IHP - Innovations for High Performance Microelectronics: Theory-8

3.5. 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.



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 O_i is shifted to too high energies by LDA, as this approximation over-binds oxygen atoms in the O₂ dimer constituting the electrically neutral O_i⁰ 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, O_V may exist only as O_V²⁺. 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 O_V⁺, O_V⁰, or even O_V^-. 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 Pr_V, 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.



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, O_I^-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, O_I^-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 O_I^-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 O_I^-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 PrO_x 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 PrO_x 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 PrO_x crystals are typically p-type when x is below approximately 1.75.



Energetics of O_V and O_I 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, O_I^2-, 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, O_I^2- (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 O_I^2- less and less favourable. Significant injection of the interstitials continues until the energy of the interstitial becomes positive at the end of the strained SiO_x 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 SiO_x 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, Pr_V^3-, 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 O_I^-2 is significantly lower in Pr₂O₃ than in PrO₂, in contrast to that, the formation energy of positively charged oxygen vacancies O_V⁺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 O_V⁺² energy in PrO₂) is apparently associated with the tendency of Pr oxides to form intermediate PrO_x phases which are structurally equivalent to PrO₂ with an (ordered) array of oxygen vacancies.



Two distinct O_V 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 O_I^-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.

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