Ab initio calculation of valley splitting in monolayer δ-doped phosphorus in silicon
© Drumm et al.; licensee Springer. 2013
Received: 16 October 2012
Accepted: 26 January 2013
Published: 27 February 2013
The differences in energy between electronic bands due to valley splitting are of paramount importance in interpreting transport spectroscopy experiments on state-of-the-art quantum devices defined by scanning tunnelling microscope lithography. Using vasp, we develop a plane-wave density functional theory description of systems which is size limited due to computational tractability. Nonetheless, we provide valuable data for the benchmarking of empirical modelling techniques more capable of extending this discussion to confined disordered systems or actual devices. We then develop a less resource-intensive alternative via localised basis functions in siesta, retaining the physics of the plane-wave description, and extend this model beyond the capability of plane-wave methods to determine the ab initio valley splitting of well-isolated δ-layers. In obtaining an agreement between plane-wave and localised methods, we show that valley splitting has been overestimated in previous ab initio calculations by more than 50%.
The study of the quantum properties of low-dimensional and doped structures is central to many nanotechnology applications[1–15]. Quantum devices in silicon have been the subject of concentrated recent interest, both experimental and theoretical, including the recent discussion of Ohm’s law at the nanoscale. Efforts to make such devices have led to atomically precise fabrication methods which incorporate phosphorus atoms in a single monolayer of a silicon crystal[17–20]. These dopant atoms can be arranged into arrays or geometric patterns for wires[16, 22] and associated tunnel junctions, gates, and quantum dots[24, 25] - all of which are necessary components of a functioning device. The patterns themselves define atomically abrupt regions of doped and undoped silicon. While silicon, bulk-doped silicon, and the physics of the phosphorus incorporation are well understood, models of this quasi-two-dimensional phosphorus sheet are still in their initial stages. In particular, it is critical in many applications to understand the effect of this confinement on the conduction band valley degeneracy, inherent in the band structure of silicon. For example, the degeneracy of the valleys has the potential to cause decoherence in a spin-based quantum computer[28, 29], and the degree of valley degeneracy lifting (valley splitting) defines the conduction properties of highly confined planar quantum dots.
The importance of understanding valley splitting in monolayer δ-doped Si:P structures has led to a number of theoretical works in recent years, spanning several techniques, from pseudo-potential theories via planar Wannier orbital bases, density functional theory (DFT) via linear combination of atomic orbital (LCAO) bases[31, 32], to tight-binding models[33–37] and effective mass theories (EMT)[38–40]. We note that several of these papers are based upon the assumption that the effective masses of δ-doped P in Si remain unchanged from bulk-doped values[38, 39], an assumption which has been challenged[30, 33]. Others assume doping over a multi-atomic plane band[33, 38] which no longer represents the state of the art in fabrication. There is currently little agreement between the valley splitting values obtained using these methods, with predictions ranging between 5 to 270 meV, depending on the calculational approach and the arrangement of dopant atoms within the δ-layer. Density functional theory has been shown to be a useful tool in predicting how quantum confinement or doping perturbs the bulk electronic structure in silicon- and diamond-like structures[41–45]. The work of Carter et al. represents the first attempt using DFT to model these devices by considering explicitly doped δ-layers, using a localised basis set and the assumption that a basis set sufficient to describe bulk silicon will also adequately describe P-doped Si. It might be expected, therefore, that the removal of the basis set assumption will lead to the best ab initio estimate of the valley splitting available, for a given arrangement of atoms. In the context of describing experimental devices, it is important to separate the effects of methodological choices, such as this, from more complicated effects due to physical realities, including disorder.
In this paper, we determine a consistent value of the valley splitting in explicitly δ-doped structures by obtaining convergence between distinct DFT approaches in terms of basis set and system sizes. We perform a comparison of DFT techniques, involving localised numerical atomic orbitals and delocalised plane-wave (PW) basis sets. Convergence of results with regard to the amount of Si ‘cladding’ about the δ-doped plane is studied. This corresponds to the normal criterion of supercell size, where periodic boundary conditions may introduce artificial interactions between replicated dopants in neighbouring cells. A benchmark is set via the delocalised basis for DFT models of δ-doped Si:P against which the localised basis techniques are assessed. Implications for the type of modelling being undertaken are discussed, and the models extended beyond those tractable with plane-wave techniques. Using these calculations, we obtain converged values for properties such as band structures, energy levels, valley splitting, electronic densities of state and charge densities near the δ-doped layer.
The paper is organised as follows: the ‘Methods’ section outlines the parameters used in our particular calculations; we present the results of our calculations in the ‘Results and discussion’ section and draw conclusions in the ‘Conclusions’ section. An elucidation of effects modifying the bulk band structure follows in Appendices 1 and 2 to provide a clear contrast to the properties deriving from the δ-doping of the silicon discussed in the paper. The origin of valley splitting is discussed in Appendix 3.
Density functional theory calculations have been carried out using both plane-wave and LCAO basis sets. For the PW basis set, the Vienna ab initio simulation package (vasp) software was used with projector augmented wave[46, 47] pseudo-potentials for Si and P. Due to the nature of the PW basis set, there exists a simple relationship between the cut-off energy and basis set completeness. For the structures considered in this work, the calculations were found to be converged for PW cut-offs of 450 eV.
Localised basis set calculations were performed using the Spanish Initiative for Electronic Simulations with Thousands of Atoms (siesta) software. In this case, the P and Si ionic cores were represented by norm-conserving Troullier-Martins pseudo-potentials. The Kohn-Sham orbitals were expanded in the default single-ζ polarized (SZP) or double-ζ polarized (DZP) basis sets, which consist of 9 and 13 basis functions per atom, respectively. Both the SZP and DZP sets contain s-, p-, and d-type functions. These calculations were found to be converged for a mesh grid energy cut-off of 300 Ry. In all cases, the generalized gradient approximation PBE exchange-correlation functional was used.
Eight-atom cubic unit cell equilibrium lattice parameters for different methods used in this work
For tetragonal cells, the k-point sampling was set as a 9 × 9 × N Γ-centred MP mesh as we have found that failing to include Γ in the mesh can lead to the anomalous placement of the Fermi level on band structure diagrams. N varied from 12 to 1 as the cells became more elongated (see Appendix 1). We note that, as mentioned in the work of Carter et al., the large supercells involved required very gradual (<0.1%) mixing of the new density matrix with the prior step, leading to many hundreds of self-consistent cycles before convergence was achieved.
Although it has been previously found that relaxing the positions of the nuclei gave negligible differences (<0.005 Å) to the geometry, this was for a 12-layer cell and may not have included enough space between periodic repetitions of the doping plane for the full effect to be seen. Whilst a 40-layer model was optimised in the work of Carter et al., this made use of a mixed atom pseudo-potential and is not explicitly comparable to the models presented here. We have performed a test relaxation on a 40-layer cell using the PW basis (vasp). The maximum subsequent ionic displacement was 0.05 Å, with most being an order of magnitude smaller. The energy gained in relaxing the cell was less than 37 meV (or 230 μ eV/atom). We therefore regarded any changes to the structure as negligibly small, confirming the results of Carter et al.[31, 32], and proceeded without ionic relaxation.
Single-point energy calculations were carried out with both software programs; for vasp, the electronic energy convergence criterion was set to 10−6eV, and the tetrahedron method with Blöchl correction was used. For siesta, a two-stage process was carried out: Fermi-Dirac electronic smearing of 300 K was applied in order to converge the density matrix within a tolerance of one part in 10−4; the calculation was then restarted with the smearing of 0 K, and a new electronic energy tolerance criterion of 10−6 eV was applied (except for the 120- and 160-layer DZP models for which this was intractable; a tolerance of 10−4 eV was used in these cases). This two-stage process aided convergence as well as ensuring that the energy levels obtained were comparably accurate across methods. In addition, for each doped cell thus developed and studied, an undoped bulk Si cell of the same dimensions was constructed to aid in isolating those features primarily due to the doping.
Results and discussion
Analysis of band structure
Once converged charge densities were obtained, band structures were calculated along the M–Γ–X high-symmetry pathway (as shown in Appendix 1), using at least 20 k-points between high-symmetry points. For comparative purposes, the band structures have all been aligned at the valence band maximum (VBM).
The Fermi level for the doped system is also shown, clearly being crossed by all three of these bands which are therefore able to act as open channels for conduction.
In the smallest cells (<16 layers), less than three bands are observed. This is likely due to the lack of cladding in the z direction, leading to a significant interaction between the dopant layers, raising the energy of each band. Whilst the absolute energy of each level still varies somewhat, even with over 100 layers incorporated, we find that the Γ1–Γ2 values are well converged with 80 layers of cladding for all methods (see Figure5). Indeed, they may be considered reasonably converged even at the 40-layer level (0.5 meV or less difference to the largest models considered). The differences between the energies of the second and third band minima (Γ2–δ splittings) are also shown in Figure5 and show good convergence (within 1 meV) for cells of 80 layers or larger.
The Fermi level follows a similar pattern to the Γ- and δ-levels. In particular, the gap between the Fermi level and Γ1 level does not change by more than 1 meV from 60 to 160 layers.
Given that the properties of interest are the differences between the energy levels, rather than their absolute values (or position relative to the valence band), in the interest of computational efficiency, we observe that using the DZP basis with 80 layers of cladding is sufficient to achieve consistent, converged results.
Valley splitting values of 1/4 ML P-doped silicon obtained using different techniques
Planar Wannier orbitala
Tight binding (4 K)b
Tight binding (4 K)b
Tight binding (300 K)b
DFT, SZP basis set a
DFT, SZP: ordered b
DFT, SZP: random disorder b
DFT, SZP:  direction alignment b
DFT, SZP: dimers b
DFT, SZP: random disorder b
DFT, SZP: clusters b
DFT, SZP:  direction alignment b
DFT, SZP: ordered, M=4b,c
DFT, SZP: ordered, M=6b,c
DFT, SZP: ordered, M=10b,c
SZP, M=9 (this work)b,c
PW, M=9 (this work)b,d
DZP, M=9 (this work)b,c
Our results show that valley splitting is highly sensitive to the choice of basis set. Due to the nature of PW basis set, it is straightforward to improve its completeness by increasing the plane-wave cut-off energy. In this way, we establish the most accurate valley splitting value within the context of density functional theory. Using this benchmark value, we can then establish the validity and accuracy of other basis sets, which can be used to extend the system sizes to that beyond what is practical using a PW basis set. As seen in Table2, the valley splitting value converges to 93 meV using 80-layer cladding. The DZP localised basis set gives an excellent agreement at 99.5 meV using 80-layer cladding (representing a 7% difference). On the other hand, our SZP localised basis set gave a value of 145 meV using the same amount of cladding. This represents a significant difference of 55% over the value obtained using PW basis set and demonstrates that SZP basis sets are unsuitable for accurate determination of valley splitting in these systems.
Density of states
The Fermi energy exhibits convergence with respect to the amount of cladding, as reported above. It is also notable that the eDOS within the bandgap are nearly identical regardless of the cell length (in z). This indicates that layer-layer interactions are negligibly affecting the occupied states and, therefore, that the applied ‘cladding’ is sufficient to insulate against these effects.
Electronic width of the plane
Calculated maximum donor-electron density, ρ max , and FWHM
Our results differ from a previous DFT calculation which cited an FWHM of 5.62 Å for a 1/4 ML-doped, 80-layer cell calculated using the SZP basis set (and 10 × 10 × 1 k-points). We note that those values were taken from the unfitted, untransformed donor-electron distribution and represent an approximately 15% underestimation in comparison with the DZP result. The peak height is not shown in the work of Carter et al., but the value from another work (1.7 × 1021 e/cm3) is a factor of 0.44 smaller than the peak we observe here. This may be due, to some extent, to the larger width of the SZP model leading to an associated lowering of the peak density.
In this article, we have studied the valley splitting of the monolayer δ-doped Si:P, using a density functional theory model with a plane-wave basis to establish firm grounds for comparison with less computationally intensive localised-basis ab initio methods. We found that the description of these systems (by density functional theory, using SZP basis functions) overestimates the valley splitting by over 50%. We show that DZP basis sets are complete enough to deliver values within 10% of the plane-wave values and, due to their localised nature, are capable of calculating the properties of models twice as large as is tractable with plane-wave methods. These DZP models are converged with respect to size well before their tractable limit, which approaches that of SZP models.
Valley splittings are important in interpreting transport spectroscopy experiment data, where they relate to families of resonances, and in benchmarking other theoretical techniques more capable of actual device modelling. It is therefore pleasing to have an ab initio description of this effect which is fully converged with respect to basis completeness as well as the usual size effects and k-point mesh density.
We have also studied the band structures with all three methods, finding that the DZP correctly determines the δ-band minima away from the Γ point, where the SZP method does not. We show that these minima occur in the Σ direction for the type of cell considered, not the δ direction as has been previously reported. Having established the DZP methodology as sufficient to describe the physics of these systems, we then calculated the electronic density of states and the electronic width of the δ-layer. We found that previous SZP descriptions of these layers underestimate the width of the layers by almost 15%.
We have shown that the properties of interest of δ-doped Si:P are well converged for 40-layer supercells using a DZP description of the electronic density. We recommend the use of this amount of surrounding silicon, and technique, in any future DFT studies of these and similar systems - especially if inter-layer interactions are to be minimised.
Subtleties of bandstructure
Regardless of the type of calculation being undertaken, a band structure diagram is inherently linked to the type (shape and size) of cell being used to represent the system under consideration. For each of the 14 Bravais lattices available for three-dimensional supercells, a particular Brillouin zone (BZ) with its own set of high-symmetry points exists in reciprocal space. Similarly, each BZ has its own set of high-symmetry directions. Some of these BZs share a few high-symmetry point labels (or directions), such as X or L (δ or Σ), and they all contain Γ, but these points are not always located in the same place in reciprocal space.
A simple effect of this can be seen by increasing the size of a supercell. This has the result of shrinking the BZ and the coordinates of high-symmetry points on its boundary by a corresponding factor. Consider the conduction band minimum (CBM) found at the δ valley in the Si conduction band. This is commonly located at in the δ direction towards X (also Y, Z and their opposite directions). Should we increase the cell by a factor of 2, the BZ will shrink (BZ→BZ’), placing the valley outside the new BZ boundary (past X’); however, a valid solution in any BZ must be a solution in all BZs. This results in the phenomenon of band folding, whereby a band continuing past a BZ boundary reenters the BZ on the opposite side. Since the X direction in a face-centred cubic (FCC) BZ is sixfold symmetric, a solution near the opposite BZ boundary is also a solution near the one we are focussing on. This results in the appearance that the band continuing past the BZ boundary is ‘reflected’, or folded, back on itself into the first BZ. Since the new BZ boundary in this direction is now at, the location of the valley will be at, as mentioned in the work of Carter et al.. Each further increase in the size of the supercell will result in more folding (and a denser band structure). Care is therefore required to distinguish between a new band and one which has been folded due to this effect when interpreting band structure.
Band folding in the z direction
Energy levels of tetragonal bulk Si structures
in k z
All methods considered in Table4 show the LUMO at Γ (folded in along ± k z ) approaching the CBM value as the amount of cladding increases; at 80 layers, the LUMO at Γ is within 1 meV of the CBM value. It is also of note that the PW indirect bandgap agrees well with the DZP value and less so with the SZP model. This is an indication that, although the behaviour of the LUMO with respect to the cell shape is well replicated, the SZP basis set is demonstrably incomplete. Conversely, pairwise comparisons between the PW and DZP results show agreement to within 5 meV.
It is important to distinguish effects indicating convergence with respect to cladding for doped cells (i.e. elimination of layer-layer interactions) from those mentioned previously derived from the shape and size of the supercell. Strictly, the convergence (with respect to the amount of encapsulating Si) of those results we wish to study in detail, such as the differences in energy between occupied levels in what was the bulk bandgap, provides the most appropriate measure of whether sufficient cladding has been applied.
Here, we discuss the origins of valley splitting, in the context of phosphorus donors in silicon. Following on from the discussion of Si band minima in Appendices 1 and 2, we have, via elongation of the supercell and consequent band folding, a situation where, instead of the sixfold degeneracy (due to the underlying symmetries of the Si crystal lattice), we see an apparent splitting of these states into two groups (6 → 2 + 4, or 2 Γ + 4 δ minima).
We now consider what happens in perfectly ordered δ-doped monolayers, as per the main text. Here, we break the underlying Si crystal lattice symmetries by including foreign elements in the lattice. By placing the donors regularly (according to the original Si lattice pattern) in one  monolayer, we reduce the symmetry of the system to tetragonal, with the odd dimension being transverse to the plane of donors. This dimension can be periodic (as in the supercells described earlier), infinite (as in the EMT model of Drumm et al.) or extremely long on the atomic scale (as the experiments are).
Immediately, therefore, we expect the same apparent 2 + 4 breaking of the original sixfold degenerate conduction band minima. Of course, as we have introduced phosphorus (which has one more electron and one more proton than silicon), this next band (still actually sixfold degenerate in bulk silicon) will be occupied and will now be influenced by the new potential. The sub-bands interact differently with the potential, thanks to the different curvatures in their dispersion relations and drop by different amounts into the bandgap. As discussed in detail in Drumm et al., the filling of these sub-bands is partial rather than complete (or absent) and is governed by both the energy of their minima and their respective effective masses. We now have an actual breaking of the sixfold degeneracy into a true 2 + 4 system.
If we still look closer, we might expect these lower degeneracies to spontaneously break - nature, after all, is said to abhor degeneracy. Indeed, this does occur, but for this special case of δ-doped Si:P, the effect is enhanced by the strong V-shaped potential about the monolayer due to the extra charge in the donor nuclei. Consideration of odd and even solutions to the effective mass Schrödinger equation for this sub-band leads to their superposition(s) and subsequent energy difference. This is enhanced further in the Kohn-Sham formalism, as evidenced in previous sections. (The four δ minima also split but on a far-reduced scale not visible using current DFT techniques.) We thus expect, in the DFT picture, to see 6 →2 + 4→1 + 1 + 4 sub-band structure, namely the Γ1, Γ2 and δ bands. The valley splitting which is the main focus of this paper is the energy difference between the Γ1 and Γ2 band minima due to the superposition of solutions.
The authors acknowledge funding by the ARC Discovery grant DP0986635. This research was undertaken on the NCI National Facility in Canberra, Australia, which is supported by the Australian Commonwealth Government. We thank Oliver Warschkow, Damien Carter and Nigel Marks for their feedback on our manuscript.
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