Calculation of Transport Parameters of SiOz Polymorphs

Silica polymorphs have been analyzed, that are similar to the amorphous phase of SiO2; specifically, the α- and β-quartz, and the α- and β-cristobalite have been addressed. Two different ab initio methods have been used to calculate the full-band structure and the electronic properties of the polymorphs. In particular, the Hartree-Fock (HF) and Density-Functional Theory (DFT) approaches have been selected. A comparison of the energy-band branches and DOS shapes is reported, showing the similarities of the electrical properties that characterize the polymorphs under study. Finally, some relevant properties are compared for the analyzed set of polymorphs.


I. INTRODUCTION
Among insulators, silicon dioxide plays a major role in the solid-state technology.Especially in recent years, experimental and theoretical inves- tigation have focused on microscopic properties of this material related with reliability problems.
To accurately analyze the physics of damage and breakdown in VLSI devices, the microscopic properties of SiO2 must be described, as well as those of the Si-SiO2 interface.As far as the electron injection into the gate oxide is con- cerned, the description of the electron-energy distribution in an MOS device is fundamental for a physically-based modeling.The carrier- distribution function in both Si and SiO2 regions can be calculated by solving the Boltzmann Transport Equation (BTE).As the amorphous SiO2 shows a number of electronic and optical properties in common with some silica polymorphs, it can be treated as a polar crystal.The SHE method, which solves the BTE basing on an expansion of the distribution function in spherical harmonics, proved to be a valuable tool for the analysis of semiconductor devices, in particular for hot-carrier effects [1].The SHE method has been applied to silicon and silicon dioxide separately: in particular, for silicon a numerical *Corresponding author.Tel.: + 39 051 2093016, Fax: + 39 051 2093073, e-mail: mrudan@deis.unibo.itfull-band structure and the main scattering mechanisms have already been incorporated [2], while for silicon dioxide the collision-matrix contributions have been taken into account [3] in the frame of an analytical energy-band structure similar to that of [4].The analysis of [3] shows that an accurate description of the dispersion relation and the density of states in energy in SiO2 is necessary.For this reason the numerical calculation of such properties based on a full-band structure is addressed here.
Earlier calculations of the band structure for some polymorphs were based mostly on an empirical type of calculation (mixed-basis [5], pseudopotential [6]).In [7] a self-consistent LDA approach has been used, based on the assumption of a locally-homogeneous electron gas.In this work, ab initio calculations of the full-band structure have been worked out by means of two different techniques: Hartree-Fock (HF) and Density-Functional Theory (DFT), with the aim of overcoming some of the approximations adopted in [5][6][7].Accurate calculations of both valence-and conduction-band structures, and of the corresponding densities of states (DOS), have been carried out.Moreover, the numerical results have been analyzed with particular emphasis on the parameters that influence the transport properties.First, the energy gaps have been compared: the HF method provides a smaller spread of values than the DFT for the poly- morphs under study, but the values are over- estimated with respect to the experimental ones.Then, the effective masses for the minimum of the conduction band have been extracted from the numerical data.Such calculation has been carried out only to the purpose of comparing our results with theoretical and experimental data shown in the literature, not to extract further information to be used in the calculation of the transport properties.Indeed, our aim is that of incorporat- ing the numerical full-band structure within the transport solver SHE without any simplifica- tion, similarly to what we have already done for silicon [2].

II. THEORY AND COMPUTATIONAL DETAILS
The calculation of the band structure and DOS can be dealt with by methods similar to those used for analyzing molecular structures.Among these are the HF and DFT methods which are im- plemented in the code CRYSTAL98(C), adopted in this work.In the HF method the N-electron problem is reduced to a system of coupled single- electron equations that minimizes the total energy of the system.In the DFT level, the system is approximated by N non-interacting electron equa- tions that give the same electron density.In the exchange-and correlation-potential terms, the Becke and Lee-Yang-Parr extensions for the local density approximation have been adopted.The crystalline space groups for the set of silica polymorphs are taken from [8].The lattice parameters were calculated as in [9], starting from the experimental values by Wickoff [8] and minimizing the HF unit cell energy.In Table I the crystalline symmetries and the lattice param- eters of the SiO2 polymorphs are reported.
Finally, the basis sets, that are the sets of normalized Gaussian functions used by the solver to calculate the atomic orbitals, have been chosen among the optimized sets listed in [9]: the choice of the basis set plays an essential role in the description of the electronic structure, in particular for an accurate determination of the conduction- band eigenvalues.In Table II, the adopted basis  sets are reported for the silicon and oxygen atoms, respectively.More details of the adopted approach are given in [10].
Figures and 2, respectively.The branches have suitably been grouped in order to present the result in a compact form; the origin of the energy scale coincides with the maximum of the valence band or the minimum of the conduction band as the HF and DFT values of the energy gap are different.The number and shape of the bands differ from one polymorph to another due to the different characteristics of their unit cells.The density of states in energy per cell (DOS) is reported in Figure 3 for c-and fl-quartz, and in Figure 4 for c-and fl-cristobalite.It is worth noting that each polymorph is characterized by a different number of branches for a given energy interval: as the total density of states is the sum III.RESULTS   As output of the HF and DFT calculations, the ground-state energy per unit cell, the pattern of the highest-valence and lower-conduction bands, and the densities of states have been analyzed.In Table III the minimum energy values by HF calculations are reported, compared to the DFT ones: it is worth noting that the basis-set coefficients have been optimized in order to minimize the ground-state energy in the HF level.The bands have been calculated along the high- symmetry lines of the first Brillouin zone specific to each polymorph [11].The energy interval investigated in this work ranges from 2-3eV below the maximum of the highest-valence band to about 6eV above the minimum of the lowestconduction band: the number of branches is rather large.In order to give sensible examples of the obtained results, the calculated band struc- tures of c-quartz and fl-cristobalite are shown in full-band structure calculated with HF method (continuous lines) and with the DFT method (dashed lines). of the densities related to each branch, c-cristobalite (with a 12-atom unit cell) higher DOS values than /3-cristobalite (6-atom unit cell).The energy band and DOS calculated with the two methods are usually similar in shape.The DFT eigenvalues are in general under study for both HF and DFT levels: a slightly lower than the HF ones.Moreover, the difference less than 15% is obtained in both sets energy-band gap is usually overestimated by the of data, this showing that the silica polymorphs HF method and underestimated by the DFT one have a number of electronic properties in with respect to the experiments.On the other common, due to the similar microscopic density.hand, the calculated energy-band gaps for the In order to quantitatively compare the results investigated polymorphs differ from each other given by the two methods for the set of by less than eV.In Figure 5, the normalized polymorphs, the effective masses of the lowest- energy-gap values EGAP (polymorph)/EGAp(C-conduction band have been calculated along the quartz) are reported for the set of polymorphs three main directions.In Figure 6 the effective masses, obtained as a geometrical average be- tween the three values along the main directions, are reported for the set of polymorphs under investigation.The values of the masses calculates by the DFT method are rather similar, this confirming that the silica polymorphs are similar.The results given by the DFT approach are in good agreement with the values reported in [4][5][6][7].In contrast a discrepancy is found in the HF data, in particular for the c-cristobalite: this is probably due to the adopted basis set, that has been optimized for c-quartz.
IV. CONCLUSIONS A selected set of silica polymorphs have been analyzed, with the aim of taking into account forms that are similar to the amorphous phase of SiO2.The investigation has been carried out in the frame of the HF and DFT approximations, using CRYSTAL98 (C).The energy-band branches and DOS shapes have been calculated, and the effective masses have been computed as well.Even if the HF and DFT methods used in the analysis usually provide energy-gap values that differ from the experimental ones, the band structure and DOS are in good agreement with results by other methods, and can be indicated as a good descrip- tion of the electronic properties for the material under study.

FIGURE 3 FIGURE 2
FIGURE 3 Density of states in energy for c-and fl-quartz.Solid lines: HF; dashed lines: DFT.

FIGURE 4
FIGURE 4 Density of states in energy for c-and flcristobalite.Solid lines: HF; dashed lines: DFT.

FIGURE 5 FIGURE 6
FIGURE5 Normalized energy-gap values of the silica polymorphs.

TABLE II
Adopted basis sets for silicon and oxygen