Energy levels in Si and SrTiO3-based quantum wells with charge image effects

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

ABSTRACT

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

I. INTRODUCTION

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 [1]. 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 [2], 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. [5] 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 SiO_x with x < 2. Chambers et al. [6] 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. [7] show that the TiO₂ can not be grown directly on Si because of thermodynamic instabilities, leading to TiSi_x 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 [8].

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 V_T(z) given by

where V₀(z) is the potential energy due to the conduction (or valence) band offset and V_im (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. [8]). In this case, we can write f() in the general form

where J₀ (qR) is the Bessel function of the zeroth order, A_q (z)is a function determined by the boundary conditions of f() at the interfaces. The solution for the image potential V_im (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 [9].

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 V_T(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. [7], 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. [13]. 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.

Acknowledgement

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

Publication Dates

• Publication in this collection
  06 July 2006
• Date of issue
  June 2006

History

• Received
  04 Apr 2005