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PHYSICAL REVIEW B 71, 165328 共2005兲

Surface passivation method for semiconductor nanostructures Xiangyang Huang, Eric Lindgren, and James R. Chelikowsky Department of Chemical Engineering and Materials Science, Institute for the Theory of Advanced Materials in Information Technology, Digital Technology Center, University of Minnesota, Minneapolis, Minnesota 55455 共Received 10 November 2004; revised manuscript received 28 December 2004; published 28 April 2005兲 A common problem in modeling the electronic structure of quantum dots is the construction of a passivating agent, which can be used to render the surface of the dot electronically inert. We demonstrate that fictitious, “hydrogenlike” pseudoatoms may be used for surface passivation for II-IV and III-V semiconductor nanostructures. Using pseudopotentials constructed within density functional theory, we provide a recipe for the construction of these passivating atoms and provide physically motived criteria for obtaining an optimal passivation. To establish the validity of this approach, we apply our procedure to passivate GaAs quantum dots. DOI: 10.1103/PhysRevB.71.165328

PACS number共s兲: 73.21.La, 73.22.⫺f, 71.15.⫺m, 78.67.Hc

Semiconductor nanoparticles have been the subject of intensive study owing to their unusual electrical and optical properties.1,2 These nanoparticles provide a unique opportunity to study the properties at nanometer scale and to reveal the underlying physics occurring at reduced dimensions. The properties of nanoparticles are often dependent on their size owing to the role quantum confinement. As such, semiconductor nanoparticles are promising building blocks in nanotechnologies because one can use their size to tailor desired properties. For example, at small length scales the band structure of a material is no longer quasicontinuous. Rather the electronic energy levels may be described by discrete quantum energy levels and can be adjusted as desired by size modifications. In this manner, an optically inactive material such as crystalline silicon can be made optically active at the nanoscale. A deep understanding of the physics of semiconductors at the nanoscale is necessary to provide the fundamental science for the development of nano-optical and electronic device applications. This understanding can best be obtained by utilizing ab initio approaches. These approaches can provide valuable insights into nanoscale phenomena without empirical parameters or adjustments extrapolated from bulk properties. To date, ab initio approaches have centered on understanding colloidal nanostructures composed of group IV semiconducting elements such as silicon and germanium, while nanostructures of technology important III-V and II-VI heteropolar semiconductors such as GaAs or ZnSe have received relatively less attention. In contrast to the group IV nanostructures, work on modeling III-V and II-VI semiconductors has been impeded by the lack of a simple surface passivation scheme. At small length scales, surface properties become more important as the surface to volume ratio increases. The surface of a semiconductor nanoparticle often contains electronically active states because of unsaturated surface bonds or dangling bond states. Surface passivation aims to rebond these dangling bonds with some passivation agent while maintaining neutrality of the whole system. A good surface passivation will remove the localized surface states from the band gap, but will not change the intrinsic behavior of the highest energy occupied molecular orbital 共HOMO兲 and the lowest energy unoccupied molecular orbital 共LUMO兲. 1098-0121/2005/71共16兲/165328共6兲/$23.00

For group IV semiconductors, hydrogen is often used as a passivating agent, both experimentally and theoretically. This choice is reasonable since the electronegativity of the hydrogen atom is close to that of Si and Ge and the H-Si or H-Ge bond provides a natural structural and electronic termination of the nanostructure. However, unlike the bonding in group IV semiconductors, bonds in III-V and II-VI are heteropolar with significant charge transfer and rehybridization occurring at the surface. Hydrogen atoms are not expected to be appropriate passivating agents for different atomic species occurring on the surface. Indeed, in experiment, organic molecules such as trioctylphosphine 共TOP兲 or trioctylphosphine oxide 共TOPO兲 are often used to passivate nanoparticles.1,2 Owing to the complexity of these passivation agents, a good understanding of the atomic structure is not available, nor is it easy to calculate such structures owing to the numerous degrees of freedom of these agents. A simpler atomistic model is highly desirable for this problem and some progress has been made. For example, Wang and Zunger proposed an empirical “ligand potential” for quantum dots of CdSe. In their model, they placed positive 共negative兲 short-range electrostatic potentials near the surface anion 共cation兲 atoms. They chose a parameterized ligand potential to have a Gaussian form and fixed the potential parameters by considering the electronic states on surfaces.3 There are some notable problems associated with this approach. For example, in the case of nonstoichiometric CdSe quantum dots 共Cd-centered or Se-centered dots兲, they found this procedure did not completely remove some of the states from the gap and had to artificially shift the Fermi level.4 Another problem concerns fitting data to surface states. In general, the surface of a dot cannot be simply described by a single surface plane structure. It is not clear that a single ligand potential can be used for a variety of dangling bonds. Moreover, the parameterization is empirical and no criteria exist to optimize the passivation procedure. Another approach to passivation, was proposed. In his investigation of the electronic structure of a polar semiconductor surface, Shirashi5 proposed a new slab model approach in which fictitious hydrogens, H*, were introduced to passivate the GaAs surface. In this way, the charge transfer and interaction between two surfaces of the slab were avoided and the terminated irrelevant surface states were

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HUANG, LINDGREN, AND CHELIKOWSKY

FIG. 1. Ionic potentials for hydrogen and fictitious hydrogens H* with a nuclear change Z of 0.75 and 1.25. The pseudopotentials were generated using a radius cutoff of 1.5 a.u.

found to leave from the band gap region due to the formation of the covalent bonds. Their fictitious hydrogen potentials were based on fractionally charged hydrogen atoms, i.e., a fractional proton charge with a corresponding fractional electron charge. The choice of the fractional proton charge is based on a simple chemical consideration of a covalent bond. In GaAs, the formal valence of Ga is three and As is five. As such, 3 / 4 of an electron from each Ga atom and 5 / 4 of an electron from each As atom combine and form covalent bonds between them 共see Fig. 1兲. Therefore, the fractional 3 / 4 共5 / 4兲 charged fictitious atoms are considered as an appropriate choice to terminate the surface As 共Ga兲 dangling bonds. Recently, Wang and Li proposed a similar model. They also used fictional hydrogen-like atoms to passivate III-V and II-VI semiconductor quantum dots.6 In their work, they used a nuclear charge of 共8 − m兲 / 4 to passivate a surface atom with formal valence charge m. This choice of nuclear charges is also based on the simple chemical consideration of the covalent bond. This method, as others, treats all elements with the same valence as identical, e.g., it does not distinguish between O, S, or Se. In this paper we used a similar scheme for surface passivation. However, we propose quantitative criteria for choosing the passivating potentials. Our method is particular to each material and each atomic species present. We base our criteria by a metric determined through an analysis of group IV semiconductors. We illustrate our method by considering GaAs quantum dots. We solve for the electronic structure of quantum dots using density-functional theory within the local-density approximation. Specifically, the Kohn-Sham equations are solved directly in real space based using a high-order finite difference algorithm.7 This real-space algorithm is especially suitable for investigating localized systems such as quantum dots. For example, realspace algorithms do not require the use of supercells to create artificially periodic systems as would be required for plane wave bases. This eliminates any cell-cell interactions and easily allows calculations for charged systems. We choose a uniform, cubic grid. As for plane waves, the grid is unbiased and convergence is controlled by only one parameter: the grid spacing. Pulay force contributions are not present as the basis is not attached to the atomic positions.7 We employ ab initio pseudopotentials, which are well

FIG. 2. Calculated charge density 共兩␺共r兲2兩兲 for Si130H98: 共a兲 HOMO and 共b兲 LUMO.

suited for uniform grids as they contain no singularities or short wavelength oscillations. Our pseudopotentials were generated using the Troullier-Martins prescription.8 With this procedure, the Hamiltonian is sparse and the eigen problem can be solved using a variety of standard techniques. We solve the eigen problem using the Arnoldi process, which is implemented in the ARPACK package.9 For large systems, this method can speed up matrix diagonalization by at least 30% compared to the generalized Davidson method. We carefully checked convergence of the energy eigenvalues with respect to the size of the spherical boundary domain and the grid spacing. At the boundary domain, we demand the wave functions vanish. For all quantum dot calculations, we required at least a 6 a.u. separation between the outmost passivating atoms and the spherical boundary. We used a grid spacing of 0.6 a.u. 共1 a . u . = 0.529 Å兲. This roughly corresponds to a plane wave cut off of 25 Ry. Our choice of grid spacing and boundary conditions ensure the eigenvalues are converged to better than ⬃0.01 eV. Before discussing the general issue of surface passivation for group III-V and II-VI semiconductors, let us consider a typical group IV quantum dot. Figure 2 illustrates the structure of a Si130 dot whose surface is terminated with hydrogens. In order to check the quality of the passivation, we

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FIG. 3. 共Color online兲 HOMO-LUMO gap of ZnSe and GaAs quantum dots, Eg, as a function of fractional charge ␩. Surface II 共or III兲 and VI 共or V兲 atoms of each dot are rebonded with fictitious hydrogens H* with nuclear charges of 1 + ␩ and 1 − ␩, respectively. The maximum gap for the same value ␩ for different dot sizes.

display the charge densities of HOMO and LUMO wave functions in this figure. The HOMO and LUMO densities are found to be strongly localized within the dot with little contribution from the surface region. An analysis of the states whose energy resides near the gap confirms that the absence of any surface localized states, i.e., occupied states localized on surface atoms are located at least 1 eV below the gap and empty states are located much higher above the gap. The absence of surface localized states near the energy gap is consistent with a passivated quantum dot. We construct quantum dots for group III-V and II-VI systems from a bulk fragment. For the III-V and II-VI semiconductors considered here, we assume zinc blende structures, unless otherwise noted. We take spherical fragments of materials with the center of mass of each quantum dot being taken either at the bond center of Ga-As 共or Zn-Se兲 or at geometry center of Ga3As3 共or Zn3Se3兲 ring. The quantum dots so constructed are stoichiometric; they consist of equal numbers of anions and cations. By carefully choosing the radius of the dot, we obtained structures that do not have any surface atoms with more than two dangling bonds. Two different kinds of fictitious hydrogen atoms were introduced to passivate these dangling bonds. To keep the net charge on the dot neutral, we required one species of our fictitious atoms to have a nuclear charge of 1 + ␩ and a valence electron charge of −共1 + ␩兲, where ␩ is a positive number. These atoms are bonded with cation atoms. The other species has a nuclear charge of 1 − ␩ and a valence electron charge of −共1 − ␩兲. These species are bonded to the anion

FIG. 4. Calculated charge density for Ga65As65H*98: 共a兲 HOMO and 共b兲 LUMO. Ga, As, and fictitious hydrogen atoms are represented by large dark circles, large gray circles, and small circles, respectively.

atoms. Ionic pseudopotentials for the hydrogen and for the fictitious atoms 共with nuclear charges of 0.75 and 1.25兲 are illustrated in Fig. 1. By varying ␩, we can simulate quantum dots in different passivating environment such as those in a colloidal environment. Unless otherwise mentioned, all dots studied in this paper are passivated using this passivation method. We need to choose a value of the fractional charge, ␩, that will optimally passivate the dot. Figure 3 shows the HOMOLUMO gap of a dot as a function of ␩. The most striking feature of Fig. 3 is that a gap opens and becomes a maximum at the same value of ␩ for all three quantum dots considered. For example, all three ZnSe quantum dots have a maximum gap at ␩ = 0.5, for GaAs, the value is ␩ = 0.25. We find this result holds for other II-VI and III-V systems. For example, we found InP quantum dots also have a maximum gap at ␩ = 0.25; CdSe dots 共with a wurtzite structure兲 open a maximum gap at ␩ = 0.5. We note that the fractional charge we obtain is consistent with that used in Refs. 5,6. We argue that the passivation for GaAs quantum dots is optimal when the HOMO-LUMO gap of the dot becomes a maximum. We make this argument in part because the gap of a passivated dot should be dominated by the intrinsic bonds

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FIG. 5. Calculated contour plots of valence charge density for 共a兲 Si130H98 and 共b兲 Ga65As65H*98 quantum dots.

within the bulk and not depend on the extrinsic surface bonds, i.e., not with the bonds between the surface atoms and the passivating agents. To confirm this, we studied a bigger * dot, Ga65As65H98 . H* denotes either of two kind of fictitious atoms. The bond lengths of H*-Ga and H*-As were determined from two model systems, GaH*4 and AsH*4, in which bond lengths are fully optimized. For Ga a H* with a nuclear charge of 1.25 is used. For As, the nuclear charge is taken to be 0.75. We found the gap of the dot to be insensitive to the bond length between surface atoms and fictitious hydrogen atoms: a 10% change in the bond length results in change of less than 0.1 eV in the gap. Figure 4 shows the calculated

FIG. 6. HOMO-LUMO gap Eg as a function of the GaAs dot diameter D. The horizontal dashed line indicates the calculated gap for bulk GaAs in the zinc blende structure.

FIG. 7. Calculated DOS’s of GaAs quantum dots with surface passivation, comparing with bulk. A Gaussian function of 0.05 eV was used for finite broadening. Calculations for bulk were done with a 12⫻ 12⫻ 8 Monkhorst-Pack k-point mesh 共Ref. 11兲 and an energy cutoff of ⬃25 Ry.

HOMO and LUMO charge densities for the passivated dot. Comparing the case of a Si quantum dot 共see Fig. 2兲 to the GaAs dot, we find similar results. The HOMO and LUMO states are mostly located within the interior of the dot with negligible surface overlap. The occupied surface states are removed from the gap to deep below the Fermi level 共more than 6 eV兲. The empty surface states are displaced to higher energies and are also removed from the gap. Both the HOMO and LUMO seem to have the p character around As atoms. Since the origin of the dot is taken at the bond center of Ga and As atoms, the HOMO level does not have a strict triple degeneracy. A splitting of about 0.02 eV occurs. The same amount of splitting is also found in the same size of the Si quantum dot. As GaAs is more ionic than Si, the electronic properties reflect a different behavior. Figure 5 shows the charge den* quantum dots. sity contour plot for Si130H98 and Ga65As65H98 Similar to the bulk materials, the bond charge forms between nearest neighboring atoms in the Si dot. In GaAs dots, the bond charge shifts to the As sites and as a result, the charge is more localized on As sites. It should be noted that the fictitious atom H* bonded with the Ga atom “extracts” charge from the Ga atom and results in a charge localized toward the H* site. The other fictitious atom bonded with the As atom shows a different feature: the bond charge is located at the midpoint between As and H*.

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FIG. 9. Calculated DOS’s of passivated GaAs quantum dots: 共a兲 Ga-centered Ga19As16H*36; 共b兲 As-centered Ga19As16H*36; 共c兲 Ga-As bond centered Ga19As19H*42.

FIG. 8. Calculated charge density for Ga65As65H*98 with levels located at 共a兲 7.93 eV 共b兲 6.37 eV below the “valence band” edge 共HOMO兲. Ga, As, and fictitious hydrogen atoms are represented by large dark circles, large gray circles, and small circles, respectively.

We can also examine our passivation method with respect to small GaAs dots and examine their evolution to quantum dots. We consider the following clusters and small quantum dots: Ga1As1,Ga3As3,Ga10As10,Ga19As19,Ga41As41,Ga65As65, and Ga171As171. Except for Ga1As1, no other clusters have a triple bond with passivation atoms. Figure 6 shows the evolution of HOMO-LUMO gaps of GaAs dots as a function of size. As seen, as the size of a dot increases, the gap is found to decrease due to quantum confinement effect. This illustrates the robust nature of our passivation scheme, i.e., the passivation is effective across a broad size range. The calculated density of states 共DOS兲 of GaAs dots are shown in Fig. 7, the DOS of bulk GaAs are also presented for comparison to illustrate the evolution of electronic states with dot size. The energy zero is chosen to be the high occupied state. For a dot containing more than a hundred atoms or so, the density of states begins to resemble the crystalline phase and “bandlike” features begin to appear. For example, as the dot size increases, the gap decreases whereas the width of the upper “incipient” valence band increases. Some notable differences do exist, e.g., some extrinsic states appear

within the “antisymmetric gap”10 between the lowest valance band and the upper valance band features. We illustrate the charge densities corresponding to energy levels located at * . These states 7.93 and 6.37 eV in Fig. 8 for Ga65As65H98 arise from bonds between the dot and the passivation atoms and have no contribution from the interior of the dot. A further charge density analysis confirms that states the fictitious hydrogens are mainly located at this region and have negligible effect on states near the HOMO-LUMO gap. Finally, we illustrate the passivation method for nonstoichiometric cases. We consider two quantum dots: one cen* , and the other centered at As tered at Ga atom, Ga19As16H36 * atom, Ga16As19H36. The calculated DOS’s are shown in Fig. 9. The HOMO-LUMO gaps of the two dots are calculated to be 3.76 and 3.64 eV, which is between that of the stoichio* * 共3.77 eV兲 and Ga19As19H42 metric cases of Ga10As10H30 共3.53 eV兲. Overall, the DOS’s are similar for the two dots. A major advantage of our passivation method is that there is no artifact of “missing-excess states,” which is an issue in “ligand potential” passivation methods, as in the work of Wang and Zunger.3 This artifact is discussed by Rabani et al.4 They noted that in the approach of Wang and Zunger, the position of the Fermi level must be shifted to correct for the lack of stoichiometry. For example, in the case of Cd20Se19, the work of Wang and Zunger placed the Fermi level within the bulk band gap and did not yield a gap characteristic of a properly passivated system. In summary, we have systematically examined a new surface passivation method for semiconductor nanoparticles and quantum dots. By using optimized fictitious hydrogen atoms, we are able to achieve a good surface passivation for both III-V and II-VI quantum dots. We illustrate this method for

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GaAs quantum dots. We believe this study can be widely used in an ab initio study of II-VI and III-V nanostructures. This work was funded in part by the National Science Foundation under Contract No. DMR-0130395 and

P. Alivisatos, Science 271, 933 共1996兲. M. Heath, Acc. Chem. Res. 32, 38 共1999兲. 3 L.-W. Wang and A. Zunger, Phys. Rev. B 53, 9579 共1996兲. 4 E. Rabani, B. Hetenyi, B. J. Berne, and L. E. Brus, J. Chem. Phys. 110, 5355 共1999兲. 5 K. Shiraishi, J. Phys. Soc. Jpn. 59, 3455 共1990兲. 6 L.-W. Wang and J. Li, Phys. Rev. B 69, 153302 共2004兲. 7 J. R. Chelikowsky, N. Troullier, and Y. Saad, Phys. Rev. Lett. 72, 1240 共1994兲. 1 A. 2 J.

DMR-0325218 and the U.S. Department of Energy under Contract No. DE-FG02-89ER45391 and DE-FG0203ER15491. The calculations were performed at the Minnesota Supercomputing Institute and at the National Energy Research Scientific Computing Center 共NERSC兲.

Troullier and J. L. Martins, Phys. Rev. B 43, 1993 共1991兲. Lehoucq, D. Sorensen, and C. Yang, ARPACK User’s Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods 共SIAM, Philadelphia, 1998兲. 10 M. L. Cohen and J. R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors, 2nd ed. 共Springer-Verlag, Berlin, 1988兲, p. 104. 11 H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 共1976兲. 8 N. 9 R.

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