1. Approach
The calculations are done by the ab initio pseudopotential plane wave code fhi96md. A typical simulation uses Local Density Approximation (LDA) for exchange and correlation energy, nonlocal pseudopotentials in Trouller-Martins scheme with 40 Ry cutoff for plane waves, and samples the Brillouin zone either at (0, 0, 0) or at various special k-point sets, e.g., corresponding to the (1/4, 1/4, 0) point from the first Brillouin zone of Si(001) 3×3 surface cell. Generalized Gradient Approximation (GGA) is occasionally used to envision the accuracy of LDA results, but since the number of the atomic configurations to be considered is large and our computational resources are limited, the bulk of the work uses LDA. Calculations for bulk properties are typically done in supercells consisting of about 100 atoms (unless larger cells are necessary due to the crystal structure), while films are modeled by periodically repeated slabs consisting of usually six Si layers and up to four layers of the oxide. The Si substrate is terminated on one side by hydrogen and the slabs are separated by about 10 Å of vacuum.
Because of the open f-shell of Pr atoms, a key problem in calculations involving Pr is the construction of a reliable Pr pseudopotential. It turns out that in practice two different Pr pseudopotentials are needed: a pseudopotential with two core f electrons for trivalent Pr(III) which corresponds to pr ions in +3 charge state, and with only one f electron for tetravalent Pr(IV) which corresponds to Pr ions in +4 ionic charge. We calibrate the pseudopotential energy difference such that the experimental difference in formation enthalpies of Pr2O3 and PrO2. Similarly, the chemical potential of Pr metal is not calculated solely from first principles but obtained from the computed total energy of Pr2O3 and the experimental formation enthalpy of this compound, whereby the chemical potential of O2 is adjusted such that the experimental formation enthalpy of SiO2 is reproduced by the calculation (the required correction is of the order of 0.3 eV). The fundamental bulk properties (lattice constant, bulk modulus) of Pr2O3 and PrO2 obtained with the Pr pseudopotentials used here are in agreement with experimental data; the discrepancies are well within the range typical for LDA calculations.
Defect formation energies in a compound (and, generally, impurity formation energies) depend on the chemical potentials of the components. In particular, for native defects in Pr oxides we have, for X = (Pr, O):
X interstitial: |
| Gf (XI) = G°f (XI) - µ(X), |
X vacancy: |
| Gf (XV) = G°f (XV) + µ(X), |
equilibrium with Pr2O3: |
| µ(Pr) + 1.5 µ(O) = G°f (Pr2O3), |
where G°f (X) is the standard (i.e., corresponding to room temperature and atmospheric pressure) free energy of formation for species X. Since we compute total energies at zero Kelvin, we use for calibration and comparison with experiment the corresponding formation enthalpies rather than free energies.
The important regimes of the chemical potential of oxygen are:
Pr2O3 in contact with Pr metal: |
| µ(O) = G°f (Pr2O3) / 3, |
Pr2O3 in contact with SiO2 on Si: |
| µ(O) = G°f (SiO2) / 2, |
Pr2O3 in contact with PrO2: |
| µ(O) = 2 G°f (PrO2) - G°f (Pr2O3) / 2, |
PrO2 in contact with air: |
| µ(O) = 0. |
Since point defects are usually charged, we must also consider the dependence of the defect formation energy on the electron chemical potential, that is, on the Fermi energy EF:
positive charge, n+ > 0: |
| Gf (n+, EF) = Gf (n+, 0) + n+ EF |
negative charge, -n- < 0: |
| Gf (n-, EF) = Gf (n-, 0) - n- EF. |
This means that the formation energy of charged defects in a dielectric in electrical contact with the Si substrate is determined by the position of the Fermi level in the substrate and by the valence band offsets between Si and the dielectric. Since the latter is affected by the electrical dipole moment at the interface between the dielectric and the substrate, the chemical character and electrical quality of the interface may have a noticeable effect on the defect formation energies and, consequently, on the defect population in the dielectric film. In order to estimate “ideal” band offsets, that is, the band arrangement due solely to the dipole moment that appears at the interface as the result of different band structures and dielectric constants in both materials, we apply the Charge Neutrality Level (CNL) alignment model, which is known to yield reasonable results for interfaces between a range of materials and is often invoked also in the context of high-k films grown on Si interfaces. Due to the well-known band gap problem in DFT calculations, the band gaps used in the calculation of CNL are taken from experiment; the band topology is obtained from ab initio DFT band structure.