Journal of Metastable and Nanocrystalline Materials Vol. 23 (2005) pp 129-132 online at http://www.scientific.net © (2005) Trans Tech Publications, Switzerland
Optical and Electron Correlation Effects in Silicon Quantum Dots S.K.Ghoshal1, K.P. Jain2, and R. J. Elliott3, 1
Department of Applied Physics, Guru Jambheshwar University, Hisar, Haryana-125001, India.
[email protected] 2
Physics Department, Indian Institute of Technology, New Delhi-110067, India.
3
Department of Theoretical Physics, 1 Keble Road, Oxford OX13NP, England, UK.
Keywords: Nanostructure, quantum dots, photoluminescence, pseudo-potential, confinement, radiative recombination, passivation and HOMO-LUMO states.
Abstract. We study (through computer simulation) the variation of the band gap as a function of sizes and shapes of small Silicon (Si) dots using pseudo-potential approach. We have used empirical pseudo-potential Hamiltonian and a plane wave basis expansion and a basic tetrahedral structure. It is found that the gap decreases for increasing dot size. Furthermore, the band gap increases as much as 0.13eV on passivation the surface of the dot with hydrogen. So both quantum confinement and surface passivation determine the optical and electronic properties of Si quantum dots. Visible luminescence is probably due to radiative recombination of electrons and holes in the quantum confined nanostructures. The effect of passivation of the surface dangling bonds by hydrogen atoms and the role of surface states on the gap energy as well as on the HOMO-LUMO states has also been examined. We have investigated the entire energy spectrum starting from the very low lying ground state to the very high lying excited states for silicon dots having 5, 18, 17 and 18 atoms. The results for the size dependence of the HOMO-LUMO gap and the wave functions for the bonding-antibonding states are presented and the importance of the confinement and the role of hydrogen passivation on the confinement are also discussed. Introduction The discovery of luminescence from silicon nanostructures and porous silicon has attracted much attention in recent years [1-5]. It has been surmised that these nanostructures of silicon have a direct band gap and emit light from violet to red depending upon the size of nanostructures. With a large surface-to-volume ratio, the surface effects become more enhanced therefore, geometry plays an important role. Surface effects as well as the quantum confinement effects control the optical and electronic properties of these materials. It is possible that the luminescence properties may be related to different silicon compounds such as a – Si:H, polysilane, SiHx and surface defect states [3-6]. Semiconductor quantum dots have attracted intense theoretical and experimental investigation over last one decade [2, 3]. The importance of such an investigation stems from the fact that the modeling of novel materials requires a fundamental understanding of the electronic structure including the role played by surfaces having different geometries, disorder, in-homogeneity and so on. However, in spite of intensive experimental and theoretical studies, no conclusive argument has been given on the mechanism of the efficient light emission from Si nanostructures [7, 8]. In this paper we investigate the effect of quantum confinement on the energy gap in silicon dots of different sizes and geometry and the role of hydrogen passivation for this system. The dependence of energy gaps on the size and geometry of the silicon dots is examined in detail. Surface passivation of the dot with hydrogen atoms is modeled through appropriate hydrogen pseudo-potential [5].
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Methodology Normally a large number of plane-wave basis states are required to adequately describe the electronic wave functions of the silicon dot. Instead we use the empirical pseudo-potential approximation, which allows the wave functions to be expanded using a much smaller number of plane-wave basis states. We know that most of the physical properties of solids are dependent on the valence electrons to a much greater extent than on the core electrons. The pseudo-potential approximation exploits this by removing the core electrons and by replacing them and the strong ionic potential by a weaker pseudo-potential that acts on a set of pseudo wave functions rather than the true valence wave functions [1, 5]. We therefore use the empirical pseudo-potential method (EPM) for the electronic structure of silicon quantum dots of different sizes. Following Wang and Zunger an appropriate form of the local empirical pseudo-potential for silicon as well as for hydrogen is used. The total screened potential in the EPM is approximated by the superposition of atomic pseudo-potentials. The dot wave functions are expanded in a large basis of plane waves with a bond length a = 5.43A° and two different values of cutoff energies (5 Ry and 10 Ry) is used for the plane wave expansion and the same cutoff is used for the potential energy too. Our interest here is to find the size dependence of the gap energy and the near band gap solutions i.e. the separation between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). Now we solve the effective single-particle Schrödinger equation for different sizes of the quantum dot. Results A five atom dot with tetrahedral arrangements joined by a single bond without and with hydrogen atoms at the surface gave an energy gap ~ 1.997 eV and ~ 2.07 eV respectively. In each case we kept the fixed bond length a = 5.43 A° between two silicon atoms. As we passivate the surface of 5-atom dot with hydrogen i.e. Si5H12, the occupied number of level changes and the emerged value of the gap also show a change. Following a similar procedure, we examine the 8, 17 and 18 atom dot without and with hydrogen atoms at the surface. We found an energy gap ~ 1.82 eV and ~ 1.89 eV for 8 atom, ~1.79 eV and ~ 1.85 eV for 17 atom, and ~ 1.693 eV and ~ 1.824 eV for 18 atom dot without and with hydrogen respectively. In each case we kept the fixed cubic cell size a = 3.2 A° between two silicon atoms. The value of the gap energy as observed from the energy spectrum decreases as the size of the dot increases implies weaker confinement for larger sizes. As we increase the number of atoms in the dot and change its geometry, the degeneracies are increasingly lifted. This is because the grouping-up tendency of the energy levels in a single energy value becomes less (effect of deconfinement). We see that for smaller sizes the confinement is more and the degeneracies are also higher. As one saturates the surface dangling bonds with hydrogen atoms, there is an enhancement of the gap energy ~ 0.07 eV for 5 atom cluster and ~ 0.13 eV for 18 atom dot. The degeneracies are also lifted because the presence of hydrogen at the surface imparts an interaction potential that acts as a perturbation. For relatively larger dot due to sp2 and sp3 hybridization de-localization of the electron is more and the energy spectrum is more non-degenerate. For smaller dots this effect of delocalization near the surface is less and hence the degeneracies are more pronounced. We present the results of our simulation for the variation of gap energy for different sizes of the dot in Fig. 1 without hydrogen-passivation (diamond symbol) and with hydrogen-passivation (filled square). We find the gap energy increases as the size of the dot decreases that confirms the stronger confinement for the smaller sizes. Higher gap with passivation is due to the localization of electron cloud arising from the sp3 hybridization of the silicon-dangling bond with the hydrogen atom. These results are in conformity with the earlier observation [5-9]. Finally, in Fig. 2 we
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compare our results (plus and solid circles) for the gap energy as a function of the dot size with other calculations. Here we have calculated the effective size of the dot under linear mapping d = Na (A°) to get a scaling of our data points. Ours : Without Hy drogen (eV )
Ours : With Hy drogen (eV )
Gap Ene r g y (e V )
2.5 2 1.5 1 0.5 0 0
5
10 Nu m b e r of Ato m s
15
20
Figure 1. Plot of the HOMO-LUMO gap energy (in eV) versus the number of atoms in different dots. Solid line (diamond): for the free surface and the solid line (square): when the surface dangling bonds are passivated with hydrogen atoms. EMA ( e V )
RK F ( e V )
W Z (eV )
O u r s : W ith o ut Hy d r o g e n ( e V )
O u r s : W ith Hy d r o g e n ( e V )
5
Energy Gap (eV)
4 .5 4 3 .5 3 2 .5 2 1 .5 0
10
20 30 40 Ef f e c t iv e S iz e d ( A n g s t r o m )
50
60
Figure 2. LUMO-HOMO band gap versus the effective sizes from Wang and Zunger (triangles), multi band effective mass result (stars), the result of the method of Rama Krishna and Friesner (square boxes) and our result without hydrogen (plus) and with hydrogen (solid circles).
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Next, we turn our attention to the wave function both for the low-lying states as well as the pair of states near the band gap (HOMO and LUMO states) for all different sizes. We calculate the ground state as well as excited state wave function for all dots using periodic boundary conditions. The real part of the ground state wave function has very strong oscillation near the core region, is highly asymmetric and then decays to a very small constant value beyond a critical distance equal to the size of the box. The imaginary part of the wave function oscillates and then rapidly goes to zero. We have calculated the wave function for the HOMO and LUMO states as a function of r and both the bonding and the anti-bonding states show strong oscillation as before near the core region and are highly asymmetric. The wave function for the HOMO and LUMO states for 8 atoms dot with hydrogen at the surface behaves in a different manner. Here, for 8 atom cluster nature of oscillation of the wave functions are different from that of the 5 atom cluster and clearly reflects the effect of geometry of the structure. This result is expected because the bonding and the antibonding states are having different symmetry and the effect of passivation is also different for the two cases. The surface states contribute differently for the HOMO and LUMO states. Here, the nature of oscillation of the wave functions are also different from that of the 5 and 8 atom cluster and clearly reflects the effect of geometry of the structure as we mentioned earlier. Our results show that the HOMO-LUMO states are localized and bulk-like instead of surface-like [9-11]. Summary We have used the empirical pseudo-potential method to calculate electronic structure of silicon quantum dots of different sizes with 5, 8, 17 and 18 atoms to examine the quantum size effects on gap energy. In the present treatment we neglect the role of multi-band coupling. Our results show that the gap energy increases as the size of the dot decreases implies the stronger confinement for smaller dots. This result is in conformity with the earlier observation of Wang and Zunger. The most interesting and remarkable result we find for the HOMO-LUMO gap energy as we vary the size of the dot. We have observed that the change in the energy gap on passivation as a function of the dot size is more prominent for larger dots. The presence of hydrogen lifts the degeneracies of the eigenvalue spectrum and results in an enhancement of the gap energies. The basic characteristics of the quantum dot wave functions pattern indicate that in all cases, the HOMO and LUMO states are localized in the interior of the quantum dot with zero amplitude on the surfaces. Recently Wolkin et al found that the PL characteristics are determined by both quantum confinement and surface passivation, which determined by the electronic states of the silicon quantum dots [11]. The above study indicates that the presence of hydrogen at the surface contributes surface states near the gap to enhance the gap energy and hence the confinement which is in conformity with the experimental results. The passivated surface states play an important role as far as the photoluminescence from finite size dots are concerned. References [1] L.W. Wang and A.Zunger: J. Phys. Chem., 98, (1994), p. 2158. [2] I. Vasiliev, S. Ogutt and J.R. Chelikowsky: Phys. Rev. Lett., 86, (2001), p. 1813. [3] R. E. Millar and V.B. Shenoy: Nanotechnology 11, (2000), p. 139. [4] T. Takagahara and K. Takeda: Phys. Rev. B., 53, (1996), p. R4205. [5] L.W. Wang and A. Zunger: J. Phys. Chem, 98, (1994), p. 2158. [6] Y. Kanemitsu: Physics Reports, 263, 1 (1995) and references therein. [7] S.B. Zhang, C.Y. Yeh and A. Zunger: Phys. Rev. B., 48, (1993), p. 11204. [8] X. D. Zhu: Appl. Phys. Lett., 74, (1999), p. 525. [9] T. Takagahara: Phys. Rev. B., 47, (1993), p. 4569. [10] F. Buda, J. Kohanoff and M. Parrinello: Phys. Rev. Lett., 69, (1992), p. 1272. [11] M.V. Wolkin, J. Jorne, P.M. Fauchet and C. Delrue: Phys. Rev. Lett., 82, (1999), p. 197.
