Preferential Positioning, Stability, and Segregation of Dopants in

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Preferential positioning, stability and segregation of dopants in hexagonal Si nanowires Michele Amato, Stefano Ossicini, Enric Canadell, and Riccardo Rurali Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b04083 • Publication Date (Web): 04 Jan 2019 Downloaded from http://pubs.acs.org on January 5, 2019

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Preferential positioning, stability and segregation of dopants in hexagonal Si nanowires Michele Amato,∗,† Stefano Ossicini,‡,¶ Enric Canadell,§ and Riccardo Rurali§ †Laboratoire de Physique des Solides (LPS), CNRS, Univ. Paris-Sud, Université Paris-Saclay, Centre scientifique d’Orsay, F91405 Orsay cedex, France ‡”Centro S3 ”, CNR-Istituto di Nanoscienze, Via Campi 213/A, 41125 Modena, Italy ¶Dipartimento di Scienze e Metodi dell’Ingegneria, Centro Interdipartimentale En&Tech, Universitá di Modena e Reggio Emilia, Via Amendola 2 Pad. Morselli, I-42100 Reggio Emilia, Italy §Institut de Ciència de Materials de Barcelona (ICMAB–CSIC), Campus de Bellaterra, 08193 Bellaterra, Barcelona, Spain E-mail: [email protected]

Abstract We studied the physics of common p- and n-type dopants in hexagonal-diamond Si –a Si polymorph that can be synthesized in nanowire geometry without the need of extreme pressure conditions– by means of first-principles electronic structure calculations and compared our results with those for the well-known case of cubic-diamond nanowires. We showed that i) as observed in recent experiments, at larger diameters (beyond the quantum confinement regime) p-type dopants prefer the hexagonal-diamond phase with respect to the cubic one as

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a consequence of the stronger degree of three-fold coordination of the former, while n-type dopants are at a first approximation indifferent to the polytype of the host lattice; ii) in ultrathin nanowires, because of the lower symmetry with respect to bulk systems and the greater freedom of structural relaxation, the order is reversed and both types of dopant slightly favor substitution at cubic lattice sites; iii) the difference in formation energies leads, particularly in thicker nanowires, to larger concentration differences in different polytypes, which can be relevant for cubic-hexagonal homojunctions; iv) ultra-small diameters exhibit, regardless of the crystal phase, a pronounced surface segregation tendency for p-type dopants. Overall these findings shed light on the role of crystal phase in the doping mechanism at the nanoscale and could have a great potential in view of the recent experimental works on group IV nanowires polytypes.

Keywords Nanowires, hexagonal diamond silicon, 2H-Si, dopants, density functional theory, formation energy Considerable progresses in the growth of hexagonal-diamond (2H) group IV nanowires (NWs) –also known as lonsdaleite NWs– have once again proved the emerging role of crystal phase engineering in the design of novel nanostructures with well tailored properties. 1–3 Over the past few years, several experimental works demonstrated the growth of 2H-Si and Ge NWs by employing different approaches including crystal structure transfer method, 4,5 strain induced transformation process, 6,7 as well as plasma-assisted vapour-liquid-solid growth. 8 Although hexagonal phases of Si are only stable in its bulk form at pressures larger than 12 GPa, all these studies conclusively demonstrated that, at the nanoscale, 2H-Si NWs can exist both as pure wires 8 and as extended domains in polytypic heterostructures. 7 Driven by this evidence, many theoretical and experimental works were dedicated to the description of the main features of this novel phase NWs. 9–15 As for the electronic and optical properties, it was shown that ultra-thin 2H-Si NWs can have direct band gap and an increased optical 2


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absorption with respect to 3C-Si NWs. 9,14 The same behaviour is still maintained in larger diameter NWs and can be enhanced by applying strain, opening great promises for photovoltaic applications. 10,16 Band alignment in hexagonal-cubic axial NWs was studied highlighting the possibility to switch the carrier localization by playing on the diameter. 9,17 On the other hand, the vibrational and thermal properties of 2H NWs were also investigated, 18,19 revealing a strongly reduced thermal conductivity with respect to 3C NWs which could be exploited in the field of thermoelectrics. 20,21 Nevertheless, despite the great interest in this subject, little information exists on the role of dopants in 2H-Si NWs and of the possible difference with 3C-Si NWs. On the experimental side, the only work which addressed the doping of hexagonal-diamond Si NWs was performed by Fabbri et al. 22 who investigated by coupling High Resolution Electron Microscopy (HRTEM) and electron diffraction, how a dopant type can affect the long-term stability of a given NW phase. On the other hand, there is a lack of theoretical works on this issue. This is instead a crucial aspect of the physical properties of a nanostructure, in particular in view of its implementation into electronic, thermoelectrics and optical devices. 23–26 Indeed, in the past many studies focusing on Si NWs and Si nanocrystals (see for example Refs. 27–37 ) made evident the role of the reduction of the size on the nature of the doping mechanism and its differences with respect to the bulk case. Quantum confinement, dielectric mismatch, crystal symmetry modifications and surface segregation of impurities are the main phenomena that must be taken into account when a nanostructure is doped and the physics that describes these effects changes with respect to the standard picture of doping in bulk semiconductors. It is hence evident how the presence of a novel phase, like the hexagonal one, could further modify and complicate the understanding of this scenario. In this letter we elucidate, by applying first-principles calculations in the framework of Density Functional Theory (DFT), the role of B and P impurities in 2H-Si NWs. In a first step the stability of these dopants in bulk 2H- and 3C-Si is carefully addressed through formation energy analysis and rationalized with symmetry and structural arguments. Our considerations are supported by a model developed analyzing the effects of other group III (Al, Ga), group IV (C, Ge) and group

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V (N, As) dopants. Then, we focused on the behaviour of B and P in ultra-thin NWs, which is presented outlining the role of size and the surface in lowering the formation energy of an impurity. The main message of this letter is two-fold: i) First, for bulk systems and larger diameter NWs, we demonstrate that, thanks to the lower symmetry of the 2H phase with respect to the 3C one, the former represents the ideal p-type doping environment to fully host the electronic and structural characteristics of trivalent impurities, ii) In the case of ultra-thin NWs, this behaviour is modified and n- and p-type doping are favored in the 3C phase. Moreover, in this case, the dopant stability strongly depends on the distance of the impurity from the surface of the wire. All these considerations shed light on which are the most favorable doping conditions in 2H-Si NWs and how the doping process may be optimized in the case of polytypic junctions. All the calculations have been performed in the framework of Density Functional Theory (DFT) using the total energy code S IESTA. 38 Local Spin Density Approximation (LSDA) was adopted for the exchange and correlation functional. Norm-conserving pseudopotentials of Troullier-Martins type were used to account for core electrons, while valence electrons were described through an optimized double-ζ polarized basis set. 39 All the ground state geometries discussed in this work were relaxed with a conjugate gradient algorithm. Convergence on the density matrix during a selfconsistent cycle was set to 10−4 eV. As for the doped systems, we studied substitutional B and P atoms as common p- and n-type dopants in group IV bulk and NWs. To corroborate the generality of our conclusions we extended our study to other group III (Al, Ga), group IV (C, Ge) and group V (N, As) impurities. The predictive power of DFT-based calculations in NWs is witnessed by several publications that tackled quantum confinement, 40–43 dielectric confinement, 28 impurity segregation, 32,44 even though in most of these cases the wires studied have smaller diameters than those grown experimentally because of computational limitations. However, as we have done here, large diameter NWs can also be modeled as bulk systems. In the following we first study impurity doping in bulk 2H-Si as an effective model for middleand large-diameter NWs where quantum and surface effects are negligible; these NWs, with diameters in the 20-100 nm range are those that are routinely grown and thus are more relevant for

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the applications. Next we move to ultra-thin 2H-Si NWs as a useful limiting case where the large surface-to-volume ratio plays an important role and where the formation energy is strongly radial dependent. Having a size comparable with the Bohr radius of Si, these systems allow us focusing on more fundamental physics related with quantum confinement and excitonic effects. In both cases we compare our results with 3C-Si. Preliminary calculations were performed with bulk 2H- and 3C-Si in order to determine the equilibrium lattice constants and total energies. As shown in the top panel of Fig. 1, the 3C structure presents an ABC stacking along the h111i direction, while an ABAB atomic ordering characterizes the 2H structure along the [0001] direction (more details about the two crystal structures can be found in Ref. 45 ). We relaxed both atomic coordinates and lattice parameters until the ˚ and the stress less than 0.1 GPa. The optimized lattice force on each atom was less than 0.01 eV/A ˚ while the in-plane lattice constant, a2H , and the c parameter for bulk 3C-Si, a3C , was 5.389 A, ˚ and 6.274 A, ˚ respectively. The computed values are in parameter of the 2H structure were 3.793 A good agreement with previous experimental and theoretical studies, 10,45–48 in which it was shown that lattice mismatch between the in-plane constants of the two phases should be of the order of 0.5 %. The difference in the cohesion energy was estimated to be 103 meV, with the cubic phase more stable with respect to the hexagonal one. For the doped bulk systems we studied group III (B, Al, Ga), group IV (C, Ge) and group V (N, P, As) atoms as substitutional impurities. The host bulk crystal was represented with a 4 × 4 × 4 supercell (512 atoms) in the case of 3C-Si and with a 6 × 6 × 3 (432 atoms) for 2H-Si. Both choices appeared to be sufficient for eliminating the spurious interaction between periodic images of impurities. The Brillouin zone was sampled with a 2 × 2 × 2 uniform k-grid. As for NWs, h111i 3C and [0001] 2H structures with a diameter of 2 nm were considered (see bottom panel of Fig. 1). The lattice constants in the direction of growth were optimized with the same tolerance criteria adopted for bulk systems. Both phases show a slight compression with respect to the bulk values of the order of 0.4 % for cubic NWs and 0.2 % for hexagonal ones. To simulate the host crystal, a supercell made by four primitive cells and six primitive cells was used

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for 3C- and 2H-Si NWs, respectively. A 1 × 1 × 9 k-grid was used in the case of the primitive cell of both phases whereas sampling was reduced to a 1 × 1 × 2 grid in the case of the doped supercells. Surface states were eliminated passivating the dangling bonds with hydrogen atoms, as it is common in this type of calculations. 49,50 We first studied the stability of B and P impurities in bulk systems. The analysis was conducted by performing total energy calculation on the relaxed systems and evaluating the formation energy (E f orm ) of the defect. E f orm is a fundamental quantity for investigating the energetic stability of impurities in a solid: it represents the energetic cost/gain of creating a defect in an otherwise perfect system 51 and thus can be used to compute important quantities such as the impurity equilibrium concentrations, 52–54 solubilities, 55,56 or diffusivities, 57,58 just to give some examples. In the formulation due to to Zhang and Northrup 52 it reads

D E f orm = Etot − ∑ ni µi

(1)

i

D is the total energy of defected system, n and µ are the number of atoms and the where Etot i i

chemical potential of a given chemical species and the sum runs over all the chemical species present in the system. The calculation of E f orm for a given impurity through this equation requires only three ingredients: a total energy calculation of the perfect crystal supercell, which divided by the number of atoms gives the chemical potential of the host material, µSi in our case; a calculation D ; a calculation of the elemental forms of of the supercell containing the defect, which yields Etot

the impurities, providing µB or µP . We performed these calculations and evaluated this quantity for B- and P-doped 3C- and 2H-Si. Our results are reported in Table 1. As can be deduced from the reported values (see Table 1), while there is no clear preference for P atoms to be in one host crystal or the other (there is only a minor difference between E f orm values of 40 meV), 2H-Si bulk is the favorite environment for B atoms. Indeed the gain in E f orm with respect to the bulk 3C-Si is of 0.27 eV meaning that p-type doping is favored in 2H systems. The equilibrium concentration of an impurity/defect i can be computed as

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E f orm (Di ) [Di ] = Ns exp kB T

(2)

where Ns is the available host lattice site concentration, E f orm (Di ) is the formation energy of defect i, kB is the Boltzmann constant and T is the temperature. Therefore, the difference in B concentration of a 2H- and a 3C-Si sample doped in the same conditions is [D2H B(Si) ] [D3C ] B(Si)

= exp

∆E 2H−3C f orm

!

kB T

(3)

If one takes 0.27 eV for the difference in formation energies, ∆E 2H−3C f orm , as obtained from our calculations, Eq. 3 predicts a concentration of B substitutional impurities that is 5 × 104 larger in 2H-Si than in 3C-Si. These are equilibrium concentrations and thus need suitable annealing cycles to be observed. Yet, they are a strong indication that even at growth time one should observe a step-like impurity concentration profile in B-doped cubic-hexagonal junctions. Interestingly, these findings are in good agreement with the experimental observations of Fabbri and co-workers. 22 In their work, by employing High Resolution Electron Microscopy (HRTEM) and electron diffraction, they demonstrated that doping species can have a different effect on the structural stability of a given phase. In particular, they showed that B atoms have a major efficacy in retaining the hexagonal-diamond phase while P atoms reveal a clear preference for the cubic-diamond one. The explanation that Fabbri et al. gave of their results is based on a simplified kinetic model taking into account the different solubility of B and P in the gold catalyst, which could induce different growth regimes and hence influence the type of crystal nucleation. From this perspective, our theoretical conclusions based on formation energy arguments (hence describing the thermodynamic stability of dopants at the equilibrium 52 ) confirm and complement their experimental observation and the related discussion. B atoms may favor the hexagonal phase with respect to the cubic ones not only because of their low solubility in the gold catalyst (that retains the hexagonal nucleation) but also because, at the equilibrium, the three-fold coordination environment is ideal for their chemical stability.

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In order to provide an explanation for the increased stability of B in 2H-Si we proceeded to carefully analyze the structural reorganization of atoms around the impurity in the host crystal. In Table 1 the nearest neighbour distances of the impurity after relaxation are reported. Looking at the values of the first neighbour distances for the pure systems (see the Supporting Information), it is straightforward to recognize a clear difference in the symmetry between the two phases. In particular each Si atom in the 3C-Si lattice has four equivalent tetrahedral bonds (Td symmetry), while the 2H-Si lattice has a lower symmetry with one longer bond along the [0001] direction and three shorter and equivalent bonds in the plane (C3v symmetry). By analyzing the modifications of bond lengths when impurities are added into the system (see Table 1), one can make the followings considerations: i) In the case of 3C-Si, the presence of B and P impurities preserves the Td symmetry of the pure systems. As already shown in previous works, 27,59–63 while Si-P bond lengths are only equally shortened by 0.3 % with respect to the pure bulk values, in the case of B doping there is noticeable uniform contraction of bond lengths around the impurity (around 12 %). This is due to the smaller radius and the lack of an electron of B atoms when compared to Si, which results in a tendency to favor chemical bonds with a significant geometrical distortion, ii) In the case of 2H-Si, both B and P dopants adapt to the C3v environment of the hexagonal crystal with one longer bond along the c crystal axis and three shorter equal bonds in the plane defined by the other lattice vectors, a and b. The variation of bond lengths with respect to pure systems are again more pronounced for B than for P. The reason why B impurities have lower formation energies in 2H structures with respect to 3C one is related to the natural tendency of B atoms to form trivalent neutral compounds. As a matter of fact, while it is known that B chemistry is closer to that of Si and C than other group III elements, it is also true that B atoms are one electron deficient with respect to Si. 64 This means that they will prefer host crystal environments in which they can build geometries close to the trigonal planar ones, as in the case of 2H lattices. This is a quite common behaviour of group III elements aiming at optimizing bonding given their relatively small number of electrons. 65 Conversely, they will favor less substitution at a 3C-Si, which would force a tetrahedral configuration. On the other

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hand, P atoms, which have a slightly smaller covalent radius than Si, do not show any preference for the crystal phase (the difference in E f orm of P atom between 2H and 3C is only 40 meV) and induce only a minimal geometrical distortion. This means that the extra electron brought in by the dopant, which necessarily fills an anti-bonding state, does not introduce any differential effect neither between the two structures nor between the different bonds of the 2H structure. This physical behaviour evokes previous theoretical calculations on Si nanostructures (see for example Refs. 27,44,66 ) in which it was demonstrated that B atoms prefer low-symmetry lattice sites close to the surface, while P atoms are stable even in the inner lattice sites. In order to further check the robustness of these arguments and to extend these conclusions, we performed additional calculations on similar substitutional neutral dopants, i.e. Al and As atoms, p- and n-type dopants in Si. Note that now both dopants have a larger covalent radius than Si. This is a major difference especially in the case of B substitution. Whereas the electronic effect (i.e. the preference for a 3+1 type local bonding stemming from the occurrence of just three valence electrons in the dopant) is still operative, the effect of the atomic radius works in a very different way. Now the central cavity must expand so that in 2H-Si the Al substituent will push more strongly in the direction of the weaker bond and the distances with the four nearest neighbors will tend to equalize as if the substitution was occurring in the very symmetric 3C-Si. In this behaviour the tendency of Al atoms to favor close-packed structures (like the face-centered-cubic bulk Al) can be recognized. 65 Thus, in this case the energetic preference cannot be deducted from a simple examination of the distances. Conversely, since the additional electron of a group V dopant does not induce any bond differentiation, no differences are expected for the As substitution with respect to the P one. The calculated results reported in Table 2 for E f orm and bond lengths are in agreement with the above expectations. No specific optimal doping condition among the two phases exists for n-type dopants (As atoms) and the substituent environment of the pure phases is kept. For p-type dopants (Al atoms) the 2H phase is again preferred (with a gain of nearly 0.3 eV), while the substituent surrounding is symmetric in both 3C and 2H phases. It is hence clear that three parameters must be considered to understand the doping mechanism: the valence and size of the

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impurity and the symmetry of the host crystal. We expanded on these considerations and tried to generalize this behaviour by carefully considering the effect of other group III, IV and V substitutional impurities in both bulk 3C- and 2H-Si. The goal was studying, given the valence of the impurity, the effect of a different atomic radius. Results of E f orm calculations for these systems are reported in Table 3. First, observing the table, it is possible to deduce a general tendency for E f orm that is strictly related to the number of valence electrons of the impurity and that confirms our previous considerations about B and P dopants in Si. If only the valence of the dopant is taken into account, one could, at a first approximation, conclude that: for group III impurities (first block of Table 3), dopants will always favour an hexagonal crystal with respect to a 3C one in order to share their three valence electrons in a trigonal-like chemical environment. Group IV impurities (second block of Table 3), as expected, have no clear preferences of one phase over the other, because of the same electronic valence of the host atoms as well as the similar atomic radius. The same behaviour could be, in principle, recognized also in the case of group V impurities (third block of Table 3), where there is only a slight preference for 3C systems. 3C Group III : E 2H f orm < E f orm 3C Group IV : E 2H f orm ≈ E f orm

(4)

3C Group V : E 2H f orm > E f orm

However, these considerations do not fully explain our results because also structural and symmetry effects can play a role. For instance, a noticeable exception to the trend of Eq. 4 is represented by the case of N which, despite the five valence electrons, favors substitution in a 2H crystal by nearly 0.1 eV. This exception, similarly to the case of Al, is due to the fact that the difference in E f orm between the two phases is a complex function of three variables: the valence of the impurity, VI ; the difference between the impurity atomic radius (rD ) and the Si atom radius (rSi ), indicated with ∆r; the degree of C3v symmetry of the impurity in the crystal, SC3v (which can be calculated as the difference between the average of the three bond lengths in the plane, dk , and the bond length 10


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perpendicular to the basal plane, d⊥ ). For a given impurity in a given host crystal, E f orm can hence be expressed as: 2H 2H E 2H f orm = E f orm (VI , ∆r, SC3v )

E 3C f orm

= E 3C f orm (VI ,

∆r,

(5)

SC3C3v )

Eq. 5 can be simplified by observing that the symmetry term, SC3v , is negligible in the case of a 3C crystal, as shown in Fig. 2, where the bond lengths of the impurity with the first neighbours after relaxation are reported. As can be seen there, depending on the atomic radius, impurities in 3C-Si induce local expansions or contractions of bond lengths, while preserving the original Td symmetry (see left panel of Fig. 2). On the other hand, a certain degree of local C3v symmetry around the impurity can be observed in the case of 2H crystals (see right panel of Fig. 2). A more compact way to analyze the results of Table 3 is to refer to the difference in E f orm between the two phases, ∆E 2H−3C f orm :

2H 3C 2H 2H 3C ∆E 2H−3C f orm = E f orm − E f orm = E f orm (VI , ∆r, SC3v ) − E f orm (VI , ∆r)

(6)

In order to give a qualitative idea of the dependence of E f orm on the electronic and structural variables, we reported in Fig. 3 the results for ∆E 2H−3C f orm (obtained from values of Table 3), together with the variation of ∆r and SC2H3v moving from the top to the bottom in a given chemical group, i.e. increasing the atomic radius. The figure must be analyzed in this way: the more ∆E 2H−3C is f orm negative the more the 2H phase is preferred with respect to the 3C one; the more ∆r is negative, the smaller the dopant is with respect to the Si atom; the more SC2H3v is negative, the more C3v -like the dopant local symmetry is. Interestingly, though on different scales, trends reported within the same group (hence for the same number of valence electrons) are qualitatively similar for the three quantities. The noticeable exception is again N, highlighting the role that the differences in the atomic radius can play. Clearly, it would be very useful to know the relative weights of the different factors that determine E f orm . To this end, if we make the simplifying assumption that the electronic and structural 11


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contributions are additive, we can rewrite Eq. 6 as:

2H 3C 2H 2H 3C ∆E 2H−3C f orm = E f orm (VI ) − E f orm (VI ) + E f orm (∆r, SC3v ) − E f orm (∆r)

(7)

3C E 2H f orm (VI ) − E f orm (VI ) can be computed by comparing the formation energies in a 3C and a 2H

lattice without allowing for geometry optimization, i.e. freezing the system in the ideal bulk Si positions and thus preventing any contribution from ∆r and SC2H3v . Therefore, Eq. 7 can be recast as:

2H−3C 2H 2H 3C ∆E 2H−3C f orm − ∆E f orm, unrelax = E f orm (∆r, SC3v ) − E f orm (∆r)

(8)

2H−3C We calculated ∆E 2H−3C in f orm, unrelax for all the impurities and plotted it, together with ∆E f orm 2H Fig. 4. The difference between the two curves is an estimation of the term E 2H f orm (∆r, SC3v ) −

E 3C f orm (∆r) in Eq. 8 and hence it accounts for the role of structural relaxation and symmetry in the stability of the dopant. The figure clearly shows that for most of the considered large radius elements (Al, Ga, Ge, P, As), regardless of the group, the dominant contribution to E f orm derives from the valence of the dopant (indeed, for these elements differences between the two curves are negligible meaning that structural variables play a minor role on the dopant stability in one phase rather than the other). This explains why, though with different degree of structural relaxations (as in the case discussed above of B and Al), group III elements will favor the hexagonal environment, while group IV and V elements will have no clear preference for the host crystal. Furthermore, in the case of smaller atoms (B, C, N), the contribution to E f orm coming from the structural variables can be relevant. Fig. 4 highlights once more the anomaly of N, where the structural relaxation is responsible for an increased stability (the gain in E f orm is around 0.1 eV) in the 2H phase which is not observed for the other group V elements considered. This specific behaviour can be tracked down to the known tendency of N atoms to share only the three p valence electrons in covalent bonding, while keeping the two s electrons in a lone pair state. Moving to the effect of the NWs size on the doping features described above and their connection with the crystal phase, we decided to perform simulations on h111i 3C and [0001] 2H

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ultra-thin Si NWs with a diameter of 2 nm (whose cross sections are presented in the bottom panel of Fig. 1). These systems have a two-fold importance: i) They represent a reliable model for Si NWs that can be experimentally fabricated both in the 2H and 3C phase, 8,67 ii) Their study can contribute to the fundamental understanding of the role of crystal phase in the doping process where quantum phenomena can be relevant and a large surface-volume ratio plays an important role. We first started by analyzing the structural properties of pure 2H- and 3C-Si NWs. As mentioned before, both types of NWs show a slight compression of the lattice constant in the growth direction with respect to the bulk values of the order of 0.4 % for 3C NWs and 0.2 % for 2H ones. The values of the first neighbours bond length distances for a Si inner atom and Si subsurface one (the most external Si atom not linked to H atoms, i.e. Si-5 in Figs. 1(c,d) in both 2H and 3C NWs) are reported in the Supporting Information. Looking at these data, we can note a common behaviour for both phases consisting in a slight compression of surface bond lengths, while they remain essentially unaltered with respect to bulk values for Si atoms in the innermost part of the NW. This surface relaxation of the wire was already demonstrated in the case of ultra-thin 3C-Si NWs 49 and is related to the higher capability of surface atoms to accommodate the stress with respect to core atoms. Furthermore, it seems that both 3C- and 2H-Si NWs preserve the global symmetry of the corresponding bulk systems (Td symmetry for the 3C phase and C3v for the 2H one). Next we addressed the study of neutral B and P impurities at the inner site in both NW phases. Results of calculations are reported in Table 4. As can be seen there, the one-dimensional confined geometry reverses the trend observed for bulk systems when B impurities are added into a NW. In this case the ideal host environment is represented by the 3C-Si lattice. Yet, it is worth noting that ∆E 2H−3C is in this case lower and amounts to 0.1 eV, leading through Eq. 3 to a value of f orm 3C −2 (to be compared with 5 × 104 for bulk systems). P impurities also [D2H B(Si) ]/[DB(Si) ] of 1.3 × 10

favor substitution at a 3C-Si lattice site by a similar amount of energy. Therefore, if in the case of medium to large diameter NWs, which can be modelled by means of bulk supercells, p-type

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doping was favored in the 2H phase and n-type doping was almost indifferent to the crystal phase, in ultra-thin NWs for both B and P dopants the favored phase is the cubic one. Also in this case the observed behavior can be rationalized from the analysis of the impurity first-neighbor distances after relaxation (see Table 4). We have already discussed that B favors a C3v symmetry, while P prefers Td . However, it turns out that, because of the possibility to laterally release strain, impurities in NWs can more easily force a given local atomic environment. Therefore, B induces a C3v first-neighbour arrangement in both 2H and 3C NWs, as well as P can stabilize a very symmetric, Td -like local structure, not only in 3C, but also in 2H NWs (see Table 4). Noteworthy, the short B-Si bond lengths induced in the 3C NWs are very similar to those of the already prepared bulk 2H lattice, indicating that even the innermost region of the NW provides a more favorable environment for the development of a three-fold coordination for the electron deficient B (note that because of the very short B-Si distances the induced rearrangement must in that case go further away than the first coordination sphere). In other words, the higher rigidity of the bulk lattice was the main driving force determining the considerably higher stability of p-type impurities in 2H-Si. Nevertheless, this constraint is much less effective as the diameter shrinks down, and eventually vanishes. This was already clearly demonstrated in the case of B-doping of Si nanocrystals 27 highlighting the tendency for trivalent impurities in nanostructures to induce a trigonal-like local chemical environment. Another fundamental aspect of the investigation of dopant behaviour in semiconductors nanostructures is the analysis of the effect of dopant position on its stability. As is known, segregation of impurities in NWs has been the subject of an intense experimental investigation aiming at evaluating the dopant distribution in the wire. Many experimental works agree on the fact that both B and P atoms prefer to occupy lattice sites near the surface of the wire rather than lattice core ones. However, the origin of this behavior is still under debate because of the complex physics behind the growth and doping processes. For example, Schitlz et al., 68 employing atom probe tomography showed that in B-doped Si NWs grown by standard Vapor-Liquid-Solid (VLS) growth, unintentional Vapor-Solid (VS) processes may arise inducing a different kinetic dopant incorporation and

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hence a non uniform doping profile along the wire. Koren et al., 69 through a Kelvin probe force microscopy analysis, confirmed the same segregation behavior for P doped Si NWs, demonstrating that, due to the high diffusivity of P atoms during the VLS growth, this behavior can also be observed in the absence of VS growth. Moreover, Fukata et al. 70 studied the segregation of B and P atoms in Si NWs during thermal oxidation by coupling micro-Raman scattering and electron spin resonance measurements. They found that, though both impurities tend to segregate to the surface oxide layer, B atoms preferentially segregate in the surface oxide layer while P atoms tend to accumulate in the Si region around the interface of the Si NW. It is hence clear that different variables play a crucial role in this effect: the kinetics of the process, the diffusivity of the impurities as well as the morphology of the NW. From the theoretical point of view, this issue has already been the subject of a detailed scrutiny in 3C-Si and 3C-Ge NWs 49 revealing that, because of the lower symmetry and the significant structural flexibility with respect to the bulk, impurities in NWs tend to prefer specific lattice sites to lower their E f orm . With the aim of shedding light on this aspect, we performed for both B and P-doped 3C- and 2H-Si NWs calculations of E f orm as a function of the impurity radial position (RD ). We labeled the inner core site as 1 and we considered different positions towards the NW surface. In particular, as clearly shown in the bottom panel of Fig. 1, the position labelled with 1 corresponds to the center of the wire, while position 6 identifies the outermost site of the wire lattice, where the atom is linked also to H atoms. Notice that in this case E f orm was computed applying the formalism presented in Ref., 71 which was especially developed for one-dimensional nanosystems. The difference of this approach with respect to the one of Zhang and Northrup consists in taking into account in the E f orm calculation both the variation of the Si chemical potential due to the reduction of the size as well as the passivation of the wire with hydrogen. More details of this approach can be found in Ref., 71 while results of calculation are presented in Fig. 5 for p-type doping (top panel) and n-type doping (bottom panel). Again, also these results reflect the fact that P favors a more symmetric first-neighbor structure, whereas B prefers lower symmetry local environments, where it is easier to fulfill the tendency to achieve a three-fold coordination. More specifically, looking at the figure,

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one can draw several conclusions: i) For both the phases, B atoms prefer to occupy surface lattice sites as a consequence of the higher freedom of structural distortion they can have in this part of the wire (leading to larger values of SC3v ). As discussed above, the similar stability of B atoms between the two phases for inner atomic positions can be ascribed to the change of symmetry in B-Si bonds due to the NW geometry that totally changes the picture described for bulk systems, ii) The dependence of E f orm on the dopant position is inverted in the case of P atoms. Here, as a consequence of the noted tendency for P atoms to prefer more symmetric environments, E f orm is lower in the inner sites with respect to surface ones (noticeably, there is also a reduction of Td symmetry associated to this effect). However, it is worth mentioning that differences in E f orm for P atoms between lattice site 1 and 5 are less than 50 meV. The higher stability for P dopants in the 3C phase than in the 2H one reflects the behaviour already observed in large diameters NWs. The same results can be conveniently represented in terms of the segregation energy (ES ) of the impurity in the NW, which is defined as the difference between the total energy of the NW with an impurity at a given lattice site and a reference energy, usually chosen as the one of the NW with the impurity in the center of the cross section. 44,72 An advantage of this formalism is that, as the chemical potential of the dopant cancels out, the segregation energy of B and P can be compared directly and its values do not depend on the specific choice, which always involves a certain degree of arbitrarity, of the chemical reservoir from where the impurity is assumed to come, e.g. monoatomic gas, suitable molecular precursor... Results of our calculations are reported in the Supporting Information for p-type and n-type doping in both 3C-and 2H-Si NWs. For B atoms the segregation energy, regardless of the NW crystal phase, is negative when the impurity occupies the surface lattice sites meaning that dopants will tend to migrate toward the surface. For P atoms the trend is opposite and impurities have a tendency to occupy inner lattice sites. It is worth noting that, for both phases, the energy difference between the most and less favored radial configuration is much larger in the case of P atoms than in the case of B atoms (almost twice as large). Though the instability in the lattice site 6 is due in both cases to the presence of a direct bond with two hydrogen atoms, this means that the driving force that keeps P atoms in the innermost part of

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the NW is stronger than the surface segregation exhibited by B atoms. These findings reproduce quite well the experimental observations of Fukata et al., 70 pointing out again the different degree of segregation of the two dopants. Indeed, though we found that the segregation effect is more pronounced for B than for P atoms, the formation energy variations along the cross section for P are less than 50 meV. This means that while for B both surface and subsurface sites are preferable, in the case of P positions along the cross section up the surface site are almost equally probable (while the energy barrier for the surface site in the case of P is too high to be overcome). The structural model based on H-termination that we have used for the surface is a usual choice in these kind of calculations 49 mostly because of its simplicity. Nevertheless, we observe that the removal of the native oxide followed by H passivation of the surface dangling bonds by means of a hydrofluoric acid dip has also been reported experimentally. 67 It is a common procedure for NW surface study and has also been successfully extended to the case of methyl-terminated NWs. 73,74 More realistic models involving an amorphous shell of silicon dioxide have been reported by Koleini and coworkers, 75 who found that impurities are more stable in the wires core in the absence of coordination defects at the Si/SiOx interface, a fact that further stresses the role of the local atomic environment to determine the stability of a defect. Stability and segregation of dopants at Si/SiO2 interfaces have been further studied by Kim et al. 76 and by Fukata et al. 77 In summary, we performed ab initio DFT calculations to investigate the role of the crystal phase on p-type and n-type dopants stability in 2H- and 3C-Si NWs. Calculations for bulk systems and larger diameter NWs revealed a clear preference for p-type dopants to be in the 2H phase as a consequence of the stronger tendency of this structure to stabilize a three-fold coordination around the impurity. The robustness of our considerations is supported by a model which takes into account both structural and electronic effects in the stability of group III, group IV and group V dopants in 2H-Si. Our results confirm and complement the experimental observations of Fabbri et al. 22 where it was shown that B-doping helped retaining the hexagonal phase in 10 nm Si NWs. Additionally, although clean 2H-3C axial or radial interfaces have not been reported yet, the cubic/hexagonal multi-domains reported by Vincent and co-workers 7 are suitable samples to verify the accumula-

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tion of B-dopant on the hexagonal side of a 2H/3C grain boundary. The theoretical predictions could be verified by means of atom probe tomography measurements. 78 In the case of ultra-thin NWs, because of the lower symmetry with respect to bulk systems and the greater freedom of structural relaxation, there is only a moderate preference for both p- and n-dopants for 3C NWs. Furthermore, at such small diameters, both phases can induce segregation phenomena for p-type dopants due, again, to tendency of trivalent atoms in group IV compounds to create a trigonal-like chemical environment, associated, in some cases, with large geometrical distortions. On the other hand, we show that segregation does not take place in the case of n-type dopants because of the high energy barriers that prevent diffusion to the surface. The novelty of these findings will likely impact future experimental works concerning the study of dopants in group IV polytype NWs.

Supporting Information Available Tables containing bong lenghts for pure Si bulk and NW systems. Calculations of segregation energy as a function of the impurities positions for both 2H and 3C NWs.

This material is

available free of charge via the Internet at http://pubs.acs.org/.

Acknowledgement M.A. greatly acknowledges the Transnational Access Programme of the HPC-EUROPA3 (project HPC17PB9IZ). Part of the high-performance computing (HPC) resources for this project were granted by the Institut du developpement et des ressources en informatique scientifique (IDRIS) under the allocation A0040910089 via GENCI (Grand Equipement National de Calcul Intensif). This work was supported by the ANR HEXSIGE project (ANR-17-CE030-0014-01) of the French Agence Nationale de la Recherche. We also acknowledge financial support by the Ministerio de Econom´ıa, Industria y Competitividad (MINECO) under grants FEDER-MAT2017-90024-P and FIS2015-64886-C5-4-P, the Severo Ochoa Centres of Excellence Program under Grant SEV-20150496, the Generalitat de Catalunya under grants no. 2017 SGR 1506. S. O. acknowledges sup18


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port/funding from University of Modena and Reggio Emilia under project ”FAR2017INTERDISC”.

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(60) Hakala, M.; Puska, M. J.; Nieminen, R. M. First-principles calculations of interstitial boron in silicon. Phys. Rev. B 2000, 61, 8155–8161. (61) Wang, L. G.; Zunger, A. Phosphorus and sulphur doping of diamond. Phys. Rev. B 2002, 66, 161202. (62) Zhou, Z.; Steigerwald, M. L.; Friesner, R. A.; Brus, L.; Hybertsen, M. S. Structural and chemical trends in doped silicon nanocrystals: First-principles calculations. Phys. Rev. B 2005, 71, 245308. (63) Nichols, C. S.; Van de Walle, C. G.; Pantelides, S. T. Mechanisms of dopant impurity diffusion in silicon. Phys. Rev. B 1989, 40, 5484–5496. (64) DeFrancesco, H.; Dudley, J.; Coca, A. Boron Reagents in Synthesis; Chapter 1, pp 1–25. (65) Kaxiras, E. Atomic and electronic structure of solids; Cambridge University Press, 2003. (66) Schoeters, B.; Leenaerts, O.; Pourtois, G.; Partoens, B. Ab-initio study of the segregation and electronic properties of neutral and charged B and P dopants in Si and Si/SiO2 nanowires. J. Appl. Phys. 2015, 118, 104306. (67) Ma, D.; Lee, C.; Au, F.; Tong, S.; Lee, S. Small-diameter silicon nanowire surfaces. Science 2003, 299, 1874–1877. (68) Schlitz, R. A.; Perea, D. E.; Lensch-Falk, J. L.; Hemesath, E. R.; Lauhon, L. J. Correlating dopant distributions and electrical properties of boron-doped silicon nanowires. Appl. Phys. Lett. 2009, 95, 162101. (69) Koren, E.; Elias, G.; Boag, A.; Hemesath, E.; Lauhon, L.; Rosenwaks, Y. Direct measurement of individual deep traps in single silicon nanowires. Nano Lett. 2011, 11, 2499–2502. (70) Fukata, N.; Ishida, S.; Yokono, S.; Takiguchi, R.; Chen, J.; Sekiguchi, T.; Murakami, K. Segregation behaviors and radial distribution of dopant atoms in silicon nanowires. Nano Lett. 2011, 11, 651–656. 25


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(71) Rurali, R.; Cartoixà, X. Theory of Defects in One-Dimensional Systems: Application to AlCatalyzed Si Nanowires. Nano Lett. 2009, 9, 975–979. (72) Galicka, M.; Buczko, R.; Kacman, P. Segregation of impurities in GaAs and InAs nanowires. J. Phys. Chem. C 2013, 117, 20361–20370. (73) Assad, O.; Puniredd, S. R.; Stelzner, T.; Christiansen, S.; Haick, H. Stable scaffolds for reacting Si nanowires with further organic functionalities while preserving Si- C passivation of surface sites. J. Am. Chem. Soc. 2008, 130, 17670–17671. (74) Haick, H.; Hurley, P. T.; Hochbaum, A. I.; Yang, P.; Lewis, N. S. Electrical characteristics and chemical stability of non-oxidized, methyl-terminated silicon nanowires. J. Am. Chem. Soc. 2006, 128, 8990–8991. (75) Koleini, M.; Colombi Ciacchi, L.; Fernandez-Serra, M. V. Electronic transport in natively oxidized silicon nanowires. ACS Nano 2011, 5, 2839–2846. (76) Kim, S.; Park, J.-S.; Chang, K.-J. Stability and segregation of B and P dopants in Si/SiO2 core–shell nanowires. Nano Lett. 2012, 12, 5068–5073. (77) Fukata, N.; Kaminaga, J.; Takiguchi, R.; Rurali, R.; Dutta, M.; Murakami, K. Interaction of boron and phosphorus impurities in silicon nanowires during low-temperature ozone oxidation. J. Phys. Chem. C 2013, 117, 20300–20307. (78) Perea, D. E.; Hemesath, E. R.; Schwalbach, E. J.; Lensch-Falk, J. L.; Voorhees, P. W.; Lauhon, L. J. Direct measurement of dopant distribution in an individual vapour–liquid–solid nanowire. Nat. Nanotech. 2009, 4, 315.

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Nano Letters

(a)

[0001]

B

(b)

C

A

B

B

A

A

C

B

A

B A

Figure 1: Top panel: lateral view of the crystal stacking for bulk 2H-Si along the [0001] direction (a) and bulk 3C-Si along the h111i direction (b). Bottom panel: cross section of a [0001] 2H-Si NW (c) and of a h111i 3C-Si NW (d) with a diameter of 2 nm. The red circles show the different lattice sites where substitutional dopants were considered. The position labelled with 1 corresponds to the center of the wire, while 6 identifies the outermost site of the wire lattice. Green and white spheres represent Si and H atoms, respectively.

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Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Table 1: The calculated formation energy (E f orm ) for B and P impurities in bulk 3C-Si and bulk 2H-Si and the nearest neighbour distances of B and P impurities after relaxation. For 2H systems, the first three bond lengths corresponds to bonds between atoms in the basal plane of the hexagonal cell. The fourth bond length is instead the one along the c direction. Impurity

˚ E f orm (eV) Bond length (A)

Crystal structure Cubic-diamond (3C)

-6.530

2.052 2.052 2.052 2.052

Hexagonal-diamond (2H)

-6.798

2.044 2.046 2.046 2.066

Cubic-diamond (3C)

-5.356

2.327 2.327 2.327 2.327


-5.316

2.316 2.317 2.317 2.339

B

P

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Nano Letters

Table 2: The calculated formation energy (E f orm ) for Al and As impurities in bulk 3C-Si and bulk 2H-Si and the nearest neighbour distances of Al and As impurities after relaxation. For 2H systems, the first three bond lengths corresponds to bonds between atoms in the basal plane of the hexagonal cell. The fourth bond length is instead the one along the c direction. Impurity


Crystal structure Cubic-diamond (3C)

-2.773

2.395 2.395 2.395 2.395


-3.016

2.389 2.394 2.394 2.395

Cubic-diamond (3C)

-2.805

2.422 2.422 2.422 2.422


-2.742

2.415 2.416 2.416 2.432

Al

As

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Table 3: The calculated formation energy (E f orm ) for group III (B, Al, Ga), IV (C, Ge) and V (N, P, As) impurities in bulk 3C-Si and bulk 2H-Si. Group Impurity Hexagonal-diamond (2H) Cubic-diamond (3C) III

B Al Ga

-6.798 -3.016 -2.124

-6.530 -2.773 -1.880

IV

C Ge

-7.471 -3.988

-7.447 -3.992

V

N P As

-4.358 -5.316 -2.742

-4.256 -5.356 -2.805

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bulk 3C-Si

B C N Al P Ga Ge As

bulk 2H-Si

2.4

2.4 pure 3C-Si

Length (Å)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

pure 2H-Si

2.3

2.3

2.2

2.2

2.1

2.1

2

2

d1

d2

d3

d4

d1

d2

d3

d4

Figure 2: The nearest neighbour distances of group III, group IV and group V impurities after relaxation. While the four bonds lengths are equivalent in the case of 3C crystal (left panel), for 2H systems (right panel), the first three bond lengths (d1 , d2 , d3 ) corresponds to bonds between atoms in the basal plane of the hexagonal cell. The fourth bond length, d4 , is instead the one along the c direction.

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2H-3C

0.1

∆E form (eV) ∆r (nm) 2H

SC (Å) 3V

0

-0.1

-0.2

-0.3 Group III

B

Al

Group IV

Ga

C

Ge

Group V

N

P

As

Figure 3: ∆E 2H−3C f orm (obtained from values of Table 3) (orange circles line) together with the variation of ∆r (green squares line) and SC2H3v (violet diamonds line) moving from the top to the bottom in a given chemical group.

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Nano Letters

0.1

∆E 2H-3C (eV) form 2H-3C ∆Eform, unrelax (eV)

0

~ 0.1 eV -0.1

-0.2

-0.3 Group III

B

Al

Group IV

Ga

C

Ge

Group V

N

P

As

2H−3C Figure 4: ∆E 2H−3C f orm (obtained from values of Table 3) (orange circles line) and ∆E f orm, unrelax (blue squares line) moving from the top to the bottom in a given chemical group.

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Table 4: The calculated formation energy (E f orm ) for B and P impurities in 3C- and 2H-Si NWs and the nearest neighbour distances of B and P impurities. For 2H systems, the first three bond lengths corresponds to bonds between atoms in the basal plane of the hexagonal cell. The fourth bond length is instead the one along the c direction. Impurity


NW structure Cubic-diamond (3C)

-6.338

2.048 2.048 2.048 2.055


-6.230

2.052 2.052 2.052 2.063

Cubic-diamond (3C)

-5.058

2.328 2.332 2.332 2.333


-4.907

2.345 2.345 2.346 2.347

B

P

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-6 2H NWs 3C NWs

-6.1

1 Eform (eV)

-6.2

2

3 6

-6.3

4 -6.4

5

-6.5 -6.6

B doping

-6.7 -6.8

0

2

4

6

8

10

RD (Å) -4.2

P doping

-4.3

6

2H NWs 3C NWs

-4.4 -4.5

Eform (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

-4.6 -4.7 -4.8

5

4 1

2

0

2

3

-4.9 -5 -5.1

4

6

8

10

RD (Å)

Figure 5: Top (bottom) panel: Formation energy (E f orm ) as a function of the distance (RD ) of the B (P) atom from the center of the wire for both 2H- and 3C-Si NWs. The labels correspond to the dopant positions indicated in the bottom panel of Fig. 1. 35


Nano Letters 1 2 3 4 5

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Preferential Positioning, Stability, and Segregation of Dopants in

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