NANO LETTERS
Controlling the Charge of a Specific Surface Atom by the Addition of a Non-Site-Specific Single Impurity in a Si Nanocrystal
2006 Vol. 6, No. 1 61-65
Torbjo1 rn Blomquist* and George Kirczenow Department of Physics, Simon Fraser UniVersity, Burnaby, British Columbia, Canada V5A 1S6 Received October 6, 2005; Revised Manuscript Received November 17, 2005
ABSTRACT We show, by use of self-consistent calculations, that the charge on a radical surface site (RSS) on a hydrogen-terminated silicon nanocrystal can be controlled by the addition of a non-site-specific dopant atom. An RSS is a surface defect where the H-termination is missing. This new effect has important implications for future hybrid semiconductor/molecular nanoelectronics. We also calculate the energy and wave function of the RSS state and its nanoscale interactions with the dopant atom.
The exponential increase in the speed and capacity of semiconductor technology, described by Moore’s law,1 has been enabled through continued miniaturization. Today, stateof-the-art semiconductor devices have a feature size of 65 nm. The atomicity of matter puts an ultimate limit to this miniaturization, and molecular electronics has been proposed as a replacement for silicon-based semiconductors. However, silicon technology has been refined to such an extent that other technologies cannot easily compete mainly due to the ease of mass production of silicon-based electronics. We are therefore most likely to see a hybrid technology, combining molecular and silicon electronics.2 One issue in the development of hybrid technology is understanding silicon surfaces on a nanoscale, both their electrical and chemical properties. Hydrogen-terminated surfaces are potentially of practical importance because of their good electronic properties and their chemical stability. Radical surface sites (RSSs) are silicon sites on the surface that lack H-termination and are associated with dangling bonds. Radicals are known from chemistry to be extremely reactive. RSSs are essential in the synthesis of hybrid devices,3,4 they may be used to control conduction through molecules on silicon samples,4 and their electrical properties may be useful in the operation of devices, e.g., chemical sensors. Despite the availability of experimental data, there has been a lack of theoretical treatment of RSSs; to date, only one publication exists that discusses doping in H-terminated silicon crystals with an RSS, by Zhou et al.5 Our results agree with the highly accurate timedependent density functional theory calculations by Zhou et * Corresponding author:
[email protected]. 10.1021/nl051995s CCC: $33.50 Published on Web 11/25/2005
© 2006 American Chemical Society
al. but extend to an order of magnitude larger nanocrystals which are experimentally accessible; this also reduces confinement effects of the finite size which allows us, for the first time, to draw conclusions about large silicon substrates, which could form the basis for hybrid silicon/ molecular nanoelectronics. We find that the RSS acts as an electronic acceptor; we calculate the wave function of this acceptor state and explore the nanoscale interactions with donor and acceptor impurities. We predict striking charge transfer effects; an entire electron is transferred from a donor any where in the nanocrystal atom to the RSS, a novel effect with implications for new hybrid semiconductor/molecular nanodevices. Acceptor impurities also affect the charge on the RSS in a non-site-specific but more subtle way. These charge transfer effects imply the very desirable ability to precisely control the charge on a specific surface site by the addition of an impurity without the necessity to control the location of the impurity. This is highly desirable since it suggests that a surprisingly simple new way to control the electronic properties of nanodevices at the atomic scale is possible. The energy of the RSS state is shown to strongly depend on the doping type of the nanocrystal, also with important implications for future devices. Our results are consistent with experiments on doped and H-terminated Si substrates with RSSs;4,6,7 this is the first theory to definitively identify an observed density of states peak for H-terminated Si surfaces7 with a specific RSS state. The quantitative understanding of RSS energy levels, charging, and wave functions developed here for the first time should be
important for engineering semiconductor/molecular hybrid nanotransistors and nanosensors. We require a technique that can handle a large number of atoms, yet with sufficient sophistication to capture the relevant physics. Hu¨ckel and tight-binding (TB) models allow for calculations on very large systems but do not include any self-consistent electron-electron interaction. Ab Initio calculations include electron-electron interactions but are limited to relatively small systems. We have developed a self-consistent Poisson-Schro¨dinger (PS) TB model that can handle large systems and includes Coulomb interaction between electrons. Our PS model is based on an existing nonorthogonal TB model8 that reproduces the band structure of silicon very well, with a band gap of 1.01 eV, and also produces reasonable values for electron and hole masses. We have also developed parameters for H, P, and Al, by fit to density functional theory (DFT) calculations, which lets us explore doped and H-terminated silicon systems.9 Other selfconsistent TB models10 have been proposed reviously, but we have specifically tailored our model to accurately reproduce the electronic structure of silicon. The current implementation of our model has an O(n2) scaling, allows for calculations on systems containing up to ∼1000 Si atoms, and converges in 15-25 iterations with a 1 meV eigenenergy precision. Our model has the potentail to achieve O(n) scaling with new algorithms.11 We begin with a brief summary of our PS model, the details can be found in ref 9. In the TB model, the hopping and overlap matrix elements between different sites are separated into a distance-dependent part (nonzero for distances up to 12.5 au) and an angle-dependent part (of the standard Slater-Koster form12). The on-site potential of an orbital of type l ∈{s, p, d} on site i is given as hil ) Alpil + Bl + Clqi + Vi
(1)
where Al and Cl are based on valence orbital ionization energies,13 Bl is a fitting parameter, 0 e pil e 2 is the Mulliken population of the orbital (the parameters are optimized for use with Mulliken population analysis),13 qi is the total charge on the atom, and Vi is the electric potential at the site. The Schro¨dinger equation Hψ ) ESψ is solved self-consistently with the electric potential
Vi )
F(r)
qj
1
+ ∫S dr ∑ j*i |r - r | |r - r | r
i
j
(2)
i
where r is the relative dielectric constant due to core polarization (core electrons are not included in the TB model) in Si, qj is the net charge on site j, ri is the position of site i, S is the structure’s surface, and F(r) is the surface polarization charge, due to the core polarization. We have used three different nanocrystal models, all shaped like a half sphere and with an RSS near the center of the flat surface. Our first nanocrystal Si552H237 has a Si(100)2 × 1:H surface, our two other nanocrystals Si534H255 and Si847H343 have Si(111):H surfaces; see Figure 1. Except 62
Figure 1. Geometries of the three silicon nanocrystal models used. Silicon atoms are large purple spheres and hydrogen atoms are small gray spheres. The RSSs (missing hydrogen atoms) can be seen near the center of the top surface of each nanocrystal. Table 1. Band Gap, Energy of RSS State Relative to the Valence Band (VB) Edge, the Occupancy of the Eigenstate Related to the RSS, and the Mulliken Charge on the RSSa system Si552H237 PSi551H237 AlSi551H237 @ 100 K Si534H255 PSi533H255 AlSi533H255 @ 30 K Si847H343 PSi846H343 AlSi846H343 @ 30 K
gap (eV) level (meV) occupancy charge (e) 1.458 1.391 1.388 1.538 1.462 1.582 1.438 1.350 1.430
206 796 17 132 546 43 93 522 30
1.00 2.00 0.61 1.00 2.00 0.00 1.00 2.00 0.01
-0.03 -0.16 -0.01 -0.06 -0.19 -0.03 -0.06 -0.19 -0.04
a Calculations are at zero temperature unless otherwise stated. Geometries as in Figure 1. Dopants are positioned near the center of the nanocrystal. Undoped and P-doped nanocrystals are in ground state even at room temperature due to large HOMO-LUMO gap.
for the top Si layer, these nanocrystal models have not been geometry-relaxed, we used a bulklike geometry for the silicon and terminated the silicon surface with hydrogen. The Si-H bonds are taken to be 1.48 Å, equal to the SiH4 bond. The geometry of the Si(100)2 × 1:H surface is based on a DFT relaxation for a crystal that is an order of magnitude smaller than the ones used here. We have also doped some of these nanocrystals by substituting a Si atom with Al or P. Our Poisson-Schro¨dinger calculations for the undoped nanocrystals show that the RSS state appears as a singly occupied level 200 meV above the VB edge for the Si(100)2 × 1:H surface and 90 meV for the Si(111):H surface in the larger nanocrystal. We find that the RSS behaves very much as an acceptor and we can see this when we add an extra electron to the cluster by doping it with phosphorus; the extra electron increases the occupancy of the RSS state; see Table 1. We can also see that the energy of the RSS state increases with occupancy, although there is also a dependence on the position of the impurity in the nanocrystal, which will be discussed later in the text. The low energy of the unoccupied RSS state however makes it easy to thermally excite and increase the occupancy at room temperature; see Table 1 which lists energies for nanocrystals with impurities near Nano Lett., Vol. 6, No. 1, 2006
Figure 2. The on-site Mulliken probabilities of the RSS state as function of the site’s radial coordinate (origin defined to be the RSS) in the undoped Si847H343 crystal. Si sites are shown as black plus signs and H sites as blue circles. The radial part of the effective mass solution is shown as a black line, i.e., |ψ(r,0)|2 with B ) 93 meV; see eq 4.
their center (7-10 Å below the surface). The energy of the RSS state is given relative to the VB edge. There is however some uncertainty in the Al-doped cluster to where the band edge is; we know that there should be a 3-fold degenerate state related to the Al acceptor, the degeneracy is however broken by the lack of symmetry in the cluster and the wave functions related to the Al-acceptor state cannot easily be distinguished from VB top states in this small system. For the Al-doped nanocrystals, we have therefore defined VB edge as the next energy level below the RSS state; this would at 0 K be the HOMO. Small energy level spacing near the Fermi energy poses a problem for convergence; our calculations for Al-doped nanocrystals have therefore been at nonzero temperature. The behavior of the RSS state in our model is consistent with DFT calculations performed on small Si clusters (Si66H64);5 our work however extends to nanocrystals that are an order of magnitude larger. The finite size of the present nanocrystals widens the band gap to ∼1.4-1.5 eV, but we do not anticipate this to have a large effect on the RSS state. The RSS state is mainly a linear combination of VB states (except for P-doped nanocrystals, where the level is near midgap; it is also very localized to the vicinity of the RSS, as shown in Figure 2. We therefore do not expect the energy of the RSS state relative to the VB edge to change much with nanocrystal size (except for P-doped nanocrystals, where we expect it to stay near midgap). Our calculations show that undoped nanocrystals have a charge on the RSS of ∼-0.03e for the Si(100)2 × 1:H and -0.06e for the Si(111):H surface. For comparison, in a fully terminated undoped nanocrystal, each SiH pair carries a Mulliken charge of 0.014e and 0.025e for the Si(100)2 × 1:H and Si(111):H surfaces, respectively. In doping the crystals, we find that the charge on the RSS depends on the type of the dopant (donor or acceptor) but is not sensitive to the position of the dopant (unless it is a (next) nearest neighbor site to the RSS); see Figure 3. Aluminum-doped Nano Lett., Vol. 6, No. 1, 2006
Figure 3. (upper panel) The binding energies of of the RSS state as function of impurity distance from RSS. The solid black curves and circles show energies for the xSi551H237 crystals [Si(100)2 × 1:H surface], the dashed blue curves and squares show energies for the xSi533H255 crystals [Si(111):H surface], and the dash-dot red curves and diamonds show energies for the xSi847H343 crystals [Si(111):H surface]. The upper curves show energies for P-doped nanocrystals, the lower show energies for Al-doped clusters. The dotted black curves show least-squares fits to B ) B0 - C/d; see main text for details. (lower panel) The Mulliken charge on the RSS for the same nanocrystals and impurity positions.
nanocrystals exhibit an RSS charge of -0.04e or smaller (the next nearest neighbor impurity position is an exception with positive charge). The net charge on the RSS is due to both the occupation of the RSS state and the polarization of the VB electrons. For phosphorus-doped nanocrystals, we find that the RSS carries a Mulliken charge of -0.16e on the Si(100)2 × 1:H surface and -0.19e on the Si(111):H surface. The charge is remarkably insensitive to the position of the impurity (especially on the Si(111):H surface), with the exception of the nearest neighbor site. The effective dielectric constant at a surface is reduced to rs ) (r + 1)/2 (for silicon r ) 11.8);14 we therefore expect an elementary point charge to be shielded to 0.16e. This matches the extra charge on the RSS exactly for P-doped nanocrystals, as compared to the RSS on an undoped nanocrystal. Thus, it is tempting to assume that the extra electron is completely localized to the RSS; however, a detailed analysis shows that the RSS wave function has a 44-49% probability at the RSS. L. Liu et al. have experimentally observed negatively charged RSSs on the Si(100)2 × 1:H surface with negative doping and uncharged RSSs on surfaces with positive doping.6 This is consistent with our results. Figure 3 shows how the RSS state energies and the RSS Mulliken charge depend on the position of the impurity. In Figure 3, a function B ) B0 - C/d has been fitted to the energies of the P-doped crystallites; here d is the distance between the RSS and the impurity site and B0 and C are 63
fitting constants. For the PSi552H237 crystallite, we get B0 ) 902 meV and C ) 1277 meV Å, and for the PSi534H255 crystallite, B0 ) 679 meV and C ) 1124 meV Å. The ionized impurity appears as a point charge. By assuming the RSS state to be completely localized to the RSS, we calculate the Coulomb energy of the extra electron on the RSS, due to the impurity to be E ) e2/4πr0d ) 1219 meV Å/d, which is in remarkably good agreement with the above fitted functions. For the PSi847H343 crystallite, we estimate B0 ) 645 meV, assuming C ) 1219 meV Å. This is in good agreement with capacitance experiments on n-type Hterminated Si surfaces, where a density of states peak has been observed at 650 meV above the VB edge.7 Our result is the first detailed calculation to associate this peak with RSSs. From the above results, we draw the conclusion that in the ground state, an RSS on a H-terminated phosphorusdoped silicon nanocrystal will always be doubly occupied and thereby negatively charged. This charging has been observed with scanning tunneling microscopy for H-terminated silicon surfaces on heavily doped substrates.6,4 On lightly doped substrates and with a negative sample bias, however, the tunneling rate from the RSS state into the tip can be larger than the rate at which the RSS state is repopulated from the bulk, and the RSS will appear singly occupied and uncharged. Also, if the density of acceptor states (RSSs) on the surface is very high, not all of these will be occupied due to the formation of a Schottky barrier.15 The RSS state energy in the Al-doped nanocrystals does not fit as well to a d-1 type curve, because of hybridization between the RSS state and the Al acceptor state due to their similar energies. The behavior in this case is more complex and not predictable without detailed calculations. We find amphoteric behavior of the RSS state only when the Al acceptor and the RSS are very close. The complex behavior of the RSS state does however not affect the charging of the RSS; it is independent of the position of the dopant, unless it is on a (next) nearest neighbor site; see Figure 3. The RSS wave function, away from the RSS can be estimated analytically in the effective mass approximation (EMA). We neglect the Coulomb potential; it is small already at the nearest neighbor site compared to the binding energy (the energy difference between the VB edge and the RSS state). We solve -
p2 2 ∇ ψ(r,θ) ) -Bψ(r,θ) 2m
(3)
where m ) 0.31me is the heavy hole mass in the current model and B the binding energy of the RSS state. The state is bound to the crystal; we therefore apply the boundary conditions ψ(r,π/2) ) 0 and ψ(r0,θ) ) A cos θ, where A is a constant and r0 ) 2.35 Å is a cutoff radius chosen to be equal to the nearest neighbor distance in silicon. The ground state is ψ(r,θ) ) C 64
e-κr(κr + 1) κ2r2
cos θ
(4)
where C is a constant and κ ) (2mB)1/2/p. We can relate this analytic result to our Poisson-Schro¨dinger calculations by using the PS binding energy for B and the Mulliken onsite probability for the RSS for the probability Pc of finding the electron/hole within the r0 radius, which allows us to normalize the wave function as ∫r>r0|ψ|2 dV ) 1 - Pc. We have plotted the on-site probability for the RSS state as function of the site’s distance from the RSS in Figure 2. We have also plotted the radial part of the EMA solution |ψ(r,0)|2. The EMA solution fits very well with the decay rate beyond 7 Å. The EMA curve would ideally follow the top envelop of the PS result, but there is some uncertainty to the cutoff radius r0 and the charge inside r0. We have also a finite size effect, the EMA has a 5.1 × 10-3 probability for the wave function beyond the 20 Å radius. The RSS wave function in the doped nanocrystals is very similar, with a much faster decay for the P-doped nanocrystals and a slower decay for the Al-doped nanocrystals, in qualitative agreement with our EMA solution. We do not however get as good a fit in these two cases between the EMA solution and the PS solution, because of uncertainty of the binding energy in the Al-doped case and because the mid-band-gap state in the P-doped nanocrystals is formed from both conduction and valence band states. To conclude, we have studied radical surface sites on hydrogen-terminated silicon nanocrystal surfaces. We have shown that the RSS behaves electronically as an acceptor. We predict that the occupation and energy of the RSS state and the charge on the RSS depend remarkably strongly on the doping of the nanocrystal: RSSs on n-type nanocrystals are negatively charged and have doubly occupied RSS states, due to a charge transfer of an entire electron from the donors to the RSS state. The charge on the RSS is strikingly independent of the position of the dopant. For undoped nanocrystals we find that the RSS state is singly occupied and the RSS is approximately uncharged. We show that the energy level of the RSS state depends on the occupation of the state, which in turn depends on the presence and nature of dopants in the vicinity; it is very close to the VB edge when unoccupied, at single occupation the energy level increases to 100-200 meV and at double occupation the energy level appears near the center of the band gap. Our results are consistent with DFT calculations for small silicon clusters5 and with experimental observations.4,6,7 We have presented the first theory that definitively identifies an observed density of states peak of H-terminated Si-surfaces7 with a specific RSS state. Our results have important implications for future hybrid semiconductor/molecular nanoelectronics: A remarkably simple new way to control the electronic properties of nanodevices at the atomic scale. We can precisely control the charge on a specific surface site on a nanoparticle by the addition of an impurity without the necessity to control the position of the impurity. Acknowledgment. We thank R. Hill and R. Wolkow for discussions. This work was supported by NSERC and the Canadian Institute for Advanced Research. Nano Lett., Vol. 6, No. 1, 2006
References (1) Moore, G. E. Electronics 1965, 38 (8), 114-117. (2) Joachim, C.; Gimzewski, J. K.; Aviram, A. Nature 2000, 408, 541548. Rakshit, T.; Liang, G. C.; Ghosh, A. W.; Datta, S. Nano Lett. 2004, 4, 1803-1807. Hersam, H. C.; Guisinger, N. P.; Lyding, J. W. Nanotechnology 2000, 11, 70-76. (3) Wolkow, R. A. Jpn. J. Appl. Phys. 2001, 40, 4378-4383. Lopinski, G. P.; Wayer, D. D. M.; Wolkow, R. A. Nature 2000, 406, 48-51. Hofer, W. A.; Fisher, A. J.; Lopinski, G. P.; Wolkow, R. A. Chem. Phys. Lett. 2002, 365, 129-134. Kruse, P.; Johnson, E. R.; DiLabio, G. A.; Wolkow, R. A. Nano Lett. 2002, 2, 807-810. Pitters, J. L.; Wolkow, R. A. J. Am. Chem. Soc. 2005, 127, 48-49. (4) Piva, P. G.; et al. Nature 2005, 435, 658-661. (5) Zhou, Z.; Friesner, R. A.; Brus, L. J. Am. Chem. Soc. 2003, 125, 15599-15607. (6) Liu, L.; Yu, J.; Lyding, J. W. Mater. Res. Symp. Proc. 2002, 705, Y6.6.1-Y6.6.6. (7) Koyanagi, S.; Hashizme, T.; Hasegawa, H. Jpn. J. Appl. Phys. 1996, 35, 946-953. Yoshida, T.; Hasegawa, H. Jpn. J. Appl. Phys. 2000, 39, 4504-4508. Hashizume, T.; Koyanagi, A.; Hasegawa, H. J. Vac. Sci. Technol., B 1996, 14, 2872-2881.
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(8) Bernstein, N.; Mehl, M. J.; Papaconstantopoulos, D. A.; Papanicolaou, N. I.; Bazant, M. Z.; Kaxiras, E. Phys. ReV. B 2000, 62, 44774487. Bernstein, N.; Mehl, M. J.; Papaconstantopoulos, D. A.; Papanicolaou, N. I.; Bazant, M. Z.; Kaxiras, E. Phys. ReV. B 2002, 65, 249902(E). (9) Blomquist, T.; Kirczenow, G. Nano Lett. 2004, 4, 2251-2254. Blomquist, T.; Kirczenow, G. Phys. ReV. B 2005, 71, 045301-1045301-9. For surface properties, the following parameters were found to be optimal: for core polarization r ) 1.25, Si-H hopping parameters Ss ) -0.262 Ry and Sp ) -0.33 Ry. (10) Frauenheim, TH.; et al. Phys. Status Solidi B 2000, 217, 41-62. (11) Niklasson, A. M. N. Phys. ReV. B 2003, 68, 233104-1-233104-4. (12) Harrison, W. A. Electronic Structure and the Properties of Solids; Dover: New York, 1989. (13) McGlynn, S. P.; Vanquickenborne, L. G.; Kinoshita, M.; Carroll, D. G. Introduction to Applied Quantum Chemistry; Holt, Rinehart and Winston, Inc.: New York, 1972. (14) Bechstedt, F.; Del Sole, R. Solid State Commun. 1990, 74, 41-44. (15) Bardeen, J. Phys. ReV. 1947, 71, 717-727.
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