Effective electrical resistivity in a square array of oriented square inclusions.

The continuing miniaturization of optoelectronic devices, alongside the rise of electromagnetic metamaterials, poses an ongoing challenge to nanofabrication. With the increasing impracticality of quality control at a single-feature (-device) resolution, there is an increasing demand for array-based metrologies, where compliance to specifications can be monitored via signals arising from a multitude of features (devices). To this end, a square grid with quadratic sub-features is amongst the more common designs in nanotechnology (e.g. nanofishnets, nanoholes, nanopyramids, µLED arrays etc). The electrical resistivity of such a quadratic grid may be essential to its functionality; it can also be used to characterize the critical dimensions of the periodic features. While the problem of the effective electrical resistivity ρ eff of a thin sheet with resistivity ρ 1, hosting a doubly-periodic array of oriented square inclusions with resistivity ρ 2, has been treated before (Obnosov 1999 SIAM J. Appl. Math. 59 1267-87), a closed-form solution has been found for only one case, where the inclusion occupies c = 1/4 of the unit cell. Here we combine first-principle approximations, numerical modeling, and mathematical analysis to generalize ρ eff for an arbitrary inclusion size (0 < c < 1). We find that in the range 0.01 ≤ c ≤ 0.99, ρ eff may be approximated (to within <0.3% error with respect to finite element simulations) by: [Formula: see text] [Formula: see text] whereby at the limiting cases of c â†’ 0 and c â†’ 1, α approaches asymptotic values of α = 2.039 and α = 1/c - 1, respectively. The applicability of the approximation to considerably more complex structures, such as recursively-nested inclusions and/or nonplanar topologies, is demonstrated and discussed. While certainly not limited to, the theory is examined from within the scope of micro four-point probe (M4PP) metrology, which currently lacks data reduction schemes for periodic materials whose cell is smaller than the typical µm-scale M4PP footprint.

Wafer-scale graphene quality assessment using micro four-point probe mapping.

Micro four-point probes (M4PP) provide rapid and automated lithography-free transport properties of planar surfaces including two-dimensional materials. We perform sheet conductance wafer maps of graphene directly grown on a 100 mm diameter SiC wafer using a multiplexed seven-point probe with minor additional measurement time compared to a four-point probe. Comparing the results of three subprobes we find that compared to a single-probe result, our measurement yield increases from 72%-84% to 97%. The additional data allows for correlation analysis between adjacent subprobes, that must measure the same values in case the sample is uniform on the scale of the electrode pitch. We observe that the relative difference in measured sheet conductance between two adjacent subprobes increase in the transition between large and low conductance regions. We mapped sheet conductance of graphene as it changed over several weeks. Terahertz time-domain spectroscopy conductivity maps both before and after M4PP mapping showed no significant change due to M4PP measurement, with both methods showing the same qualitative changes over time.

Terahertz wafer-scale mobility mapping of graphene on insulating substrates without a gate.

We demonstrate wafer-scale, non-contact mapping of essential carrier transport parameters, carrier mobility (µdrift), carrier density (Ns), DC sheet conductance (σdc), and carrier scattering time (τsc) in CVD graphene, using spatially resolved terahertz time-domain conductance spectroscopy. σdc and τsc are directly extracted from Drude model fits to terahertz conductance spectra obtained in each pixel of 10 × 10 cm2 maps with a 400 µm step size. σdc- and τsc-maps are translated into µdrift and Ns maps through Boltzmann transport theory for graphene charge carriers and these parameters are directly compared to van der Pauw device measurements on the same wafer. The technique is compatible with all substrate materials that exhibit a reasonably low absorption coefficient for terahertz radiation. This includes many materials used for transferring CVD graphene in production facilities as well as in envisioned products, such as polymer films, glass substrates, cloth, or paper substrates.

Electrically continuous graphene from single crystal copper verified by terahertz conductance spectroscopy and micro four-point probe.

The electrical performance of graphene synthesized by chemical vapor deposition and transferred to insulating surfaces may be compromised by extended defects, including for instance grain boundaries, cracks, wrinkles, and tears. In this study, we experimentally investigate and compare the nano- and microscale electrical continuity of single layer graphene grown on centimeter-sized single crystal copper with that of previously studied graphene films, grown on commercially available copper foil, after transfer to SiO2 surfaces. The electrical continuity of the graphene films is analyzed using two noninvasive conductance characterization methods: ultrabroadband terahertz time-domain spectroscopy and micro four-point probe, which probe the electrical properties of the graphene film on different length scales, 100 nm and 10 µm, respectively. Ultrabroadband terahertz time-domain spectroscopy allows for measurement of the complex conductance response in the frequency range 1-15 terahertz, covering the entire intraband conductance spectrum, and reveals that the conductance response for the graphene grown on single crystalline copper intimately follows the Drude model for a barrier-free conductor. In contrast, the graphene grown on commercial copper foil shows a distinctly non-Drude conductance spectrum that is better described by the Drude-Smith model, which incorporates the effect of preferential carrier backscattering associated with extended, electronic barriers with a typical separation on the order of 100 nm. Micro four-point probe resistance values measured on graphene grown on single crystalline copper in two different voltage-current configurations show close agreement with the expected distributions for a continuous 2D conductor, in contrast with previous observations on graphene grown on commercial copper foil. The terahertz and micro four-point probe conductance values of the graphene grown on single crystalline copper shows a close to unity correlation, in contrast with those of the graphene grown on commercial copper foil, which we explain by the absence of extended defects on the microscale in CVD graphene grown on single crystalline copper. The presented results demonstrate that the graphene grown on single crystal copper is electrically continuous on the nanoscopic, microscopic, as well as intermediate length scales.

Theoretical analysis of a dual-probe scanning tunneling microscope setup on graphene.

Experimental advances allow for the inclusion of multiple probes to measure the transport properties of a sample surface. We develop a theory of dual-probe scanning tunneling microscopy using a Green's function formalism, and apply it to graphene. Sampling the local conduction properties at finite length scales yields real space conductance maps which show anisotropy for pristine graphene systems and quantum interference effects in the presence of isolated impurities. Spectral signatures in the Fourier transforms of real space conductance maps include characteristics that can be related to different scattering processes. We compute the conductance maps of graphene systems with different edge geometries or height fluctuations to determine the effects of nonideal graphene samples on dual-probe measurements.

Graphene conductance uniformity mapping.

We demonstrate a combination of micro four-point probe (M4PP) and non-contact terahertz time-domain spectroscopy (THz-TDS) measurements for centimeter scale quantitative mapping of the sheet conductance of large area chemical vapor deposited graphene films. Dual configuration M4PP measurements, demonstrated on graphene for the first time, provide valuable statistical insight into the influence of microscale defects on the conductance, while THz-TDS has potential as a fast, non-contact metrology method for mapping of the spatially averaged nanoscopic conductance on wafer-scale graphene with scan times of less than a minute for a 4-in. wafer. The combination of M4PP and THz-TDS conductance measurements, supported by micro Raman spectroscopy and optical imaging, reveals that the film is electrically continuous on the nanoscopic scale with microscopic defects likely originating from the transfer process, dominating the microscale conductance of the investigated graphene film.

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.

A variable probe pitch micro-Hall effect method.

Hall effect metrology is important for a detailed characterization of the electronic properties of new materials for nanoscale electronics. The micro-Hall effect (MHE) method, based on micro four-point probes, enables a fast characterization of ultrathin films with minimal sample preparation. Here, we study in detail how the analysis of raw measurement data affects the accuracy of extracted key sample parameters, i.e., how the standard deviation on sheet resistance, carrier mobility and Hall sheet carrier density is affected by the data analysis used. We compare two methods, based primarily on either the sheet resistance signals or the Hall resistance signals, by theoretically analysing the effects of electrode position errors and electrical noise on the standard deviations. We verify the findings with a set of experimental data measured on an ultrashallow junction silicon sample. We find that in presence of significant electrical noise, lower standard deviation is always obtained when the geometrical analysis is based on the sheet resistance signals. The situation is more complicated when electrode position errors are dominant; in that case, the better method depends on the experimental conditions, i.e., the distance between the insulating boundary and the electrodes. Improvement to the accuracy of Hall Effect measurement results is crucial for nanoscale metrology, since surface scattering often leads to low carrier mobility.

Electrical characterization of single nanometer-wide Si fins in dense arrays.

This paper demonstrates the development of a methodology using the micro four-point probe (µ4PP) technique to electrically characterize single nanometer-wide fins arranged in dense arrays. We show that through the concept of carefully controlling the electrical contact formation process, the electrical measurement can be confined to one individual fin although the used measurement electrodes physically contact more than one fin. We demonstrate that we can precisely measure the resistance of individual ca. 20 nm wide fins and that we can correlate the measured variations in fin resistance with variations in their nanometric width. Due to the demonstrated high precision of the technique, this opens the prospect for the use of µ4PP in electrical critical dimension metrology.

Electrical Homogeneity Mapping of Epitaxial Graphene on Silicon Carbide.

Epitaxial graphene is a promising route to wafer-scale production of electronic graphene devices. Chemical vapor deposition of graphene on silicon carbide offers epitaxial growth with layer control but is subject to significant spatial and wafer-to-wafer variability. We use terahertz time-domain spectroscopy and micro four-point probes to analyze the spatial variations of quasi-freestanding bilayer graphene grown on 4 in. silicon carbide (SiC) wafers and find significant variations in electrical properties across large regions, which are even reproduced across graphene on different SiC wafers cut from the same ingot. The dc sheet conductivity of epitaxial graphene was found to vary more than 1 order of magnitude across a 4 in. SiC wafer. To determine the origin of the variations, we compare different optical and scanning probe microscopies with the electrical measurements from nano- to millimeter scale and identify three distinct qualities of graphene, which can be attributed to the microstructure of the SiC surface.

Breakthrough in current-in-plane tunneling measurement precision by application of multi-variable fitting algorithm.

We present a breakthrough in micro-four-point probe (M4PP) metrology to substantially improve precision of transmission line (transfer length) type measurements by application of advanced electrode position correction. In particular, we demonstrate this methodology for the M4PP current-in-plane tunneling (CIPT) technique. The CIPT method has been a crucial tool in the development of magnetic tunnel junction (MTJ) stacks suitable for magnetic random-access memories for more than a decade. On two MTJ stacks, the measurement precision of resistance-area product and tunneling magnetoresistance was improved by up to a factor of 3.5 and the measurement reproducibility by up to a factor of 17, thanks to our improved position correction technique.

Atomic Layer Deposition of Ruthenium with TiN Interface for Sub-10 nm Advanced Interconnects beyond Copper.

Atomic layer deposition of ruthenium is studied as a barrierless metallization solution for future sub-10 nm interconnect technology nodes. We demonstrate the void-free filling in sub-10 nm wide single damascene lines using an ALD process in combination with 2.5 Å of ALD TiN interface and postdeposition annealing. At such small dimensions, the ruthenium effective resistance depends less on the scaling than that of Cu/barrier systems. Ruthenium effective resistance potentially crosses the Cu curve at 14 and 10 nm according to the semiempirical interconnect resistance model for advanced technology nodes. These extremely scaled ruthenium lines show excellent electromigration behavior. Time-dependent dielectric breakdown measurements reveal negligible ruthenium ion drift into low-κ dielectrics up to 200 °C, demonstrating that ruthenium can be used as a barrierless metallization in interconnects. These results indicate that ruthenium is highly promising as a replacement to Cu as the metallization solution for future technology nodes.

Graphene mobility mapping.

Carrier mobility and chemical doping level are essential figures of merit for graphene, and large-scale characterization of these properties and their uniformity is a prerequisite for commercialization of graphene for electronics and electrodes. However, existing mapping techniques cannot directly assess these vital parameters in a non-destructive way. By deconvoluting carrier mobility and density from non-contact terahertz spectroscopic measurements of conductance in graphene samples with terahertz-transparent backgates, we are able to present maps of the spatial variation of both quantities over large areas. The demonstrated non-contact approach provides a drastically more efficient alternative to measurements in contacted devices, with potential for aggressive scaling towards wafers/minute. The observed linear relation between conductance and carrier density in chemical vapour deposition graphene indicates dominance by charged scatterers. Unexpectedly, significant variations in mobility rather than doping are the cause of large conductance inhomogeneities, highlighting the importance of statistical approaches when assessing large-area graphene transport properties.

Accurate microfour-point probe sheet resistance measurements on small samples.

We show that accurate sheet resistance measurements on small samples may be performed using microfour-point probes without applying correction factors. Using dual configuration measurements, the sheet resistance may be extracted with high accuracy when the microfour-point probes are in proximity of a mirror plane on small samples with dimensions of a few times the probe pitch. We calculate theoretically the size of the "sweet spot," where sufficiently accurate sheet resistances result and show that even for very small samples it is feasible to do correction free extraction of the sheet resistance with sufficient accuracy. As an example, the sheet resistance of a 40 microm (50 microm) square sample may be characterized with an accuracy of 0.3% (0.1%) using a 10 microm pitch microfour-point probe and assuming a probe alignment accuracy of +/-2.5 microm.