Unveiling Stable Group IV Alloy Nanowires via a Comprehensive

Aug 28, 2013 - Group IV nanowires (NWs) such as silicon and germanium nanowires (SiNWs ... electronic properties, e.g., band gap, can be tuned by vary...
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Unveiling Stable Group IV Alloy Nanowires via a Comprehensive Search and Their Electronic Band Characteristics Man-Fai Ng* and Teck L. Tan* Institute of High Performance Computing, Agency for Science, Technology, and Research, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore S Supporting Information *

ABSTRACT: By means of density functional theory calculations, the cluster expansion method, and Monte Carlo simulations, we identify the stable spatial configurations (ground states) for ⟨100⟩ CSi, GeSi, and SnSi alloy nanowires (NWs) across compositions. In particular, we find that stable configurations of GeSiNWs and SnSiNWs exhibit core−shell segregation tendencies, while those of CSiNWs favor ordering. Moreover, we show compositional ranges where the band gaps are expected to vary linearly with composition, allowing predictable band gap fine-tuning. We also predict composition ranges where the spatial separation of near-band gap states are imminent, making it possible for electron−hole charge separation. By addressing both the issues of stability and the compositional trend of electronic band structure, our work should prove useful for designing alloy NWs of smaller dimensions. KEYWORDS: Alloy nanowire, silicon, germanium, tin, cluster expansion, Monte Carlo, density functional theory, band structure

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the alloy NWs; core−shell, zinc-blende (mixed), abrupt (clustered), and random configurations are four common atomic configurations used in first-principles theoretical models of group IV binary NWs, which demonstrate very different electronic properties.19−28 However, a more comprehensive evaluation of alloy NW structural stability is found lacking in the literature. Previous compositional studies of GeSiNWs utilize idealized models of fixed configuration, e.g., core−shell configurations at discrete compositions20,21 or random configurations.23 To aid in the design of alloy NWs that are structurally stable across alloy composition, it is important to systematically evaluate the relative stability of the myriad alloy configurational possibilities, not just a few idealized configurations, and identify the stable ones for property predictions. One approach is to employ cluster expansion (CE) Monte Carlo (MC) simulations29−37 to carry out a comprehensive and direct search for the stable configurations of the group IV alloy NWs. The CE method allows stable configurations at all compositions to be identified and their electronic properties compared, lifting the restriction on fixed configuration type and allowing the incorporation of alloy stability into our studies. By constructing reliable CE Hamiltonians from hundreds of density functional theory (DFT) calculated energies, we explore the configuration−stability relationship of ⟨100⟩ CSiNWs, GeSiNWs, and SnSiNWs (at a fixed diameter) across composition. We identify ground state configurations for the NWs; stable configurations in GeSiNWs and SnSiNWs are core−shell-like, while those of CSiNWs are ordered. We further

roup IV nanowires (NWs) such as silicon and germanium nanowires (SiNWs and GeNWs, respectively) have emerged into an important class of nanomaterials over the past decade. Together with their unique electronic and chemical properties stemming from quantum confinement and high surface-to-volume ratio, the compatibility of these NWs to current silicon electronics industry and bioapplications makes it special. In fact, the NWs have been utilized successfully in many areas such as advanced electronics1−4 and energy related research.5,6 To further extend their applicability, these single-element NWs may be alloyed with other elements to fine-tune their electronic properties. In fact, group IV alloy NWs are increasingly being researched upon recently because their electronic properties, e.g., band gap, can be tuned by varying the composition ratio7 and the interface between the alloy materials.8 Furthermore, alloying would provide an alternative to property tuning via size effect,9 especially in cases where minimal variation in NW size is critical. Various synthesis techniques such as laser ablation10 and vapor−liquid−solid chemical vapor deposition (VLS-CVD)11−14 have been used successfully to grow group IV alloy NWs (e.g., GeSiNWs and CSiNWs) with the desired composition via controlling the precursor gas types, pressure, and temperature. Applications of alloy NWs to thermoelectric devices,15 field-effect transistors16,17 and pressure sensors18 have also been suggested. While the group IV alloy NWs obtained from experiments are usually expressed in terms of composition ratios, e.g., GexSi1−x where 0 ≤ x ≤ 1, the exact distribution of the elements within the NW are difficult to determine. However, the sitespecific ordering of the atoms strongly impacts the properties of © 2013 American Chemical Society

Received: August 9, 2013 Published: August 28, 2013 4951

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Figure 1. Cross-section views of the NWs: C24H24, Si24H24, Ge24H24, and Sn24H24, respectively, and selected stable alloy NW configurations with ⟨100⟩ growth orientation. The numbers 0, 4, 8, 12, 16, 20, and 24 indicate the total number of Si atoms in each configuration. Atoms lying within the red and yellow zones are defined as the core and the shell atoms, respectively. Black, orange, green, gray, and white balls represent C, Si, Ge, Sn, and H atoms, respectively.

In the CE method,42,43 the energy of each alloy NW configuration, σ, is expanded in terms of cluster (cl) correlation functions, ϕcl(σ) = Πk∈clξk. The CE Hamiltonian29,30,44 is written as

determine the electronic band characteristics of the stable configurations and show how the electronic band gaps vary with alloy composition. Interestingly, we discover that for GeSiNW and SnSiNW, the variation of band gap with composition is linear at some composition ranges, which to our best knowledge has not yet been reported in the literature for fixed size NW. Examining the electronic density of states, we discuss the possibility of (electron−hole) charge separation within the stable NW configurations. The present work illustrates how one could efficiently and reliably predict stable alloy NW configurations, enabling us to characterize NW electronic properties across composition. Geometry optimizations are performed using DFT within the generalized gradient approximation (GGA) in the form of Perdew, Burke, and Ernzerhof exchange-correlation functional38 implemented in Vienna Ab-initio Simulation Package (VASP, version 4.6).39,40 The core−valence interaction is described by the projector augmented wave (PAW) method.41 The cutoff energy for the planewave expansion is set at 400 eV for GeSiNWs and SnSiNWs and 450 eV for CSiNWs. Monkhorst−Pack sampling with 1 × 1 × 8 k-point grids is used. NWs are constructed by cleaving the bulk Si along the ⟨100⟩ growth orientation; various binary alloy configurations are then constructed by substituting Si atoms with C, Ge, or Sn atoms at the desired positions. The NW, consisting of 24 atomic (substitutional) sites and 24 H atoms passivating the outer shell, is put in a large unit cell with more than 12 Å vacuum spacing in lateral x- and y-directions to minimize spurious interactions between the neighboring images. The NW axis lies along the z-direction; the unit cell in the zdirection is 4 atomic-layer thick (around 4 to 7 Å depending on the alloy system, see Supporting Information). The alloy NWs and the unit cell lattices are fully relaxed until the absolute value of the forces acting on each atom is less than 0.01 eV/Å. Band structure and density-of-states (DOS) calculations are performed on the self-consistent charge densities, using 1 × 1 × 19 k-point grids for DOS calculations. Even though the DFT method underestimates the true band gap, we note that the calculations are sufficiently reliable for predicting band gap trends as is done elsewhere.19−22,25−28

E CE(σ ) =

∑ Vclϕcl(σ ) cl

(1)

where σ is a vector of {ξ1,ξ2,...,ξn} denoting the occupation of each binary alloy NW configuration, where ξk takes the value of 1 if site k of the NW is occupied by a Si atom and 0 otherwise (H atoms used for passivation are excluded). Vcl is called the effective cluster interaction (ECI). In practice, all but a finite number of ECI are close to zero; thus, a properly truncated CE Hamiltonian will predict energies accurately. Only symmetry distinct ECI in the truncated CE needs evaluation, where a finite learning set of DFT energies is utilized. For each of the three alloy NW systems, an independent CE is constructed using DFT calculated energies, E DFT(σ). We use the Thermodynamic Toolkit (TTK)29,30,44,45 for constructing our CE and ground state search. TTK creates a list of symmetryunique clusters in the NW that are ranked according to a physical hierarchy; clusters involving fewer sites and shorter spatial extent are physically more important (i.e., pair interactions are more significant than triplet interactions etc., and shorter range pairs are more important than longer ranged pairs). Truncated CEs are constructed from DFT energies of relaxed NW configurations via structural inversion. Only CEs that are locally complete are considered,46 where each cluster in the truncated CE has all its subclusters included. A truncated CE with high predictive capability is constructed from a large cluster pool using leave-one-out cross-validation (CV1) as the selection criterion.47 The search for ground state configurations for each alloy NW system is conducted iteratively. An initial learning set of ∼128 high symmetry configurations is used to construct the CE, and via simulated annealing using Monte Carlo,44 ground states at each possible alloy composition are predicted. Any new configurations not in the learning set are evaluated by DFT, and the energies are added to the learning set. Subsequent CE learning sets thus contain a higher proportion of low-energy 4952

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configurations. The above iterations stop when the final CE does not predict new configurations. The final ECI and cluster sets for each alloy NW are given in the Supporting Information. The CV1 errors for GeSiNWs, SnSiNWs, and CSiNWs are 0.1, 0.4, and 4.0 meV per atom, respectively. Shown in Figure 1 are the optimized hydrogenated ⟨100⟩ CNW, SiNW, GeNW, and SnNW with diameters of 0.80, 1.20, 1.26, and 1.44 nm, respectively; optimized lattice constants along z are 0.39, 0.57, 0.59, and 0.67 nm, respectively. We shall focus on the alloying pattern along the radial direction, and the core and shell regions of the NW are shown in Figure 1. We show in Figure 1 selected stable NW configurations at different composition ratios for each of the three alloy systems. The complete list of stable configurations is presented in the Supporting Information. Importantly, stable configurations of GeSiNW and SnSiNW show core−shell type arrangements, while stable CSiNW configurations exhibit an ordered arrangement with unlike atoms forming neighbors. The DFT formation energies of various alloy configurations for GeSiNWs, SnSiNWs, and CSiNWs are shown in Figure 2. For each alloy system, X−Si (where X is Ge, Sn, or C), the formation energy19 of a configuration, σ, containing m X atoms and n Si atoms is defined as 1 ⎡ DFT m EDFT(X ) ⎢ E (σ ) − m+n N⎣ ⎤ n EDFT(Si)⎥ − ⎦ m+n

E f (σ ) =

(2)

For our NW unit cell, m + n = 24 (substitution sites) and N = 48 (total atoms including H). The Ef of initial guess configurations are higher than those obtained subsequently from simulated annealing. The ground state hull is constructed by joining the ground states (denoted by circles); it is energetically favorable for states above the hull to phase separate into the adjacent ground states at T = 0. At finite T, configurations above the ground state hull are accessible, with those closer to the hull being assessed more frequently. The Ef of random solutions (obtained from CE) are positive at most compositions; hence, configurations near the hull (negative Ef) are stable against disordering. For GeSiNW, the 1-body ECIs at the shell sites are ∼0.15 eV lower than those at the core, indicating that Si prefers occupying the shell. The first few pair ECIs are weakly negative (with magnitude below 0.02 eV), indicating a weak preference for clustering, i.e., alike atomic species tend to locate next to one another. The combined effects of the ECIs energetically favor segregation type Ge(core)−Si(shell) configurations (see Figure 1). In fact, the perfect core−shell structure (50 atomic percentage (at %) Si) have the lowest Ef; other high symmetry configurations that are commonly used as theoretical models at 50% Si, e.g., reverse core−shell, abrupt, and mixed GeSiNW configurations,19 have higher Ef. Because the magnitude of pair and multisite (3-site clusters and beyond) ECIs of GeSiNW are the weakest among the three alloy systems, the GeSiNW configurations have the smallest spread in Ef. Weak multisite ECIs result in a Ef−composition plot that is highly symmetric about 50 at % Si. The ECIs (see Supporting Information) for the SnSiNW energetically favor the formation of core−shell-like configurations too (Sn occupies the core). However, the ECIs are several times larger in magnitude than their counterparts for the GeSiNW. Hence, the spread in Ef is wider than that for Ge-

Figure 2. DFT formation energies (Ef) versus composition for relaxed configurations in the (a) GeSi, (b) SnSi, and (c) CSi alloy NWs. Configurations from the initial guess have relatively high Ef and are marked with × or squares. Low-energy configurations predicted during the ground state search (via cluster expansion) are marked with +. Configurations marked with squares undergo excessive relaxation and are not used in the cluster expansion fit. Ground state configurations are marked with circles, and the solid line joining them forms the ground state hull. Dotted lines give the Ef of the fully disordered states (fully random solution). 4953

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up to the core−shell configuration (m = n = 12) (XmSin, where X = C, Ge and Sn), at which, a crossover to direct band gap is observed. In the direct gap region, the dependence with composition is nonlinear, and reduced quantum confinement effect (RQCE) is observed for one configuration (i.e., its band gap is lower than that of GeNW and SiNW). This nonlinear variation, RQCE, and direct to indirect band gap transitions are also observed in other studies using ⟨110⟩ and ⟨111⟩ core− shell NWs.19−21 Unlike GeNW, SnNW has a direct band gap of lower magnitude. This allows compositional tuning of SnSiNW over a range of 1.00 eV (compared to 0.16 eV for GeSiNW); band gaps are direct across all compositions. The band gap increases linearly with the addition of Si to SnNW up to the core−shell configuration at n = 12. Further addition of Si atoms leads to a nonlinear dependence between the band gap and composition. CSiNW may be used as a wide band gap material and a tunable range of 1.56 eV owing to the large band gap difference between C and Si NWs. However, the compositional trend for the band gap is less straightforward. Starting from CNW, the band gap decreases almost linearly up to 6 Si atoms after which there is a sharp increase. The band gaps of CSiNW containing 8 to 12 Si atoms are relatively constant; these ordered configurations have high resemblance to the zinc-blende structure. Further addition of Si leads to large lattice distortion (see Figure 1) and the (zinc-blende) ordering is destroyed. The band gap decreases sharply with at % Si, and RQCE (where the NW band gap is lower than that of SiNW) is observed for n > 15. Our results show that alloying SiNW with C and Sn allows a band gap tuning across 2.40 eV. The continuous change from low (SnNW) to high band gap (CNW) occurs with reasonable distortion from the ideal ⟨100⟩ SiNW structure. Although one might tune the band gap directly using a CSnNW, control over the alloy structure will be more difficult in view of the huge lattice mismatch between C and Sn. We further examine how each individual alloying element in the NWs contributes to their electronic band properties via an analysis of their projected density-of-states (PDOS). This provides an indication of whether the valence band maximum (VBM) and conduction band minimum (CBM) are spatially localized (in different regions of the NW). Such spatial localization of the bands would suggest the possibility of (electron−hole) charge separation in the alloy NW, which might be useful in key renewable energy applications such as in water splitting and solar cells.50 Recent theoretical calculations have pointed to a possible charge separation in ⟨110⟩ GeSiNWs. It was shown in Ge(core)−Si(shell) NWs (∼4 nm in diameter) that the VBM is confined to the Ge core and the CBM to the Si shell;28 similar band localization was also observed for ⟨110⟩ GeSiNWs (∼1 nm in diameter) with a half−half configuration.22 In our case (⟨100⟩ NW with a ∼1 nm diameter), the PDOS in Figure 4 indicates that states near both the VBM and CBM are largely localized on the Ge core atoms of the core−shell ground state, Ge12Si12; hence, no charge separation is expected. For Ge (Si) rich configurations, the states near the VBM and CBM are dominated by the majority species. Interestingly for Ge8Si16, the states near the VBM are localized on the minority species, Ge (core), while the CBM states are contributed (almost) equally by Ge and Si. As the PDOS from Si is expected to increase with at % Si, we expect that at 16 < n < 24, the CBM would become

SiNW. The larger size mismatch between Sn and Si (compared to Ge and Si) results in a higher degree of relaxation away from the ideal ⟨100⟩ NW lattice positions. The presence of stronger multibody ECIs results in a less symmetric Ef−composition plot. CSiNW, however, has strong pair and multisite ECIs (an order of magnitude larger than those of GeSiNW, see Supporting Information). The nearest-neighbor (NN) pair ECIs are mostly positive, and the second NN pair ECIs are all negative (magnitudes range from 0.02 to 0.5 eV), favoring ordered configurations where Si and C atoms are nearest neighbors (see Figure 1). The configuration with the lowest Ef (50 at % Si) resembles the zinc-blende structure, exhibiting an ABAB stacking along the NW axis. The zinc-blende structure is also a polytype of the SiC line compound in the bulk C−Si phase diagram.48 As C and Si has the largest size mismatch among the three systems, we observe huge distortions away from the ideal ⟨100⟩ NW lattice positions for some configurations, which have high Ef and are not used in the training set for the CE. Large multisite ECIs, largely a result of significant structural relaxation, lead to a highly asymmetric Ef− composition plot. In addition, we note that the affinity between the alloying element and the passivating H atoms plays a role in determining the segregation profile of the NWs. As the bond strength between group IV elements with H increases in this order, C > Si > Ge > Sn,49 we expect Si to segregate to the surface for both GeSiNW and SnSiNW, in agreement with our findings. For CSiNW, the strong ordering tendency (as revealed by CE) overwhelms the tendency for C atoms to segregate to the surface, promoting ordered configurations instead. Next, we examine the electronic properties of the stable alloy NWs. The plots of band gap as a function of composition ratios are shown in Figure 3. The most stable configuration predicted at each given composition is used for the characterization. Starting from pure GeNW, which has an indirect band gap, the band gap of GeSiNW increases linearly with the addition of Si

Figure 3. Plots of DFT electronic band gap as a function of stable configuration at each composition for CSiNWs, GeSiNWs, and SnSiNWs. Arrows mark the CNW, SiNW, GeNW, and SnNW and the most stable configurations of the alloy NW. All band gaps are direct (D) except for Ge-rich configurations, where band gaps are indirect (I). 4954

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localized at the Sn core, while a good proportion of the CBM are localized on Si (away from the core), suggesting the possibility of some charge separation for n > 12. For CSiNWs, the states near the VBM and CBM are in general dominated by the majority species. Strikingly, for C12Si12 (zinc-blende-like configuration), the states near the VBM and CBM are localized on C and Si, respectively. However, because the C and Si atoms are not spatially separated in the zinc-blende configuration (c.f., core−shell structure), there will be no charge separation, as the localized VBM and CBM are adjacent to each other. Spatial arrangement of atoms is an important factor for effective charge separation. In conclusion, via the use of DFT calculations, the cluster expansion method, and Monte Carlo simulations, we examined extensively the configuration−stability relationship of ⟨100⟩ CSiNWs, GeSiNWs, and SnSiNWs at a fixed diameter and characterize their electronic properties across composition. We identified the stable alloy NW configurations for the three systems over the entire composition range. While stable configurations of GeSiNWs and SnSiNWs exhibit core−shell segregation tendencies, those of CSiNWs favor an ordering pattern with C and Si being nearest neighbors. In addition, we revealed the electronic properties of these stable alloy NWs, demonstrating how the band gap may be tuned via alloying and suggesting composition ranges where the alloy NW may act as effective charge separator. Interestingly, the band gap varies linearly with composition for the Ge (Sn) rich regions in the GeSiNWs (SnSiNWs) and in a nonmonotonic way in the Si rich region. For CSiNW, the variation is nonmonotonic for the entire composition range. The present work thus illustrates how one could efficiently and reliably predict stable alloy NW configurations for electronic properties characterization across composition. Such efforts will help create better alloy NW models for explaining or predicting experimental observations.

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ASSOCIATED CONTENT

S Supporting Information *

Effective cluster interactions for all alloy nanowires; list showing all stable alloy nanowire configurations. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the A*STAR Computational Resource Centre (ACRC) in Singapore through the use of its high performance computing facilities.

Figure 4. Plots of projected density-of-state (PDOS) of the constituent atoms in (a) GeSiNWs, (b) SnSiNWs, and (c) CSiNWs with different configuration ratios. States near the CBM are enlarged in the insets for clarity. The Fermi level is set to 0 eV. Labeled in green are NW configurations that could act as possible electron−hole charge separators.

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