The continued decrease of feature sizes in integrated circuits and the corresponding decrease in the interconnect half-pitch below the bulk electron mean free path λ 1,2 results in an increase in the line resistivity [3][4][5] due to electron scattering at surfaces 6,7 and grain boundaries. 8- 10 This resistivity increase causes an interconnect signal delay that renders further downscaling of Cu interconnects problematic 1,11 and has resulted in the search for alternative conductor options 12 and the introduction of Co in the first level(s) of back-end metallization. 12,13 The resistivity size effect is typically described by semiclassical models based on the Boltzmann transport equation, namely the Fuchs and Sondheimer (FS) model for surface scattering 14,15 and the Mayadas and Shatzkes (MS) model for grain boundary scattering. 16 Approximate forms of both models predict an additive resistivity contribution that is proportional to the product ρoλ, where ρo is the bulk resistivity and λ is the bulk mean free path which is 40 nm for Cu at room temperature. 17 Therefore, the search for alternative materials for narrow interconnect lines has focused on metals with a ρoλ product that is smaller than the ρoλ = 6.7 × 10 -16 Ωm 2 for Cu, 12,17 since they are expected to exhibit a correspondingly higher conductivity in the limit of narrow wires. This argument neglects variations in the surface scattering specularity and grain boundary reflection coefficient which depend on the conductor material and the surface/interface structure. [18][19][20][21][22] Some promising metals include W, 23 Ru, 24 Co, 6 and Rh, 25 with ρoλ = (10.8/18.9, 5.1, 12.2, and 4.5) × 10 -16 Ωm 2 , respectively. Co and Ru have received the most attention as possible Cu replacement, because they combine a relatively low ρoλ with a large electromigration resistance, improving the reliability and facilitating a reduction in the barrier/adhesion layer thickness which further reduces the line resistance. 17,[26][27][28][29][30] More specifically, while Cu requires a Ta(N) layer (≥ 2 nm) to prevent diffusion into the low-k dielectric, Co needs only an approximately 1-nm-thick TiN barrier 27 and Ruthenium just a 0.3nm-thick TiN adhesion layer to suppress delamination during chemical mechanical polishing. 28,31 Compounds with large Fermi surfaces have the potential to outperform the conductivity of narrow Cu lines despite their tendency for enhanced electron scattering caused by their smaller Brillouin zone in comparison to elemental metals. Compounds are promising if they have a small ρoλ product to limit the resistivity size effect and a large cohesive energy which promises thermal stability without the need for diffusion barriers. However, the search for suitable compounds is challenging because of the vast number of possible materials and also because the bulk resistivity of many compounds is not well established. Some proposed materials include ordered intermetallic compounds like NiAl, Cu2Mg, and CuAl2, [32][33][34][35] transition metal germanides 2 and MAX-phase conductors. 36 MAX-phase materials exhibit a layered crystal structure with the general formula Mn+1AXn where n = 1, 2, 3, M is an early transition metal, A is an A-group element and X is carbon or nitrogen. 37,38 They combine metallic and ceramic properties including high melting points, good electrical and thermal conductivity, high thermal stability and ductility. [37][38][39][40][41] Their resistivity is anisotropic, with some reported in-plane resistivities < 10 μΩ cm, 37,39 which is comparable to that of elemental metals. Sankaran et al. 36 have employed first-principles calculations to determine the ρoλ product for a large number of MAX phases and have recently reported 170 compounds with ρoλ values below that for Cu, and have proposed their potential use as conductor in narrow interconnect lines. We note that a predicted low ρoλ is no guarantee for a low resistivity scaling 12 and that experimentally determined ρoλ values for elemental metals have been reported to be more than 50% larger than the first-principles predictions for W, 23 Rh, 25 Co, 6 and Mo, 12 and even more than two times larger for Nb, 42 Ni, 7 and Ta. 12 Thus, it is important to measure the resistivity scaling for MAX phase materials in order to confirm or correct the theoretical predictions and to evaluate their potential as interconnect metal.
In this paper, we report on measurements of the resistivity size effect in epitaxial Ti4SiC3(0001) layers, a MAX-phase material with a previously predicted ρoλ= 3.1 × 10 -16 Ωm 2 that is more than two times smaller than that of Cu and also exhibits a 2.5 times larger cohesive energy, 36 yielding an expected small resistivity size effect and expected good electromigration performance. 43 We use epitaxial layers without grain boundaries such that the measured ρ vs thickness d can be directly analyzed with the FS model for electron surface scattering without confounding effects from electron scattering at grain boundaries which would be described by the MS model. Our measured ρ at room temperature and 77 K indicates, in fact, a very small resistivity scaling and data fitting yields an effective temperature-independent ρoλ = 3.8 2.1 × 10 -16 Ωm 2 . Our new first-principles calculations predict ρoλ = 19.3 × 10 -16 Ωm 2 in the basal plane. This is much larger than the prediction from the previous report and also larger than the experimental value which is determined by assuming completely diffuse electron surface scattering. Thus, we attribute the small resistivity scaling primarily to 80% specular electron surface scattering which may be facilitated by the layered crystal structure and confirms the potential of MAX phase materials as potential conductor for narrow interconnect lines.
All Ti4SiC3(0001) films were deposited onto single-side polished 10×10×0.5 mm 3 Al2O3(0001) substrates in a three-chamber ultra-high vacuum magnetron sputtering system with a base pressure of 10 -7 Pa. 44 The substrates were cleaned with consecutive 10 min ultrasonic baths in trichloroethylene, acetone, isopropyl alcohol, and deionized water, blown dry with nitrogen, and attached to a Mo sample holder with colloidal silver paint. They were subsequently introduced into the deposition system and degassed in vacuum at 1000 ℃ for one hour. The substrate temperature was kept at 1000 ℃ for the subsequent depositions done in 3 mTorr 99.999% Ar, using Ti (99.995%), Si (99.999%) and C (99.999%) targets that were facing the continuously rotating substrate at 9, 23 and 9 cm distances and at -45°, 0° and 45° tilts. The targets were sputter cleaned for 5 min with a shutter covering the substrate, followed by deposition of a 12-nm-thick TiC(111) nucleation layer using 360 and 120 W applied to the Ti and C targets for 4 min. Then, deposition of Ti4SiC3 layers was initiated without interruption by adjusting the magnetron power to 360, 60 and 90 W for the Ti, Si and C targets, respectively, yielding a Ti4SiC3 deposition rate of 0.08 ± 0.01 nm/s. The deposition time was varied to obtain a series of samples with a Ti4SiC3 thickness d = 5.8 -92.1 nm, as determined by X-ray reflectivity (XRR) measurements.
Samples were cooled in vacuum for 10-15 h to reach room temperature (293 K). Subsequently, they were transferred in vacuum to an attached analysis chamber for in situ transport measurements 45,46 using a linear four-point probe with 1 mm inter-probe spacings operated with 1-100 mA. Next, the samples were removed from the vacuum system through a load lock that was vented with dry nitrogen and submerged in liquid nitrogen immediately within 1-3 seconds after air exposure to minimize possible surface oxidation prior to ex situ transport measurements at 77 K with a similar four-point probe but with both sample and measurement tips submerged in liquid nitrogen. The samples were warmed up to 293 K in a desiccation chamber to minimize ice build-up, followed by ex situ resistivity measurements at room temperature. The sheet resistance was obtained from the measured voltage-to-current ratio using a geometric correction factor. 47 The Ti4SiC3 resistivity was determined by subtracting the conductance of the nucleation layer from the overall conductance using a parallel conductor model. The conductance of the 12-nm-thick TiC nucleation layer was measured from a sample without MAX-phase overlayer and was 2 -25 times smaller than the Ti4SiC3 conductance, such that the experimental uncertainty associated with the thickness or composition of the nucleation layer results in a negligible expected uncertainty of ± 3 % in our reported Ti4SiC3 resistivity.
X-ray diffraction (XRD) and XRR analyses were conducted using a PANalytical X'pert PRO MPD system with a Cu Kα source. -2 scans were taken with a parallel beam with a divergence <0.055° using a parabolic mirror and a scintillation point detector. rocking curves were obtained using a hybrid mirror with a Ge(220) two bounce monochromator which yields a Cu Kα1 beam (λ = 1.5406 Å) with a 0.0068° divergence, and a PW3018/00 PIXcel line detector with 3 channels open corresponding to a 0.04° acceptance angle to detect a fixed 2 corresponding to the expected value for the Ti4SiC3 00010 reflection. φ-scans were obtained with fixed 2 values and χ tilts to detect asymmetric reflections from the substrate, the nucleation layer, and the Ti4SiC3 layer, and using a point-focus optics with a polycapillary lens providing quasi-parallel Cu Kα X-rays with a divergence of less than 0.3°. XRR patterns were obtained using parallel x-rays from a parabolic mirror and were analyzed using the PANalytical X'Pert Reflectivity software. Data fitting was done first for a sample with a TiC nucleation layer but without a Ti4SiC3 layer, yielding a TiC thickness of 12.0 ± 0.2 nm. Subsequently, the XRR patterns from the other samples were analyzed by fixing the TiC nucleation layer thickness to 12.0 nm and the densities of Al2O3 and TiC to the known values of 4.0 and 4.9 g/cm 3 , while the roughness at all interfaces and the thickness and density of the MAX phase layer are free fitting parameters. XPS analyses were performed using Al Kα radiation (1486.6 eV) in a PHI 5000 Versaprobe system with a hemispherical analyzer and an 8-channel detector. The sample surfaces were sputter etched using a 3 keV 2mA Ar + beam incident at 45° relative to the surface normal over a 1 mm 2 area, yielding a 1.4 nm/min sputter rate. Ion beam sputtering was done in 15 s steps, corresponding to etching of 0.35 nm per step, followed by acquisition of highresolution spectra of Ti 2p 3 , Si 2p, C 1s and O 1s peaks using a pass energy of 50 eV and a step size of 0.2 eV. The layer compositions were determined from the measured area under the curves after Shirley background subtraction and using the sensitivity factors within the PHI MultiPak software package.
First-principles calculations and Fermi surface integration to determine the ρoλ product were done following our previously described procedures, 17,23 using the Vienna ab initio simulation package (VASP) with a plane wave basis set with an energy cutoff of 500 eV, the Perdue-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) exchange correlation functional, 48 the projector-augmented wave method, 49 and a pseudo potential for Ti that includes only the first two shells such that Ti 3s, 3p, and 3d electrons are explicitly calculated. The atomic positions and the size and shape of the 16-atom Ti4SiC3 unit cell were iteratively relaxed, yielding lattice parameters a = 3.077 Å and c = 22.67 Å. Self-consistent calculations using a Γ-centered 44×44×8 k-point grid were employed to determine the charge distribution, which was subsequently kept fixed for calculations with a finer 240×240×32 kpoint mesh. The chosen 1.8×10 6 k-points yield a computational accuracy for the ρoλ products that is converged to ±1%. The ρoλ product is directly calculated by numerical integration over the Fermi surface for different transport directions, 12 more specifically for transport along [101 ̅ 0 ,, [112 ̅ 0 ,, and [0001,. We note that the calculations are done for T = 0 K and that including a Fermi-smearing of kT = 0.026 eV (300 K) reduces the calculated in-plane ρoλ value by 4%. In order to verify the computational approach, ρoλ was determined completely independently using a different computational approach as described in Ref. 35. Briefly, after a self-consistent DFT calculation, we use Wannier interpolation 50 to rapidly evaluate energies and velocities on a much denser 384×384×1536 k-point mesh for calculating the ρoλ Brillouin zone integral. These calculations were performed in JDFTx 51 using the PBEsol 52 GGA exchange-correlation functional and GBRV pseudopotentials 53 with a 680 eV plane-wave cutoff. As noted later, the two independent computational approaches yield ρoλ values which are within 1% of each other.
Figure 1 shows typical XRD and XRR results from a 37.4-nm-thick Ti4SiC3(0001)/TiC(111)/Al2O3(0001) layer. The -2 pattern in Fig. 1(a) has a strong doublepeak feature at 41.68° and 41.79° ascribed to the Al2O3 0006 substrate reflections of the Cu Kα1 and Kα2 x-rays and layer peaks at 35.88° and 39.93° attributed to the TiC 111 and the Ti4SiC3 00010 reflections, respectively. The minor peak at 37.49° is due to the Al2O3 0006 substrate reflection of residual Cu Kβ radiation. No other peaks can be detected over the entire measured 2 = 10-80° range except the higher order TiC 222 peak at 76.03°. This indicates a preferred out-of-plane orientation with Ti4SiC3(0001) || TiC(111) || Al2O3(0001). The Ti4SiC3 00010 peak position yields an out-of-plane lattice constant c = 2.260 nm determined using the weighted average wavelength of Kα1 and Kα2 x-rays, in good agreement with the reported c = 2.2606 nm from bulk Ti4SiC3, 54 indicating a negligible (<0.01%) strain. The Ti4SiC3 00010
full-width at half maximum (FWHM) peak width in the -2 scan is 0.26°, indicating an outof-plane x-ray coherence length 55 of 36 nm. This value is nearly identical to the layer thickness d = 37.4 nm from the XRR analysis, implying no detectable peak broadening due to crystalline defects. Similarly, the 2.5 times less intense TiC 111 peak is 0.71° wide, yielding a 13 nm outof-plane coherence length that matches the TiC nucleation layer thickness of 12.0 nm. Similar agreement is found for all samples in this work (not shown). The inset in Fig. 1(a) shows an rocking curve of the Ti4SiC3 00010 reflection. It exhibits a FWHM of 0.60°, confirming the crystalline alignment of the Ti4SiC3 and Al2O3 z-axes and indicating an in-plane x-ray coherence length of 22 nm. This value is 1.3 times larger than what has previously been reported for 500-nm-thick Ti4SiC3 layers deposited by magnetron co-sputtering on Al2O3(0001) with a 20-nm-thick TiC nucleation layer, 56 suggesting a higher crystalline quality for our layers. We note that the measured rocking curve width is within 0.5-0.8° for all Ti4SiC3(0001) layers in our study. Thus, it is nearly identical for all samples, indicating a crystalline quality and mosaicity that is independent of the layer thickness d = 5.8 -92.1 nm. 0001) is in agreement with previous reports suggesting that TiC epitaxy is possible despite the 11.2% lattice mismatch between TiC(111) and Al2O3(0001). 57 In contrast, the inplane mismatch from TiC(001) to Ti4SiC3(0001) is only 0.8% and epitaxy is expected since the unit cell of Ti4SiC3 consists of two TiC blocks with opposite stacking that are separated by a plane of Si atoms. Correspondingly, growth of Ti4SiC3(0001) on TiC(111) can lead to a single crystal Ti4SiC3 layer despite the two-domain microstructure of TiC(111), because the Ti4SiC3(0001) layer effectively contains blocks of both TiC(111) domains. We note that all samples presented in this paper exhibit similar XRD φ-scans (not shown) as those in Fig. 1(b), confirming that all Ti4SiC3 layers in this study are epitaxial single crystals. Figure 1(c) shows typical XRR results from the same sample as the XRD data in Figs. 1(a) and (b). The measured intensity is plotted in a logarithmic scale vs the scattering angle 2. It shows characteristic interference fringes which are used to determine the layer thickness by curve fitting as described in the previous section. The result from curve fitting is plotted as dashed line and is shifted by a factor of 10 for clarity purposes. This analysis yields a thickness of 37.4 0.6 nm for the Ti4SiC3(0001) layer, in reasonable agreement with the nominal d = 38.4 nm determined from deposition rate calibrations. It also provides values for the root-mean-square (RMS) roughness of the Ti4SiC3 surface and the Ti4SiC3-TiC and TiC-substrate interfaces of 0.7 0.2 nm, 0.1 0.1 nm and 0.5 0.2 nm, respectively. We note that all three roughnesses are quite small, but that the Ti4SiC3-TiC interface is particularly smooth, with a roughness below the XRR detection limit. We attribute this to the structural similarity of TiC and Ti4SiC3, which only differ by the Si layer that separates TiC-blocks within the MAX-phase structure. Correspondingly, we envision the following growth process: the 12.0-nm-thick TiC(111) nucleation layer prior to Ti4SiC3 deposition exhibits a 0.4 0.1 nm surface roughness, as measured by XRR from a sample with a nucleation layer without MAX phase. Subsequent Ti4SiC3 deposition leads to continued growth of TiC which is interrupted by atomically smooth Si layers. That is, even with an initially rough TiC surface, the Si layer within Ti4SiC3 may be atomically smooth. This Si layer, in turn, defines the interface between TiC and Ti4SiC3 since it is the only structural difference between these two materials, leading to an effectively atomically smooth interface. Such a smooth interface may result in the low electron scattering and promisingly low resistivity scaling of Ti4SiC3, as discussed below. We also note that we use XRR curve fitting to explore the possibility for the formation of an oxide layer on the Ti4SiC3 top surface. Adding a 1-nm-thick oxide layer to the XRR curve fitting procedure leads to a reduced fitting quality, implying that a possible surface oxide layer has a negligible (<1 nm) thickness. Similar XRR measurements and analyses are done for all samples, yielding layer thicknesses d = 5.8 0.2, 13.6 0.3, 37.4 0.6, 79.5 0.8 and 92.1 0.8 nm and RMS surface roughnesses of 0.5 0.2, 0.6 0.3, 0.7 0.2, 0.6 0.2 and 0.7 0.3 nm. Thus, the surface roughness is independent of d = 5.8 -92.1 nm, which may be due to the layered crystal structure which facilitates atomically smooth terraces.
Figure 2 shows selected regions of an XPS spectrum from a 21.1-nm-thick Ti4SiC3 film. It is obtained after cyclic sputter etching 10 nm into the layer to minimize effects of surface contamination from air exposure. The XPS spectrum shows the Si 2p peak at 99.0 eV attributed to the -4 valence state of Si in the Ti4SiC3 layer. 58,59 The C1s peak is at 281.8 eV, in good agreement with 281.6 eV reported for carbon in TiC, 60 confirming the -4 valence state and the same local bonding environment of C in Ti4SiC3 as in TiC. The Ti 2p1/2 and 2p3/2 peaks are at 460.7 and 454.7 eV, exactly matching the position of the reported Ti 2p doublet-peak feature at 460.7 eV and 454.7 eV in TiC. 61 Analysis of the spectra as a function of sputter depth (not shown) indicates that oxygen contamination can be detected only for the first two sputter cycles corresponding to < 1 nm of etching. This suggests that the detected oxygen is primarily due to H2O adsorption during air exposure while chemical oxidation of the Ti4SiC3 layer is negligible, consistent with the XRR results. The relative peak intensities remain constant between the 3 rd and 60 th sputter cycle, corresponding to ~20 nm of etching, indicating the absence of a detectable composition gradient associated with possible surface segregation. The composition is determined from the average of the 3 rd -60 th spectra, using the area under the peaks corrected by sensitivity factors. This yields measured atomic composition with 48.5 % Ti, 11.8 % Si, and 39.7 % C, in good agreement with the expected 50 : 12.5 : 37.5 for stoichiometric Ti4SiC3. The measured Ti/Si ratio of 4.12 and C/Si ratio of 3.38 are 3 % and 13 % above the expectation for a stoichiometric composition, which is attributed to selective sputtering of Si by the Ar + ion beam, caused by the lower bond energy and good mass-match with Ar. 62,63 Further etching into the TiC nucleation layer results in C 1s and Ti 2p peaks (not shown) with the same binding energies as shown in Fig. 2 for Ti4SiC3, indicating the same chemical environment and valence states for Ti and C in TiC and Ti4SiC3, as expected based on the structural similarity. The composition of the TiC nucleation layer is measured as 52.9 % Ti and 47.1 % C, indicating approximately stoichiometric TiC and confirming the previously reported negligible preferential sputtering in TiC. 62 Figure 3 is a plot of the resistivity ρ of epitaxial Ti4SiC3(0001) layers vs their thickness d = 5.8 to 92.1 nm measured at 293 and 77 K. The red data points are the in situ measured values while the open grey squares indicate ρ after air exposure. We note that the plotted ρ is in the in-plane direction parallel to the (0001) surface, and that the data is corrected for the conductance from the 12-nm-thick TiC(111) nucleation layer, which however only contributes 4 -33 % to the overall measured sheet conductance such that this correction is expected to lead to a negligible uncertainty in ρ. The room-temperature resistivity of the thickest (d = 92.1 nm) layer is 35.2 0.4 cm. ρ remains approximately constant with decreasing thickness down to d = 13.6 nm, but shows a detectable 7 % increase to ρ = 37.5 1.1 cm for d = 5.8 nm. This increase is attributed to electron surface scattering, as discussed below. The plotted ex situ measured ρ matches the in situ values with changes in ρ of less than 1 % for all layers with d ≥ 37.4 nm. However, air exposure causes a detectable 3.9 and 3.6 % resistivity increase for d = 13.6 and 5.8 nm, respectively. This increase indicates a transition to more diffuse electron surface scattering during surface oxidation, similar to what has previously been reported for Cu, 46,64 Co, 65 Ni, 7 and Ru, 66 and is quantified below. The low temperature resistivity measured in liquid nitrogen at 77 K, plotted as blue circles in Fig. 3, suggests a slight 15% increase with decreasing layer thickness, from ρ = 9.5 0.2 cm for d = 92.1 nm to 9.6 0.2, 10.1 0.2, 10.4 0.3, 11.0 0.4 cm for d = 79.5, 37.4, 13.6, 5.8 nm, respectively. However, we note that this increase is only approximately two times larger than the experimental uncertainty, such that the following quantification of the resistivity size effect has a large relative uncertainty.
The dotted, solid, and dashed lines in Fig. 3 represent lower limit, best fit, and upper limit resistivity predictions from the semiclassical Fuchs-Sondheimer model. 4,14,15 The curves are obtained assuming completely diffuse surface scattering (specularity parameter p = 0) and mean-free paths λ = 0.5, 1.1 and 1.7 nm at 293 K and λ = 1, 3 and 5 nm at 77 K. We note that the two parameters p and λ in the FS model typically cannot be determined independently from data fitting, because for any choice of p within the physically meaningful range [0,1, there exists a λ within [λmin, ∞, such that an experimental ρ vs d data set is equally well described as with another (p, λ) pair. 6,66 Here λmin represents the lower bound of possible mean-free paths and is obtained by setting p = 0. Correspondingly, as first step, we set p = 0 and plot predicted curves for different λ values, where the latter effectively is the lower bound (λmin) or alternatively the mean free path under the assumption of completely diffuse surface scattering at both the upper and lower Ti4SiC3 surfaces. The red solid curve through the room-temperature data points is for λ = 1.1 nm and uses an in-plane bulk resistivity ρo = 35.1 cm, which is the average of the in situ measured resistivity of the two thickest d = 79.5 and 92.1 nm Ti4SiC3 samples. This bulk resistivity is lower than the previously reported 50 cm for Ti4SiC3. 56 We attribute our lower ρo to the good crystalline quality, as evidenced by the narrow -rocking curves for both the TiC nucleation layer and the Ti4SiC3 MAX phase layer. The plotted solid curve describes the measured ρ well. However, because of the very small resistivity size effect which is nearly as large as the experimental uncertainty, a considerable range of λ values yield curves that are consistent with the measured ρ vs d. This is illustrated with the dotted and dashed lines for λ = 0.5 and 1.7 nm, which are minimum and maximum curves consistent with the experimental uncertainty. Thus, this data analysis yields λ = 1.1 0.6 nm and a corresponding ρoλ = (3.8 2.1) ×10 -16 Ωm 2 for Ti4SiC3 at room temperature. The measured resistivity from the air exposed samples is slightly higher and is quite well described by the dotted λ = 1.7 nm curve. However, as discussed below, the increase during air exposure is better described by a decrease in p than an increase in λ, since λ represents the bulk electron mean free path which, by definition, should be independent of surface oxidation. Data analysis at 77 K is performed in a similar approach, assuming p = 0 and setting ρo = 9.6 cm to the average of the d = 92.1 and 79.5 nm layers. This bulk resistivity is 3.7 times smaller than at room temperature, which is due to the reduced electron-phonon scattering at 77 K. Correspondingly, λ is larger at 77 K than at room temperature, as indicated by the solid line for λ = 3.0 nm and the dotted and dashed lines for λ = 1.0 and 5.0 nm, yielding an overall λ = 3.0 2.0 nm at 77 K. This corresponds to a ρoλ = (2.9 1.9) ×10 -16 Ωm 2 , which is equal to the room temperature ρoλ = (3.8 2.1) ×10 -16 Ωm 2 within experimental uncertainty. That is, ρoλ is temperature independent, as expected from classical transport models. 14,15 The inset in Fig. 3 shows a plot of the Fermi surface of Ti4SiC3, as determined from density functional calculations. The Brillouin zone shape is very anisotropic. It is 8 times wider within the layer plane than along the z-axis, because the real space lattice constant along the hexagonal axis c = 2.267 nm is much larger than a = b = 0.3077 nm in the basal plane. The plotted colors indicate the Fermi velocity vf, which ranges from approximately 5×10 4 to 5×10 5 m/s, with an average vf = 2.3×10 5 m/s. This is five times smaller than vf = 11.1×10 5 m/s for Cu. 17 Numerical integration over the Fermi-surface with the appropriate weights yields a calculated ρoλ = 19.3 ×10 -16 Ωm 2 for transport in the basal plane and ρoλ = 13.6 ×10 -16 Ωm 2 for transport along the hexagonal [0001, axis. We note that the basal plane value is independent (numerically converged to ±1%) of the transport direction within the basal plane, which is attributed to the 6-fold rotational symmetry. 24 Our calculated ρoλ = 19.3 ×10 -16 Ωm 2 is approximately 6 times larger than what has been previously reported. 36 This large deviation motivates us to carefully verify the convergence of our calculations as well as to determine ρoλ using a completely independent first principles calculation with a different density functional code, Fermi surface integration scheme, and approach to determine the electron velocity distribution, as described in Section II. This independent calculation yields ρoλ = 19.2×10 -16 Ωm 2 in the basal plane and 13.7×10 -16 Ωm 2 along the [0001, axis, in excellent agreement (< 1% deviation) with the above 19.3×10 -16 Ωm 2 and 13.6×10 -16 Ωm 2 , respectively. Thus, we have confidence in the accuracy of our calculations but do not know the detailed reason for the large deviation with respect to the previously published work. 36 We note that the convergence with respect to the k-point mesh is quite slow and speculate that the deviation may also be related to the way the anisotropy is treated by the previous study. More specifically, we perform a 3D integration over the entire Fermi surface because all electrons contribute to the current along a given transport direction (defined by the electric field) even if their velocity is not parallel to the transport direction. Conversely, consideration of only the electrons with (Fermi-) velocities along the transport direction (or in this case within the transport plane) would yield a distinctly different ρoλ value.
The calculated ρoλ = 19.3 ×10 -16 Ωm 2 is five times larger than the experimental ρoλ = (3.8 2.1) ×10 -16 Ωm 2 that is obtained by assuming completely diffuse surface scattering. This suggests that the Ti4SiC3(0001) surfaces exhibit a considerable probability for specular electron scattering. To quantify this effect, we use the calculated ρoλ and the measured roomtemperature ρo to determine a theoretical λ = 5.5 0.1 nm. We use now this calculated roomtemperature λ as a constant within the FS model and consider the scattering specularity p as a variable quantity. This yields p = 0.80 for the in situ room-temperature data. That is, the red solid line in Fig. 3 represents the FS prediction for λ = 5.5 nm and p = 0.80, while, as discussed before, it also represents case for λ = 1.1 nm and p = 0. Similarly, the dotted and dashed lines correspond to the FS prediction for λ = 5.5 nm and p = 0.89 and 0.69, respectively. These two p values define the range of the surface scattering specularity that is consistent with the in situ measured resistivity and the theoretical λ = 5.5 nm. Thus, p = 0.79 0.10, or simply p = 0.8 if implying the uncertainty with significant figures. That is, 80% of electrons scatter specularly at the Ti4SiC3(0001) surface or the Ti4SiC3(0001)/TiC(111) interface. This value is quite high in comparison to p = 0 reported for most metal-interfaces 3,67,68 and p = 0-0.7 for metal surfaces in vacuum. 6,24,64 We attribute the high scattering specularity at the Ti-4SiC3(0001)/TiC(111) interface to the structural similarity between TiC(111) and Ti4SiC3(0001), causing this interface which is defined by the first Si monolayer to be atomically smooth, as indicated by our XRR analyses.
We now consider the electron surface scattering specularity of air exposed samples. Similar to previous reports, 21,68 we attribute the resistivity increase during air exposure to a decrease in the scattering specularity p1 of the air-exposed top surface, while the scattering at the layer-bottom is unaffected by air exposure. Correspondingly, we keep λ = 5.5 nm and p2 = 0.80 fixed but decrease p1 to obtain the red dashed curve which describes the air exposed data well. This analysis yields p1 = 0.61. That is, both in situ and ex situ room temperature resistivity can be described using a constant mean free path λ = 5.5 nm and a constant scattering specularity p2 = 0.80 at the bottom surface of the layer, while the specularity at the top surface decreases from p1 = 0.80 in vacuum to p1 = 0.61 after air exposure. We note that this analysis disregards the possibility that p1 ≠ p2 for the as-deposited samples. Removing this restriction would result in opposite shifts of p1 and p2 but would yield the same specularity change Δp1 = -0.19 during air exposure. Similar air-exposure analyses have previously been reported for Cu(001) and Co(0001) surfaces and have yielded Δp1 = -0.6 and -0.55, respectively. 6,21 This indicates that air exposure of Ti4SiC3(0001) has a smaller effect on electron surface scattering than for the case of Cu(001) and Co(0001).
Analysis of the 77 K data with the calculated ρoλ yields a theoretical λ = 20.3 0.4 nm at 77 K. The solid blue line in Fig. 3 can be obtained using this λ-value and a p = 0.85, while the dotted and dashed lines correspond to p = 0.95 and 0.75 with λ = 20.3 nm. That is, we determine an average specularity p = 0.85 0.10 for electron scattering at the interfaces between Ti4SiC3 and liquid N2 and between Ti4SiC3 and TiC. This value is similar to p determined from the in situ data, suggesting that the Ti4SiC3/liquid N2 and the Ti4SiC3/vacuum interfaces exhibit comparable electron scattering, and that the short 1-3 s air exposure between sample removal from the deposition system and immersion into liquid N2 does not significantly alter the Ti4SiC3 surface. This reinforces the XRR and XPS results which indicate a good resistance against surface oxidation of our Ti4SiC3 samples.
XRD, XRR and XPS analyses show that sputter deposition from Ti, Si, and C targets onto TiC(001)/Al2O3(0001) substrates leads to stoichiometric, smooth, epitaxial Ti4SiC3(0001) layers. The in situ measured resistivity vs thickness indicates a small effective roomtemperature electron mean free path λ = 1.1 0.6 nm when assuming diffuse surface scattering. This value is 35 times smaller than that of copper, indicating the small resistivity scaling of Ti4SiC3 which is also evident from the only 7% resistivity increase when the layer thickness is reduced to 5.8 nm. The corresponding ρoλ = (3.8 2.1) × 10 -16 Ωm 2 is in good agreement with ρoλ = (2.9 1.9) ×10 -16 Ωm 2 obtained from measurements at 77 K, indicating that ρoλ is independent of temperature as expected from classical transport models. First-principles calculations yield ρoλ = 19.3 × 10 -16 Ωm 2 in the basal plane. This value is five times larger than the effective experimental values obtained by assuming diffuse surface scattering and corresponds to a room-temperature λ = 5.5 0.1 nm. Correspondingly, we interpret the low resistivity scaling to specular surface scattering with an average specularity parameter p = 0.79 0.10 in vacuum and p = 0.85 0.10 in liquid N2 at 77 K. Air exposure causes a moderate resistivity increase which is attributed to a decrease in the scattering specularity Δp1 = -0.19 of the top surface. The overall results confirm the previously predicted small resistivity scaling for Ti4SiC3 but attribute it to specular surface scattering rather than a small electron mean free path. Ti4SiC3 has a relatively large bulk resistivity ρo = 35.1 0.4 cm such that it is not competitive as conductor for narrow high-conductivity interconnects. However, its small measured resistivity scaling is promising and indicates that MAX phase materials have potential for interconnect applications if a compound with a low ρo can be synthesized. Figure 1.
Preprint: Minghua Zhang, Sushant Kumar, Ravishankar Sundararaman and Daniel Gall, "Resistivity scaling in epitaxial MAX-phase Ti4SiC3(0001) layers," J. Appl. Phys. 130, 034302 (2021).