Interface relaxation within the friction model
3.9. The density of misfit defects at the "perfect" interface between Pr2O3 and Si(001)
Reloadable charge can be trapped by electrically neutral defects which have electron transition levels in the forbidden energy range. A Si dangling bond, if not associated with Pr, acts as this kind of trap. Also oxygen vacancy in Pr2O3 introduces a reloadable trap state. Life time of charge localized on such trap sites at the interface or in an ultrathin film is limited because the carrier can tunnel away to the substrate. Fixed charge can exist even in an ultrathin film when the relevant transition levels are degenerate with Si bands. Examples of such fixed charge traps in Pr2O3 are Pr vacancy, Pr interstitial, and O interstitial. These defects have been briefly discussed in the Section on native point defects.
We will now roughly estimate the lower limit of the interfacial density of misfit Si dangling bonds. We approximate the film/interface/substrate system by a sum of three components:
1. completely relaxed film with rigid lattice,
2. partially relaxed interfacial layer of a given thickness, with lattice constant matching a certain multiplicity of the substrate lattice constant, and
3. rigid substrate.
The strained layer is assumed to be compressed, so that one expects formation of dangling bonds in the substrate (the substrate has more surface orbitals than needed for bonding with the layer). Moreover, we assume that all atomic planes in the layer relax in identical way. We also ignore misfit defects between the strained layer and the relaxed film, because the need to create such defects only increases the effective stiffness of the layer, facilitating the formation of Si dangling bonds. Elastic energy of this system can be written in terms of elastic constants of the film (energy stored in the layer), force constant for deviations from the ideal alignment of "connecting points" between the layer and the substrate (interface ``friction'' energy), and the matching period of the substrate and the strained layer. The energy change upon relaxation enabled by an array of mismatch defects should be compared to the formation energy of such a defect.
The computed friction force constant is small (ks = 1.4 eV/Å), an order of magnitude smaller than typical bond stretching constants in covalent crystals. Assuming that the interfacial Pr2O3 is described by the bulk elastic constants, that the lateral relaxation proceeds along one direction, and approximating the plane-strain force constant of the film by
kf = (B Alatt) / [ 3(1-ν)(1-2ν) ] = 28.31 eV/Å
where B = 280 Gpa and Alatt = 11.07 Å are the computed values and ν is the Poisson ratio assumed to be 1/3, we conclude that while a 2 monolayer (ML) thick film cannot relax through interfacial defect formation, a 3 ML film stores enough energy to create one misfit defect (with formation energy of 1‑2\,eV) per about 15 lattice sites (2·1013 cm-2).
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Interface relaxation within the friction model |
The diagram illustrates the outcome of this interface friction model of dangling bond creation at Pr2O3/Si(001) interface. The crosses indicate the strain at each connection node. Blue crosses refer to strain in the film, red crosses indicate the strain in the interfacial plane. The interface friction constant has been obtained for the chemically sharp interface 0(SiPr) (cf. the Section on the interfacial structures). Note that the strain field induced by the defect (node 0) extends only to 20-30 nodes.
This result means that a Pr2O3 film grown directly on Si(001) would induce an unacceptably high density of interfacial defects. Therefore, an interfacial silicate layer seems to be necessary. The stress may be relieved in the intercalate planes without formation of interfacial dangling bonds. However, the Si content in the silicate should be kept low enough (Pr/Si ratio about 1) to eliminate the hazard of formation of Si-Si bridges in the film.
Computational Resources
Ab initio calculations are done on the IBM Regatta p690+ supercomputing cluster located in Forschungszentrum Jülich. The computing time within our project hfo06 (18000 POWER+ CPU hours/year + 30% due to Unicore bonus) is granted by von Neumann Institute for Computing and allows us to scan up to some hundred of different defect structures yearly, including a detailed investigation of those of tem, which turn out to be the most relevant for our study.