Crystal orientation mapping in the transmission electron microscope was used to quantify the twin boundary length fraction per unit area for five Ta38Si14N48/SiO2 encapsulated Cu films with thicknesses in the range of 26–111 nm. The length fraction was found to be higher for a given twin-excluded grain size for these films compared with previously investigated SiO2 and Ta/SiO2 encapsulated films. The quantification of the twin length fraction per unit area allowed the contribution of the twin boundaries to the size effect resistivity to be assessed. It is shown that the increased resistivity of the Ta38Si14N48 encapsulated Cu films compared with the SiO2 and Ta/SiO2 encapsulated films is not a result of increased surface scattering, but it is a result of the increase in the density of twin boundaries. With twin boundaries included in the determination of grain size as a mean-intercept length, the resistivity data are well described by 2-parameter Matthiessen's rule summation of the Fuchs-Sondheimer and Mayadas Shatzkes models, with p and R parameters that are within experimental error equal to those in prior reports and are p = 0.48(+0.33/−0.31) and R = 0.27 ± 0.03.

The classical resistivity size effect is the phenomenon wherein the resistivity of a conductor increases when its dimensions are of the order of the electron mean-free path.1–22 In a polycrystalline thin film, the two electron scattering mechanisms that contribute to this resistivity increase are surface scattering and grain-boundary scattering. The semiclassical models most commonly used to describe these two contributions are the Fuchs-Sondheimer (FS) model21 and the Mayadas Shatzkes (MS) model,22 respectively. In the FS model, the resistivity increase is a result of diffuse scattering of conduction electrons at the conductor's exterior surfaces with a probability of 1–p, where p (0p1) is a specular scattering coefficient that is typically inferred from experimental data. The FS expression for the resistivity, ρFS, of a thin film is21 

ρFS=ρi[1(32k)(1p)1(1t31t5)1exp(kt)1pexp(kt)dt]1,
(1)

where k = h/λ, h is the film thickness, λ is the temperature-dependent electron mean free path, and ρi is the bulk resistivity of the metal. Thus, the resistivity increase predicted by this model is ΔρFS=ρFSρi.

In the MS model, each grain boundary that is perpendicular to the direction of the current flow is treated as an internal surface, and when a conduction electron collides with the grain boundary, it has a probability of transmission or reflection that is quantified by a reflection coefficient, R. This coefficient takes values between zero and one and is commonly fitted to the experimental data. The resistivity of the film, ρMS, is given by22 

ρMS=ρi[132α+3α23α3ln(1+1α)]1,
(2)

where α=(λg)R(1R) and g is the average grain size. Using the MS model, the increase in resistivity due to grain-boundary scattering is ΔρMS=ρMSρi.

Prior work by Sun et al.2–4 on SiO2 and Ta/SiO2-encapsulated nanometric Cu films showed grain-boundary scattering to be the dominant mechanism contributing to the resistivity increase related to the classical size effect. These studies also showed a weaker, but still significant, contribution from surface scattering. Furthermore, it was also observed that a simple summing of the FS model for surface scattering with the MS model for grain-boundary scattering provided the best quantitative description for the classical resistivity size effect in the films.

One question that was not answered in the Sun et al.2–4 analysis, however, was whether twin boundaries in Cu contribute significantly to the measured resistivity. Twin boundaries, denoted as Σ3 boundaries in the coincident site lattice (CSL) description of grain boundaries,23 are special interfaces with a misorientation of 60° about the ⟨111⟩ crystallographic axis. A Σ3 boundary is classified as a coherent twin boundary if the boundary plane is {111}; otherwise, the boundary is an incoherent Σ3 boundary. Due to the difficulty of characterizing twin boundaries in Cu at the nanoscale, this contribution was ignored in these earlier studies. The later study of Barmak et al.1 made use of advances in crystal orientation mapping in the transmission electron microscope that allowed quantification of the twin boundary content. They then re-analyzed the resistivity data of Sun et al.4 using the mean linear intercept, including twin boundaries, as the measure of grain size and showed that the conclusions of Sun et al.4 held, but with model parameters of surface specularity p = 0.48 and grain boundary reflection coefficient R = 0.26 compared with p = 0.55 and R = 0.43 when twin boundaries were excluded.

In the current work, the resistivity of a series of five Ta38Si14N48 encapsulated Cu films with thicknesses in the range of 26–111 nm is investigated. Crystal orientation mapping in the transmission electron microscope is used to quantify the twin boundary length fraction per unit area. For the same twin-excluded grain size, this twin length fraction is higher for the Ta38Si14N48 encapsulated films studied here compared with the SiO2 and Ta/SiO2 encapsulated films.4,24 With twin boundaries included in the determination of grain size as mean-intercept length, the resistivity data are well described by p and R parameters that are within experimental error equal to those previously reported in Ref. 1. Furthermore, it is shown that exclusion of the twin boundaries results in a worse fit to the experimental resistivity data. It is also demonstrated that the increased resistivity of the Ta38Si14N48 encapsulated Cu films compared with the SiO2 and Ta/SiO2 encapsulated films is not a result of increased surface scattering, but it is a result of the increase in the density of twin boundaries.

Detailed descriptions of the thin-film deposition technique and thickness and resistivity measurements are available elsewhere.2–4 Briefly, the Cu films were sputter-deposited on Si (100) substrates with a 150 nm thick layer of thermally grown SiO2 and cooled to −40 °C by contact with a liquid nitrogen cooled Cu plate. The Cu layer was DC-sputtered from a 99.9999% pure Cu target. Prior to the deposition of the Cu layer, a 20-nm thick layer of SiO2 followed by a nominally 2 nm-thick layer of Ta38Si14N38 were sputter deposited to form an oxide/nitride bilayer underlayer. Following the deposition of the Cu layer, the Ta38Si14N38 and the SiO2 layers were sputter deposited in reverse order to the underlayer deposition to form a bilayer overlayer, thereby fully encapsulating the Cu layer. The encapsulated films were annealed in Ar + 3% or 4% H2 at ambient pressure at 600 °C for 30 min, which, based on prior studies,2–4 resulted in single grains through the thickness of the film. This was confirmed for the current samples via cross-sectional transmission electron microscopy. The aim of the encapsulation was to provide nearly identical scattering surfaces for the Cu film, and, in addition, to minimize voiding in the film during annealing.

The thickness of the Cu layer and the roughness of the Cu/Ta38Si14N48 interfaces were measured using X-ray reflectivity experiments at the Stanford Synchrotron Radiation Lightsource, in the manner of our previous studies.4 The sheet resistance was measured at room temperature and 4.2 K using a dipping van der Pauw geometry four-point probe.4 

Plan-view TEM samples were prepared using a back-etching technique, by thinning initially with a HF + HNO3 solution, and subsequently with a diluted HF solution.25 Void fraction in the films was measured using high angle annular dark field imaging (HAADF).25,26 Crystal orientation mapping in the TEM was used to determine whether boundaries were twin or non-twin boundaries and this was also one of the methods to obtain grain size (including and excluding twin boundaries). Since the orientation mapping experiments in the transmission electron microscope are described in detail in Ref. 26, only a brief description is given here. Mapping was performed on an ASTAR™ (NanoMEGAS, Brussels, Belgium) system installed on a FEI Tecnai F20 TEM (FEI Corporation, Hillsboro, OR, USA) with a field emission gun and an accelerating voltage of 200 kV at a probe size of ∼1 nm. Scan step sizes were chosen to be roughly consistent with the corresponding estimated grain sizes excluding twin boundaries and ranged between 2.5 nm and 12.5 nm for all five samples. Each orientation map contained 250–300 steps along both x and y directions. Approximately 30 fields of views with ∼100–200 grains in each field of view were collected for each sample. Fig. 1 shows a representative color-coded inverse pole figure map in the film normal direction for one of the Cu films.

FIG. 1.

(a) Representative color-coded inverse pole figure map along the film normal direction from the 111.0 nm-thick Cu film and (b) color-code for the inverse pole figure map.

FIG. 1.

(a) Representative color-coded inverse pole figure map along the film normal direction from the 111.0 nm-thick Cu film and (b) color-code for the inverse pole figure map.

Close modal

Conical dark field (CDF) imaging of the samples in the TEM was the second method used to obtain an independent measure of the grain size excluding twin boundaries for comparison with the values obtained by crystal orientation mapping. Since this technique for grain size measurement has not been described elsewhere, it will be presented in detail. CDF was used instead of hollow cone dark-field (HCDF) imaging used in the study of SiO2 and Ta/SiO2 encapsulated-Cu films reported by Sun et al.4 for reasons presented below.

In HCDF imaging, the tilted incident beam is rotated about the optic axis so that multiple dark field images from the various beam orientations sampled are integrated into one image. The objective aperture is inserted to limit the diffracted beams contributing to the image contrast. The intensity of a point in a HCDF image is the sum of the intensity of electrons diffracted for each of the incident beam orientations as the beam is rotated about the optic axis. A HCDF image provides better contrast than the bright field (BF) image for grain boundary identification because the number of diffracted beams contributing to the image is limited.4,27 At the same time, since the imaging is not limited to a single diffracted beam forming the image, the number of grains appearing bright in a HCDF image is larger, allowing more grain boundaries to be identified. Though HCDF imaging is better than bright and dark field imaging to calculate the grain size, even with this technique, the image contrast is still complex and the identification of grain boundaries is difficult. It is also necessary to take HCDF images at different specimen tilts as the information from a single image is not usually sufficient for grain boundary identification.27 Typically, three tilts are used at −2°, 0°, and +2° of sample tilt, and care has to be taken to ensure that the same field of view is imaged. However, even when the sample is in the eucentric position, imaging the same field of view at different sample tilts requires significant effort. Also, to obtain high quality HCDF images, it is necessary to perform all the alignment procedures required for high-resolution TEM. These stringent requirements in terms of alignment and retention of the same field of view, in addition to the experience needed in grain boundary identification, make HCDF imaging a challenging technique. Nevertheless, the technique can be successfully utilized if other techniques are not available.4 

In CDF imaging implemented in the TSL™ automated crystallography in the TEM (ACT) system that was used in the current study, the incident beam is tilted and rotated about the optic axis, similar to HCDF imaging. However, in CDF, a large number of dark field images under different diffraction conditions are collected as the beam is tilted and rotated around the optic axis. Unlike HCDF imaging, the tilt of the incident beam is not fixed and dark field images are collected for a range of beam tilts. These advantages of the CDF technique make it a simpler technique for grain size measurement. In the current study, the CDF images were acquired using the TSL™ ACT system installed on a Philips CM12 TEM operating at 120 kV. A 10 μm objective aperture, the smallest available in the microscope, was used for most of the observations. This objective aperture size was used to maximize the diffraction contrast. However, in cases where the image intensity was low due to a large sample thickness, the next larger aperture, which is 20 μm in size, was used. A 100 μm condenser aperture, which is the maximum available aperture size, was used to maximize the intensity of the illumination. CM Delphi Tutor made remote computer control of the TEM possible. The images were collected using a retractable Gatan 780 wide-angle CCD camera, which was computer controlled by Gatan Digital micrograph. The digital camera had a 12 bit dynamic range. The smallest tilt angle used to collect the dark field image was 0.48°. This was the smallest beam tilt for which diffracting grains were observed in the dark field images. Most of the grains could be identified by collecting dark field images from the first two or three Bragg peaks closest to the transmitted beam. Therefore, unlike in orientation imaging using CDF scanning, it was not necessary to collect dark field images at large beam tilts. This meant that the total scan time was greatly reduced and was usually less than 5 min. Therefore, specimen drift and microscope instability during data collection were not an issue. The largest tilt angle used was 1.00°. The chosen number of tilt steps between the largest and the smallest tilt angles was 6. A rotation step size of 4° was chosen. The choice of beam tilt and rotation step sizes ensured that a sufficient number of dark field images were collected, while the experiment time was sufficiently short that specimen drift was negligible. For each scan, around 500 dark field images were collected at a resolution of 250 × 250 pixels. The images were collected at this resolution because of limited memory on the older computer of the CM12 microscope. The magnification used was such that the smallest grains were at least 5 pixels wide, while there were approximately 100 grains in the image. To determine the grain size using the CDF images, each dark field image was added layer by layer into Adobe Photoshop and grain boundaries were traced using the pen tool. The white level of the dark field images was adjusted so that grain identification became easier. Care was taken to sample adequate number of grains across the size range.

In addition for measuring the grain size of the Ta38Si14N48 encapsulated films, the CDF method was used to obtain the grain size for a SiO2 encapsulated sample obtained from the same deposited film for which the grain size had been measured by HCDF and reported in the resistivity study of Sun et al.4 This was done to ensure that grain sizes measured by the different techniques (imaging and mapping) could be considered as equal within experimental error, thereby allowing the current study to be directly compared with those of Sun et al.4 and Barmak et al.1 for SiO2 and Ta/SiO2 encapsulated Cu films.

Grain sizes using the traced grain maps obtained from the CDF and HCDF images and the reconstructed grain boundaries of the crystal orientation maps (see Section II B) were calculated using Image J.28 For a collection of grains N grains, Sun et al.2–4 used the equivalent circle diameter, DA¯, of mean area, A¯=Ai/N

DA¯=4A¯π,
(3)

as the measure of grain size excluding twin boundaries.2–4 Approximately 1000 grains were measured per sample, except for the SiO2-encapsulated film, for which fewer grains (approximately 200) were measured for comparison with prior reports.2–4 The edge grains were excluded in determining grain size. The error on grain size is given at 95% confidence for the number of grains measured, as detailed in Ref. 29.

Table I compares the grain size obtained by two different observers using the CDF method. The grain size value is equal, within experimental error, for the two observers, thereby providing confidence in the user-to-user consistency of the CDF grain size measurement protocol. Table II compares the grain size for a SiO2-encapsulated Cu film studied by Sun et al.4 and measured in the current study by both the CDF and HCDF methods. Given the equality of grain size, within experimental error, for this film for these two methods, it can be concluded that the twin-excluded grain sizes measured in the current study for the Ta48Si14N48 encapsulated Cu films can be directly compared with the twin-excluded grain sizes of the SiO2 and Ta/SiO2 encapsulated Cu films measured by the HCDF method used by Sun et al.4 

TABLE I.

Comparison of the grain size excluding twin boundaries measured by two different observers using the conical dark field (CDF) imaging technique for a Ta38Si14N48/SiO2 encapsulated Cu film annealed at 600 °C for 30 min.

DA¯CDF (nm)
Cu layer thickness (nm)Observer 1Observer 2
38.2 ± 0.4 74 ± 4 76 ± 4 
DA¯CDF (nm)
Cu layer thickness (nm)Observer 1Observer 2
38.2 ± 0.4 74 ± 4 76 ± 4 
TABLE II.

Comparison of the grain size excluding twin boundaries of a SiO2 encapsulated Cu film annealed at 150 °C for 30 min examined by the hollow cone dark field (HCDF) imaging technique, DA¯HCDF, and reported in Refs. 4 and 25 with the results obtained in the current study on electron transparent samples made from the same film, and examined using conical dark field (CDF) imaging, DA¯CDF. The number of grains measured is given in parenthesis.

FilmCu thickness (nm)DA¯HCDF (nm)DA¯CDF (nm)
SiO2/Cu/SiO2 35.3 54 ± 2 (1363) 58 ± 7 (200) 
FilmCu thickness (nm)DA¯HCDF (nm)DA¯CDF (nm)
SiO2/Cu/SiO2 35.3 54 ± 2 (1363) 58 ± 7 (200) 

Analysis of the crystal orientation maps was carried out using the TSL™ OIM data analysis software, as described in the previous work, to identify grains and the types of boundaries separating adjoining grains.24,26 An example of the reconstructed boundary map is displayed in Fig. 2, which shows coherent and incoherent Σ3 boundaries, as well as non-Σ3 boundaries. However, as in the previous study of Barmak et al.,1 in determining the impact of twin boundaries on resistivity, no distinction was made between coherent and incoherent Σ3 boundaries, and boundary segments were simply categorized as either Σ3 or non-Σ3. Reconstructed grain-boundary maps from the five samples without twin boundaries were also extracted by excluding all boundaries within a tolerance of 8° of the Σ3 misorientation.26 These maps were used to obtain the grain size excluding twin boundaries, DA¯OM, using Image J, as described in Section II A. Approximately 1000 grains excluding twin boundaries were measured per sample.

FIG. 2.

Representative reconstructed boundary network obtained from crystal orientation maps of the 39.4 nm-thick Cu film. Here, thick black lines are coherent twin boundaries and thick red lines are incoherent twin boundaries. Thin black lines represent other types of boundaries.

FIG. 2.

Representative reconstructed boundary network obtained from crystal orientation maps of the 39.4 nm-thick Cu film. Here, thick black lines are coherent twin boundaries and thick red lines are incoherent twin boundaries. Thin black lines represent other types of boundaries.

Close modal

To account for the presence of twin boundaries, the mean intercept length was used as a measure of grain size, because, as noted by Barmak et al.,1 twin boundaries do not always meet at triple junctions and grain shapes are no longer equiaxed. The mean intercept length is defined as

L¯all=πAiCi,
(4)

where Ai (Ci) is the area (perimeter) of each of the grains in the aggregate.

As in Refs. 1 and 4, the Bayesian information criterion (BIC) in addition to sum-squared errors (SSEs) were used to evaluate the goodness-of-fit of the FS + MS model to the room-temperature and liquid helium temperature resistivities of the Ta38Si14N48/SiO2 encapsulated Cu films of the current study. The BIC incorporates a penalty term for an increased number of adjustable parameters, and, in the work of Sun et al.4 in which 9 models were compared, the BIC proved to be a suitable criterion for comparing models with different numbers of adjustable parameters. The FS + MS model was found to be the model with the fewest number of adjustable parameters (the surface specularity, p, and grain boundary reflectivity, R) to adequately describe the resistivity data of SiO2 and Ta/SiO2 encapsulated Cu films.

The BIC, with the assumption of normally distributed errors, is

BIC=2×ln(L)+aln(n)=nln(SSEn)+nln(2π)+n+aln(n),
(5)

where L is the overall likelihood (i.e., the product of the likelihoods for each of the measurements), a is the number of adjustable or fitting parameters, n is the number of experimental measurements, and

SSEn=σ2=i=1n(ρiexperimentρimodel)2n,

where ρiexperiment is the experimental and ρimodel is the calculated resistivity. For the formulation of the BIC given in Eq. (5), “good” models have negative BIC's.

To determine the optimal values of the adjustable parameters p and R, a forward modeling approach was employed to arrive at the global minimum of SSE.1 The values of p and R were varied in steps of 0.01 from 0.01 to 0.99 and the errors, E=ρiexperimentρimodel, were calculated for each of the experimental resistivity results over the full grid of p and R. The error values for the minimum SSE were used to determine the BIC using Eq. (5). The experimentally measured resistivity of bulk Cu, ρi, as a function of temperature was used throughout to calculate the temperature-dependent mean-free path (arising from phonon scattering) for the conduction electrons using the relationship: ρiλ = 6.6 × 10−16 Ω m2.4 

The independent and coupled errors in p and R in the FS + MS model were determined via the bootstrap resampling method of statistical analysis using 10 000 re-samplings on the n squared error values to obtain the SSE at the 95% confidence level.1 Briefly, in one resampling step employing this method, a new set of n squared error values was selected randomly, one at a time, from the original set and a new SSE is calculated using this new set. The resampling was repeated 10 000 times and the distribution of SSEs was plotted, and the SSE at the 95th percentile was employed to determine the errors on p and R.1 

Table III lists the thicknesses of the Cu layers, the roughness of the upper and lower interfaces with the Ta38Si14N48 encapsulation layers, and the area percent of voids in the films. For all the films, the voiding was well below the limit that would affect the calculated film resistivities.4 Table III also gives the grain sizes excluding twin boundaries, DA¯OM and DA¯CDF, for all five films, where the values are seen to be equal, within experimental errors, for the two methods. Table III additionally lists the mean linear intercept, including twin boundaries, L¯all, for the five films, and film resistivities at both room and at liquid He temperatures.

TABLE III.

For Cu films encapsulated in Ta38Si14N48 and annealed at 600 °C, thickness, root mean square roughness of the upper, r1, and the lower, r2, Cu/encapsulant layer interfaces, void fraction, grain size data as equivalent circle diameter of mean area excluding twin boundaries from crystal orientation mapping,22DA¯OM, and CDF techniques, DA¯CDF, with the number of grains measured given in parenthesis, grain size as the mean intercept length including twin boundaries L¯all using crystal orientation mapping in the transmission electron microscopy, and 293 K and 4.2 K resistivity.

Thickness (nm)r1/r2 (nm)Void (%)DA¯OM (nm)DA¯CDF (nm)L¯all (nm)ρ at 293 K (μΩ cm)ρ at 4.2 K (μΩ cm)
25.6 ± 0.7 0.8/2.1 1.4 51 ± 2 (4539) 52 ± 3 (974) 27.5 2.99 1.30 
39.4 ± 1.0 1.1/2.0 1.0 87 ± 4 (1576) 90 ± 4 (1035) 37.6 2.99 1.03 
56.8 ± 0.1 0.9/1.2 0.9 106 ± 3 (7997) 112 ± 6 (1082) 48.7 2.75 0.81 
72.4 ± 0.1 0.9/1.2 0.7 147 ± 5 (4718) 156 ± 8 (953) 59.8 2.60 0.64 
111.0 ± 0.3 0.9/1.9 0.7 315 ± 16 (1256) 315 ± 19 (525) 93.4 2.25 0.38 
Thickness (nm)r1/r2 (nm)Void (%)DA¯OM (nm)DA¯CDF (nm)L¯all (nm)ρ at 293 K (μΩ cm)ρ at 4.2 K (μΩ cm)
25.6 ± 0.7 0.8/2.1 1.4 51 ± 2 (4539) 52 ± 3 (974) 27.5 2.99 1.30 
39.4 ± 1.0 1.1/2.0 1.0 87 ± 4 (1576) 90 ± 4 (1035) 37.6 2.99 1.03 
56.8 ± 0.1 0.9/1.2 0.9 106 ± 3 (7997) 112 ± 6 (1082) 48.7 2.75 0.81 
72.4 ± 0.1 0.9/1.2 0.7 147 ± 5 (4718) 156 ± 8 (953) 59.8 2.60 0.64 
111.0 ± 0.3 0.9/1.9 0.7 315 ± 16 (1256) 315 ± 19 (525) 93.4 2.25 0.38 

Figure 3 compares the twin boundary length fraction for the Ta48Si14N48-encapsulated Cu films with two SiO2-encapsulated Cu films investigated previously.1,4 It is clearly seen that, for a given grain size, the Ta48Si14N48-encapsulated Cu films have significantly larger length fractions of twin boundaries. The increased twin boundary length fraction gave rise to an increase in film resistivity for a given film thickness, if equivalent circle diameter of mean area, DA¯, was used as a measure of grain size. However, as will be shown, by using the mean intercept length including twin boundaries, L¯all, which accounts for the increased density of twin boundaries, the resistivity of the Ta48Si14N48-encapsulated Cu films is satisfactorily described using the FS + MS model with the same values of p and R that were used for the SiO2 and Ta/SiO2 encapsulated films.

FIG. 3.

Twin boundary length fraction per unit area, including both coherent and incoherent Σ3 boundaries, as a function of grain size excluding twin boundaries, DA¯OM, for Ta38Si14N48 encapsulated films is compared with the Σ3 fraction as a function of grain size excluding twin boundaries, DA¯HCDF, for two SiO2 encapsulated films reported by Sun et al.4 and Darbal et al.24 

FIG. 3.

Twin boundary length fraction per unit area, including both coherent and incoherent Σ3 boundaries, as a function of grain size excluding twin boundaries, DA¯OM, for Ta38Si14N48 encapsulated films is compared with the Σ3 fraction as a function of grain size excluding twin boundaries, DA¯HCDF, for two SiO2 encapsulated films reported by Sun et al.4 and Darbal et al.24 

Close modal

Table IV compares the mean squared error (MSE), the BIC, and the optimal values of surface specularity, p, and grain boundary reflection coefficient, R, along with the errors for the models for the SiO2 and Ta/SiO2 encapsulated Cu films reported previously1 and for the combined resistivity data of all three types of encapsulation layers (SiO2, Ta/SiO2, and Ta38Si14N48/SiO2) considered in this current work. The values in the table allow the impact of exclusion or inclusion of twins, as well as the impact of different specularity parameters for different surfaces, to be evaluated. The first two rows in the table for SiO2 and Ta/SiO2 encapsulated films show that excluding or including twin boundaries does not impact the MSE or the BIC (to any significant degree). However, inclusion of the twin boundaries does impact the value of R (0.26 vs. 0.43), as would have been expected and discussed in prior work.1 When the data for all three encapsulation layers (SiO2, Ta/SiO2, and Ta38Si14N48/SiO2) are combined, exclusion or inclusion of twin boundaries has a very significant impact not only just on R but also on the MSE and the BIC. For the 2-parameter FS + MS model, exclusion of twin boundaries results is an increase of MSE from 0.01 to 0.02 and a significant degradation of the BIC from −73.1 to −52.8. Therefore, clearly twin boundaries do scatter electrons and should be included in the description of the resistivity data. Furthermore, examination of the last two rows of the table for a 3-parameter FS + MS(caps) that optimizes the surface specularity for the Ta38Si14N48/SiO2 encapsulation layer separately from the Ta/SiO2 and SiO2 encapsulation layers shows a degradation (an increase) in the BIC compared with the 2-parameter FS + MS model, whether twins are excluded (−50.7 vs.−52.8) or included (−70.0 vs. −73.1). Since the magnitude of the increase in BIC is >2 in both cases, the 2-parameter FS + MS model wherein a single specularity parameter is used for all three encapsulants provides a better description of the resistivity data. It should also be noted that the surface specularity parameter for the Ta38Si14N48/SiO2 encapsulant is higher than the SiO2 and Ta/SiO2 encapsulants in the 3-parameter FS + MS(caps) model compared with the 2-parameter FS + MS model. In other words, the higher resistivity of the Ta38Si14N48/SiO2 films for a given grain size is not a result of having a surface with a higher fraction of diffuse scattering, whether twins are excluded or included.

TABLE IV.

Film type, model type, the number of adjustable model parameters, a, twins included or excluded, the number of resistivity values used in obtaining the model parameters, n, sum squared errors (SSEs) and mean squared error (MSE), the Bayesian information criterion (BIC) calculated using Eq. (4), optimized model parameters of surface specularity, p, and grain boundary reflection coefficient, R, and independent and coupled errors in these parameters. The measure of grain size used for determining optimal model parameters is the equivalent circle diameter of mean area, DA¯, when twins are excluded, and the mean linear intercept, including twin boundaries, L¯all, when twins are included. The errors in the values of p and R where given are determined via the bootstrap resampling method. See text for more detail.

Model parameters
FilmsModelNo. of parameters, aTwinsNo. of data points, nSSE (MSE) (μΩ cm2)BICIndependentCoupledReferences
Cu/Ta/SiO2, Cu/SiO2 FS + MS Excluded 44 0.48 (0.01) −66.2 p = 0.55, R = 0.43 … 1 and 4 
FS + MS Included 44 0.48 (0.01) −66.5 p = 0.48 ± 0.13 p = 0.48 (+0.32/−0.23) 
R = 0.26 ± 0.02 R = 0.26 (+0.03/−0.02) 
Cu/Ta/SiO2, Cu/SiO2, Cu/Ta38Si14N48 FS + MS Excluded 54 1.03 (0.02) −52.8 p = 0.56, R = 0.43 … This work 
FS + MS Included 54 0.70 (0.01) −73.1 p = 0.58 (+0.16/−0.14) p = 0.58 (+0.33/−0.31) This work 
R = 0.27 ± 0.01 R = 0.27 ± 0.03 
FS + MS(caps) Excluded 54 0.99 (0.02) −50.7 pSiO2,Ta/SiO2 = 0.55, pTa38Si14N48 = 0.62 … This work 
R = 0.43 
FS + MS(caps) Included 54 0.69 (0.01) −70.0 pSiO2,Ta/SiO2 = 0.57, … This work 
pTa38Si14N48 = 0.65R = 0.27 
Model parameters
FilmsModelNo. of parameters, aTwinsNo. of data points, nSSE (MSE) (μΩ cm2)BICIndependentCoupledReferences
Cu/Ta/SiO2, Cu/SiO2 FS + MS Excluded 44 0.48 (0.01) −66.2 p = 0.55, R = 0.43 … 1 and 4 
FS + MS Included 44 0.48 (0.01) −66.5 p = 0.48 ± 0.13 p = 0.48 (+0.32/−0.23) 
R = 0.26 ± 0.02 R = 0.26 (+0.03/−0.02) 
Cu/Ta/SiO2, Cu/SiO2, Cu/Ta38Si14N48 FS + MS Excluded 54 1.03 (0.02) −52.8 p = 0.56, R = 0.43 … This work 
FS + MS Included 54 0.70 (0.01) −73.1 p = 0.58 (+0.16/−0.14) p = 0.58 (+0.33/−0.31) This work 
R = 0.27 ± 0.01 R = 0.27 ± 0.03 
FS + MS(caps) Excluded 54 0.99 (0.02) −50.7 pSiO2,Ta/SiO2 = 0.55, pTa38Si14N48 = 0.62 … This work 
R = 0.43 
FS + MS(caps) Included 54 0.69 (0.01) −70.0 pSiO2,Ta/SiO2 = 0.57, … This work 
pTa38Si14N48 = 0.65R = 0.27 

In summary, the best description of the resistivity data for all three film types is given by the 2-parameter FS + MS model with twin boundaries included. Furthermore, the optimal values of the model's parameters, p = 0.48 and R = 0.27, are equal within experimental error for the combined data of three film types and for the previously reported two film types. The independent and coupled ranges of error at 95% confidence on the SSE using the bootstrap method are also given for p and R in Table IV. As can be seen from the table and the figure, the error range for p is significantly larger than that for R, in agreement with the previous reports.1,12,18

The values of R = 0.26, including twin boundaries, and R = 0.43, excluding twin boundaries, compare well with the range of calculated values obtained by Feldman et al.30 based on a Green's function formalism for twin and non-boundaries in Cu. The values of p = 0.48 and R = 0.26 obtained in the current study also compare well with the values reported by Josell et al.18 of p = 0.55 and R = 0.20 based on a chi-squared fit of model predictions to the resistivity data of Cu lines. However, despite these favorable comparisons, there is one issue that is worth addressing, as noted in the Introduction. In most, if not all studies, except this work and that of Sun et al.,4 the grain size is either not measured and assumed equal to a sample structural dimension (e.g., film thickness or line width) or is not measured for statistically significant populations. Moreover, in cases where grain size is measured, it is unclear which measure of grain size is used. Beyond the measures indicated above, namely, DA¯ and L¯, a third measure is the mean equivalent circle diameter, D¯, wherein the equivalent circle diameter is found for each grain, from which the mean diameter for the population of grains is obtained. Using these different measures, different values for R will be found.

Thickness, grain size for statistically significant populations, and resistivity were measured for five Ta38Si14N48/SiO2 encapsulated Cu films. Grain sizes excluding twin boundaries were obtained using both imaging and crystal orientation mapping methods in the transmission electron microscope to allow direct comparison of resistivity data obtained here with those in the previous reports of SiO2 and Ta/SiO2 encapsulated films. Crystal orientation mapping also allowed the twin density and the twin boundary length fraction in the Ta38Si14N48/SiO2 encapsulated films to be measured. It was found that the twin boundary length fraction per unit area in these films was higher for a given twin-excluded grain size than previously studied SiO2 and Ta/SiO2 films. Forward modeling of the resistivity data showed that the resistivity data of the Cu films for all three encapsulation layers (SiO2, Ta/SiO2, and Ta38Si14N48/SiO2) are satisfactorily described by a 2-parameter, Matthiessen's rule addition of the Fuchs-Sondheimer surface scattering and Mayadas Shatzkes grain boundary scattering model, FS + MS, with a single surface specularity parameter, p, and a single grain boundary reflection parameter, R, that are within experimental error equal to those reported for SiO2 and Ta/TaSiO2 encapsulated films. The values are p = 0.48(+0.33/−0.31) and R = 0.27 ± 0.03 obtained by minimization of sum squared errors.

It was also shown that exclusion of twin boundaries resulted in a significantly poorer description of the resistivity data as measured by the mean squared error and the Bayesian information criterion. Furthermore, it was shown that the increased resistivity of the Ta38Si14N48/SiO2 encapsulated films compared with the SiO2 and TaSiO2 films for a given grain size was not a result of a difference in the surface specularity parameter for the Ta38Si14N48/SiO2 encapsulation.

Financial support from the SRC, Tasks 1292.008, 2121.001, and 2323.001, and from the MRSEC program of the NSF under DMR-0520425 was gratefully acknowledged. Use of the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515. We acknowledge Madeleine N. Kelly (Carnegie Mellon) for assistance with some data processing.

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