VOLUME 4, NUMBER 10, OCTOBER 2004 © Copyright 2004 by the American Chemical Society
Silicon-based Molecular Electronics Titash Rakshit, Geng-Chiau Liang, Avik W. Ghosh, and Supriyo Datta* School of Electrical and Computer Engineering, Purdue UniVersity, West Lafayette, Indiana 47907 Received April 15, 2004; Revised Manuscript Received June 1, 2004
ABSTRACT Molecular electronics on silicon has distinct advantages over its metallic counterpart. We describe a theoretical formalism for transport in semiconductor−molecule heterostructures, formally combining a semiempirical treatment of bulk silicon with a first-principles description of the molecular chemistry and its bonding with silicon. Using this method, we demonstrate that the presence of a semiconducting band-edge can lead to a novel molecular resonant tunneling diode (RTD) that shows negative differential resistance (NDR) when the molecular levels are driven by an STM potential into the semiconducting band gap. The NDR peaks show a clear polarity reversal, appearing for positive bias on a p-doped and negative for an n-doped substrate, a feature that is in agreement with recent experiments by Guisinger et al.1,2
Traditionally, molecular electronic efforts, both theoretical3,4 and experimental,5 have been driven by thiol-gold chemistry to molecules bonded to gold substrates. However, several recent experiments have demonstrated the feasibility of attaching various molecules on silicon surfaces.1,2,6 The development of molecular electronics on silicon is particularly important for two reasons. First, it will enable the development of integrated devices that can utilize the powerful infrastructure provided by the silicon-based I-C industry. Second, unlike gold, silicon has a band gap that allows the design of a new class of resonant tunneling devices with possible applications in logic7 and low-power memory.8 In view of these significant potential payoffs, we believe it is worthwhile at this time to develop models for electron transport in such silicon-based molecular devices. In this paper we present a theoretical framework for silicon-based molecular electronics and outline the physics of RTDs based on such structures. Modeling transport through molecules on silicon presents a considerable theoretical challenge compared to metal substrates, owing to the complex electrostatics and surface physics of silicon: the 10.1021/nl049436t CCC: $27.50 Published on Web 09/08/2004
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depletion width in silicon extends over several layers even for a highly doped substrate, making a first-principles treatment of the overall electrostatics quite prohibitive. Furthermore, molecular devices on silicon require a formulation that can account for the band gap, surface reconstruction, and corresponding surface states. A common problem encountered is that standard quantum chemical basis sets that describe well the molecular chemistry provide a very poor description of the semiconductor bandstructure. We provide a scheme to integrate the two distinct systems and present results that combine ab initio treatments of the molecule with a semiempirical description of the silicon bands. The same approach can be used to integrate the molecule with more advanced treatments of silicon. Our results show that we can expect to see NDR using molecules such as TEMPO (2,2′,6,6′-tetramethyl-1-piperidynyloxy) or styrene9 on p-silicon at modest (2-3 V) positive substrate voltages, when the NDR action is determined by levels based on the highest occupied molecular orbitals (HOMO). A schematic band diagram is shown in Figure 1. Recent experimental observations1,2 seem in reasonable
Figure 1. Schematic description of molecular RTD involving a silicon band edge (CB: conduction band, VB: valence band, Vac: vacuum, L: LUMO, H: HOMO) and an STM tip. Bottom: (a) Equilibrium band alignment in p-Si molecule-metal heterostructure corresponds to a flat Fermi energy EF (dashed line) near the silicon VB edge. (b) For positive substrate bias V, the levels move up until the HOMO levels leave the silicon valence band into the band gap, leading to a sudden drop in transmission (Figure 3) and a corresponding NDR in the I-V (Figure 4). (c) For negative bias, there is no NDR.
agreement with these results. By contrast, it appears that much larger negative voltages will be needed to observe these effects on n-type silicon where NDR would take place via the lowest unoccupied molecular orbitals (LUMO). Experimentally, however, the peaks appear earlier than theoretically expected for molecules on n++-Si.1,2 This is an observation that deserves careful attention, since relatively little work has been reported to date for LUMO-based conduction. Overall Formalism. We study conduction through nanoscale devices by coupling an electronic structure calculation with a nonequilibrium Green’s function (NEGF)-based treatment of quantum transport.4,10 The active device (which may include a few contact atoms to which it is bonded) is described within an appropriate basis set by a molecular Hamiltonian H and overlap matrix S, the electrostatic potential by U, while the contacts enter our treatment through energy-dependent, self-energy matrices Σ1,2(E). The NEGF equations can be summarized as G ) [ES - H - U - Σ1 - Σ2]-1 T ) Trace[Γ1GΓ2G†], Γ1,2 ) i[Σ1,2 - Σ†1, 2] I ) (2e/h)
∫ dE T(E)[f1(E) - f2(E)]
(1)
where f1,2 are the contact Fermi functions. The NEGF prescription further allows us to formally include incoherent scattering in the device. In our present treatment, we include that (when necessary) using phenomenological, phaserandomizing Bu¨ttiker-probes.11,12 While the above transport formalism is quite general, the details of the various ingredients such as H, U, and Σ1,2 vary with varying geometry, device bandstructure, or contact 1804
Figure 2. Results for unreconstructed Si(100) DOS in a hybrid basis: (a) bulk DOS in (I) effective mass theory (EM) and (II) extended Huckel theory (EHT) with parameters fitted to match the bulk silicon bandstructure (ref 15). The surface DOS match in energy domain for (III) EHT and (IV) 6-31 g* bases. (b) The surface DOS match in real space as well, plotted here at the Fermi energy, a0 denoting the Bohr radius.
surface physics. In the following, we describe our theoretical models for these quantities, explaining in this context the significant theoretical challenges that a silicon substrate poses in contrast to metal contacts. Modeling the Silicon Substrate, Σ1(E). We have developed a method by which a bulk silicon bandstructure calculated using a suitable semiempirical scheme (such as effective mass (EM),13 tight-binding,14 or extended Huckel (EHT)15 theories) can be connected up with a detailed density functional (DFT) treatment of the molecule, without introducing spurious basis-transformation related artifacts at their interface. We accomplish this connection by solving a boundary value problem for the real-space Green’s function g(r b,r b′;E) matched on a surface silicon atom.16 We start with an appropriate semiempirical description of the bulk silicon bandstructure and use a three-dimensional real-space recursive formalism along the surface normal to get the semiempirical silicon surface Green’s function g.4 A slab of relaxed silicon layers of a known, optimized geometry can then be incorporated to precisely include the effect of surface reconstruction.17 The semiempirical surface Green’s function is matched in the ab initio basis set16 and the corresponding silicon self-energy Σ1 ) τgτ† obtained using the coupling matrix τ between the bulk and surface silicon atoms (the latter being incorporated into the device Hamiltonian). We thus have an ab initio description of the molecule and silicon surface chemistry, interfacing with a semiempirical treatment that quantitatively captures the bulk and surface properties of the silicon crystal (Figure 2). It is well known18 that unpassivated silicon has surface states in the band gap, as in Figure 2. Such silicon surface states would influence the STM current, provided they can be replenished from the bulk through inelastic processes occurring in the contact. We ignore such effects in this paper. In any case, these surface states are eliminated on hydrogenpassivation or on bonding the molecule to the surface silicon dimers.19 Modeling the tip, Σ2(E). In any STM measurement on a molecule, it is important to deconvolve the tip and the Nano Lett., Vol. 4, No. 10, 2004
molecular density of states (DOS) so that one can unambiguously identify features intrinsic to the molecule. For a known geometry of the contact, we can compute its self-energy with atomistic detail and even include parts of it directly within the device Hamiltonian;4 however, such a model is not useful at this point, given the lack of knowledge of the experimental tip structure and its position on the surface, as well as the robustness of the shape of the experimental I-V characteristics with variations in the tip structure.1 For a tungsten tip it is believed20 that the tunneling current is dominated by the d-electrons of loose atoms at its end. We model the tipsample interaction by evaluating the largest coupling matrix element in LDA/LANL2DZ between the styrene HOMO wave function and the d-electrons of a single tungsten atom sitting at varying distances from it. The coupling is then combined with an overall tip DOS ∼ 0.25 /eV to give the tip self-energy Σ2(E). We find that the coupling can vary considerably depending on the lateral position of the tip atom relative to the molecule, due to accidental symmetries in the overlap matrix elements. This needs further consideration both from theory and experiment, since the current level is influenced substantially by the STM tip. Modeling the Molecular Hamiltonian, H, and the Electrostatic Potential, U. In our calculations, the molecular Hamiltonian is computed with Gaussian 9821 using LDA/631g*. Molecular RTD action hinges on the bias dependence of the corresponding molecular levels and wave functions, the silicon band-bending, and the tip electrochemical potential relative to the bulk Si band edges (Figure 1). Under bias, the molecular levels move relative to the silicon band edge because (1) a substantial part of the applied bias drops across the tip-to-substrate gap due to the large dielectric constant of silicon relative to the molecule and the vacuum gap; and (2) the molecular field is poorly screened by the absence of metal-induced gap states (MIGS) that would normally exist with metal substrates.22 At a critical voltage determined by the electrostatics, the HOMO level in Figure 1 reaches the bulk silicon valence band edge, after which it enters the Si band gap and its transmission drops abruptly, leading to a prominent NDR in the I-V of the molecule. Note that the nontriVial part of the physics inVolVes the slippage of the leVels past the silicon band edge. Although one may expect the levels to be pinned to the substrate by strong chemisorption and by surface states, the NDR is established by the elimination of these surface states by the molecule tying up the dimer dangling bonds, as well as by the tip-induced molecular potential drop that drives the levels past the bulk band edge. Thus the self-consistent potential U is the most important part of the modeling. In our geometry, the applied potential divides among the molecule, the vacuum gap between the STM and the molecule, and the band bending in the substrate. The band bending affects both the electrostatic boundary conditions for Poisson’s equation and the quantum boundary conditions entering through the contact self-energy. Given that a density functional treatment of a 2 nm atomistic, doped silicon depletion layer is quite computationally prohibitive at this time, and given that the physics of the depletion layer is Nano Lett., Vol. 4, No. 10, 2004
expected to be dominated by the substrate doping and the work function difference with the STM material, we coarsegrain the electrostatic boundary conditions using a continuum approximation that is standard in silicon device physics (MEDICI23). The corresponding voltage drops across the silicon depletion layer and the vacuum gap are thereafter used as boundary conditions for a fully rigorous, atomistic treament of Poisson’s equation across the molecule, evaluated within Gaussian 98. The band-bending is further validated against calculations based on the depletion-width approximation,24 as well as surface photovoltage (SPV) measurements on hydrogen-passivated silicon.25 Band bending turns out, however, to be relatively unimportant for degenerately doped silicon, since a ∼1-2 nm wide depletion layer is quite transparent to electron tunneling, so that the NDR is determined by the bulk silicon band-edges rather than the surface potential. Results. We use our hybrid-basis formalism to obtain the coupling of a molecule to silicon for a given bonding geometry and a surface structure. The actual bonding geometries vary26 and are sometimes controversial,27 so we will adopt specific geometries that can be benchmarked with existing literature. Thermal effects are included through the lead Fermi functions, but not included in the molecular geometry, guided by the experimental observation of strong anchoring1 and limited molecular mobility. We start by simulating transport through styrene bonded to a p-doped H-passivated c(4 × 2) Si(100) surface, for which the geometry observed experimentally and from geometry optimization involves a single bond between the molecule and a surface silicon atom (Figure 4a). The isolated bonding is expected to reduce the role of reconstruction, which we ignore for this specific paper. We leave detailed description of reconstruction for future articles,17 with the observation that this will be important for clean Si(100) where the molecule straddles the entire silicon dimer. Our bonding geometry, LDA energy levels, and wave functions of styrene agree well with previously published first-principles results for styrene on Si(100):H.28 Figure 3 shows a color plot of the transmission under bias of styrene on p-Si, described within the EM model. The EM model is used for simplicity (Figures 3 and 4b,c) and describes the band-edge properties correctly (Figure 2), where most of our physics lies (a more detailed EHT model, Figure 4a, gives a similar result). For increasing positive substrate voltage, the molecular levels rise until a HOMO level reaches the silicon valence bandedge EV and its transmission gets cut off thereafter. This leads to an NDR in the I-V, calculated for an STM tip almost touching the molecule.29,30 For negative bias, there is no such molecular NDR, since the HOMO levels move deeper into the valence band and the LUMO levels lie outside the bias window. An increased tip-to-sample vacuum gap (2 Å) causes a smaller voltage drop across the molecule, postponing the onset of the NDR peaks (dashed line). The NDR peak position from a level E0 is roughly given by (EV - E0)/η, where η denotes the average molecular potential under unit applied bias (η ∼ 0.5 for an STM touching the molecule, decreasing approximately linearly with increasing 1805
Figure 4. Calculated I-V curves for styrene on p-doped H:Si(100) with an STM almost touching the molecule, using (a) EHT and (b) EM basis for Si. The calculations are non self-consistent at this time (ref 29). Increasing the air gap to 2 Å (dashed line) decreases the current and postpones the NDR. (c) I-V for TEMPO on p-Si(100) (EM) and (d) styrene on n-doped H:Si(100) (EM) (ref 31). The NDR reverses polarity on reversing doping, although in calculations for this specific geometry, we had to lower the LUMO levels artificially to bring the NDR into the bias window.
Figure 3. Calculated transmission T(E, V) (color plot) through styrene on hydrogen-passivated, degenerately doped p-silicon (NA ) 5 × 1019 cm-3). Styrene transmission peaks shift under bias and disappear when they reach the valence band edge EV. The transmissions are shown below at specific bias voltages near the NDR to clearly show the vanishing of the conducting peaks. Voltage values are (a) 1 V, (b) 2.5 V, and (c) 3.4 V.
vacuum gap). The polarity dependence of the NDR in the experiments by Guisinger et al.1,2 on TEMPO and cyclopentene on p+-Si is in agreement with the mechanism considered here; however, the location and nature of the peaks need further work to identify the experimental mechanism unambiguously, especially for n-type substrates.31 For molecules such as styrene on n-silicon,9 NDR arises from LUMO levels at negative substrate bias. Figure 4d shows the I-V for n-styrene (ND ) 5 × 1019 cm-3). The LDA LUMO levels for styrene are a few volts higher than the CB edge,28 pushing the first NDR peak to a much higher negative bias voltage than is observed experimentally. Although the polarity reversal is consistent with our theory, the quantitative mismatch between theoretical and experimental peak values is something that clearly requires further theoretical work. The mismatch can be traced back to the molecular HOMO-LUMO gap being much larger than the silicon band gap. Electron addition levels are usually controversial and could differ from theoretical predictions due to various effects associated with charging, correlation, or image effects.20 In addition, there do exist reports showing a significant reduction in molecular HOMO-LUMO gap 1806
upon adsorption on a doped substrate,32 although the theoretical mechanism for such a drastic reduction is still unclear. The origin of NDR in our simulations is qualitatively different from the NDR mediated by tunneling through localized atomic states arising from a sharp structure in the tip DOS.33 NDR due to a tip-to-level resonance would show up either for both bias directions for both p and n-type samples (if the tip structure interacts with both HOMO and LUMO levels) or for one bias direction for only one of the samples (if it interacts with just one of them). Furthermore, such NDRs are likely to be observed even for the bare silicon surface, in contrast with our theoretical simulations as well as experiments from the Hersam group. Although it is hard at this time to rule out alternate mechanisms for NDR, such as due to conformational changes of the molecule driven by large fields, the consistent observation of a clear polaritydependent one-sided NDR for molecules on silicon, for positive substrate bias on p-Si and negative on n-Si for various tips1,2 seems quite suggestive of a mechanism that is mediated by interactions between the molecular levels and the silicon band-edge. Furthermore, the NDR is expected to be attenuated for monolayers since parts of the tip at different distances from the molecules produce different voltage drops across the molecular layer, leading to an averaging of the peak positions. Such an attenuation for a monolayer has also been observed by the Hersam group.34 In our simulations, the multiple NDR peaks arise from the multiplicity of levels sequentially crossing the relevant band edge,35 which effectively acts as a filter and produces peaks in the current-voltage, rather than in the conductancevoltage curves. The precise identity of the levels in the Nano Lett., Vol. 4, No. 10, 2004
specific experiment1 is currently under investigation.36 The peak positions are expected to vary with varying tip-sample separation, consistent with experimental variations. The polarity-dependent NDR opens up the possibility of employing a combination of p- and n-type experiments on a single molecule to map out the relevant addition and removal levels responsible for the NDR. For instance, the experimental peak voltages ∆n,p and Fermi energies EFn,Fp for n- and p-Si can be used to infer a conductance gap ≈ η(∆n + ∆p) + (1 η)(EFn - EFp), taking into account tip-substrate bandalignment effects at equilibrium. Self-consistent charging can bring additional physics through band-filling, such as generating a hysteresis in the I-V due to the abrupt drop in substrate coupling at the NDR position. Such hysteresis effects are observed in RTDs37 but seem to be compromised by the low charge transfer in these molecules near the band edge. For realistic device parameters, our self-consistent Poisson equation does generate a hysteresis in the I-V; however, the hysteresis is very weak, and is expected to be washed out by scattering. Acknowledgment. We are grateful to M. Hersam for liberally sharing their unpublished experimental results with us. We also thank M. S. Lundstrom, M. Hybertsen, C-F. Huang, and D. Janes for useful discussions and T. Kazmi for help with generating the figure for the table of contents. This work was supported by ARO-DURINT, NSF, and the Semiconductor Technology Focus Center on Materials, Structures and Devices. Author list is reverse alphabetical; all authors contributed equally. References (1) Guisinger, N. et al. Nano Lett. 2004, 4, 55. (2) Guisinger, N. et al. Nanotechnology 2004, 15, S452. (3) Emberly, E. G.; Kirczenow, G. Phys. ReV. B 1998, 58, 10911. Yaliraki, S. N.; Ratner, M. A. J. Chem. Phys. 1998, 109, 5036. Taylor, J. et al. Phys. ReV. B 2001, 63, 245407. Di Ventra, M. et al. Phys. ReV. Lett. 2000, 84, 979. Ghosh, A. W. et al. cond-mat/0212166. (4) Damle, P. S.; Ghosh, A. W.; Datta, S. Phys. ReV. B Rapid Commun. 2001, 64, 201403 R. Damle, P. S.; Ghosh, A. W.; Datta, S. Chem. Phys. 2002, 281, 171. (5) Reed, M. A., et al. Science 1997, 278, 252. Chen, J. et al. Science 1999, 286, 1550. Porath, D., et al. Nature 2000, 403, 635. Reichert, J., et al. Phys. ReV. Lett. 2002, 88, 176804. Cui, X. D., et al. Science 2001, 294, 571. Tian, W., et al. J. Chem. Phys. 1998, 109, 2874. (6) Wolkow, R. A.; Jpn. J. Appl. Phys. 1 2001, 40, 4378. Hersam, M. C., et al. Nanotechnology 2000, 11, 70. Lenfant, S., et al. Nano Lett. 2003, 3, 741. (7) Mathews, R. H., et al. Proc. IEEE 1999, 87, 596. (8) Seabaugh, et al. IEDM’98 Technol. Dig.; pp 429-432. (9) Experimentally, however, styrene tends to desorb under bias for p-Si substrates,1 while cyclopentene and TEMPO do not. (10) Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge University Press: Cambridge, 1995.
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