Hall effect measurement for precise sheet resistance and thickness evaluation of Ruthenium thin films using non-equidistant four-point probes Open Access

Since the invention of the integrated circuit, microelectronics technology has developed at a break-neck speed to the current level, where extremely sophisticated, highly complex systems can be squeezed into minute microelectronic chips with a functionality that affects our every-day life. This development has been supported not only by groundbreaking developments in fabrication technology and ingenuous circuit design but also by significant advances in metrology. Metrology is increasingly important in production for process control, in research to support development of new technology, and in science to understand limitations and opportunities imposed by small size effects, which may eventually slow down the speed of new technology developments in the microelectronics industry.

A key metrology task is the characterization of the electronic transport properties of thin films of semiconductors and metals,¹ to provide direct measurements of sheet resistance, sheet carrier density and carrier mobility; key parameters for the functionality and performance of microelectronic devices. Traditionally, sheet resistance measurements have been done using macroscopic four-point probes,^1,2 however, the development of the micro four-point probe,^3,4 which offers significant advantages compared to macroscopic probes, has created new metrology opportunities for characterizing transport properties of materials.⁵ 

Sheet carrier density and carrier mobility can be extracted from Hall effect measurements,¹ which traditionally are done on structured samples such as cloverleafs or small squares in van der Pauw configuration.⁶ The traditional methods can be used to measure mobilities well below 1 cm²/(Vs).⁶ The traditional methods, however, require significant sample preparation, e.g., a sample piece must be cut out and contacts deposited on the corners of the sample. The micro Hall effect method has been demonstrated as a powerful alternative.⁷ Here a micro four-point probe is positioned on a thin film sample in proximity of an electrically insulating boundary. Hall effect measurements are then performed with a constant magnetic flux density (B_z) perpendicular to the sample surface. Each measurement takes just 10 s, and as a result this method is particularly convenient in process development and for process control. The direct result of the micro Hall effect measurement is the sheet resistance (R₀) and the Hall sheet resistance (R_H) from which the Hall mobility (μ_H) and the Hall sheet density (N_HS) can be calculated.

The micro Hall effect method was developed for four-point probes with equidistant electrodes.⁷ However, here we show, theoretically, that it is advantageous to use a probe with non-equidistant electrodes, in agreement with other geometry dependent studies where precision is improved by optimizing electrode distances.^8–10 We verify the theoretical prediction in a series of measurements on thin Ruthenium films (3-30 nm) where sheet resistance, Hall sheet resistance, Hall mobility and Hall sheet carrier density are determined.

A key element in extracting Hall effect information from the four-point probe measurement is determination of the distance (y₀) from the probe to the insulating straight boundary, and thus the procedure after the four-point probe measurements can be separated into two steps; 1) extraction of the probe-boundary distance y₀ and the sheet resistance R₀; 2) extraction of the Hall sheet resistance (R_H) based on the probe position from step 1). From R₀ and R_H the Hall mobility and Hall sheet carrier density can be calculated using⁷ 

where Z is the carrier type (Z = ±1) and e is the unit charge.

Previously, multiple methods for determining the distance y₀ have been demonstrated. The distance y₀ can be found by fitting a model to a series of measurements recorded at different distances to the boundary, where the distances between measurement positions are known.⁷ It has also been shown that it is possible to use only two measurement positions, that can be either far apart¹¹ or closely spaced.¹² A particularly convenient single engage method to determine y₀ using two four-point sub-probes (with different pitch) of a 7 point probe has also been developed.¹³ 

However, for many samples determining the distance between the probe and the insulating boundary is not the main source of error. The main source of error is often dependent on how accurately R_H can be determined. This is especially true for thin film metal samples where R_H ≪ R₀.

In this study we focus on methods to reduce the effects of the noise sources affecting R_H. The study will only use the single engage method of determining y₀ but the findings may be applied the the other methods as well. Instead of being limited to using only equidistant four-point probes, we will investigate how the precision will be affected by using non-equidistant four-point probes to determine the Hall sheet resistance.

For this study a 7 point probe with equidistant electrode spacings will be used like the ones shown in Fig. 1. For each measurement only 4 electrodes are used, two for passing a current through the sample and two for measuring a voltage. A set of 4 of the 7 electrodes is called a sub-probe. The sub-probe sketched in Fig. 1 uses pins 1,3,5 and 7 and will for simplicity be called sub-probe 1357. Figure 1 also shows how the electrode pitches a, b and c of a sub-probe are defined, a is the distance between the first and second electrode used, b is the distance between the second and third electrodes and c is the distance between third and forth electrodes. The electrode configurations B and B’ shown in Fig. 1 are the ones commonly used for Hall measurements to determine R_H. The A and A’ configurations are used only when determining y₀ and R₀ and are therefore not illustrated.

When the probe is placed parallel to the insulating boundary at a distance (y₀) and a current (I₀) is passed through the sample, the theoretical average resistance in the B and B’ configuration is

where

Similarly, the theoretical difference between the resistances in the B and B’ configurations is

Equation (2) shows that the average resistance $RBB′¯$ only consists of the log terms and is proportional to R₀ for R_H ≪ R₀. Likewise, the resistance difference ΔR_BB′ consists only of arctan terms and is proportional to R_H.

When determining ΔR_BB′ the two main sources of error are electrode position error and electrical noise. The electrode position error arises from the fact that the electrode will not land at their nominal position. The electrical noise arises from the signal to noise ratio of measurement electronics.

The position error can be quantified by calculating the sensitivity of ΔR_BB′ to the movement of each electrode. This sensitivity is found by calculating the partial derivatives ∂ΔR_BB′/∂x_i, where x_i is the position of the i’th electrode. Any position error due to movements of electrode 1 or 4 will effectively result in a change of width, which will be compensated for when the distance to the insulating boundary is calculated. Thus, only position errors related to electrode 2 and 3 are of interest. By using that x₂ − x₁ = a, x₃ − x₂ = b, x₄ − x₃ = c the sensitivities for electrode 2 and 3 can be calculated as

These equations show that for an equidistant four-point probe (a = b = c) the sensitivity to movement of electrode 2 and 3 becomes 0.

From the partial derivatives the relative standard deviation (σ_rel) of ΔR_BB′ can be estimated by

where σ_x is the standard deviation of the position error for each electrode (the standard deviation is assumed identical for all electrodes).

In Fig. 2(a) the relative standard deviation of ΔR_BB′ is plotted as function of a and b normalized with the probe width (W = a + b + c). The plot is calculated for the following settings y₀/W = 4/60 and σ_x/W = 0.1/60. The symbol × corresponds to an equidistant probe (a = b = c = W/3) and the symbol + corresponds to the non-equidistant probe a = W/3, b = W/6 and c = W/2. Figure 2(a) shows that the smallest relative error is obtained for the equidistant probe.

The other important source of error is the electrical noise. The electrical noise is composed of two terms; a part related to the signal to noise ratio (SNR) of the measurement and a part related to the minimum possible resistance the electronics can resolve (R_min). Since ΔR_BB′ is the difference between 2 measurement configurations repeated N times the relative standard deviation of ΔR_BB′ for electrical noise will have a prefactor of $2/N$⁠. Each measurement will have a standard error of $RBB′¯/SNR+Rmin$⁠. Thus it is possible to estimate the relative standard deviation of ΔR_BB′ from

From experiments, similar to those of Kjaer et al.,¹⁴ we have found that SNR=10000 and the minimum resistance (R_min) that can be resolved in a single measurement is roughly 30 μΩ. In Fig. 2(b), the relative standard deviation of ΔR_BB′ due to electrical noise is plotted as function of a and b. For the plot, the following parameters were used: R₀=5 Ω, R₀/R_H = 5000, y₀/W = 4/60. This plot shows that the lowest relative standard deviation is obtained for the non-equidistant probe.

In order to determine experimentally which sub-probe yields the lowest standard deviation for Hall measurements a series of measurements were performed on Ruthenium (Ru) thin film samples of varying thicknesses (3-30 nm). All samples were prepared by atomic layer deposition of Ru on TiN (0.3 nm)/SiO₂ (90 nm)/Si, as described by Wen et al.¹⁵ and Popovici et al.¹⁶ The Ru area density N_RBS of the samples was measured by Rutherford backscattering spectrometry (RBS).¹⁷ The Ru film thickness d was calculated from N_RBS assuming a bulk Ru density of n_Ru ≃ 7.4 × 10²² cm⁻³, i.e., d = N_RBS/n_Ru. Prior to the M4PP measurements the samples were cleaved in order to get straight insulating boundaries to perform Hall measurements at.

The measurements were performed using a microHall-M300 tool from CAPRES A/S and a micro 7-point probe (M7PP) with all electrode distances fixed at 10 μm. A magnetic field with the flux density B_z = 600 mT was applied perpendicular to the sample. The probe was placed nominally 4 μm from the insulating boundary during measurements. For each sample, 7 engages were performed parallel to the insulating boundary, thus keeping the distance between the probe and insulating boundary constant. At each point of the linescan A, A’, B and B’ configurations were measured for sub-probes 1234, 4567 and 1357 in order to be able to determine the distance to the edge and the sheet resistance of the sample. The reason both sub-probes 1234 and 4567 were recorded was to be able to eliminate any small rotational misalignments between the probe and the insulating boundary. Furthermore, during the same engage, 6 B and 6 B’ configurations were measured using both sub-probes 1357 (equidistant) and 1347 (non-equidistant) in order to determine the Hall sheet resistance.

From the measurements R₀ and R_H were determined and in Fig. 3(a) the results are plotted as function of sample thickness; the error bars correspond to one standard deviation. In order to relate the measured R₀ and R_H to resistivities the resistance values were multiplied with sample thickness.

From R₀ and R_H the Hall mobility μ_H and the Hall sheet carrier density N_HS can be calculated from Eq. 1. The calculated values of the N_HS and μ_H are plotted as function of sample thickness in Fig. 3(b), where the error bars correspond to one standard deviation. The blue dashed line through the Hall sheet carrier density points is the corresponding linear fit. The Hall sheet carrier density is expected to be proportional to the sample thickness, but from the linear fit it is seen that there is an offset, which could suggest that all samples are ∼ 3.7 nm (∼ 2.7 × 10¹⁶ Ru atoms/cm²) thicker than the nominal value, which is however not the case, since the uncertainty on the measured Ru area density is less than 3%. The offset is not caused by the 0.3 nm of TiN, since the resistivity is much larger than that of Ru, and due to exposure to air before deposition of Ru the TiN is oxidized and thus has an even higher resistivity; as a result TiN contributes negligibly to the transport. In other metal systems, such as Cu,¹⁸ the Hall coefficient has been shown to depend on thickness and this was ascribed to band structure effects. However, in the case of Ru, we suggest that the offset can be explained by the fact that Ru is a compensated metal with an equal number of holes and electrons contributing to the transport and with much higher hole mobility (μ_p) than electron mobility (μ_n).¹⁹ A detailed analysis shows that use of Eq. 1 then results in μ_H = (μ_p − μ_n) and N_HS = P(μ_p + μ_n)/(μ_p − μ_n) if transport and Hall mobilities are identical; here P is the hole sheet density. As a result the calculated Hall sheet carrier density overestimates the real sheet carrier density by the factor (μ_p + μ_n)/(μ_p − μ_n). This factor is expected to increase as the film thickness decreases, due to a longer mean free path for holes than for electrons, which makes the hole mobility more sensitive to film thickness than the electron mobility. The sheet density measured at thicker films is consistent with approximately one hole per Ru atom, i.e., a hole concentration of p ≃ 7.4 × 10²² cm⁻³, which is 2.6 times higher than that calculated from the bulk Hall coefficient reported by Hurd²⁰ but in better agreement with the thin film measurements of Steeves.¹⁹ 

In Fig. 3(c) the relative standard deviations of N_HS and μ_H are plotted as function of sample thickness. From this plot it is observed that using sub-probe 1347 (non-equidistant) results in relative standard deviations that are 2-4 times lower than those obtained using sub-probe 1357 (equidistant).

From Fig. 3(c) it is also observed that the relative standard deviations are similar to those calculated in Fig. 2. This means the models can potentially be used to design a future probe, such that the influence of position errors and electrical noise are minimized.

In conclusion, we have shown that for samples with R₀ ≫ R_H, it is possible to reduce the relative standard deviation of Hall sheet carrier density and Hall mobility by changing the inter electrode spacing while keeping the total width of the probe constant. By using four point probes with electrode spacings of 20, 10 and 30 μm compared to probes with equidistant electrode spacings of 20 μm, it is possible to reduce the relative standard deviation by a factor of 2-4. Furthermore, an analysis of noise sources is presented, which can be used to design probes that are optimized for measurements on samples with a specific R₀/R_H ratio and a given standard deviation of position error. From measurements on Ruthenium samples ranging in thickness from 3 nm to 30 nm, it is shown that the Hall sheet carrier density is almost linearly dependent on sample thickness. Thus, the micro Hall effect measurement method can also be used to monitor variations in thickness of thin metal films.

This work was financially supported by Innovation Fund Denmark grant number 5165-00042B and H2020 European project no. 688225.