Print version ISSN 0103-9733
Braz. J. Phys. vol.36 no.2a São Paulo June 2006
HETEROSTRUCTURES, QUANTUM WELLS AND SUPERLATTICES
Energy levels in Si and SrTiO3-based quantum wells with charge image effects
T. A. S. Pereira; M. G. Bezerra; J. A. K. Freire; V. N. Freire; G. A. Farias
Departamento de Física, Universidade Federal do Ceará, Campus do Pici, Caixa Postal 6030, 60455-900 Fortaleza-CE, Brazil
In the present work we develop a theoretical study to analyze how the image charges effects can modify the electronic properties in Si and SrTiO3-based quantum wells. We have used the method based on the calculation of the image charge potential by solving Poisson equation in cylindrical coordinates. The numerical results show that the electron-heavy hole recombination energy can be shifted by more than 200 meV due to the combination of charge image and SiO2 (SrTiO3) interface thickness effects.
Keywords: Energy Levels; Si; SrTiO3-based quantum wells
The SiO2 gate thickness shrinkage to less than 50 Å is driving research efforts to find alternative oxides with high-k dielectric constant to allow for physically thicker films that can limit leakage current problems . Recent research shows that SrTiO3, HfO2 and TiO2 materials are the most promising candidates to SiO2 replacement in development of new semiconductors devices. With the advances in the growth of SrTiO3 on silicon by epitaxy , the development of quantum confinement based optical devices is possible, following the steps of an original suggestion and demonstration of light emission in Si/SiO2 quantum wells (QWs) [3, 4], by simple replacement of silicon dioxide by SrTiO3. The understanding of the atomic structure at the silicon/oxide interface is still not complete. However, the ability to grow gate quality crystalline oxide films on Si in industrial scale is yet to be demonstrated. Droopad et al.  observed that the possible explanation for the formation of such an amorphous layer includes the diffusion of oxygen during the growth of the oxide layer reacting with the interfacial Si atoms. This would suggest that the interface layer is some form of SiOx with x < 2. Chambers et al.  have shown that the interest in including an interfacial layer of SiO2 suggests significant progress in order to reduce the leakage current, either because SiO2 acts as an electron tunnel barrier or because SiO2 at the interface increases the conduction band offset between SrTiO3 and Si. Tuan et al.  show that the TiO2 can not be grown directly on Si because of thermodynamic instabilities, leading to TiSix and SiO2 at the interface. However, TiO2 can be grown as an epitaxial film on SrTiO3 (001) that can be grown epitaxially on Si(001) with negligible interface reaction.
In this work, we present results on electron-hole energy recombination from confined states in abrupt Si/SiO2/SrTiO3 and Si/SrTiO3/TiO2 QWs. The aim is to search for high-k dielectric based light emission devices, which can be important in developing silicon-based technology for future nano-optoelectronic device integration. The dielectric mismatches among the materials of QWs are included through the conventional image potential for a point charge Q near an interface .
II. THEORETICAL MODEL
The carrier Hamiltonian in Si and SrTiO3-based quantum wells can be written as
with i = electron or hole and the total effective potential VT(z) given by
where V0(z) is the potential energy due to the conduction (or valence) band offset and Vim (z) is the image charge potential contribution, given by the Poisson's equation due to the presence of a point charge
The solution in the cylindrical coordinates is independent of the azimuth angle j (see detail in Ref. ). In this case, we can write f() in the general form
where J0 (qR) is the Bessel function of the zeroth order, Aq (z)is a function determined by the boundary conditions of f() at the interfaces. The solution for the image potential Vim (z) is
where (z0 ) is solution of (3) if the dielectric constant were z independent.
Using the above considerations, we solved the Schrödinger equation for the perpendicular motion:
and the eigenvalues and eigenfunctions of this equation are calculated through a matrix transfer scheme .
III. NUMERICAL RESULTS
The heterostructures used in our simulations have been a TiO2/SrTiO3/Si and SrTiO3/SiO2/Si QW. All the parameters used for SrTiO3, SiO2, TiO2 and Si are presented in Table I.
In the present calculation two effects are considered, the charge image and the interface thickness. Fig. 1 shows the confinement potential VT(z) and the wave functions of TiO2/SrTiO3/Si (left) and SrTiO3/SiO2/Si (right) heterostructures with quantum well width of 5.0 nm and SiO2 (SrTiO3) layers with thickness of 1.0 nm.
Considering TiO2/SrTiO3/Si system, and based on the work of Tuan et al. , the conduction band offset of 0.1 eV and 0.2 eV between Si and SrTiO3 and between Si and TiO2, respectively were used - see Fig. 1 (left). In this structure the attractive character of the image charges in the well region introduce a sharp and deep potential profile which can trap carriers close to the SrTiO3/Si interfaces. This fact can be seen looking through the top of electron wave function evolution in Fig. 2 (left).
For SrTiO3/SiO2/Si systems electron confinement was not observed if we consider a band offset of 0.1 eV between SrTiO3 and Si. In this structure we have used a band offset of 0.9 eV, as proposed by Zang et al. . Thus, as shown in Fig. 1 (right), there are two confined states for electrons and four confined states for heavy hole in a narrow QW with width of 5.0 nm. We also observed that in this system the SiO2/Si interface does not permit interfacial confinement due to the repulsive character of the image charge in the well region - see Fig. 2 (right).
Figure 3 shows the results for electron-heavy hole recombination energy in TiO2/SrTiO3/Si QW (top) and SrTiO3/SiO2/Si QW (bottom), as function of the QW width for SrTiO3 (SiO2) layer with thickness of 1.0 nm and 1.5 nm. Solid and dashed lines represent carrier recombination energy including the effects due to image charges.
Dashed dotted and dotted lines represent carrier recombination energy without image charges effects. The attractive character of the image potential in the well region of TiO2/SrTiO3/Si QW, Fig. 3 (top), decreases the recombination energy (solid and dashed lines), when compared with carrier recombination energy in QW without image charge effect (dashed dotted and dotted lines). The repulsive image potential in the well region of SrTiO3/SiO2/Si QW, Fig. 3 (bottom), increases the recombination energy. The SrTiO3 (SiO2) layer is expressive because it strongly affects the recombination energy of the QWs. Fig. 3 also shows that increasing the thickness of the SrTiO3 (SiO2) layer, the recombination energy blue-shift in TiO2/SrTiO3/Si by about 40 meV and in SrTiO3/SiO2/Si by about 800 meV, respectively, for a well width of 4.0 nm.
In conclusion, we have studied image charges effects in TiO2/SrTiO3/Si and SrTiO3/SiO2/Si QWs. The image charges related potential structure (sharp and deep) can trap electrons close to the SrTiO3/Si interfaces. In this system we also observed that the attractive character of the image potential inside of the well region decreases the recombination energy. For SrTiO3/SiO2/Si QW we observed electron confinement only for a 0.9 eV conduction band offset between the SrTiO3 and Si layers. This structure does not have a sharp and deep potential profile due to the repulsive character of the image potential that increases the recombination energy, consequently the carrier do not presents an interfacial confinement.
T. A. S. Pereira and M. G. Bezerra were supported by Brazilian National Research Council (CNPq). The authors also would like to acknowledge the NanoSemiMat #550.015/01-9.
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Received on 4 April, 2005