Introduction

Four-point electrical probing has become a major interdisciplinary technique used in applied and fundamental physics, semiconductor industry, geology, etc. to characterize solid materials by measuring their resistances1. This method using an on-sample setup with two outer contacts supplying and draining the current as well as the inner pair of contacts measuring the voltage drop V is applied primarily to determine the sheet resistance. The separation of current and voltage terminals eliminates the lead and contact influence on the locally probed resistances. Next, van der Pauw2,3 proposed an original procedure for determining specific electrical resistance (resistivity) ρ of the sample of arbitrary shape with four probes located on the specimen’s periphery. In such nonlocal experiments with two contacts A and B for the current IAB and the other two C and D serving to measure the voltage drop VCD, the VCD value is found in regions far from the nominal current path and is therefore controlled by the entire current distribution across the sample. The measured nonlocal resistance is determined as

$${R_{{\text{AB, CD}}}}={V_{{\text{CD}}}}/{I_{{\text{AB}}}}$$
(1)

The development of a three-dimensional heterogeneous integration technology based on the vertical stacking of thin films of different types4,5 requires a generalization of the four-terminal methodology to the case of currents flowing perpendicular to the planes (CPP) in the multilayers under study6. In particular, such samples include sandwiches consisting of non-magnetic (N) and ferromagnetic (F) metal films, for example, giant magnetoresistance devices7,8. If the electronic properties of the layers differ fundamentally, the presence of interfaces can lead to proximity effects with material properties “leaking” into each other and thereby creating new functionalities otherwise absent in isolated components9. One of the most interesting heterostructures in this regard are superconductor/ferromagnet/superconductor (S/F/S) trilayers that are at the forefront of proximitized materials research thanks to unique properties arising due to competing quantum effects of Cooper pairing and magnetic exchange interactions10,11. The product of their area and the normal CPP resistance Rn, the main characteristic of metallic S/F/S junctions in the normal (n) state, is primarily determined by the charge transport across the S/F interface12,13. Therefore, its knowledge is very important for improving corresponding superconducting circuitries, while experimental definition and interpretation of the Rn value represent a much more complex task than a similar (already standard) problem for a parallel charge flow14. For this reason, the multilayer specimen has been often modeled as an equivalent circuit with several parallel resistances6,15,16. This approach allowed the authors to leave the already standard on-sample four-probe configuration unchanged, although ignoring the role of contact resistances between layers left unattended some physical phenomena in the studied hybrids, which are discussed below.

As for СPP measurements, one of the surprising discoveries was the detection, more than fifty years ago, of nonlocal negative resistance in mesoscopic sandwiches with two metal electrodes separated by an insulating layer17. Let us emphasize that it was not the negative differential resistance dI(V)/dV < 0, which appears in a limited voltage range away from V = 0 but rather the absolute negative value of the four-probe resistance determined by Eq. (1), which is undoubtedly apparent and emerges when the difference in electrical potentials between two voltage contacts is opposite to the expected one18. Following theoretical predictions19,20previously observed negative values of the four-terminal resistance17,18,21,22,23 have been obtained for the crossed configuration of current and voltage contacts shown in Fig. 1a. The explanation for the inequality RAD, CB < 0 was based on the corresponding four-resistor model24,25,26 (Fig. 1a) or its more advanced finite-element modification17,21. This approach predicts positive RAB, CD (Fig. S1a) while RAD, CB (Figs. 1 and S1b) values can be negative, see the figures and related discussions in Supplementary Material.

Fig. 1
figure 1

Four-probe non-local configurations for resistance measurements of a non-magnetic/ferromagnetic/non-magnetic (NFN) sandwich, when the СPP charge flow is driven from source IS to drain ID and the voltage drop is measured between two other nodes bounding the related conditional resistor(s). (a) Conventional four-resistor model24,25,26 with the crossed configuration of current and voltage contacts. (b) Circuit diagram with six conditional resistors for RAB, CD measurements. (c) Schematics of the through-N/F/N sample four-terminal configuration with contacts B and D made of Pt, two NbN superconducting films with contacts A and C, and the ferromagnetic interlayer F in grey, blue, and yellow, respectively. The currents IXY mean the net currents flowing from contact X to contact Y (X, Y = A, B, C, D and X ≠ Y), the arrows correspond to the expected directions of the charge flow and, accordingly, to positive current values. If, due to the charge transport redistribution, the current between the two nodes, by which the voltage drop is determined, changes its direction (and consequently the voltage drop changes its sign), the four-probe resistance calculated using Eq. (1) will be negative.

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The conclusion RAB, CD > 0 contradicts experimental data for our three-dimensional version of the four-terminal configuration shown in Fig. 1c as well as the general theoretical approach to transport in mesoscopic systems, see below. To resolve the inconsistencies, we have revisited the four-resistor approximation used before24,25,26 and developed a simple heuristic model (Fig. 1b) that differs by introducing two additional scattering links. To experimentally reproduce its main features, a non-trivial way of connecting contacts to the sample under study was used, as well as its dimensions. Usually, the multilayer stack is patterned into pillars with designed dimensions of tens or hundreds of nanometers, see, e.g., the work27. After patterning, the pillars are passivated and a thick, highly conductive top electrode is applied to them, extending along the entire area of the pillar. This ensures the creation of an equipotential surface which largely eliminates the effects of unavoidable sample inhomogeneities28. On the contrary, our task was to create contact pads small compared to the sample size, the transverse dimensions of which were increased to 25 μm × 25 μm. The trilayers obtained were passivated with side bilayers formed by a 30 nm thick AlN layer and about 200 nm thick SiO2 oxide and two 35 nm thick platinum strips were deposited on top of them. A schematic view of the studied sandwiches and their biasing is shown in Fig. 1b, while Fig. S6 in Supplementary Material demonstrates the related cross-section. Using this approach, we found that the CPP resistance RAB, CD (Fig. 1c) of the samples with the internal structure intentionally modified using various F interlayers, turned out to be very sensitive to small local fluctuations in the normal and superconducting parameters in depth.

Results

Six-resistor model for nonlocal four-terminal probing

In mesoscopic physics, the charge transport in a multi-terminal structure is usually described by the Landauer-Büttiker approach relating the electrical resistance to the scattering properties of the system20,29. In its linear-response version, the current leaving contact X is defined by the equation equivalent of related Ohm’s formula \({I_{\text{X}}}=(2{e^2}/h)\sum\limits_{j} ( {T_{{\text{XY}}}}{V_{\text{X}}} - {T_{{\text{YX}}}}{V_{\text{Y}}})\), where TXY is the probability of charge transfer from terminal X to terminal Y, \({V_{\text{Y}}}={\mu _{\text{Y}}}/e\), µY is the related chemical potential, X, Y = A, B, C, D, and X ≠ Y. The currents IXY in Fig. 1 and S1 in Supplementary Material mean the net charge flows from contact X to contact Y: \({I_{{\text{XY}}}}=(2{e^2}/h)({T_{{\text{XY}}}}{V_{\text{X}}} - {T_{{\text{YX}}}}{V_{\text{Y}}})\). Leaving quantum analysis with related interference effects for the future, we restrict ourselves to the classical limit having the advantage to transform a material property directly into a qualitatively understandable concept. In the absence or presence of a relatively weak magnetic field, TXY and TYX probabilities coincide and the above expression for IX will depend on the \({V_{\text{X}}} - {V_{\text{Y}}}\) difference. As a result, we can introduce “resistances” describing such contributions \({R_{{\text{XY}}}}=h/(2{e^2}{T_{{\text{XY}}}})\) and then proceed with the four-probe configuration as a circuit of six resistors by applying the known Kirchhoff laws: the conservation of currents at each node and the vanishing directional sum of the voltage drops around any closed loop. Let us emphasize that RXY resistances characterize the current distributions within the circuit under consideration and cannot be determined separately. However, as will be seen below, measuring the four-probe resistance allows one to obtain a qualitative idea of the resistance of individual sections in the heterostructure studied.

To demonstrate the difference between the four-resistor schemes24,25,26 shown in Fig. 1a and the six-resistor circuit diagram in Fig. 1b, let us turn to a well-known analogy, the electrical Wheatstone bridge circuit30. In the simplest realization, it consists of two parallel branches linked by a third one located between intermediate points at which the difference in potentials, the circuit output, is measured. When all resistances are equal, the voltage drop across the third branch is zero due to the current balance. If all resistive components in our model coincide and are equal to R, then from Figs. 2a and S3, we obtain RAB, CD = RAD, CB = 0 in contrast to the previous model, Fig. 1a, which gives fundamentally different results \({R_{{\text{AB,}}{\kern 1pt} {\text{CD}}}}=R/4\) and \({R_{{\text{AD,}}{\kern 1pt} {\text{CB}}}}=0\), for details see Supplementary Material.

In the general case, the model shown in Fig. 1b contains too many parameters, so we assume the existence of two groups of resistances: four resistances Rb, identified with the bulk (inner core) contribution, and two near-surface resistances Rs. In N/F/N trilayers discussed below, the first parameter is associated with two N/F interfaces and the F film in series while the second parameter characterizes an N electrode and its interface with a Pt contact. In the case of a separate NbN layer, Rb and Rs are properties of its inner part and the near-surface area, respectively. The difference in the Rb and Rs values leads to a non-zero outcome, negative for \({R_{\text{s}}}>{R_{\text{b}}}\) and positive otherwise (Fig. 2a and Fig. S3). Moreover, small deviations of the Rs/Rb ratio from unity entail relatively large variations in RAB, CD and RAD, CB values.

Fig. 2
figure 2

Effect of the difference between near-surface Rs and bulk Rb resistances. (a) Calculated four-probe resistance RAB, CD versus the ratio Rs/Rb where \({R_{{\text{AB}}}}={R_{{\text{CD}}}}={R_{{\text{AD}}}}={R_{{\text{CB}}}}={R_{\text{b}}}\) and \({R_{{\text{AC}}}}={R_{{\text{DB}}}}={R_{\text{s}}}\). The red lines in the insets indicate the dominant directions of charge flows at the surface (\({R_{\text{s}}}<{R_{\text{b}}}\)) and in the bulk (\({R_{\text{s}}}>{R_{\text{b}}}\)). (b) Measured temperature dependence of CPP four-probe resistances RAB, CD(T) for NbN(80 nm)/F(50 nm)/NbN(80 nm) trilayers with F = Co, Ni, and NiCu (dashed-dotted, dotted, and dashed lines, respectively) compared with the same characteristic for a single NbN (160 nm) layer (solid line).

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Nonlocal four-probe resistances of N/F/N trilayers

The proposed interpretation of through-the-sample four-probe measurements was verified by changing the relationship between normal-state resistances Rs and Rb in NbN(80 nm)/F(50 nm)/NbN(80 nm) sandwiches with F = Co, Ni, and NiCu alloy (Fig. 2b). The main details of their fabrication are described in Methods. The specific resistances of the materials used can be found in the literature: ρNbN = 70–80 µΩ⋅cm31,32ρCo = 5.2–6.8 µΩ⋅cm8,14,28ρNi = 3.0–6.8 µΩ⋅cm8,33and ρNiCu = 51 µΩ⋅cm34,35. As can be seen, the resistivities of the NbN compound and the NiCu alloy are close in value while ρCo and ρNi are an order of magnitude smaller.

As noted above, in addition to the F-layer resistance, the S/F interfaces make their own (often dominant12,14) contribution to the bulk Rb resistance. It means that in NiCu-based sandwiches one should expect inequality Rs < Rb, which leads to a positive value of the CPP four-probe resistance RAB, CD according to Fig. 2a and in full agreement with the experiment (Fig. 2b). Replacing the NiCu alloy with strong ferromagnets Co or Ni, one can suppose the opposite inequality Rs > Rb. Indeed, it was found14 that bulk N-state resistances of Nb/Co/Nb trilayers are determined primarily by the doubled Nb/Co interface contributions. Using the concerned data14 and the fact that the interface resistance of the contact of two dissimilar metals is roughly proportional to the sum of their resistivities36 we expect Rb values for NbN/Co/NbN and NbN/Ni/NbN trilayers to be much lower than in NiCu-based sandwiches and, as a result, negative RAB, CD resistances, Fig. 2b.

In Fig. 2b, the CPP four-probe results obtained for three-layer hybrids are compared with charge transport data for a single 160 nm thick NbN film that is much thicker than its dirty-limit coherence length ξ of a few nm37. According to the paper31the critical temperature Tc of the N-to-S transition in NbN films is uniquely determined by the value of the product kFl where l is the electron mean free path and kF is the Fermi wave vector. Using Fig. 1b from the work31the free-electron value kF = 1.9⋅1010 m− 1, and our Tc ≈ 16 K, we find that kFl ≈ 8 and l is less than 1 nm, which is much smaller than any size in the NbN films studied. In such samples, charge transport turns into a local process that makes it probable to define a local conductivity accurately characterizing the system response to an applied electric field38. This statement justifies the possibility of separating the contributions of the bulk with electron transport controlled by structural defects and the scattering of electrons on random inhomogeneities at the conductor boundaries39.

The surface-scattering-limited regime, also known as superdiffusive, is qualitatively different from the ordinary bulk-scattering transport since in this case, the conductance is determined by a small fraction of itinerant electrons propagating at grazing angles with the film surface40,41. We speculate that just this peculiar electron transfer, which is localized in the near-surface region between two probes on the same surface, controls Rs values, while the Rb magnitudes for probes on opposite boundaries are determined by collisions with bulk defects. Although the specific intensity of the superdiffusive transfer of electrons, whose wave vectors are almost parallel to the film surface, significantly exceeds that for trajectories inside the bulk, the resistance Rs is expected to be greater than Rb due to the difference in electron paths by two orders of magnitude. This means that we are dealing with a conditional “sandwich” formed by two more resistive outer regions (NbN film surfaces) and a less resistive bulk. Immediately, following Fig. 2a, we get a negative value of RAB, CD(T > Tc) for a single NbN layer in the normal state as follows from Fig. 2b. In general, its RAB, CD(T) behavior above the critical temperature Tc is сlose to a constant value since the electrical resistance of thick NbN layers changes very little with temperature, sometimes even with a negative temperature coefficient42.

However, with the insertion of ferromagnetic interlayers, this statement is no longer true. While RAB, CD(T) is almost constant for the trilayer with a NiCu film, the other two S/F/S sandwiches demonstrate a noticeable increase in the absolute value of the four-probe resistance, which remains negative. As for the NiCu alloy, the observed RAB, CD(T) behavior can be explained by a very low change in its resistivity over a wide temperature range, similar to NbN in the normal state43. As a result, the CPP four-probe resistance of NiCu-based samples does not vary noticeably with temperature. In contrast, the temperature coefficients of Ni and Co resistances of the order of 0.005–0.007 K− 1 belong to the highest among metals33which leads to an increase in Rb and, as a consequence, to the growth in the absolute value of the four-point resistance (Fig. 2b).

When the temperature decreases below Tc, the resistances RAB, CD(T < Tc) become very small. Previous studies of Nb/F/Nb trilayers44 showed that the doubled product of the area through which the CPP current flows and its resistance ≅ 6 × 10− 11 Ω сm2 is similar for all elemental F metals studied8 and is twice as high for the alloys. Using the estimates 10− 11 Ω сm2 for the interface contribution8 and 10− 5 Ω сm for the F metal resistivity we obtain the expected value of the order of 20–30 µΩ for the residual resistance of the Nb/F/Nb trilayers at T < < Tc. Our data show a residual resistance slightly below 100 µΩ for NbN/F/NbN trilayers with F = Co and Ni and a three to four times further increase in this value for F = NiCu.

To conclude, we emphasize once again that the nonlocal implementation of the CPP four-probe technique applied to transversely inhomogeneous conducting sandwiches is capable of revealing the charge transport features hidden inside them and, thus, serving as a simple scalable diagnostic approach to characterize superconducting, spintronic, and hybrid electronic heterostructures. We expect that the developed six-resistor model will be also useful in express analyses of related non-ohmic devices with a strong dependence of transport properties on an external parameter, the role of which in the case of superconductors is played by temperature.

Methods

Sample preparation

Single NbN films and NbN-based trilayers on c-cut Al2O3 substrates were obtained by pulsed laser deposition technique in an ultrahigh vacuum chamber using an excimer KrF laser with 248 nm wavelength, the pulse duration of 35 ns, and the laser fluency of 4.94 J⋅cm− 2. NbN films were deposited at a constant temperature of 600 °C under 9.3 Pa pressure of the reactive N2 + 1% H2 atmosphere. For Co and Ni, the deposition of the ferromagnetic films took place at 200 °C in the Ar atmosphere under 4.5 Pa pressure and 5.2 Pa pressure for Ni50Cu50 alloy. After deposition, the chemical and structural properties of the samples were characterized by several analytical techniques45. Subsequently, they were patterned using optical lithography and Ar ion etching into a 25 μm × 25 μm square shape. Figure S6 in Supplementary Material shows schematically our non-magnetic/ferromagnetic/non-magnetic samples with the contact arrangement for carrying out nonlocal CPP four-probe sample measurements which, as argued in the main text of the article and Supplementary Material, strongly enhances sensitivity to inhomogeneity factors and make related experiments on hybrid trilayers the method of choice for knowing the spatial distribution of related parameters.

Electrical measurements

The CPP nonlocal four-probe RAB, CD (Eq. (1)) resistances were measured using the Physical Property Measurements System (PPMS) DynaCool (Quantum Design) by applying a 10 µA current into the NbN strip beneath the trilayer under study and draining it out of the Pt contact above it, see Fig. 1c in the main text and Fig. S6 in Supplementary Material. The average current density was about 1.5 A/cm2. In Fig. 2b, we demonstrate the non-local four-probe resistance-vs-temperature data for all-metallic hybrid sandwiches composed of two 80 nm thick NbN films and a 50 nm thick core made of three archetypal ferromagnets, Co, Ni, or NiCu alloy, which are compared with the corresponding data for single 160 nm thick NbN films.