Quantum dot modeling of semiconductors

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4 Research Signpost 37/661 (2), Fort P.O., Trivandrum , Kerala, India Advances in Condensed Matter Physics, 2009: ISBN: Editor: Ali Hussain Reshak 3 Quantum dot modeling of semiconductors Y. Al-Douri School of Microelectronic Engineering University Malaysia, Perlis (UniMAP) Block A, Kompleks Pusat Pengajian Arau Jejawi, Perlis, Malaysia Abstract The indirect energy gaps (Γ-X) are calculated using Empirical Pseudopotential Method (EPM) for semiconductors compounds. The obtained band gaps of semiconductors that form the Quantum Dot (QD) of Single Electron transistor (SET) are used to calculate the quantum dot potential within our recent model. This study has predicted materials for quantum dot. The external effects are used to test the validity of this model. Also, the energy gap and the quantum dot potential as a function of alloy concentration are studied for Ga x A l-x As and Zn x Cd 1-x Se alloys. The results are in reasonable agreement with Udipi et al. results. Correspondence/Reprint request: Dr. Y. Al-Douri, School of Microelectronic Engineering University Malaysia, Perlis (UniMAP) Block A, Kompleks Pusat Pengajian Arau Jejawi, Perlis, Malaysia

5 2 Y. Al-Douri 1. Introduction Quantum dots represent the ultimate reduction in the dimensionality of a semiconductor device. While there have been several observations of singleelectron behavior in GaAs heterostructures, efforts in Si-based devices have been limited to either lithographically defined dots [1,2] or using selfassembly techniques [3 5]. Many of experimental and theoretical works have concentrated on quantum dots. Milicic et al. [6] have developed a novel technique for the fabrication of Si quantum dot structures in which a controllable number of electrons in the dot is achieved through appropriate biasing of the top and side (depletion) gates. They presented simulation results obtained with 3D Schrödinger Poisson solver that show clear correlation between the conductance peaks with the energy level spectrum in the dots. Quantum dots are characterized by a relatively small number of free electrons (normally electrons in the conduction band of a semiconductor with an effective mass m*) confined within an island with submicrometer dimension. Udipi et al. [7] presented semiclassical simulation results for the potential energy profile and electron density distribution in a 200 nm silicon quantum dot. For the solution of the continuity equation, the efficient difference approximations, proposed by Scharfetter and Gummel [8] extended to three dimensions. In essence, they followed the two-dimensional approach due to Selberherr et al. [9] to extend two to three dimensions. Pakes et al. [10] described a series of numerical simulations using Technology Computer Aided Design (TCAD) that allow to determine, in an appropriate architecture, the capacitive coupling of a subsurface electronic charge to the Single Electron Transistor (SET) structure. They examined the variation in offset charge induced by subsurface charge motion over displacements typical of the inter-qubit spacing, to identify whether the induced charge is detectable within the current limits of fabricated SET devices. The Kane [11] proposal for a solid-state NMR quantum computer presented a potentially scalable model for a quantum computer based on materials established in conventional silicon technology. It is based on the use of 31 P nuclear spins as qubits, with donor electrons functioning to mediate the control of single-qubit operations [12] and of the interaction between adjacent qubits [13] and permit readout of the nuclear spin states. The Kane proposed device has a MOS-type structure with single 31 P donor atoms buried at a depth of order 20 nm, within a 28 Si substrate; a multi-qubit device consists of a regular array of dopants with separation approximately 20 nm.

6 Short Title 3 The main advantages of pseudopotential calculations are: (1) The computations are usually restricted to dealing with valence electrons. The core electrons are not analyzed and are projected out of the subspace considered. As a consequence, properties associated with the valence electrons, e.g. optical spectra, chemical bonding, etc., may be analyzed directly with no complications arising from core state computations. (2) The potentials are generally much weaker than atomic potentials allowing perturbative type approximations and simple basis sets, i.e., for most calculations plane waves are used for expanding the valence wavefunctions. (3) The pseudopotentials themselves are transferable. In many situations it may associate a specific potential with an atom or ion and it is found that the potential does not change with environment [14]. Wavefunctions obtained using pseudopotential are accurate in regions outside the core. This usually causes no inconvenience since it is the valence electrons which are of interest and the pseudowavefunctions give excellent charge densities which reveal the chemical bonding nature of the crystals involved. The wavefunctions can also be used for evaluating transition matrix elements for optical processes and these results are satisfactory for most purposes [14]. So band gaps, pressure dependence of electronic structure, optical constants, photoemission spectra, dielectric functions depending on frequency and wavevector, alloy properties, bonding and chemical properties, etc. can in principle be and have been evaluated for many materials using the pseudopotential approach. In this work, a new empirical model for the calculation of the quantum dot potential of a specific class of materials is established, based on the calculation of the indirect band gap (Γ X), (to best of our knowledge, empirical model has not been reported previously). Only the pseudopotential form factors are required as input, the computation of potential itself is trivial and the accuracy of the results reaches that of other calculations. This option is attractive where it allows the consideration of hypothetical structure and the simulation of experimental condition that are difficult to achieve in the laboratory, e.g. temperature or pressure effect. The detailed calculations of pseudopotential techniques are in the following section. Third section discusses the obtained results and the conclusion is in the last one.

7 4 Y. Al-Douri 2. Techniques The pseudopotential itself evolved for fifty years ago. The Phillips cancellation theorem [15] showed that valence electrons experienced a repulsive potential when they were close to the ion core. This potential arises because the valence electron wavefunctions are constrained to be orthogonal to the core states. This orthogonality can be represented as a repulsive potential which acts to keep valence electrons out of the core. The repulsive potential nearly cancels the strong attractive ionic potential leaving a net weakly attractive pseudopotential. In principle, some electronic states do not exist in the core and hence no repulsive potential can be assumed. For example in Carbon, the atomic configuration is 1s 2 2s 2 2p 6 where 1s 2 represents the core. The 2s electrons will see a repulsive potential because of the 1s 2 electrons in the core, but the 2p electrons will not. Hence pseudopotentials should depend on angular momentum, l, and these pseudopotentials are called nonlocal pseudopotential. Let ψ ki be the exact (valence) electron wave for the i th irreducible representation of the wave vector k and let φ kj be a wave function of a core state with the same wave vector (say a Bloch function). Since the valence function ψ ki must be orthogonal to the core stateφ kj, it is convenient to write: ψ ki υ a i = + ki kjφ kj (1) where a i kj * 3 ( ) = ( r) ( r) r φ φ ki υ υ kj ki kj d = (2) The orthogonality of ψ ki to the core state is insured by Eq. (2). It is expected that υ ki to be a smooth function even near a nucleus since the rapid variation of ψ ki in that region really due to the requirements of orthogonality. To determine the equation satisfied byυ ki : υ ki j a ( ) kj ki φ kiυ ki E E i H + = kj kj E (3) pseudopotential V p is now defined by V p = j a i kj φ kj ( ) υ E kj E ki ki (4)

8 Short Title 5 Then Eq. (3) takes the form: ( H ) υ = ki kiυ ki V p E + (5) The pseudopotential Hamiltonian used for this calculation is given by 2 H = + V ( r) 2m (6) where V(r) is the pseudopotential that can be expanded in reciprocal lattice vectors G. For the zinc-blende structure, this yields [16] ( S A ) ( V cos V sin ) e ig τ V r = G τ + i G τ, (7) G G G V S G and V A G are symmetric and antisymmetric pseudopotential form factors of endpoint binary compound and can be written in terms of the atomic potentials as V V S G A G = = [ V ( G) + ( G) ], 1 V [ ( G) ( G) ] V V (8) The form factors in Eq. (8) are adjusted empirically by fitting the calculated band structure to the experimental data. They depend on the magnitudes of G. As in most of the EPM calculations, cutoff value of 2 2 G = 11(2π / a) is used. The discrete form factors are taken from Cohen and Bergstresser [14]. The discrete form factors of the endpoint binary compounds are given in table 1 together with the lattice constant a. The EPM involves the fitting of the V(G)'s of Eq. (7) and (8) to experiment. The form factors are estimated or calculated and the measured structure factor is used to compute the total potential. After solution of Schrödinger equation for the eigenfunctions and eigenvalues is obtained, these are used to calculate the reflectivity, R, or modulated reflectivity, R'/R, or the density of states, etc. and comparison of these functions is made with experiment. Agreement between theory and experiment is then used as the primary criterion for determining the next set of form factors. After a few tries good agreement is usually obtained. The comparison between theory and

9 6 Y. Al-Douri experiment usually leads to correspondence of ~ 0.1 ev in the optical constants over a range in energy of about 1 Ry. Because of crystal symmetry and the weakness of the pseudopotential only a few V(G)'s are required for each band structure. The first non-vanishing form factors occur for (a/2π)g 2 = 3,8,11 in the diamond structure, and the corresponding V(G)'s yield a band structure which is accurate to a few percent over a Rydberg range. Generally three numbers (i.e. the form factors) are sufficient to determine each pseudopotential. It is impressive that the EPM works so well considering the limited input. It is remarked that not only this approach yield accurate calculated properties of the involved semiconductors, but the potentials for the elements making up the semiconducting can be used to study the properties of these composite elements. 3. Results and discussion 3.1 The quantum dot potential Single-electron-Transistor (SET) is a device that exploits the tunneling quantum effect to control and measure the movement of single electrons. Basically, it consists of an island or dot separated from source and drain electrodes by two tunnel junctions T 1 and T 2, through which electrons; e can tunnel (Fig. 1). Since, the goal of this study is to understand how qualitative concepts, such as quantum dot potential, can be related to microscopic properties as band gap calculation. This calculation is based upon pseudopotential form factors which are adjusted by a non-linear least-square method which have been already reported in the literature [17-19]. Figure 1. Basic SET structure and the equivalent circuit.

10 Short Title 7 The calculated principal band gaps are given in table 2, a good agreement is obtained in comparison to experimental data obtained by optical, ultra photoemission spectroscopy (UPS) and X-ray photoemission spectroscopy (XPS) data were also used [16,19-22] Quantum dot potential (x10-3 V) Ge Si GaAs AlSb GaP InP AlAs GaSb InAs InSb ZnS CdSe ZnSe CdTe ZnTe MgSe CdS Energy gap Γ-X (ev) Figure 2. The quantum dot potential versus the calculated indirect energy gap Γ X. An important observation for studying quantum dot potential is the clear differences between the indirect band gaps (Γ X) in going from Si to MgSe as seen in table 3. Hence, the indirect band gaps are predominantly dependent on the potential. The differences between the top valence energy level at Γ and lower conduction energy level at X have led to consider this model. The basis of the model is the indirect band gap (E g ) in terms of the top valence energy level at Γ and lower conduction energy level at X as seen in table 3. The electron velocity and mobility are not included in our model. Figure 2 displays the mapping of our calculated indirect band gaps (Γ X) versus the corresponding quantum dot potential. The fitting of these data gives the following empirical formula: P QD b =. E λ gγχ a (9)

11 8 where a b Y. Al-Douri is constant (in ev -1 ) as shown in table 4 and λ is a parameter appropriate for group-iv (λ = 6), III-V (λ = 4) and II-VI (λ = 2) semiconductors (in V -1 ). The calculated values of quantum dot potential are given in table 5 are in agreement with Udipi et al. [7] results of 1 mv. The results in table 5 show that the potential calculated in our model exhibit the same trends as those found in the values derived from Udipi et al. It may conclude that the present potential calculated in a different way than the definition of Udipi et al. is in good agreement with the empirical potential values as seen in table 5. It appears from the later table and the corresponding Fig. 2, S compounds of II-VI compounds are recommended strongly as predicted new compounds for quantum dot due to their potentials on the dot are the closest to 1 mv. In order to understand the fundamental differences between diamond and zinc-blende structure, it will concentrate on the prototypical series: Ge, GaAs and ZnSe. In this isoelectric series, it would transverse from a diamond to III- V and II-VI semiconductors. The advantage of this approach is that all three of these crystals possess nearly identical lattice constants. Any changes in their electronic structure must arise solely from the dissimilarity of the potentials for atoms 1 and 2 in Eq. (8). The symmetric part of the potential, which is an average form factor of Ga and As should resemble the form factors of Ge. This approach is used successfully in the study of zinc-blende structure. The antisymmetric part of the potential Eq. (8) is then a measure of the difference between the Ga and As potentials [14]. The results for ZnSe are probably not as accurate. There are several reasons for this situation. First, the larger band gap in ZnSe requires higher photon energies than GaAs for reflectivity measurements. Second, ZnSe is more ionic than GaAs [18,24,25]. In particular, the strong Se potential means that some of the approximations which allowed the use of plane wave basis begin to break down. Thirdly, in the Zn chalcogenides the Zn 3d level lies fairly close to the valence band maximum. It does not explicity include this level in the band calculations, and as a consequence any properties which depend on this level will not be probably accounted for. 3.2 Application study of pressure effect The most remarkable aspect of tetrahedral coordinated structures is their low density. Therefore, under pressure, a tetrahedral coordinated semiconductor can be transformed to a structure with high density. The zinc-blende structure (ZB) has the lowest minimum total energy. It is the most stable phase of these compounds at ambient pressure. If pressure is applied, the volume decreases and a transition to the b-sn (or NaCl) phase

12 Short Title 9 occurs at relatively low pressure. The investigation of chemical trends in solid state properties appears thus as an extremely useful part of new materials research. This is interesting when one tries to gain some information about the many properties of the group of binary compounds under pressure. The solid state physics regards one of the useful parts of modern materials researches, especially when one tries to gather some information about the properties of binary compounds under external effect. Therefore, it seems very important to relate the pressure effect of the compounds to the type of bond between the nearest atoms by controlling the evolution of the bond character with pressure in terms of the indirect band gap (Γ X). Thus, it is necessary to demonstrate the pressure induces the electrons to tunnel the quantum dot. The experimental data for the pressure variation of the forbidden energy gap (direct or indirect) are similar for many materials. Most of these data are found in the literature [26] and fitted by the formula: E E P) = (0) α g g p ( (10) which contains fitting parameter α. It was originally given by Varshnni [27] using the behavior of E g (P) in the vicinity of zero temperature as the best way to achieve an empirical fit. Table 1. The adjusted symmetric and antisymmetric form factors for III-V and II-VI semiconductors (in Rydberg), and the lattice constants are given (in atomic units).

13 10 Y. Al-Douri Table 2. The calculated and experimental principal band gaps for semiconductors in ev. a: Ref. [19], b: Ref. [16], c: Ref. [20], d: Ref. [21], e: Ref.[22]. Table 3. The calculated top valence energy level at Γ, lower conduction energy level at X and the indirect band gaps Γ X for semiconductors in ev. To test the validity of our model, the variation of the quantum dot potential under pressure effect for GaP and ZnTe is calculated. At normal pressure, the covalent semiconductors are four-fold coordinated. The reason

14 Short Title 11 Table 4. Constants a and b and the root mean square percentage error (RMSPE) for the empirical relation. These data are taken from Ref. [23]. Table 5. The calculated potential on the dot is corresponding to the calculated capacitance and the energy necessary to add one electron on the dot of semiconductors. that the density is so low is that nearest neighbors are bound by overlapping hybridized orbitals, which are the well-known sp 3 hybrids with tetrahedral direction. Therefore, these covalent compounds can be transformed either through chemical shifts or under pressure into a denser structure, which may be ionic or metallic. Thermodynamically, the three structures are separated by a first-order phase transition, but, microscopically, the responsibility of the

15 12 Y. Al-Douri interactions for the phase transition may be the same as for the chemical trends within the covalent structures. As pressure is applied, the volume deforms. Using our empirical formula, the potential values for the above two semiconductors for different pressures are calculated. The results are given in table 6. It is noticed that the potential decreases as pressure increases. The pressure induces large variation of quantum dot potential for GaP and does not like to ZnTe due to intrinsic properties of energy gap of the two different compounds. The pressure found in our calculations is confirmed by the variation of the top valence energy level at Γ and lower conduction energy level at X as seen in table 7. Table 6. The calculated potential on the dot at room and low temperature with the corresponding pressure. *: The used data. a: Ref. [28]. b: Ref. [29]. c: Ref. [30]. Table 7. The calculated top valence energy level at Γ, lower conduction energy level at X and the indirect band gaps Γ X (in ev) at room and low temperature for GaAs and AlAs. 3.3 Study of ternary alloys To further enhance our understanding of the electronic and optical properties of ternary materials for device application, an empirical pseudopotential band structure calculation for these alloys has been carried out. The empirical pseudopotential method [31] (EPM) has been proven to be one of the most reliable methods for band structure calculation of semiconductors. In comparison, the self consistent pseudopotential method in the local density approximation (LDA) usually underestimates the energy band [16]. In the empirical pseudopotential method, the actual atomic potential is replaced by pseudopotential and a set of atomic form factors where adjusted, so that the calculation produces energy bands as accurately as possible in the whole composition range with existing experimental data. Together with the virtual crystal approximation (VCA) the EPM offers the most effective and accurate means of obtaining the band structure of alloy.

16 Short Title E ΓΓ a Energy gap (ev) E ΓL E ΓX Ga x Al 1-x As Concentration (x) 4.5 b E ΓX 4.0 Energy gap (ev) E ΓL Zn x Cd 1-x Se 2.5 E ΓΓ Concentration (x) Figure 3. Calculated values of E ГГ, E ГX, E ГL gaps plotted as a function of alloy concentration (VCA calculation) for Ga x Al 1-x As (a) and for Zn x Cd 1-x Se (b).

17 14 Y. Al-Douri 0.80 a Quantum dot potential (x10-3 V) Ga x Al 1-x As Concentration (x) 0.88 b Quantum dot potential (x10-3 V) Zn x Cd 1-x Se Concentration (x) Figure 4. Variation of quantum dot potential with alloy concentration for Ga x Al 1-x As (a) and for Zn x Cd 1-x Se (b).

18 Short Title 15 The pseudopotential form factors for the pure components GaAs, AlAs, ZnSe and CdSe are shown in table 1. They give the reasonable agreement with experiment and other calculations for the principal energy gaps (see table 2). With the VCA the curves for the conduction Γ, X and L valleys of Ga x Al l-x As and Zn x Cd 1-x Se alloys versus concentration of Ga and Zn, respectively are given in Fig. 3. The reference energy level is the top of the valence band. It is clear that Ga x Al l-x As and Zn x Cd 1-x Se alloy is a semiconductor and the calculated variation of the energy gap was found to be linear with mole fraction for Zn x Cd 1-x Se alloy as displayed in Fig. 3. The Ga x Al l-x As alloys exhibits an indirect gap for the given composition while Zn x Cd 1-x Se alloy shows direct gap. The horizontal axis corresponds to energy, referenced by E v. The second state of the second valence band is primarily of cation s character, it changes rapidly to anion p-like at the top of the valence band. The conduction bands are more difficult to describe than the valence bands because they are more delocalised and more free-electron-like than the valence states. This is particularly true for the conduction band states far removed from the VBM. The free-electron behaviour results in more dispersive bands. Again, a nonlinear dependence of the alloy properties on the Ga concentration is observed. On the other hand, the variation of quantum dot potential for Ga x Al l-x As and Zn x Cd 1-x Se alloys is studied and the calculated quantum dot potential variation was found to be nonlinear and linear with mole fraction for Ga x Al l-x As and Zn x Cd 1-x Se alloys, respectively as shown in Fig Conclusion The EPM is used to establish a new empirical model to calculate the quantum dot potential (QD) of single-electron transistor (SET). It is concluded that the sudden variation of the quantum dot potential is an indication of an electron tunneling the quantum dot. The empirical model obtained for the potential gives results close to 1 mv in good accord with Udipi et al. [7] results, and application of pressure demonstrates the validity of our model to predict other physical properties of such compounds. The calculated values of principal energy gaps and quantum dot potential versus mole concentration of Ga x Al l-x As and Zn x Cd 1-x Se alloys are calculated with reasonable results of alloys studies. References 1. Guo, L., Leobandung, E., Zhuang, L., and Chou, S. Y., 1997, J. Vac. Sci. Technol. B 15, Zhuang, L., Guo, L., and Chou, S. Y., 1998, Appl. Phys. Lett. 72, Scott-Thomas, J. H. F., Field, S. B., Kastner, M. A., Smith, H. I., and Antoniadis, D. A., 1989, Phys. Rev. Lett. 62, 583.

19 16 Y. Al-Douri 4. Ishikuro, H., and Hiramoto, T., 1997, Appl. Phys. Lett. 71, Irvine, C., Durani, Z. A. K., Ahmed, H., and Biesemans, S., 1998, Appl. Phys. Lett. 73, Milicic N., Badrieh, F., Vasileska, D., Gunther, A., and Goodnick, S. M., 2000, Superlattices and Microstrucutres, 27, Udipi, S., Vasileska, D., and Ferry, D. K., 1996, Superlattices and Microstrucutres, 20, Scharfetter, D. L., and Gummel, H. K., 1969, IEEE Trans. Electron. Dev. ED-16, Selberherr, S., Shutz, A. and Potzl, H. W., 1980, IEEE Trans. Electron Dev. ED- 27, Pakes, C. I., Conrad, V., Ang, J. C., Green, F., Dzurak A. S., Hollenberg, L. C. L., Jamieson, D. N., and Clark, R. G., 2003, Nanotechnology, 14, Kane, B. E., 1998, Nature, 393, Wellard, C. J., Hollenberg, L. C. L., and Pakes, C. I., 2002, Nanotechnology, 13, Parisoli, F., Wellard, C. J., Hollenberg, L. C. L., and Pakes, C. I., 2002, Proc. 15th Aust. Inst. Phys. Congr. Sydney. 14. Cohen, M. L. and Chelikowsky, J. R. Pseudopotential for semiconductors, 1982, In: Handbook on semiconductors, edited by Moss, T. S. Vol. 1, North-Holland Publishing Company. 15. Phillips, J. C. and Kleinman, L. 1959, Phys. Rev. 116, Cohen, M. L., and Bergstresser, T. K., 1966, Phys. Rev. 141, Kobayasi, T., and Nara, H., 1993, Bull. Coll. Sci. Tohuku Univ. 2, Al-Douri, Y., Abid, H., Zaoui, A., and Aourag, H., 2001, Physica B 301, Al-Douri, Y., Abid, H., and Aourag, H., 2001, Physica B 305, Bechiri, A., Benmakhlouf, F. and Bouarissa, N., 2002, Mater. Chem. Phys. 77, Tsidilkovski, I. M., 1982, Band structure of semiconductors, Pergamon Press, Oxford. 22. Benkabou, F., Aourag, H., Certier, M., and Kobayasi, T., 2003, Physica B 336, Nag, B. R., 1995, J. Appl. Phys. 77, Al-Douri, Y., Abid, H., and Aourag, H., 2000, Mater. Chem. Phys. 65, Phillips, J. C. Bonds and Bands in Semiconductors, 1973, Academic Press, San Diego. 26. Adachi, S., 1992, Physical properties of III-V semiconductor compounds, Wiley Interscience, New York. 27. Varshnni, Y. P., 1967, Physica 34, Chelikowsky J. R., 1987, Phys. Rev. B 35, Zhang, S. B., and Cohen, M. L., 1987, Phys. Rev. B 35, Yu, S. C., Spain, I. L., and Skelton, E. F., 1978, Solid State Commun. 25, Al-Douri, Y., 2003, Mater. Chem. Phys. 82, 49.

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