3ω correction method for eliminating resistance measurement error due to Joule heating

Electrical four-terminal sensing at (sub-)micrometer scales enables the characterization of key electromagnetic properties within the semiconductor industry, including materials’ resistivity, Hall mobility/carrier density, and magnetoresistance. However, as devices’ critical dimensions continue to shrink, significant over/underestimation of properties due to a by-product Joule heating of the probed volume becomes increasingly common. Here, we demonstrate how self-heating effects can be quantified and compensated for via 3ω signals to yield zero-current transfer resistance. Under further assumptions, these signals can be used to characterize selected thermal properties of the probed volume, such as the temperature coefficient of resistance and/or the Seebeck coefficient. © 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0063998


I. INTRODUCTION
Here, we wish to introduce the 3ω technique 1,2 into the context of micro-four-point probe (M4PP) electrical metrology, [3][4][5][6][7][8][9] with the aim of improving the measurement accuracy of the latter via quantification and compensation for the so-called Self-Heating Effect (SHE). 10,11 The undesirable heating associated with electrical microprobing came to attention in the early 1960s, when the contact size between the metallic electrodes and the probed semiconductors dropped below ∼0.1 mm. [12][13][14] Since then, critical dimensions of semiconductor devices have shrunk by multiple orders of magnitude, while self-heating effects are often intentionally amplified. 15,16 Thus, the elimination of undesirable by-products of heating in electrical microprobing has become increasingly relevant 17 and is the main focus of the present contribution.
Traditionally, the unintentional generation of Joule heat during electrical microprobing has been mitigated via, e.g., measurement at sufficiently low 14,18 or transient/pulsed currents 17 and/or via optimization of the probing geometry. 18,19 In a recent paradigm shift, we demonstrated 9 that the seemingly undesired heating at such scales is highly reproducible and may be intentionally amplified in order to quantify the thermal properties of the material stack under test. Figure 1 summarizes the key highlights from Ref. 9, where an equidistant micro-four-point probe with a pitch of 10 μm [ Fig. 1(a)] was used to measure the sheet resistance of an ultrathin (16 nm) Pt film deposited on top of a fused silica substrate. The observed increase in the sheet resistance with the sampling current was proportional to the current squared in the 0.5-5 mA range [circles in Fig. 1 (FEM) simulations [continuous line in Fig. 1(b)]. Furthermore, by treating the thermal conductivity of the substrate as a known parameter, the temperature coefficient of resistance (TCR) of the thin film could be estimated with an excellent (<2%) precision across multiple consecutive measurements ( Table 1 in Ref. 9). A classic correction for self-heating effects involves the extrapolation of the linear trend in Fig. 1(b) to zero current 10,11 to obtain the heat-unaffected, "zero-current" resistance R 0 . However, such a correction requires two or more resistance measurements at wellseparated currents. This requirement not only prolongs the measurement time but also exposes the target to potential irreversible changes at higher currents. Noting that mainstream M4PP metrology uses lock-in amplification (LIA) to reduce electrical noise, 7 here we explore the possibility of estimating zero-current resistance by isolating and quantifying the thermally induced voltage component from higher voltage harmonics. Falling into the broad category of 1ω-2ω-3ω methods, 2 our particular M4PP measurement strategy and 3ω correction scheme are briefly outlined below.

II. THEORY
The following mathematical treatment is a recapitulation of Dames and Chen. 2 For further simplification, we shall assume a slowly varying measurement current (quasi-DC) such that any electrical and thermal lags (e.g., due to electrical and thermal capacitances) can be ignored. Thus, the location-and time-dependent temperature increment ΔT(r, t) of the probed volume instantaneously follows the Joule heat dissipation distribution, which to first order can be expressed in terms of the position-dependent sample resistivity 0 (r) and current density J(r, t) as 0 (r)|J(r, t)| 2 . Furthermore, using J(r, t) = g J (r)I(t), where I(t) is the instantaneous current and g J (r) is the measurement geometry-dependent vector function, 20 the instantaneous temperature increment can be written as ΔT(r, t) = ψ(r)I 2 (t). Note that ψ(r) is a transfer function from the current squared into temperature, which explicitly depends only on the probing geometry (and implicitly on the material properties of the probed volume).
We shall assume a linearized resistivity model (r, t) = 0 (r) [1 + αΔT(r, t)], where α is the temperature coefficient of resistivity (TCR), such that the instantaneous transfer resistance R(t) becomes: Here, R 0 is the zero-current transfer resistance, andŜ(r) is the M4PP sensitivity to a local change in resistivity, 9,20 which in this paper is defined as:Ŝ (r) = J(r) .J(r) IĨ whereJ andĨ are the hypothetical current density and intensity in an adjoint system with interchanged current and voltage assignments. 9,20 The function ΔT eff (t) is a domain-averaged "effective" temperature increment, 2,9 which for M4PP may be expressed as With this definition of Ψ as a domain-scaled and -averaged ψ(r), the instantaneous transfer resistance becomes If we use a low-frequency sinusoidal measurement current at the angular frequency ω = 2πf , the instantaneous measured voltage becomes where we have introduced the first and third harmonic resistances, defined as R 1ω = R 0 + 3 4 R 0 αΨI 2 0 and R 3ω = − 1 4 R 0 αΨI 2 0 , respectively. Note that for a positive α, the expected third harmonic voltage phase angle is φ 3ω = π, making R 3ω negative; for a negative α, φ 3ω = 0 and R 3ω is positive.
Equation (6) also shows that the zero-current transfer resistance R 0 can be recovered from harmonic resistances R 1ω and R 3ω by the simple calculation The temperature coefficient of resistivity can then be calculated from which is easily derived from Eqs. (6) and (7).

A. Samples
In this study, we focus on four industrially relevant samples, including a metallic nanoscale interconnect [ Fig. 2(a)], a metallic

B. Instrumentation
Electric resistance measurements at the microscale were performed using a CAPRES microRSP ® -A300 tool utilizing digital lock-in amplification (LIA). 7 Specifically, and in contrast to its default (factory) settings of extracting solely the first harmonic, 7 the proprietary LIA module of the A300 tool (cf., Ref. 28) was programmatically extended to extract the transfer resistance Rnω and its phase φ nω up to the third harmonic (n = 1, 2, 3) of the input current frequency. This extended LIA module was thoroughly tested against a manifold of known waveforms as well as benchmarked against an external best-in-class tool (MFLI from Zürich Instruments).
Samples (a)-(c) were characterized using an equidistant probe with a pitch of 8 μm; 7 sample (d) with a specialized current-in-plane tunneling (CIPT) probe with varying distances in the 0.5-10 μm range. 27 31 All models included coupled electric currents and heat transfer in solids modules; the simulation of sample (c) included thermoelectric effects as well. All nominal device dimensions/geometries were accurately reproduced (with domain reduction due to symmetry wherever possible), and all the experimentally obtained physical constants (Sec. III A) were assigned. Model element sizes were on average a factor of 5-10 smaller than the critical dimensions in their vicinity. The initial conditions of potential (0 V) and temperature (300 K) were supplemented by a thermal insulation on the probing surface (the upper plane of each device) and a constant temperature of 300 K on all other external surfaces. In all models, two terminals of opposite polarity were located 8 μm apart, delivering a sinusoidal current through corresponding trapezoidal (a), semicircular (b), and hemispherical (c) contact geometries (with effective contact radii further denoted as r 0 ). For the metal-semiconductor contacts in (c), an additional thermal flux at each contact (arising from a contact resistance of 50 Ω, obtained experimentally) was added, following the procedure described in Ref. 9. A time-dependent solver was used to simulate the underlying waveforms (64 time points per period) for each of the observation points in Figs. 2(a)-2(c), with numerical tolerance <10 −5 . Boundary probes (a) or point probes [(b) and (c)] were used to obtain the voltage at the approximated (a) or precise [(b) and (c)] locations of the two sensing electrodes. The voltage harmonics were then extracted using a numerical lock-in amplifier (thoroughly validated against synthetic waveforms). Convergence tests were conducted to verify that the domain size, meshing, and tolerance were adequately selected.

A. De-trending resistivity measurements from self-heating effects
The linear response of line resistance [ Fig. 2(a)], sheet resistance [ Fig. 2(b)], bulk resistivity [ Fig. 2(c)], and magnetoresistance [ Fig. 2(d)] as a function of the square of the probing current is ubiquitous in all studied materials, leading to fractional errors of up to a few percent. 32 Given that the M4PP method is generally associated with a precision and reproducibility of <0.1%, 33 self-heating errors of up to a few percent cannot be regarded as negligible and necessitate an adept correction scheme. In contrast, the proposed 3ω correction method, involving a linear combination of the first and third harmonics [Eq. (7)], yields current-insensitive "flat" trends [squares

ARTICLE
scitation.org/journal/rsi in Figs. 2(a)-2(d)], whose slopes are statistically indistinguishable from 0 and whose means overlap with the zero-current intercept that may be regressed from the uncorrected measurements. 10,11 B. Determination of thermal properties The trends of the uncorrected measurements [intentionally removed by Eq. (7) The obtained TCR estimates quoted above are in remarkable agreement with the literature. Specifically, the best fit α = 1.13 × 10 −3 K −1 for the 50 nm wide Cu nanowire promptly extends the trend emerging from wider lines (80-330 nm) 34 and is also in line with more recent findings. 35 The α = 3.34 × 10 −3 K −1 of the Ni thin film is well bracketed by the broad range reported in Ref. 36 and specifically matches the estimate for a 12.3 nm thick Ni film in Ref. 37 (estimated 3.2 ± 0.4 × 10 −3 K −1 from their Fig. 2). Finally, the best-fit α = 1.93 × 10 −3 K −1 of Si:B not only matches its theoretically expected values, 38,39 but is also within ∼6% of its direct and independent M4PP measurement on a hotplate. Our best fit contact radii (r 0 ), while technically representing a method-specific parameter rather than any useful sample-probe interaction, are nevertheless consistent with scanning electron microscopy estimates (e.g., Ref. 19).

V. CONCLUSION
The gradual miniaturization of microelectronic devices results in the increase of undesirable self-heating effects when these devices are subjected to electrical/electromagnetic probing using the M4PP. Key fingerprints of heating in response to an applied alternating current can be detected in higher harmonics of the measured voltage, 2,12,13 which are easy to isolate by means of lock-in amplification. 7 Here, we have presented the theory (Sec. II), experimental proof (symbols in Fig. 2), and numerical verification (lines in Fig. 2) for the use of 3ω voltage signals for de-trending M4PP resistance measurements from self-heating effects. The presented 3ω correction [Eq. (7)] was demonstrated on samples of broadly varying structures and dimensionalities (Fig. 2). In all studied materials, a definitive (percent-level) improvement in accuracy of M4PP measurements was demonstrated. This marks the 3ω correction as yet another qualitative breakthrough in the evolving accuracy of M4PP resistance metrology (cf., Ref. 27).
The success in reproducing both raw and de-trended M4PP observations via FEM simulations supports the applicability of our quasi-DC assumption to the low-frequency range (<50 Hz), within which routine M4PP measurements are typically performed. 8,9 While an extension of the theory for true AC is highly desirable, 2 the electrical and thermal fields arising from even the simplest fourpoint probing geometries are complex 9 and render such a mathematical treatment significantly beyond the scope of this work. It should be noted that the presented DC-limit correction has been observed to perform well even at higher frequencies of ∼400 Hz. Nevertheless, since the cut-off frequency for the proposed 3ω correction depends on a multiplicity of parameters (including, among others, the desired tolerance, probe geometry, material properties, etc.), we are currently hesitant to report a guiding cut-off frequency. Instead, we encourage to explore and set such thresholds for particular case scenarios via sensitivity analysis based on numerical modeling [cf., Figs. 2(a)-2(c)].
We believe that this study solidifies the recently demonstrated capability of the M4PP for TCR metrology, 9 extending it to a much broader range of materials, device geometries, and electrical dimensionalities ( Fig. 2). At the same time, we emphasize that in our current state-of-the-art, the thermal properties obtained from such M4PP measurements are highly model-driven and are not to be mistaken for a straightforward measurand (as transfer resistance is). Nevertheless, we believe that the prospects of the higher harmonic M4PP measurements to complement, overlap, and perhaps even crossover with scanning thermal microscopy techniques 40,41 are rather self-evident and highly promising.

DATA AVAILABILITY
The data and models that comprise Figs. 1 and 2 are available from the corresponding author upon reasonable request.