Optimizing surface defects for atomic-scale electronics: Si dangling bonds

TL;DR

This paper demonstrates how to engineer silicon surface dangling bonds using first-principles calculations, showing how sample thickness and strain can tune their electronic properties for atomic-scale electronic and quantum devices.

Contribution

It introduces a method to control dangling bond defect states on silicon surfaces through thickness and strain, unifying various experimental and theoretical findings.

Findings

Ultrathin silicon samples create isolated impurity states in the band gap.

Strain enhances the isolation of dangling bond states from bulk bands.

The study provides a unified interpretation of existing literature on silicon dangling bonds.

Abstract

Surface defects created and probed with scanning tunneling microscopes are a promising platform for atomic-scale electronics and quantum information technology applications. Using first-principles calculations we demonstrate how to engineer dangling bond (DB) defects on hydrogenated Si(100) surfaces, which give rise to isolated impurity states that can be used in atomic-scale devices. In particular we show that sample thickness and biaxial strain can serve as control parameters to design the electronic properties of DB defects. While in thick Si samples the neutral DB state is resonant with bulk valence bands, ultrathin samples (1-2 nm) lead to an isolated impurity state in the gap; similar behavior is seen for DB pairs and DB wires. Strain further isolates the DB from the valence band, with the response to strain heavily dependent on sample thickness. These findings suggest new methods for tuning the properties of defects on surfaces for electronic and quantum information applications. Finally, we present a consistent and unifying interpretation of many results presented in the literature for DB defects on hydrogenated silicon surfaces, rationalizing apparent discrepancies between different experiments and simulations.

Full text

The ability to engineer semiconducting devices at the atomic scale is key to achieving further miniaturization of electronics, and to using the quantum nature of point defects for quantum information applications. One promising atomic-scale fabrication method employs scanning tunneling microscopy (STM) to create and manipulate defects on semiconducting surfaces Zwanenburg et al. (2013). For example, dangling bonds (DBs) have been created and successfully manipulated on hydrogen-terminated Si(100) surfaces by desorbing individual H atoms from the substrate Lyding et al. (1994). Ensembles of DBs on silicon surfaces are now widely used to create atomically-precise systems of defects Haider et al. (2009); Pitters et al. (2011); Bellec et al. (2013); Schofield et al. (2013); Ye et al. (2013); Taucer et al. (2014); Kolmer et al. (2014); Labidi et al. (2015); Kawai et al. (2016); Rashidi et al. (2016); Hitosugi et al. (1998); Soukiassian et al. (2003); Mayne et al. (2006), leveraging expertise with the fabrication of silicon devices, including the ability to produce clean and regular hydrogen-terminated surfaces.

Numerous experiments have demonstrated many attractive properties and potential applications of DBs on H:Si(100). These defects interact over next-nearest neighbor distances Haider et al. (2009); Pitters et al. (2011); Schofield et al. (2013), and the charge of individual DBs can be reversibly manipulated, with given charge states persisting for hours Bellec et al. (2013). In addition, these DBs display negative differential resistance, potentially providing a new component for atomic-scale electronic circuitry Rashidi et al. (2016). Theoretical work has suggested that pairs of DBs may be used to create a charge qubit Livadaru et al. (2010). Furthermore, DBs may be assembled into specific patterns for electronics or quantum simulations Schofield et al. (2013); Soukiassian et al. (2003); Ample et al. (2011); Kepenekian et al. (2014), including one-dimensional conducting or semiconducting wires Hitosugi et al. (1999); Naydenov and Boland (2013); Ye et al. (2013). They further serve as a starting configuration for atomically-precise dopant placement Schofield et al. (2003); Ruess et al. (2004); Fuechsle et al. (2012); Weber et al. (2012); Morley (2014). Hence tuning and manipulating the properties of DBs, e.g., charge states, may lead to a promising strategy to build a flexible atomic-scale platform of defects for electronic and quantum information technology applications.

In this work, using the results of first-principles calculations we propose ways to realize DB defect states on H:Si(100) with energies within the electronic gap of bulk Si; in particular we show how to tune sample thickness and strain to obtain desired energy and charge states. While doing so, we also address existing controversies present in the literature on the properties of DB states on Si surfaces. We present a consistent interpretation of previous results, and use advanced methods to ensure our findings are robust. We first show how the thickness of the Si sample can be manipulated to alter the electronic properties of the neutral Si DB state, and we demonstrate that a stable positively-charged DB state is accessible only in thin ($1.2\text{\,}\mathrm{nm}\text{/}$) Si samples. We also present similar effects for multiple DB systems. We then turn our attention to the effect of biaxial strain, showing that the electronic response of defects to strain depends significantly on the slab thickness. We propose that by combining thickness and strain, one may engineer the properties of neutral DB defects for use in atomic-scale electronics.

Methods: We carried out density functional theory (DFT) Hohenberg and Kohn (1964); Kohn and Sham (1965) calculations with plane-wave basis sets and norm-conserving pseudopotentials Troullier and Martins (1991); Ceresoli using the Quantum ESPRESSO package Giannozzi et al. (2009); qe_ . We modeled Si DBs on an H-terminated Si(100) slab periodically repeated in two directions and having a finite number of layers in the third, with vacuum separating periodic images. All neutral DB calculations were spin-polarized. Atomic geometries were optimized until forces on the atoms were less than $0.013\text{\,}\mathrm{eV}\text{/}\mathrm{\SIUnitSymbolAngstrom}$.

We used the gradient-corrected exchange-correlation functional developed by Perdew, Burke, and Ernzerhof (PBE) Perdew et al. (1996), as well as hybrid functionals Kümmel and Kronik (2008). In particular we adopted a dielectric-dependent hybrid with the fraction of exact exchange $\alpha=0.085\approx 1/\epsilon_{\infty}^{\mathrm{Si}}$, shown to reproduce accurately the electronic properties of bulk Si Skone et al. (2014). We also used the Heyd-Scuseria-Ernzerhof (HSE) functional Heyd et al. (2003, 2006) to compare with previous work. In addition, for selected configurations we performed many-body perturbation theory (MBPT) calculations Hedin (1965); Hybertsen and Louie (1986); Aryasetiawan and Gunnarsson (1998) at the $G_{0}W_{0}$ level using the WEST software package Govoni and Galli (2015); Scherpelz et al. (2016); wes . Within $G_{0}W_{0}$ the exchange-correlation energy entering DFT is replaced by an electronic self-energy calculated using the screened Coulomb interaction and the Green’s function. WEST uses spectral decomposition techniques Govoni and Galli (2015); Wilson et al. (2008, 2009); Nguyen et al. (2012); Pham et al. (2013) and methods based on density functional perturbation theory Baroni et al. (2001) to optimize calculations for large systems Govoni and Galli (2015).

Details of geometries, calculation parameters, convergence tests, and identification of DB states, can be found in the Supplemental Material (SM).

Results: Multiple computational results have been reported in the literature for the singly-occupied, neutral DB (DB0) state, relative to the VBM of Si: $-0.3\text{\,}\mathrm{eV}\text{/}$ Wieferink et al. (2010), $0.013\text{\,}\mathrm{eV}\text{/}$ Schofield et al. (2013), $0.2\text{\,}\mathrm{eV}\text{/}$ Blomquist and Kirczenow (2006), $0.35\text{\,}\mathrm{eV}\text{/}$ Livadaru et al. (2010); Haider et al. (2008), $0.36\text{\,}\mathrm{eV}\text{/}$ Ye et al. (2013), and $0.42\text{\,}\mathrm{eV}\text{/}$ Raza (2007) (other calculations Bellec et al. (2013); Komsa and Pasquarello (2013); Kawai et al. (2016) only addressed doped systems and/or charged DBs) 111Only the lowest two values from Refs. Wieferink et al. (2010); Schofield et al. (2013) are explicitly spin-polarized; the calculations that cluster at 0.35-0.42 eV are thus unlikely to be physically accurate.. These results differ quantitatively and qualitatively: indeed, a defect state located in energy above the VBM is expected to be well-isolated electronically, whereas one below may instead hybridize with other electronic states in the material and hence may not be amenable to manipulation.

In order to rationalize the various literature values for the DB0, we calculated its electronic properties for many model slabs, differing by the number of layers and supercell lattice constant. We found that the choice of lattice constants in the plane perpendicular to the surface primarily influenced the degree of dispersion of the DB state (see SM for details). We observed instead a much more pronounced dependence of the nature of the DB state on the thickness of the slab, as shown in Figure 1(a). Its energy relative to the vacuum energy of the supercell model is roughly constant. However, the positions of the VBM and conduction band minimum (CBM) vary significantly with the number of layers in the slab. In general quantum confinement leads to a larger bandgap whose convergence toward the bulk value is very slow as a function of the slab layer number Li and Galli (2010); Fischetti et al. (2011). A similar dependence on thickness was recently found for a bare Si(100)-$p(2{\times}2)$ surface Sagisaka et al. (2017).

Figure 1(a) shows that for a slab 8 layers thick ($\approx$1.2\text{,}\mathrm{nm}\text{/}) the DB energy is $0.09\text{\,}\mathrm{eV}\text{/}$ above the VBM energy, while for a 16-layer slab ($\approx$2.3\text{\,}\mathrm{nm}\text{/}) its energy is $0.05\text{\,}\mathrm{eV}\text{/}$ to $0.19\text{\,}\mathrm{eV}\text{/}$ below the VBM energy. The DB0 state is well-isolated for 8 or fewer layer slabs, while it is mixed with bulk Si states for 16 or more layers, as shown in Figure 1(c-d). To verify these findings are robust with respect to the level of theory used, we carried out additional calculations using dielectric-dependent hybrid functionals and MBPT. Figure 1(b) shows that the position of the occupied DB state relative to the VBM is nearly the same at all levels of theory. Hybrid and $GW$ calculations significantly correct the bandgap energy found at the PBE level, but leave the DB state positions relative to the VBM and CBM unchanged.

Hence we conclude that for Si samples $1.2\text{\,}\mathrm{nm}\text{/}$ or thinner, the neutral singly-occupied DB state falls within the bulk gap, while for samples $2.3\text{\,}\mathrm{nm}\text{/}$ or thicker, it is hybridized with bulk states and resides below the VBM. We expect the DB0 state to have a much shorter coherence time for thickness $>$2.3\text{,}\mathrm{nm}\text{/}$$, impacting its behavior in quantum information applications. However, in other applications, its hybridization with bulk states of thick slabs may be beneficial by facilitating long-range interactions between point defects.

We note that these DB properties are different from those of Si DBs at a Si/\ceSiO2 interface, for which the energy of the neutral defect is found by electron paramagnetic resonance to reside in the gap of bulk Si Lenahan and Conley Jr. (1998). This difference is presumably due to the significantly different environment surrounding DBs in the two systems Raza (2007). We also stress that the behavior above is not due to a change in the net magnetization density (which is not significant as a function of layer thickness), but rather is due to changes in the energy level of the singly-occupied DB, which is an isolated defect state for thin slabs but a resonant defect state for thick slabs.

While understanding the properties of the neutral DB is important for potential quantum information applications, charge transition energy levels (CTLs) are the quantities of interest for scanning tunneling spectroscopy observations Bellec et al. (2013); Ye et al. (2013); Taucer et al. (2014); Kolmer et al. (2014); Labidi et al. (2015); Kawai et al. (2016); Rashidi et al. (2016), and applications for electronics Bellec et al. (2013); Rashidi et al. (2016) or charge qubits Livadaru et al. (2010). We obtained CTLs by computing total energies of different charge states in their respective optimized geometries (thus calculating adiabatic transitions), taking into account corrections for the Coulomb interaction of periodic images, and alignment of the electrostatic potential between configurations Freysoldt et al. (2014). For surfaces, correction methods used in bulk systems are not applicable due to the large variation in dielectric constant between the bulk and vacuum. We followed the prescription suggested in Ref. Komsa and Pasquarello (2013). Briefly, a sawtooth electric field is used to compute the $z$-dependent dielectric constant, from which a periodic electrostatic model of the charged defect is constructed. The electrostatic energy is then calculated using finite size scaling by extrapolating the energy of the model computed for cells of increasing sizes. As we were interested specifically in the properties of thin Si slabs, we kept the slab height constant during extrapolation.

Calculations at the PBE level of theory were performed to check for convergence, and consistent results were found, within $\pm$0.1\text{,}\mathrm{eV}\text{/}$$, when varying vacuum length of the supercell by a factor of 2 and horizontal supercell area by a factor of 2.7. Figure 2 shows our results using the dielectric-dependent hybrid functional. Qualitatively, the PBE results are similar to those in Figure 2, with a $0.2\text{\,}\mathrm{eV}\text{/}$ decrease in the $({+}/0)$ CTL relative to the VBM (see SM).

Figure 2 shows that for thick samples ($>$2.3\text{,}\mathrm{nm}\text{/}$$), only the $(0/{-})$ CTL falls within the bulk Si bandgap. This result is consistent with the experimental observation showing long lifetimes of both the DB0 and DB*-* charge states, when the DB is appropriately charged by an STM tip which is then removed Bellec et al. (2013). No long-lived DB*+* state was detected experimentally, even though a $p$-type Si sample was used Bellec et al. (2013). Interestingly, Figure 2 also shows that very thin samples may exhibit long-lived DB*+* states without requiring any other perturbations, making all three charge states easily accessible. Such a system may provide a flexible platform for quantum information technology applications.

Arrangements of multiple DBs lead to additional possibilities for atomic-scale electronics. We considered two prototypical multiple-DB systems, and we found that they exhibit the same properties as a single DB, as a function of thickness. Figure 3(a) shows the band structure of a neutral DB pair Livadaru et al. (2010); Pitters et al. (2011); Schofield et al. (2013). Its geometry (Figure 3(b)) is an example of that proposed for charge qubits Livadaru et al. (2010); Pitters et al. (2011). For a 4-layer slab all DB states are well-separated, while for a 16-layer slab the occupied DB states become resonant with the valence band. The presence of this resonance may lead to stronger interactions between bulk and DB states; the overall larger gap for a 4-layer system should also improve addressability of the DB pair using mid-infrared lasers Shaterzadeh-Yazdi et al. (2014).

Figure 3(c) shows the band structure of the anti-ferromagnetic DB wire in Figure 3(d) Hitosugi et al. (1999); Lee et al. (2008, 2009); Robles et al. (2012); Kepenekian et al. (2013); Naydenov and Boland (2013); Ye et al. (2013); Engelund et al. (2016). For a neutral DB wire on a 4-layer slab, both the occupied and unoccupied one-dimensional (1D) bands lie within the gap, and thus the wire may conduct either electron or hole states under suitable bias. In contrast, for a 16-layer slab, only the unoccupied 1D band lies within the gap, while the occupied 1D band is resonant with the bulk Si states. Thus, for 16 layers, hole conduction would be expected to occur through both bulk and wire states when a suitable bias is applied, removing the 1D nature of the conductivity.

A recent theoretical study raised the intriguing possibility of using strain to isolate surface states of a bare Si(100) surface, showing that the VBM is lowered in energy when biaxial tensile strain is applied in the horizontal directions Zhou et al. (2013). However, this result is not consistent with those of theoretical and experimental investigations of bulk Si under biaxial tensile strain, showing that the VBM energy increases and the bulk bandgap decreases Munguía et al. (2008); Richard et al. (2003). Our results indicate that for standard (thick) silicon surfaces, strain will not isolate surface states, in contrast to the conclusions of Ref. Zhou et al. (2013). Instead, only for thin Si slabs can strain be used to isolate surface states, a result of the remarkably different response to strain for thin slabs compared to bulk systems. Figure 4(a) shows this difference in response: the VBM position as a function of strain shows a qualitatively different trend for thick ($\gtrsim$4\text{,}\mathrm{n}\mathrm{m}$$) slabs, where it increases with strain, and thin (10-layer, $1.5\text{\,}\mathrm{nm}\text{/}$) slabs, where it decreases with strain to a thickness-dependent minimum. As a result, for thick slabs the fundamental gap remains close to that of the bulk, but for thin slabs tensile strain leads to an increase of the gap relative to that of the bulk. An unstrained 10-layer slab has a fundamental gap $0.4\text{\,}\mathrm{eV}\text{/}$ larger than that of bulk Si. In contrast, under 3% biaxial tensile strain its fundamental gap is $0.8\text{\,}\mathrm{eV}\text{/}$ larger than that of the bulk.

Thus, while in thick slabs biaxial tensile strain would not be expected to aid in isolating DB states, in very thin slabs, the opposite is true, as shown in Figure 4(b). A state which is not well-isolated in a $1.5\text{\,}\mathrm{nm}\text{/}$ unstrained slab becomes isolated for strains of 1% or more. For direct comparison with Ref. Zhou et al. (2013), results in Figure 4 used the HSE hybrid functional; calculations with PBE showed the same trend as a function of strain (see SM). Note that Ref. Zhou et al. (2013) used a 10-layer slab model, effectively reporting results applicable only to thin slabs.

In summary, we have proposed how to engineer the properties of DBs on hydrogenated Si surfaces by varying sample thickness and applied stress. We have shown that the single particle energy and wavefunction of DBs on H:Si(100)-$(2{\times}1)$ may be more readily isolated from those of bulk states in thin samples ($<$1.2\text{,}\mathrm{nm}\text{/}) than in bulk-like slabs. Specifically, in thin samples, the neutral DB state is well above the VBM, and three charge states may be stabilized. In thick ($>$2.3\text{\,}\mathrm{nm}\text{/}) samples, the neutral DB state is instead hybridized with bulk states, and the positively charged DB is not stable; for bulk samples the neutral DB state is about $0.3\text{\,}\mathrm{eV}\text{/}$ below the VBM. We verified that our results are robust with respect to the level of first-principles theory used, including semi-local and hybrid functionals and many-body perturbation theory.

Dangling bond pairs and wires showed the same response to sample thickness. Notably, thin samples allow hole conduction along isolated DB wires, whereas in thick samples conduction would also occur through the bulk. Additionally, we found that in thin samples biaxial tensile strain will further isolate the DB energy from that of the VBM. However, strain is not helpful in isolating states in thick Si samples.

We emphasize the importance of carrying out accurate calculations, numerically converged and at a high level of theory, in order to determine the properties of isolated DBs. Although numerous experimental STM studies have been performed, both tip-induced band bending and non-equilibrium charging did not allow for a clear extrapolation of results to isolated DB configurations Bellec et al. (2013); Schofield et al. (2013); Ye et al. (2013); Taucer et al. (2014); Kolmer et al. (2014); Labidi et al. (2015); Kawai et al. (2016); Rashidi et al. (2016). Furthermore, recent work has shown that nearby dopants can affect the behavior of DBs Labidi et al. (2015), which further complicates the interpretation of experimental findings. Our study of electronic properties as a function of film thickness was able to reconcile apparent discrepancies found in published results, which were reported for different numbers of layers in the slabs and sometimes interpreted as representative of bulk samples (we estimate that 60 layers are necessary for calculations to be representative of thick, bulk-like samples).

A question remains on the experimental realization of the thin films proposed here as promising platforms. Si films as thin as $3\text{\,}\mathrm{nm}\text{/}$ have been reported Liu et al. (2002); strained Si-on-insulator samples less than $10\text{\,}\mathrm{nm}\text{/}$ thick have also been fabricated Munguía et al. (2012). While the 1-2 nm slabs considered here may require new techniques, their experimental realization appears possible in the near future. Finally, we expect the results found here for DBs may be valid for several other defects when placed in thin Si slabs. This includes many defects used in quantum information applications, such as isolated phosphorous Saeedi et al. (2013); Pla et al. (2013), boron Salfi et al. (2016), bismuth Mohammady et al. (2012); Wolfowicz et al. (2013), or selenium Lo Nardo et al. (2015); Morse et al. (2016) dopants, as well as patterned surface systems Schofield et al. (2013); Soukiassian et al. (2003); Ample et al. (2011); Kepenekian et al. (2014). Indeed the electronic properties of the DBs change as a function of thickness due to the change of the VBM and CBM themselves, not because of a substantial shift of the defect level. Hence the combination of strain and thickness proposed here to isolate DB defects and stabilize multiple charge states should be generalizable to other types of defects.