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  • Introduction


1. Theoretical calculations for high-k dielectrics

The purpose of these calculations is to assist development of gate dielectrics for new generations of MOS transistors. The primary practical issue is to achieve the Equivalent Oxide Thickness (EOT) as low as 1 nm and less. EOT is associated with general features of the film morphology. In particular, the chemical composition of the film is a function of the distance from the substrate, affecting the effective dielectric constant of the film. Moreover, the density of charge traps (fixed and reloadable) in the film and at the interface must be brought down to a value comparable to that typical for industrial quality gate SiO2. Our theoretical work has so far been focused on the analysis of atomistic aspects of these issues for the family of high-k dielectrics containing praseodymium oxides: PrOx, Pr silicate, Pr aluminate, Pr titanate. In 2007, we have started calculations for other materials, as strontium tantalate. Our atomistic calculations are done with DFT framework.


Within the frame of this project, we have so far understood several essential properties of Pr oxides, silicates and silicides, starting from band structure calculations (which we conducted also for a range of other metal oxides and chalcogenides). In particular, we gained insight into the fundamental interfacial structures and interface layer formation mechanisms associated with Pr oxide and silicate growth on Si(001) and Si(111), including the film surface roughness, the substrate oxidation and intermixing, estimate of the misfit dangling bond density at the Pr2O3/Si(100) interface, the mechanism governing the stacking order of hexagonal Pr2O3 and Y2O3 on Si(111), energies and electrical behaviour of some native defects and critical impurities, and the evolution of the interfacial dipole during oxidation. We addressed the chemical reactions pertinent to silicate film growth by solid state reaction between SiO2 and a Pr-containing layer, including the energetics of Pr impurities in amorphous SiO2 and oxidation of Pr2Si2O7. We analysed the process of the interface state passivation by remote action of Ti, and the onset of interface silicide growth. We systematically investigated the native defects in bulk Pr2O3, PrO2, Pr2Si2O7, PrAlO3, and (to only certain extent) in Pr2Ti2O7 and SrTa2O6. We considered the segregation of atoms from the substrate and/or metal gate (Si, B, Ti, N) into the dielectric film and of metal atoms (Pr, Ti) into Si substrate and into SiO2 interface layer, the dissolution of H2O, and the influence of moisture on the intermixing with Si from the substrate. We formulated a consistent model of positive fixed charge formation which we argue to be valid in high-k oxides as well as in SiO2. We investigated the electrical conductivity mechanism in PrOx. Finally, we also studied the effect of carbon and nitrogen contamination of HfO2, focusing on substitutional and interstitial CN.


We begin with a few words on the theoretical method (Section Approach) and on praseodymium oxides in general (Section Bulk Pr Oxides). After that, we describe in more detail some of the results mentioned above (Section Selected Results).

2. 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.

3. Bulk Pr oxides

Fig. 1.
Fig. 1. Each Pr atom in cubic PrO2 has eight oxygen neighbours

The simplest Pr oxide is PrO2. It crystallizes in the CaF2 structure which can be visualized as the zinc blend structure with anion sites occupied by Ca (or praseodymium) atoms, and the cation and tetrahedral interstitial sites occupied by F (or oxygen) atoms. Thus, every Pr atom is eightfold-coordinated and every O atom is fourfold coordinated. All Pr atoms in PrO2 are Pr(IV) and each has transferred four electrons to oxygen atoms.

Fig. 2.
Fig. 2. Each Pr atom in cubic Pr2O3 has six oxygen neighbours

When a quarter of the oxygen atoms is removed from the crystal, this charge transfer is reduced to three electrons per metal atom and Pr2O3 with Pr(III) is formed. Every Pr atom has now six O neighbors and every O atom retains four Pr neighbors. The atomic structure of remains closely related to that of PrO2: the oxygen vacancies are ordered in a 2×2×2 simple cubic array made of Pr fcc cubes, each fcc cube containing two O vacancies. Such a cubic Pr2O3 is stable under high vacuum. When deposited on Si(001) by electron beam evaporation, it grows with the (110) axis normal to the substrate. Therefore, our calculations are done nearly exclusively for Pr2O3; the dioxide is treated mostly as a reference to calibrate the pseudopotential for tetravalent Pr. This is necessary because one cannot a priori exclude that

Pr(IV) atoms appear in certain structures of the interfacial region. However, up to now we have not identified any structures containing Pr(IV) which are stable under the conditions during growth and annealing of MBE Pr oxide layers.


The calculated lattice constant of Pr2O3 is 11.074 Å, less by 0.3% than the experimental 11.115 Å, and the bulk modulus is 280 GPa. LDA calculations yield 2.0 eV for the offset between bulk Si and bulk Pr2O3 valence band maxima.

Fig. 3.
Fig. 3. Pr(III) in PrOx may have one O neighbour more, Pr(IV) may have one O neighbour less

There are also numerous mixed phases of Pr oxides, containing both Pr(III) and Pr(IV) atoms and having a composition of n(Pr2O3)·m(PrO2). The defected fluorite (vacancy) structure of the oxide is retained in these crystals, hence at least some of the Pr atoms have seven neighbors. Typical composition of praseodymium oxide is Pr6O11, corresponding to an ordered (Pr2O3)·4(PrO2) phase.

4. Selected results

Reduction of the EOT to the target value is a major technological and scientific problem. EOT is increased by oxidation of the substrate during the growth and by the presence of a silicate layer at the interface (Pr silicate has a dielectric constant lower than that of Pr2O3, although it is higher than that of SiO2). To this end, we have addressed several issues. For example, we have:



  • established a low-energy model of the chemically sharp Pr2O3/Si interfaces; 

  • analyzed the Si(001) substrate oxidation in the pre-amorphous regime;

  • approximated the range of O chemical potential where the Pr2O3 film and SiO2 mix;



  • proposed a conceptual model for the interfacial silicate formation.



Another group of problems has to do with the need to reduce the density of charge traps in order to keep the leakage currents and threshold voltage under control. In this context, we have for example:



  • analyzed the role of native point defects in the formation of fixed charges and charge traps; 



  • studied the influence of substrate and selected impurities on charge trapping; 



  • proposed a generic atomistic model for positive fixed charge in dielectric oxides on Si 



  • studied the behavior of Ti at the interface to Si(001); 



  • estimated the misfit defect density at the interface between Pr2O3 and Si(001).

4.1 Chemically sharp Pr2O3/Si(001) and Pr2O3/Si(111) interface

A good matching between lattice spacing of the oxide and of the Si(001) substrate occurs when the (110) axis of the oxide is normal to the substrate, and the (100) axis of the film is parallel to the (110) axis of the substrate. In this configuration, each three Si atoms find a corresponding pair of Pr atoms. Two of these three Si atoms become now dimerized, which leaves the Si surface with four dangling bonds per each 3×1 unit, that is, two Si dangling bonds per each Pr atom on the oxide side. If the oxide had the composition of PrO2, this would mean that there are four oxygen atoms per these two Pr atoms; the position of these O atoms would roughly match the position of the dangling bonds and each of the interfacial Si atoms could be oxidized. However, in Pr2O3 stoichiometry there are only three O atoms available for this purpose, leaving in each 3×1 unit one Si dangling bond without an oxygen partner.


An additional important factor is a charge mismatch between these materials. Bulk Pr2O3 is strongly ionic: one can assume that each Pr atom gives three electrons away and each O atom captures two electrons. But since O atoms at the interface form bonds (predominantly covalent) with Si atoms, they are only partially involved in the charge transfer from metal atoms. As a consequence, each interfacial Pr2O3 moiety donates two electrons which cannot be localized on the existing anions if the oxide is stoichiometric.

Fig. 4. Green oxygen atoms are inserted to satisfy the electron counting rule

We find that in the ideal case of pure Pr2O3/Si(001) film (no silicate) these electrons are trapped by additional O located in the second Pr layer (the structure labeled 0(SiPr) in the energy diagram in the next Section); in the figure, these oxygen atoms are indicated by green shading. The stoichiometry of Pr oxide in the interfacial atomic layer becomes now Pr2, but the Pr atoms in this layer are in the Pr(III) state as in bulk Pr2O3, not in Pr(IV) state as in bulk PrO2. This is because the electrons needed to complete the valence shells of these additional oxygen atoms arrive from Si atoms and not from Pr atoms.

Fig. 5. Stoichiometric interface with silicon "dangling bonds"

In stoichiometric Pr2O3/Si(001), the sesquioxide composition of the interface is maintained by removing some oxygen atoms moves from Si-O-Pr sites. This creates Si-Pr "bonds", which are in this case predominantly ionic, with a negatively charged Si dangling bond stabilized by electrostatic attraction with two Pr+3 neighbors (the structure labeled 1(SiPr) in the energy diagram in the next Section). This dangling bond is not active electrically, because the electrostatic interaction with the positively charged neighbors moves the (0/-) electron transition state into the valence band of silicon. The removal of oxygen from the interface does not change the interface into metallic, because the amount of electrons taken away from Pr atoms to localized states does not change. The only difference is that some of these electrons

go now to Si dangling bonds and not to oxygen atoms. Note that the concentration of oxygen atoms in the first Pr oxide layer remains the same as in PrO2. The interface becomes metallic only when the oxygen is lost from this oxide layer.


The same electron counting rule, which results in oxygen enrichment, is responsible for this oxygen enrichment of the first oxide layer in Pr2O3/Si(001), is also responsible for the behavior of oxygen at Pr2O3/Si(111) interface. Non-metallic character of the interface is maintained by the addition of oxygen to the first atomic layer of the oxide until PrO2 stoichiometry is achieved. As in the case of Pr2O3/Si(001), the Pr atoms in this layer are Pr(III) as in bulk Pr2O3, not Pr(IV) as in bulk PrO2. This is a general rule. No matter if the film is Pr2O3, PrO2, or PrOx, the first layer of the oxide on top of silicon consists of PrIIIO2. Indeed, all valence orbitals of Si which are not saturated by Si atoms are used in covalent bonds with oxygen atoms of the oxide. In order to conserve the charge neutrality of the interface, all oxygen vacancy sites in the interface layer of cubic Pr2O3 have to be filled with oxygen; otherwise, the metal atoms in the interface layer donate electrons to silicon, the interface becomes metallic, and the interface energy increases. So exactly this amount of oxygen as is needed to fill the vacancy sites is needed to keep the interface semiconducting.


There are two possible arrangements of the first atomic layers of oxide with respect to Si(111) substrate. They differ by the stacking order of (111) layers and are named after this order as type-A and type-B interface. For A-type interface, the stacking of Pr oxide layers in the cubic film follows the stacking of Si(111) double layers, while for B-type interface there is a stacking fault at the interface. The stacking fault corresponds to 180° rotation of A-type film around an oxygen atom connected to Si. The essential difference between these configurations is the relative position of metal atoms in the first layer of the oxide and unoxidized Si atoms in the first (111) double layer. In B-type films, the metal and Si atoms are in registry, while in A-type films they are out of registry and the first in-registry Si atom belongs to the second (111) double layer. In B-type films, the metal and Si atoms are in registry, while in A-type films they are out of registry and the first in-registry Si atom belongs to the second (111) double layer.


These two geometries are associated with different strength of the electrostatic interaction between the metal atoms and the electrons in Si-Si bonds at the interface, as well as with the Si core. From ab initio calculations we obtain that type-B interface is energetically preferred over type-A interface, in agreement with experimental observation. The computed energy difference is relatively small (about 40 meV per interface Pr atom), which is reflected in experiment in the presence of a small fraction of A-type domains. During oxidation of the substrate, these domains convert to B-type, which in turn is consistent with the increase of the computed energy difference Si-Si bonds in the substrate are oxidized, and with the interpretation of the energy difference as coming predominantly from electrostatics.

4.2 Substrate oxidation in the pre-amorphous regime

During MBE deposition by electron beam evaporation from a Pr oxide source, the growing film is in contact with vapor consisting of PrO molecules and O atoms. In practice, this means that the system is exposed to an oxidizing environment. Indeed, noticeable Si oxidation takes place unless the substrate temperature is so low that the Pr oxide film grows amorphous. When the temperature is high enough for epitaxial growth (e.g., 500ºC), some oxygen atoms arrive at the substrate and react there.

Fig. 6
Fig. 6. Stability of Pr2O3/Si(001) oxidation states


We have analyzed the stability of Pr2O3/Si(001) interface structures as a function of oxygen chemical potential µ(O). In the diagram, zero of the chemical potential is chosen at the equilibrium with SiO2, while the equilibrium with O2 vapor, ½µ(O2) is far on the right-hand side (the LDA value of the oxygen energy in O2 is 4.9 eV above that in SiO2). The labels associate each structure with its characteristic feature: n(SiPr) means that there are n SiPr interfacial units per each 3×1 unit (for the atomic models of 0(SiPr) and 1(SiPr), see the previous Section), while n(SiOSi) means that there are n such units on the Si side.

Let us now go shortly through this diagram, starting from the oxygen poor (left hand side) and proceeding towards the oxygen rich (right hand side) limit. We focus on the interface structures with the lowest energies, indicated by thick solid lines. The interface labeled 4(SiPr) has four SiPr units in each 3×1 cell, that is, there is no oxygen between interfacial Si and Pr atoms. These interfacial SiPr sites are the first to be oxidized. The energy of oxygen incorporated into such a site depends on the local geometry (dimerized or undimerized Si atom) and on the total number of incorporated oxygen atoms and varies between -1.2 and -0.5 eV. S-Si bonds are oxidized next. Due to geometry constraints, O incorporated into a surface Si-Si bond has energy by about 0.6 eV higher than in SiO2. This can be viewed as the onset of SiOx formation (the energy of oxygen interstitial in bulk Si, that is, of an oxygen atom inserted into a Si-Si bond, is about 1.4 eV in this energy scale). Oxidation of subsurface Si-Si bonds leads to further stress accumulation, Si ejection, and eventually to formation of amorphized SiO2 interfacial layer. Oxidized silicon has a strong tendency to mix with Pr oxide, as discussed in the next Section.

4.3 Formation of the interfacial silicate

We computed total energies for numerous simplified models of interfacial silicates, differing in stoichiometry, atomic arrangement, and imposed lateral periodicity. Most of these structures are clearly unstable, that is, their energies fall well above the lowest energies of silicate-free film at any realistic value of oxygen chemical potential. However, some of the structures turned out to be stable for O chemical potential in the SiOx range. This is in spite of the fact that the small cell sizes used (for practical reasons) in the calculation have most certainly lead to accumulation of lateral stress in the silicate layer.

Fig. 7. Stability of Pr2O3/Si(001) interfacial silicate layers



Bold lines correspond to the structures discussed below. As in the previous energy diagram, the labels associate each structure with its characteristic feature: buck:SiO2 and flat:SiO2 refer to SiO2 molecules dissolved in an ultrathin film under a buckled and a flat surface (see the structural models below), PrOSi indicates the presence of mostly Pr-O-Si bonds in an intercalating SiO2 layer (see the structural model below) and PrOSiSi indicates that Si-Si bonds occur in a thicker intercalating SiO2 layer (see the structural model below). The structures 0(SiPr) and 4(SiOSi) are the same as in the previous Section on substrate oxidation.

Fig. 8. Substitutional SiO2 moiety in Pr2O3 bulk

The onset of silicate formation is the dissolution of a SiO2 molecule in Pr2O3. Two O-2 atoms from the Pr2O3 lattice are then substituted by the (SiO4)-4 moiety. The energy difference between the SiO2 molecule being a member of the network of amorphous SiO2 and the SiO2 molecule incorporated into bulk Pr2O3 is about 1.2 eV. This means that dissolution of SiO2 in Pr2O3 begins to be energetically favorable already at the chemical potential of oxygen about 0.6 eV above the equilibrium between Si, O and SiO2 (i.e., 0.6

eV above the energy zero in the diagram above). This value which falls within the SiOx region (in the energy diagram above, the limit of strained SiOx is indicated on the upper abscissa axis; see also the energy diagram in the previous Section on substrate oxidation).

Fig. 9. SiO2 at the interface

Calculations for a SiO2 molecule dissolved in a Pr2O3/Si(001) film yield a similar result. Silicate formation begins at µ1 = 0.8 eV when the film is made of only three monolayers of the oxide. This is a higher energy than that obtained for the bulk; also in the energy diagram of interface silicates, this structure (labelled buck:SiO2 there) has a high energy. A glance on the atomic structure suggests that the energy may be increased because the surface of this very thin film buckles due to the additional volume introduced by the molecule. This buckling is expected to lead to an increase in the surface energy of the film. We will now verify this hypothesis by making the film smoother.

Fig. 10. SiO2 at the interface

The simplest way to remove the buckling is to deposit additional Pr2O3 to cover the depressions (compare both figures); the necessary amount of oxide is half a monolayer. This smoothing lowers the fomation energy noticeably indeed. The formation energy turns out to drop by the amount equal to ΔE0.5 = E4 - E3.5 = E3.5 - E3, where Ex is the energy of the film with x atomic layers of Pr2O3. This is what one would expect if the reason for increased formation energy were an increase in surface roughness. It is also interesting to note that silicate formation begins now at µ2 = 0.3 eV, that is, at energy even lower than that in bulk material; the dissolved molecule is

more stable in the ultrathin film than in bulk (or in the middle of a thick film). The apparent reason for this is that realaxation of stresses is much easer close to the surface. The surface cannot support any stress in the normal direction, but in the stress field in the bulk has to decay in the crystal lattice. After cancelling out the energy loss due to surface buckling, we have taken the full adventage of stress relaxation at the surface. As a result, we obtain a structure which is stable already at µ(O) only 0.3 eV above amorphous SiO2 (cf. the energy diagram above and the structure labelled flat:SiO2). The price for this is, however, amorphization of the film.

Fig. 11. Single-layer intercalate

Low energy of the interfacial silicate was computed for the case when a monolayer of Si oxide was intercalated above the first Pr2O3 layer. Most of the bonds of the intercalate are of silicate character, and many bonds between silicon atoms are not oxidized. Oxidation of these bonds is not favorable energetically; the resulting stress is too high. This structure is labeled PrOSi in the energy diagram above. Its energy is low enough to allow for the formation of the intercalate from oxidized Si arriving from strained SiOx on the substrate side of the interface, but the stress induced by substrate oxidation must be somewhat higher than needed for insertion of SiO2 in the form of flat:SiO2. Nevertheless, due to the quasi-

homogeneous distribution of silicon in the film, the surface of the film does not buckle. The intercalation process is thus an alternative way to keep the surface of the film smooth during silicate formation; this time, the surface layers of the film remain crystalline.

Fig. 12. Double-layer intercalate

Also when additional two monolayers of Si suboxide are intercalated above the first Pr2O3 layer, there are Si-Si bonds in the silicate. This structure is labeled PrOSiSi in the energy diagram in the beginning of this Section; Above µ(O) = 0.5 eV, this structure is stable with respect to the loss of Si to the substrate and the loss of oxygen to the sites defining the chemical potential (also in this case this would be the SiOx interfacial layer, but again the degree of oxidation-induced stress required for intercalate formation is higher than for previously discussed structures).


To summarize, we have seen how the concentration of Si in the interfacial silicate region may be stabilized in intercalate geometries, that is, when the Si atoms are inserted in layers between Pr2O3 planes. This process makes it possible to transfer the information on the substrate geometry into the film even if silicate is produced during early phase of growth. On the other hand, not all SiSi bonds in the intercalated silicate are oxidized. Oxidation of all these bonds may be difficult without simultaneous excessive oxidation of the substrate: the energy gain is comparable for both processes and the concentration of SiSi bonds is much higher in the silicon than in the silicate. In a separate Section we will argue that incompletely oxidized silicon is a source of positive fixed charge and (in another atomic configuration) it acts as a rechargeable trap.


The intercalated interfacial interface is not stable as a bulk material; for example, the formation energy of bulk Pr2Si2O7  is negative even with respect to fully relaxed SiO2. Nevertheless, a full mixing to an amorphous and relaxed silicate may be kinetically difficult during MBE deposition of Pr2O3 on a Si substrate. A partially ordered silicate existing in the kinetic growth path would explain the recovery of the substrate-determined orientation in the Pr2O3 film grown on top of the apparently amorphous interfacial layer. This recovery is clearly visible in TEM images and in XRD rocking curves.

Fig. 13. SiO2 molecule on Pr2O3/Si(001) surface



But how does SiO2 enter into the growing film? We found that SiO2 moieties are stable on the surface of a single Pr2O3 monolayer if the oxygen potential is in the range of the oxygen energies in the surface Si-Si bonds (the labels on the lines in the energy diagram correspond to the labels on the structural models). These molecules can be overgrown with Pr2O3, leading to silicate formation from the very beginning. Comparison of energies and geometries of other structures computed by us indicates that the overgrowth process may be complicated: when the growth of a Pr2O3 plane is not yet complete and the surface is rough, or when the amount of SiO2 is not high enough to build an intercalating plane without inducing a strong deformation to the capping Pr2O3 plane, then SiO2 moieties tend to segregate to the surface. After the topmost Pr2O3 plane is closed or enough SiO2 is collected, the silica units move under the surface in order to maximize the number of Pr-O-Si bonds.

4.4 The role of native point defects in the formation of charge

As first we consider the classical native point defects in Pr oxides, namely interstitials and vacancies on both sublattices. Since the point defects are created during processing (deposition, annealing), we show the formation energies obtained for the Fermi level aligned with its position in intrinsic Si. Indeed, slightly doped Si (1016cm-3) is nearly intrinsic already for temperatures around 300°C.

Fig. 14. Electron levels of oxide lattice defects


The diagrams show electron transition states of oxygen vacancy and oxygen interstitial in Pr oxides. The black edges of the band diagram of Pr2O3 and PrO2 mark the position of topmost occupied non-f states of the perfect crystal. The dark grey edge in Pr2O3 is the estimated position of the topmost occupied f band, and the light grey edge in PrO2 is the estimated position of the topmost empty f band. The relative position of the defect levels with respect to Si bands are approximate, because they depend not only on the interface dipole but also on the distribution of fixed charge in the oxide. The electric field produced by this charge causes non-linear band bending in the oxide.


Judging from these diagrams, both defects are double charged in Pr2O3, while the charge state of oxygen interstitial in PrO2 is expected to be electrically neutral because it is located above the estimated position of the empty f-band. We should, however, keep in mind that the position of this band is only a rough estimate; in fact, more realistic band structure calculations place these localized f states higher in energy. Moreover, the position of the (0/--) transition state for Oi is shifted to too high energies by LDA, as this approximation over-binds oxygen atoms in the O2 dimer constituting the electrically neutral Oi0 defect (in this charge state, the interstitial oxygen atom is dimerized with one of the lattice oxygen atoms). This problem is restricted to this particular case, in which the change of the charge state results in a dramatic change of the defect configuration (from O monomer to O2 dimer).


In unbiased Pr2O3 films in contact with Si substrate, OV may exist only as OV2+. But when the Fermi energy (or imref for electrons) increases above the conduction band in Si, that is, when negative voltage is applied to the metal gate, OV may trap electrons on a deep state and change to OV+, OV0, or even OV-. In the latter case, the electron trapped on the vacancy is localized on d orbitals of the neighbouring Pr atom. Nevertheless, this may happen only at relatively high voltages applied across the oxide. Indeed, these defects do not introduce deep charge transition levels in the region of the Si band gap nor in the region of about 1 eV above and below. In order to enable a defect-assisted transport of carriers across the oxide, the applied voltage would have to significantly exceed 1 V. At such high voltages and under reasonable concentration of defects, the leakage current is anyway dominated by tunnelling.


Nevertheless, this does not make these defect benign. Namely, they are multiply charged (the same is true PrV, a defect which we will briefly consider further on). Now, Coulomb potential binds free charges more strongly in dielectrics than in semiconductors; this is because the band gaps of these materials are wide, which means that the refraction index (i.e., high-frequency dielectric constant) is lower than that for Si. The binding energy of a carrier trapped by a single charged centre may be tenfold higher than in Si. Since the binding energy increases as the square of the charge Z of the centre, a double (and particularly a triple) charged defect is likely to have "shallow" states which are located close to or even within the energy range of the band gap of Si. High density of these defects may thus make the film leaky due to the presence of quasi-hydrogen like states on multi-charged centres.

Fig. 15. Energetics of defect formation in Pr oxides


We have therefore a good reason to investigate the formation process of native defects. Formation energies of various charged defects in Pr oxides depend on the chemical potential of oxygen as shown in the diagram. In this diagram, the Fermi level corresponds to intrinsic silicon, the band offset between Si and Pr2O3 is taken as 2.2 eV, and 1.2 eV is used for the valence band offset between Si and PrO2. The energy of the intrinsic NcO defect (a valence alternation defect in which the oxidation state of a single Pr atom is by one less than in perfect bulk) is estimated as a half of that of the oxygen vacancy. The SiNcO defect is associated with Si and discussed in a separate Section. The calculated formation energy of G-type Pr2Si2O7 (per Si atom) is also displayed.


Consider, for example, the case of oxygen interstitial, OI-2. Its formation energy decreases with increasing µ(O): it exceeds 1.0 eV for µ(O) corresponding to equilibrium between Pr2O3, oxygen and Pr metal, and becomes negative when µ(O) approaches that of SiO2. Thus, when the Pr2O3 film remains in the contact with any oxygen vapor and when the Fermi level in the film is still determined solely by the band offset to the silicon substrate, oxygen is inserted into the film in the form of negatively charged interstitial atoms, OI-2. In particular, this means that first layers of Pr oxide tend to reduce the silicon substrate, stealing from it the oxygen that may have partially oxidized it before the film growth set on. This effect is at least partially hindered kinetically when the source of oxygen is a well-formed oxide, that is, when removal of an oxygen atom from the source material is associated with the creation of a defect) oxygen vacancy) there. If this vacancy cannot be easily annihilated at the growth temperature, its formation energy must be included in the energy balance.


The concentration of negatively charged oxygen increases until the energy loss due to electrostatic repulsion compensates the energy gain due to the formation of OI-2. A rough estimate gives the upper limit for this concentration in the range of 1013cm-2. High-temperature annealing should thus create a negative fixed charge in films deposited on Si, even if the processing is done in oxygen-poor N2.


In principle, self-compensation of these charges by conversion of some of the Pr atoms to Pr(IV) (i.e., Pr+4) is possible. However, it can take place only when the energy lost due to the electrostatic repulsion exceeds the energy gain due to the capture of electrons from the Fermi level in the semiconductor to oxygen acceptors. (More precisely, the excess electrons captured at the OI-2 acceptor may arrive either locally from Pr(III) (i.e, Pr+3) neighbours or from the outside of the dielectric. In Pr2O3 films grown on Si the latter source is more favorable. As the oxygen content x in PrOx increases towards 1.75, the f shell of Pr(III) experiences more and more repulsion from the electrostatic charge of O atoms and becomes a more and more competitive source of the electrons.) Thus, the film is expected to remain negatively charged even if it is partially oxidized to PrOx with x increased towards 1.75, i.e., even when the film becomes a mixture of the majority component Pr2O3 (with Pr+3 ions) and the minority component PrO2 (with Pr+4 ions). Such a negative charge, typical for our as-grown Pr2O3/Si(001) MBE films, is also consistent with the fact that PrOx crystals are typically p-type when x is below approximately 1.75.

Fig. 16. Energetics of OV and OI formation in Pr2O3 close and far from the Si substrate


The next diagram illustrates the influence of Fermi level on the energetics of charged defects, using oxygen interstitials and vacancies in Pr2O3 as an example. The Fermi level in the left panel is aligned with its position in intrinsic Si. Slightly doped Si (1016cm-3) is nearly intrinsic already for temperatures around 300°C, that is, the left panel corresponds to the situation when a perfect Si is brought into contact with a perfect Pr2O3 at the temperature at which Si is intrinsic. In the right panel, the Fermi level is at the position at which the formation energy of negatively charged oxygen interstitial vanishes on a thermodynamically stable boundary between Pr2O3 and PrO2. (Note that this is an additional approximation, because in reality the transition between Pr2O3 and PrO2 occurs through a complicated series of intermediate oxidation states, and the phase stable at standard conditions is not PrO2, but Pr6O11.)


The data in the left panel show that silicon is not oxidized by Pr2O3 in the reaction in which oxygen vacancies are created in the oxide (or at least that such a reaction does not happen directly): the formation energy of oxygen vacancy in any charge state is strongly positive (strong energy loss) even if the silicon would be oxidized to perfect SiO2. On the other hand, the formation energy of double negatively charged oxygen interstitial, OI2-, is negative, meaning that Pr2O3 may decompose SiO2 film grown on Si into silicon (re-grown at the silicon surface) and negatively charged oxygen interstitials, OI2- (injected into the Pr2O3 film) if the temperature is high enough so that the kinetic barriers on the reaction pathway can be overcome. The inserted interstitials are double acceptors; neutral interstitials are stable only if the Fermi level is very close to the valence band of Pr2O3, and positively charged interstitials are unstable, as typical for this kind of defect in metal oxides. These acceptors build up negative charge, which bends the bands of Pr2O3 upwards, reducing the valence band offset, bringing the Fermi level closer to the valence band of Pr2O3, and making the formation of OI2- less and less favourable. Significant injection of the interstitials continues until the energy of the interstitial becomes positive at the end of the strained SiOx regime. This happens when the bands of Pr2O3 move upwards by approximately 0.5 eV. If a dipole moment of this magnitude would be generated across a 1 nm SiO2 interface layer between a poor SiO2/Si interface and oxygen interstitial atoms located in Pr2O3 close to this interfacial layer, it would require the interfacial defect density of about 1×1013/cm2, a concentration about one order of magnitude higher than the interface defect density directly after thermal oxidation of Si.


At this moment the formation of oxygen vacancies in the upper part of the film (in the reaction in which an O2 molecule desorbs to the ambient vapour) is, naturally, still unfavourable (the panel on the right, high values of µ). But if the oxygen atom released from the lattice site when the vacancy is formed may move to the silicon substrate and oxidize it there, then energy is gained (the panel on the right, µ approaching that of SiO2). This can be easily achieved if an oxygen interstitial close to the interface moves into silicon, its neighbour sitting further from the interface moves into its place, and so on, until the last of the interstitials in the chain is replaced by the oxygen atom released from the site at which the vacancy is created. In this way, oxygen is transported from the surface region of Pr2O3 into the Si substrate until both materials are separated by a SiO2 or SiOx interfacial layer that efficiently separates both materials chemically. The reduction of the surface SiO2 layer which may take place in the initial phase of growth reverts thus eventually into substrate oxidation.


Negatively charged Pr vacancies, PrV3-, are also stable at higher µ(O), already under UHV conditions. In other words, when Pr vacancies are created during deposition, they cannot be annealed out by Post Deposition Annealing (PDA). Their presence may, on the other hand, stabilize the presence of oxygen vacancies, which have the opposite charge state and may also be produced during growth. The computed (ab initio) formation energy of Frenkel pairs in the oxygen sublattice of Pr2O3 is low, only 1.7 eV.


Finally, we briefly turn our attention to other Pr-based dielectrics. As seen in the right-hand side of the energy diagram, the formation energy of oxygen interstitials OI-2 is significantly lower in Pr2O3 than in PrO2, in contrast to that, the formation energy of positively charged oxygen vacancies OV+2 is significantly higher in Pr2O3 than in PrO2. The first effect is due to the fact that oxygen interstitials in Pr2O3 occupy these sites in the lattice from which oxygen is removed when the stoichiometry of the oxide changes from PrO2 to Pr2O3. In PrO2 these low-energy sites are already filled with oxygen and the interstitial atoms have to be placed in less convenient locations where they experience a remarkable compressive stress from the crystalline neighborhood. The second effect (the decrease of the OV+2 energy in PrO2) is apparently associated with the tendency of Pr oxides to form intermediate PrOx phases which are structurally equivalent to PrO2 with an (ordered) array of oxygen vacancies.

Fig. 17. Two distinct OV configurations in Pr2Si2O7


The important difference between native point defects in Pr2O3 and in Pr2Si2O7 is that the oxygen vacancy is not a charged defect in the latter material (apart from the possibility that the vacancy created by taking away an oxygen atom from between two Si atoms is a precursor of the defect family similar to E' in SiO2). This is because O atoms in Pr2Si2O7 have either only Si neighbors or both Si and Pr neighbors. The O vacancy created by removal of an O atom from between two Si atoms (panel on the left) results in a formation of an electrically neutral Si-Si bond. The removal of an O atom from a site where it had, besides Pr neighbors, also a Si neighbo results in the formation of a Si dangling bond: the upper one of the red Si atoms is threefold-coordinated (panel on the right). Although this dangling bond does capture an electron, this electron comes locally from the metal atoms: the negative charge localized now at the Si dangling bond is exactly the same charge that was collected from the metal atoms by the removed O atom. The electron occupies a state located about 1 eV below the valence band of Si (given the computed valence band offset between Si and Pr2Si2O7, amounting to 2.7 eV). This situation is similar to the appearance of SiPr bonds at the incompletely oxidized interface between Si(001) and Pr2O3


Another important difference between Pr2O3 and Pr2Si2O7 is that the formation energy of OI-2 is significantly higher in the silicate and becomes negative only close to the O2 extreme of the chemical potential of oxygen. The reason for this is similar as it was in the case of PrO2: in contrast to Pr2O3, there is no interstitial site in the silicate that would be naturally suited for oxygen to fill.

4.5 The influence of substrate and some impurities on charge trapping

Fig. 18. H2O in Pr2O3 right after its dissociation

Moisture is definitely the factor that has to be examined in the context of fixed charge formation. As other rare earth oxides, Pr2O3 readily absorbs water, to the extent that it is easily converted to a hydroxide. When a water molecule is dissolved in Pr2O3, it dissociates into (OH)I- interstitial (the oxygen atom becomes fourfold coordinated: with one H connected to it, and three nearest Pr neighbors) and H+. The latter becomes attached to a lattice oxygen atom, forming a defect which may be termed a substitutional OH group, (OH)O+; the affected oxygen atom is now fivefold coordinated, with one impurity H atom and four Pr neighbors from the lattice. Both OH

groups are single negatively charged, but the substitutional group replaces in the lattice a double negatively charged oxygen atom, hence it acts as an effective positive charge. These defects do not introduce any localized electron states in the gap of Pr2O3 and, since they have the opposite charge, a dissolved H2O molecule cannot act as a fixed charge.


Nevertheless, it is possible that the charge balance is affected by defect reactions in the film. For example, if the oxygen atom from dissociated H2O is used to oxidize silicon in the substrate to (fully relaxed) SiO2, the (OH)I- interstitial becomes converted to (OH)O+. Two positive fixed charges are thus created by each H2O molecule taking part in such a reaction. We calculated that this oxidation reaction is energetically favorable by about 0.3 eV per H atom in p-type Si. This means that, in principle, one cannot exclude that such processes take place in ultrathin Pr2O3/Si films exposed to moisture. The calculated formation energy of (OH)I- in Pr2O3 is also negative within a broad range of O potential in the oxidizing regime (we assume here that the chemical potential of H corresponds to that in H2O remaining in thermodynamic equilibrium with H, that is, that the sum of 2µ(H) and µ(O) yields the formation energy of water), meaning that water from air would be a source of negative fixed charge in Pr2O3. The formation energy of (OH)O+ is sufficiently small only when the chemical potential of oxygen approaches the UHV range. Although the formation energy of (OH)O+ is negative in equilibrium with SiO2, this does not seem to have a direct relevance to the fixed charge formation. Positively charged defects might be formed in this way if, for example, a water-contaminated oxide is sealed with a Si layer and then annealed.


Boron is the traditional acceptor used in MOSFET channels and in polysilicon gates. It is not a direct source of fixed charge in Pr oxides. Although boron atoms strongly segregate from Si to Pr2O3, they substitute Pr in the lattice. As BPr, B atoms are isovalent impurities. They are electrically neutral and introduce no localized states, at least not in the hazardous energy region within approximately one eV to the band gap of Si. Nevertheless, boron segregation may be responsible for fixed charge generation is by kick-out of Pr interstitials, PrI+3 by B interstitials, BI+. The kick-out is energetically favorable by 0.8 eV when the Fermi level is aligned with that of intrinsic Si. This means that each B atom that makes it to the oxide produces one PrI+3 ion. If the annealing of an uncapped layer takes place in an atmosphere containing enough oxygen to oxidize these ions to Pr2O3, this effect is irrelevant. However, if the annealing takes place under a capping layer (as it is likely to be the case during technological processing), the Pr interstitial atoms may remain unoxidized in the film, producing a fixed charge that is particularly problematic because it may move across the dielectric in the electric field and render the device unreliable by causing a hysteresis in its CV characteristics.


Hazardous positively charged interstitial metal atoms may be also injected from such metal films as titanium. This injection becomes energetically unfavorable (by at least 2 eV) when the Ti source is oxidized to TiO2. Nevertheless, nanosize inclusions of TiO2 in Pr2O3 and in Pr2Si2O7 may trap positive charge. Similarly to the charge trapped on SiPr, this positive charge is re-loadable and can be neutralized by electrons. This means that such inclusions act as Trap Assisted Tunneling or Poole-Frenkel centers, contributing to the leakage current flowing across the dielectric.

4.6 A generic atomistic model for positive fixed charge in dielectric oxides on Si

The origin of intrinsic fixed charges is unclear even in SiO2, although its relation to the excess of silicon has been recognized. We argue that the positive fixed charge in SiO2 comes from a triple-coordinated oxygen atom that is associated with a Si dangling bond arising from an incompletely oxidized Si atom injected into the oxide during the process of thermal oxidation. We adapt a similar model to explain the appearance of fixed charges in Pr2O3 and in transition metal oxides and rare earth metal oxides in general.


To begin with, note that in silicon oxynitride, each Si atom is bonded to four oxygen or nitrogen atoms, each O atom is bonded to two Si atoms, and each N atom is bonded to three Si atoms. This saturates all the valances; the oxynitride is an insulator. Similarly, a nitrogen atom can be incorporated into the amorphous SiO2 network without generating any localized states when it becomes bonded to sp3 orbitals of three Si neighbors. Each Si atom remains fourfold and each O atom remains twofold coordinated, but the impurity N atom is threefold coordinated.

Fig. 19. SiNcO+ (or O3+) configuration in SiO2 resembles that of nitrogen in SiO2


This is the white atom in the figure. One can view this configuration as a Si dangling bond (e.g., of the Si atom painted in pink) saturated by a N atom substituting a nearby O site. Imagine now that this threefold-coordinated, group-V nitrogen atom is now replaced by a group-VI oxygen atom. This oxygen atom has one electron more than needed to saturate the valences of Si, and therefore the additional electron is donated to the conduction band (CB) or bonded only weakly on the defect. Since the CB bottom of SiO2 is 3 eV above the CB bottom of the Si substrate, the electron is then transferred to the substrate and a fixed charge is formed in the SiO2 film. We name this defect a Silicon-related Nitrogen Coordinated Oxygen (SiNcO) defect. Formally, it is a so-called valence alternation defect and is labelled as O3+, with the subscript 3 indicating that the valence of oxygen inreased to 3, and the "+" sign designating the charge of the defect in the standard notation. We use here the name SiNcO as it allows us to give the common name to a family of conceptually similar, positively charged defects based on valence alternated oxygen but with various number of neighbors, as is common for oxygen in high-k oxides.


We have verified that SiNcO is indeed a donor which donates an electron to the Si substrate on which the SiO2 film is grown. Our first attempts to estimate the formation energy of this defect indicate that it is noticeably (be 1.5-2 eV) smaller than the formation energy of a Si dangling bond in SiO2.


We now return to the case of Pr2O3. We computed that the presence of oxygen in the ambient (as during post-deposition annealing) promotes dissolution of Si atoms from the substrate into the Pr2O3 film. In particular, energy is gained when a Pr atom is replaced by a Si atom taken from the substrate and subsequently oxidized to Pr oxide. In an otherwise perfect lattice, such a substitutional SiPr has a dangling Si bond, which introduces electron transition states in the the upper part of the Si band gap. In an amorphous network or at a grain boundary, this dangling bond may arrange itself next to an oxygen atom from the oxide, thus forming a SiNcO similar to that in SiO2. We will now estimate the formation energy of such a defect.

Fig. 20. Two model configurations of SiNcO+ in Pr2O3


Building a model of such a configuration in order to verify the conjunction that such a defect would act as a fixed charge would be a tedious task. Instead, we have placed an interstitial SiH3 molecule in two configurations in crystalline Pr2O3: in the perfect crystal and in a Pr2O3 void created in an otherwise perfect crystal. In both cases, the defect behaves as expected: it delivers a positive fixed charge. However, the formation energies ESi-NcO (with respect to Si-Si bond in disilane, Si2H6) is very different: when computed for the Fermi level corresponding to that of intrinsic Si, it amounts to 2.4 eV in the first configuration, and to -1.2 eV in the second configuration. The difference comes apparently from the high compressive stress in the first configuration: the Si atom is forced to a site close to three Pr atoms (note that in spite of that, a regular bond with the oxygen atom is formed). In contrast to that, the second configuration allows the Si atom to find a place reasonably distant from the Pr neighbors without compromising the Si-O bond length. This leads to a significant lowering of the formation energy in the second configuration. We will treat these two formation energies as the upper and lower bound estimate of the bond energy between the dangling bond of Si and an NcO atom in Pr2O3.


In order to estimate the SiNcO formation energy we still need an estimate of the energy of a regular bond between Si atom and an O atom in Pr2O3. For this purpose, we will use the energy computed in Section on Formation of the interfacial silicate for the SiO2 molecule dissolved in Pr2O3. The formation energy 4ESi-OPr of this defect with respect to SiO2 is about 1.2 eV and corresponds to the formation of four regular Si-O bonds in Pr2O3. Since the SiNcO defect has three such bonds and one SiNcO bond, we are now in a position to estimate its formation energy E:


E = ESi-NcO + 3 ESi-OPr

Fig. 21. Energetics of defect formation in Pr oxides


The result has been already been plotted as a function of the chemical potential of oxygen in the energy diagram of native point defects; here we repeated this diagram again for convenience The two thin red dash-dot line correspond to the upper and lower bounds, while the line labeled SiNcO+ corresponds to their arithmetic average. We see that SiNcO+ is the energetically most favorable defect in the regime of oxygen chemical potentials already slightly (about 0.5 eV) above the equilibrium with SiO2, if the average value is taken as the estimate of its formation energy. Even in the pessimistic case (the upper bound estimate) the formation energy of SiNcO is only slightly higher than the formation energy of the oxygen interstitial Oi-2 when the chemical potential of oxygen approaches the equilibrium between Pr2O3 and PrO2. What is even more significant, both formation energies (the upper limit for SiNcO+ and the energy of Oi-2 are clearly negative in this limit, meaning that these defects are formed spontaneously.


We expect that the SiNcO formation energy is low also in other metal oxides. The reason is that the low energy of SiNcO is caused by the oxidation of the Si atom. Even when the SiNcO formation energy is estimated from quite unfavorable atomic configurations (SiO2 interstitial in Pr2O3 and SiH3 interstitial in the perfect Pr2O3 lattice, the fact that Si-O bonds are formed overweighs the geometrical constraints associated with the particular defect site. Since the presence of metal neighbors to these O atoms affects the strength of this bond only moderately (as proven by the fact that amorphous silicates stable up to several hundred degrees C can be obtained not only with "silifilic" rare-earth oxides but also with "silifobic" materials such as HfO2 and TiO2, the appearance of SiNcO as a major positive fixed charge source in, e.g., HfO2/Si(001), seems plausible. More to the point, HfO2 is purposely grown on Si in amorphous form, and this is the amorphous host that is the natural environment for the SiNcO atomic configuration.

4.7 The behavior of Ti at the interface to Si

We observed that when a Pr silicate film containing a high concentration of rechargeable interface states (e.g., with every atom in ten or in hundred interfacial atoms being a source of such a state) is covered with a Ti layer and subsequently annealed even in a relatively low temperature of 200°C, most of the interfacial states disappear. The mechanism by which titanium passivates the interface states is not immediately obvious.

Fig. 22. TiSi and H-saturated Si dangling bond at the interface between Si(001) and SiO2


Electrically active interface states at the interface between Si(001) and a SiO2 of a good electrical quality are due to Si dangling bonds. How can a Ti atom passivate such a defect? A Si dangling bond has an odd number of electrons, hence a Ti atom cannot passivate it directly as it has an even number of electrons. Indeed, a Ti atom (green) substituting an interfacial Si atom with a dangling bond (n the left-hand side of the model, such a red Si atom, but with the dangling bond saturated by the black H atom, is visible) has similar electrically active states in the gap as a Si dangling bond.

Fig. 23. Electron states of TiSi and of unsaturated Si dangling bond

In the diagram, electron transition states introduced by such a substitutional Ti atom (top) are compared to the states introduced by an unsaturated Si dangling bond (bottom). The pink areas correspond to the valence and conduction bands of the Si substrate.


Therefore, Ti atom cannot passivate the dangling bond by substitution; neither a passivation by attachment (as done by H) works. One can imagine that Ti acts indirectly, through a strain field which induces recombination of dangling bond pairs. Still, this means a significant re-arrangement of the atomic structure, requiring a long-range reconstruction of the oxidized Si and for that reason it may be difficult to realize, given that the passivation occurs already at 200°C.


We propose therefore that the electrically active defects passivated by Ti are "excess" defects (excess in comparison to the typical dangling bond states) associated with atomic-scale variations of Si density at the interface, such as dimers protruding from a locally flat Si(001)/SiO2 interface. Titanium dissolves these defects remotely, first by adsorbing losely bonded interfacial oxygen, and second by expelling silicon from the silicate film and causing regrowth of a more perfect interface.

Fig. 24. Si dimer at the interface between Si(001) and SiO2


Let us consider the interface dimer defect. In the model of this structure, the dimer atoms are the two semi-transparent pink and red atoms. The purple atom of the dimer disturbs the coordination of the purple Si atoms and the greenish O atoms; the properly coordinated Si atoms are red and the properly coordinated O atoms are yellow. We expect such defects at "interstitial" sites in the oxide network or at edges of small Si islands. Due to the over-coordination, electrically active states appear in the gap. This type of compression-induced defects is plausible since the oxidation of Si occurs in a constrained manner, under a silicate film. In contrast to that, the SiO2 film growing on the silicon substrate during a classical thermal oxidation process is stress-free (substantial stress exists only in the immediate vicinity to the interface). Since the volume occupied by a Si atom is about twice as large in SiO2 as in Si, oxidation leads to the appearance of compressive stress that is relaxed not only through changes in the topology of bonds, but also through ejection of Si atoms into the substrate as well as into the oxide, where the ejected atom becomes finally oxidized (if the oxidation is not complete, a positive fixed charge (SiNcO) may be formed). Naturally, the ejection into the oxide happens more easily when the oxide is stress-relaxed and can accommodate the additional compressive stress associated with the presence of the new atom. When the growth of the oxide is mechanically constrained, as is in the case of the growth of the interfacial layer or during low-temperature oxidation of Si, less Si atoms are expected to be injected into the oxide, the Si density at the interface will thus be higher, and compression-type defects should become more abundant. Indeed, it has been observed experimentally that low-temperature oxidation of Si produces interfacial defects which are apparently not due to regular Si dangling bonds.

Fig. 25. Regrown SiO2/Si(001) interface


The presence of titanium helps to reduce the density of such defects. Ti enters the silicate from the metallic Ti film, drains O atoms from the defected interfacial sites (Ti oxidizes easily) and expels Si atoms from the silicate (Pr silicate and Ti oxide do not mix easily). This results in Si regrowth in previously compressed areas; in the ideal case, the electrically active states are removed completely, as indicated in the figure. Semi-transparent atoms in this figure are now the original atoms of the Si dimer and the Si atoms expelled from the silicate. No long-range reconstruction of the oxide is needed for this process to take place. The only requirement is that the temperature is high enough for Si and O transport between the Ti-containing silicate and the interface.

Fig. 26. Interstitial Ti below SiO2/Si(001) interface


Finally, we find that when the Ti atom comes so close to the interface with Si that it is able to substitute the Si atom in the dangling bond site, then it will not stop there, but it will advance into the substrate, where it acquires an interstitial position. The Ti atom (green) does not stop at the dangling-bond site but advances into the substrate. The resulting sub-surface complex consists of the Ti interstitial atom and the Si atom with the dangling bond. (Note that the model is viewed from the direction rotated by 90° with respect to those of the previous images.) A seed of a metallic Ti silicide inclusion is formed in this way.

Fig. 27. Stability of substitutional and interstitial Ti at the interface


This formation of Ti silicate seeds is energetically favorable for all Fermi levels within the Si band gap. Such inclusions are highly undesirable as they would electrically damage the transistor.

4.8 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).

Fig. 28. 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 Juelich. 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.



Das Gebäude und die Infrastruktur des IHP wurden finanziert vom Europäischen Fonds für regionale Entwicklung, von der Bundesregierung und vom Land Brandenburg.