and electric coefficients of both samplesin a dilution fridge using a resistive heater, two RuO thermometers, and two lateral contacts. Partial data waspublished in Refs. [8 and 9]. At T ∼ . V. NERNST SIGNAL FLUCTUOSCOPY OFNb . Si . AND OTHER MATERIALS
As already noted, in previous studies the dependenceof the fluctuation contribution to the Nernst signal onmagnetic field and temperature above the critical onehas been fitted by asymptotic expressions and inter-polations between them with limited accuracy, whichalso does not allow for a consistent extraction of the ghost I IIIIIIV VI V II VIX VIII SC QF ghost temp.crossover line1.54t + - FIG. 3.
Left:
Phase diagram with the lines of the BCS second critical field h c ( t ), the ghost field h ∗ ( t ), the ghost temperature t ∗ ( h ), the mirror field h ( m ) ( t ), and the crossover line from quantum to thermal fluctuations t (qt) ( h ). The regions of qualitativelydifferent asymptotic behavior is indicated by roman numbers, which are explained in table I. The region of quantum fluctuationsis marked by “QF” – in this region the Nernst coefficient becomes negative. The shaded region is enlarged on the right , whichshows both ghost lines with a density plot of the Nernst signal. In addition the ( t, h )-gradient of N ( fl ) is indicated by a vectorfield. The ghost lines follow the vertical and horizontal gradients, respectively. N ( f l ) / N h FIG. 4. Fit of the Nernst signal for Nb . Si . (sample 1)to Eq. (9). The found fitting parameters are T c0 = 0 . H c (0) = 0 . lines. Here we use the general expression Eq. (9) for de-tailed numerical analysis & high precision fitting of ex-perimental data in the whole t - h plane without the inter-polation procedure.In Figs. 4 and 5 one can see how accurately Eq. (9)fits the experimental data of two Nb . Si . samplesusing only two fitting parameters: T c0 and H c (0). Thevalues of the fitting parameters for sample 1 are T (theo)c0 =0 . K and H (theo)c2 (0) = 0 . T .Similarly, the fits of the measurements obtained onsample 2 give the values of critical temperature and sec-ond critical field close to their experimentally estimated -3 -2 -1 N ( f l ) / N h FIG. 5. Fit of the Nernst signal for Nb . Si . (sample 2) toEq. (9), shown in half-logarithmic representation. The foundfitting parameters are T c0 = 0 . H c (0) = 0 . meanings: T (theo)c0 = 0 . K and H (theo)c2 (0) = 0 . T . Thelower values for the critical temperature are in agreementwith previous observations, that T c0 is typically overesti-mated in the experiment using traditional conveniencemethods.The dependence of the position of maximum in theNernst signal N (fl) ( h ) versus temperature for Nb . Si . is shown in the Fig. 6, which demonstrates both the nu-merically obtained theoretical curve and the values ex-tracted from the experimental data. One can see thatthe behavior of h ∗ ( t ) obtained from the numerical study ghost field (GF)mirror field (MF) GF sample 1GF sample 2MF sample 2 FIG. 6. Numerically evaluated ghost field curve h ∗ ( t ) fromEq. (9) in solid green and the mirror field in dashed green.Extracted ghost field from experimental data scaled by fittingparameters with error bars for both samples and the extractedmirror field for sample 2, when the Nernst coefficient startsto deviate from linear behavior. As one can see the error barsfor the ghost field become larger for larger temperatures sincethe maxima become very broad. of the extremum of Eq. (9) is strongly non-linear closeto T c0 , but becomes linear as function of temperaturequickly and can be described by Eq. (13). However, wenote that the error bars of the experimentally obtainedghost field become quite large due to the broadness of themaxima, such that the theoretical curve lies well withinthe error. Similar results are obtained for sample 2.Besides the two Nb x Si − x samples, we also analyzedseveral other available Nernst signal measurements usingNernst fluctuoscopy. In Fig. 7, the temperature depen- h th * (t)Eu-LSCOPCCO (x=0.17)ln(t)/h c2 FIG. 7. Fit of Eu-LSCO and PCCO to the numerically eval-uated ghost field curve h ∗ ( t ) from Eq. (9) compared to loga-rithmic dependence. dence of the normalized ghost field (scaled by H c2 (0))from two different experiments (dots and crosses) is com-pared to the numerically obtained ghost field line fromEq. (9) (solid red line), and to the empirical ∼ ln( t ) (thingray line). The experimental data on Eu-LSCO (pur- ple dots) are taken from Ref. 16 (Fig. 3b) and the dataon PCCO at doping level x = 0 .
17 (overdoped sample,green crosses) from Ref. 10 (Figure 10). One sees thatthe experimental findings fit the theoretical curve verywell, and, in particular follow the linear behavior givenby Eq. (13). N ( f l ) / N h FIG. 8. Fit of the normalized Nernst signal vs. magnetic fieldmeasurements on heavy-Fermion superconductor URu Si to Eq. (9) for different temperatures. The fitting parametersare T c0 = 1 .
14K and H c2 (0) = 1 . Insert: the ghost fieldmeasurements (blue circles) compared to the universal curvefollowing from Eq. (9) (green) and the logarithmic [Eq. (1)]fitting (red). In addition we added a linear fit to the experi-mental data (dashed blue).
Finally, we also applied the numerical Nernst fit-ting procedure to the heavy-Fermion superconductorURu Si , where we used the measured Nernst signaldata at different temperatures (Fig. 4 in that reference)and fitted N (fl) ( h ) with fitting parameters T c0 = 1 .
14 K,which is slightly lower than the empirically determinedvalue of 1 . K , and H c2 (0) = 1 .
11 T which is close to thevalues found in previous experimental works . Theresult is shown in Fig. 8. Based on these Nernst signalfittings, we extracted the positions of maxima (the valuesof ghost fields) and compared them to the experimentallyextracted values in the inset of Fig. 8. VI. DISCUSSION
We presented a complete analysis of the magnetic fieldand temperature dependence of the fluctuation inducedNernst signal in a large variety of superconductors rang-ing from conventional to unconventional ones. A com-plete expression of the fluctuation contribution to theNernst signal is obtained in the whole range of tempera-ture and magnetic field and applied to experimental databy numerical analysis. Both magnetic field and temper-ature dependence of the Nernst signal data is fitted withvery good accuracy using only two fitting parameters:the superconducting temperature T c0 and the upper crit-ical field H c2 (0).Our approach predicts a linear temperature depen-dence for the ghost critical field well above T c0 , contraryto previous heuristic arguments resulting in a logarith-mic dependence on temperature . Within the errors ofthe experimental data, this linearity is indeed observedin many superconductors far from T c0 . From a technicalpoint of view we note, that the maxima of the Nernstsignal become very shallow at large temperatures, whichmakes their extraction from experimental data very diffi-cult. Therefore the seemingly simple approach to deter-mination of the critical temperature T c0 and critical field H c2 (0) from the fitting of the ghost field should be done with care, giving high temperature points lower weight. ACKNOWLEDGEMENTS
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