On the feasibility of ab initio electronic structure calculations for Cu using a single s orbital basis

The accuracy of a single s-orbital representation of Cu towards enabling multi-thousand atom ab initio calculations of electronic structure is evaluated in this work. If an electrostatic compensation charge of approximately 0.3 electrons per atom is used in this basis representation of copper, the electronic transmission in bulk and nanocrystalline Cu compares accurately to that obtained with a Double Zeta Polarized basis set. The use of this representation is analogous to the use of single band effective mass representation for semiconductor electronic structure. With a basis of just one s-orbital per Cu atom, the representation is extremely computationally efficient and can be used to provide much needed ab initio insight into electronic transport in nanocrystalline Cu interconnects at realistic dimensions.

Copper is the current and and projected metal of choice for back-end-of-line (BEOL) interconnects used in semiconductor logic and memory technology. It has been well known that resistivity trend for copper is strongly non-linear and has been increasing rapidly with a reduction in interconnect width. A number of semi-empirical models have been formulated to explain this phenomenon. Broadly speaking, these models can be categorized as variants of the Fuchs-Sondheimer model (F-S) 1,2 of surface scattering-induced increase in resistivity or the Mayadas-Shatzkes (M-S) model 3 of grain-boundary scattering-induced increase in resistivity.
While these models provide important insights into electron transport in metals, they are fit a posteriori to experimental data with empirical parameters such as average reflectivity and specularity. Such parameteric models can be fit with a non-unique set of parameters and consequently provide very little predictive insight from a materials design perspective.
For instance, new experimental data on nanocrystalline copper requires a recalibration of the model to account for the new data. Often times, such empirical models can lead to conflicting physical insight. For instance, Graham et al. 4 fit the resistivity of sub-100 nm Cu interconnect lines to a purely surface scattering based model, while Steinhögl et al. 5 fit the resisitivity of Cu lines in the same dimensional range to a predominantly grain boundary scattering based model.
In spite of the apparent seriousness of the resistivity runaway problem and lack of understanding of the fundamental features that govern electronic transport in nanocrystalline (polycrystalline, with nanometer-range sized grains) Cu structures, there have been very few first principles based investigations on electron transport in nanocrystalline Cu [6][7][8][9][10] . Part of the reason for such a paucity of first principles based models is the cumbersome computational requirement (memory and execution time) for convergence of first principles transport calculations for realistic Cu nanocrystalline systems. This is illustrated with a simple example -The ITRS projects the smallest damascene interconnect cross sectional area of 10 × 22 nm 2 for damascene Cu in the year 2025 11 . Cu has a lattice constant of 3.61Å at room temperature. A cross sectional sliver of a length of just 1 nm with the aforementioned area of Cu at the projected ITRS dimensions contains approximately (assuming bulk density holds even for nanocrystalline samples) 18700 atoms. If one were to simulate such a sliver using What is needed then, is an ab initio method that is computationally efficient for systems containing several thousand Cu atoms while retaining an acceptable level of accuracy. In this respect, Cu possesses a unique advantage as compared to other metals such as partially d filled transition metals. Cu has a valence electron configuration of 3d 10 4s 1 . The d orbitals are spatially localized and completely filled while the s orbital is only partially filled and delocalized. An obvious question then arises -is it feasible to to represent Cu's electronic structure ab initio using only the partially filled and delocalized s orbitals while retaining acceptable accuracy? If the answer were to be in the affirmative, such a method would allow tremendous computational savings, allowing the ab initio investigation of multi-thousand atom Cu systems.
In this paper we report the results of our investigation into this question. We provide conditions for which the electronic structure of systems represented in a single s orbital representation can be compared favorably with that of DZP representations, which have in turn compared favorably to experiment in previous work 6,13,14 . Our end goal in this investigation is evaluation of the feasibility of using a reduced basis in studying electron transport in nanocrystalline Cu structures where the scattering mean free path is significantly smaller than the phonon scattering mean free path. Accordingly, we use ballistic transmission and ballistic transmission per unit area as figures of merit for comparing electronic structure throughout this paper.
The rest of this paper is organized as follows. We first describe the computational methodology used in our investigation. This is followed by a comparison of electronic structure computed using the two sets of bases for a variety of boundary conditions and polycrystalline configurations. We then conclude with a discussion on the relative accuracy of the method an its applicability in transport calculations on multi-thousand atom systems.   • Bulk with grain boundary repeated infinitely along transport direction.
• Nanowire with a single grain boundary and open boundary conditions along transport direction.
• Nanowire with grain boundary repeated infinitely along transport direction.
Since the number of possible configurations considering grain size and orientation distribution is potentially infinite and computationally cumbersome to compute in the DZP basis, we limit ourselves to a limited number of grain orientation distributions and a grain size of 1nm. An example of a structure for each boundary condition is shown in figure 5. Tables I and   II compare the transmission per unit area computed using the two sets of bases. Of special interest is the Σ3 [111]/[111] twin boundary that has the lowest specific contact resistivity of all grain boundaries 10 and is the least disordered grain boundary structure in Cu.  The cross sectional areas considered in this paper were limited to a maximum cross sectional area of 181Å 2 owing to difficulties in converging structures larger than 800-900 atoms using the DZP basis set. This is far smaller than current or projected interconnect dimensions. It is evident from the results above that as the surface area to volume ratio increases (as one goes towards bulk systems), the accuracy of transmission calculated in the s-orbital basis set improves. Additionally, it is important to note that the projected resistivity runaway It should also be pointed out that the feasibility evaluation carried out above is restricted to electronic structure. This is similar in spirit to attempts at modeling semiconductor electronic structure by use of a single effective mass. Total energy calculations using a single s-orbital give results that do not match DZP total energies since d orbitals contribute significantly to total energy in Cu.
In conclusion, we have evaluated the accuracy of a single s-orbital representation of Copper towards enabling multi-thousand atoms ab initio calculations of electronic structure.
We found that upon modification through doping, the s-orbital representation accurately reproduces key features of the bulk Fermi surface, the electronic structure of monocrystalline nanowires, bulk and nanowire single grain boundary structures and twin boundary structures. The electronic structure of multi grain boundary structures showed less comparative quantitative accuracy, but accurately matched all qualitative trends obtained through DZP calculations. The use of this representation is analogous to the use of single band effective mass representation for semiconductor electronic structure. We have found that the representation allows easy scaling to multi-thousand atom systems due to use of a reduced basis.
With a basis of just one s-orbital per Cu atom, the representation is extremely computa-