Letter pubs.acs.org/NanoLett
Optical Emission in Hexagonal SiGe Nanowires Xavier Cartoixà,† Maurizia Palummo,‡ Håkon Ikaros T. Hauge,§ Erik P. A. M. Bakkers,§ and Riccardo Rurali*,∥ †
Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain Dipartimento di Fisica and INFN, Università di Roma “Tor Vergata” via della Ricerca Scientifica 1, 00133 Roma, Italy § Department of Applied Physics, TU Eindhoven, Den Dolech 2, 5612 AZ Eindhoven, The Netherlands ∥ Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus de Bellaterra, 08193 Bellaterra, Barcelona, Spain ‡
S Supporting Information *
ABSTRACT: Recent advances in the synthetic growth of nanowires have given access to crystal phases that in bulk are only observed under extreme pressure conditions. Here, we use first-principles methods based on density functional theory and many-body perturbation theory to show that a suitable mixing of hexagonal Si and hexagonal Ge yields a direct bandgap with an optically permitted transition. Comparison of the calculated radiative lifetimes with typical values of nonradiative recombination mechanisms indicates that optical emission will be the dominant recombination mechanism. These findings pave the way to the development of silicon-based optoelectronic devices, thus far hindered by the poor light emission efficiency of cubic Si.
KEYWORDS: Nanowires, hexagonal silicon, DFT, optical emission
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olid-state electronics1−3 was possibly the greatest technological revolution of last century and it modified our lives in a way so deep that it is difficult to imagine the world we live in without it. Although the transistor that got the Nobel prize in Physics in 1956 to Bardeen, Brattain, and Shockley was based on germanium, soon after silicon took over and quickly became the most important material for electronics, accounting for over the 95% of semiconductor devices sold worldwide today. While the physical principles behind the transistor effect were the same, silicon won the battle over other semiconductors thanks to its superior material properties, particularly the existence of a native oxide, SiO2, with excellent interface quality, and because is one of the most abundant elements on earth. Efficient light emission, however, is not among the many remarkable properties of silicon and this limitation has hindered the development of silicon-based optoelectronic devices. Considerable work has been done to find optically active materials that are fully compatible with silicon technology4 but clearly the Holy Grail keeps on being the luminescence out of bulk monocrystalline silicon. The reason for the poor emission efficiency of Si is the indirect nature of its fundamental bandgap. When electrons are excited to the conduction band, their decay to the valence band with the emission of a photon must involve a transfer of momentum to the crystal lattice by means of a phonon to guarantee its conservation. The need to absorb or emit a phonon as well makes the photon emission process so slow that in practice most of the recombinations will happen non© XXXX American Chemical Society
radiatively. For this reason, tremendous research efforts have been devoted to make the bandgap of Si direct. Besides the promising recent achievements obtained with Si quantum dots,5,6 ingenious approaches based on strain,7,8 heterostructuring,9,10 or nanostructuring11,12 have been proposed but to date the goal of light emission in Si is still elusive. Recent advances in the synthetic growth of semiconducting nanowires13,14 have given access to crystal phases that in bulk are observed only under extreme conditions of pressure.15,16 In particular, hexagonal Si has been obtained through a crystalphase transfer method,17,18 where a wurtzite GaP nanowire is used as template,19 while segments of hexagonal Ge obtained by a strain-induced phase transformation have been reported within quasi-periodic nanowire homostructures.20,21 The bandgap of hexagonal Si is smaller than the one of cubic Si but is still indirect.21−24 On the other hand, hexagonal Ge and at variance with its cubic counterpart is predicted to have a direct bandgap of 0.32 eV at the Γ-point.21,22 This fact could have important implications with respect to the long-standing goal of designing a Si-based light-emitting material because it suggests that a hexagonal Si1−xGex alloy could become a direct bandgap semiconductor for sufficiently high Ge content, x. The great advantage of the alloy over pure hexagonal Ge is that if x is not too high the lattice mismatch with Si would be moderate, Received: April 6, 2017 Revised: June 22, 2017 Published: June 27, 2017 A
DOI: 10.1021/acs.nanolett.7b01441 Nano Lett. XXXX, XXX, XXX−XXX
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Nano Letters making defect-free interfaces, essential for the performances of electronic and optoelectronic devices, possible. The crystal-phase transfer method used for the synthesis of hexagonal Si19 has also been successfully used for the growth of hexagonal Si1−xGex alloys as well with Ge concentration of up to 0.77.25 Therefore, these kinds of alloys are within current growth capabilities. Yet, to date no reliable electronic or optical characterization study of these materials exists, most likely because of the challenge of obtaining pure, defect-free samples. In this Letter, we use first-principles methods based on density functional theory (DFT) and many-body perturbation theory to predict the electronic and optical properties of hexagonal Si1−xGex nanowires.26 We show that indeed the bandgap can become direct and assess which is the minimum Ge content for this to happen. We compute the optical properties demonstrating that the direct radiative transition at Γ is permitted and that it will be the dominant recombination mechanism. Further increasing the Ge content beyond the minimum threshold needed to obtain a direct bandgap allows tuning the emission at expenses of a larger lattice mismatch with pure Si, providing a way to tune the absorption spectrum for versatile photovoltaic applications. A full many-body description of the electronic and optical properties allows us quantifying the role of excitonic effect and quantitatively estimating the emission wavelength. Given the size range of the nanowires routinely grown experimentally, where quantum and dielectric confinement effects have vanished,14 we carry out our calcualtions in bulk supercells (details of the computational methods and a brief review of their reliability are given in the Supporting Information). At first, in order to scan efficiently the configuration space of the hexagonal Si1−xGex alloys we have performed a series of density functional calculations within the virtual crystal approximation (VCA)27 with the Quantum Espresso code28 (see Supporting Information for full details of the computational setup). In this conceptually simple approach to the description of an alloy, the system is considered to be made of virtual atoms whose properties are interpolated between those of the constituent elements, Si and Ge in our case. Within density functional methods, the process essentially boils down to a suitable mixing of the corresponding pseudopotentials.29 While this approach neglects real disorder, it has the advantage that any nominal composition of the hexagonal Si1−xGex alloy can be described within the four-atom primitive cell (see Figure 1a) and its good accuracy for semiconductors has been widely reported.29,30 Our results for indirect−direct transition of the bandgap with composition x varying from 0 to 1 are shown in Figure 2a. On the other hand, Figure 2b shows the linear dependence on x of the lattice parameters, as predicted by Vegard’s law, which is known to work well in SiGe-based alloys. The most important result is clearly that increasing the Ge content drags the minimum of the conduction band at Γ down until the electronic bandgap becomes direct at x ≳ 0.35. Figure 2c displays the band structure diagram for x = 0, x = 0.2, and x = 0.4 where the direct nature of the bandgap of the latter is clear. Despite the underlying somewhat crude approach, VCA works well even in complex systems such as piezoelectric perovskites.31 However, it is still an approximate method and its reliability should be assessed case by case, thus it is difficult to arrive to any conclusion in absence of experimental data or supercell calculation at the same level of the theory.
Figure 1. (a) Side view of the cubic (left, ABC) versus hexagonal (right, AB) stacking along the [111] crystal axis; the shaded area indicates the four-atom hexagonal unit cell used in the VCA calculations. (b) The 3 × 3 × 3 supercell used to account explicitly for the alloy disorder (blue and yellow spheres represent Si and Ge atoms, respectively); we show a cross-section perpendicular to the [111] crystal axis.
Therefore, in order to go beyond the VCA results, we constructed a 3 × 3 × 3 supercell of the four-atom hexagonal primitive cell and randomly populated the lattice sites with Si and Ge atoms to achieve a given target composition, thus approaching the compositional disorder of an alloy (see Figure 1b). Guided by the VCA results of Figure 2 we considered x = 0.2−0.4, using for each composition the lattice parameter predicted by Vegard’s law. While these calculations are more accurate and avoid the simplifying assumption of VCA, they have the disadvantage that the resulting band structure is multiply folded and it is far from obvious to say if the bandgap is direct or not.32 To circumvent this problem, we used a band unfolding technique, as implemented in the BandUP code,33 to recover an effective primitive cell picture of the band structure.34,35 The method is based on the evaluation of the spectral weight PmK⃗ (k ⃗) =
SC 2 ∑ |⟨ψnkPC⃗ |ψmK ⃗ ⟩|
(1)
n
|ψSC mK⃗ ⟩,
which gives the probability of an eigenstate of the Hamiltonian in the supercell representation, to have the same character of a primitive cell Bloch state of wave vector k.⃗ The results obtained36 are shown in Figure 3 and are in good agreement with the VCA. Again it can be appreciated how increasing the Ge concentration lowers the conduction band minimum at Γ, while the valence band and the conduction band minimum at M experience only minor changes. The bandgap is found to become direct at x = 0.3, a slightly lower Ge concentration than within VCA. These results are encouraging and indicate that SiGe alloying within the hexagonal crystal phase is indeed a viable way to obtain a direct bandgap. However, the possibility to observe optical activity is subordinated to having a nonzero matrix element in correspondence to the direct optical transition. Cubic Si nanowires are a textbook case in this sense: the bandgap can become direct as a result of confinement, but the folded bands maintain their character and the optical direct transition is heavily suppressed.14 Therefore, our next step was the calculation of the optical properties of a Si0.6Ge0.4 alloy with the Yambo code.37 The imaginary part of the dielectric function was computed within the Random Phase Approximation (RPA), an independentparticle description, without the inclusion of local-field effects.38,39 We conducted a careful study of the k-point sampling to achieve a converged result, finding that a 12 × 12 × B
DOI: 10.1021/acs.nanolett.7b01441 Nano Lett. XXXX, XXX, XXX−XXX
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Figure 2. VCA results. (a) DFT indirect bandgap corresponding to the transition between the valence band maximum at Γ and the conduction band minimum at M (red squares), and direct bandgap at Γ as a function of the composition x (black triangles); the fundamental bandgap is direct for x ≳ 0.35. (b) a and c lattice parameter as a function of Ge fraction. (c) Band structure diagram of hexagonal Si1−xGex for x = 0 (pure hexagonal Si), x = 0.2, and x = 0.4 (close to the minimum Ge content yielding a direct bandgap within the VCA approximation).
Figure 3. Unfolded band structures for (a) Si0.8Ge0.2, (b) Si0.7Ge0.3, and (c) Si0.6Ge0.4. The bandgap becomes direct at Γ for hexagonal Si1−xGex alloys with x ≥ 0.3.
Note that due to computational limitation we had to use a less dense k-points mesh for the GW and BSE calculations, thus the curves in the inset are to be taken only qualitatively, yet capturing the essential physics (i.e., excitonic effects only play a minor role). The fully converged RPA absorption spectrum, using the right value of the electronic bandgap provided by the GW calculation, is shown in Figure 4. Even if the hexagonal SiGe alloy under study has a direct bandgap, the absorption intensity at the onset grows smoothly. By zooming in on the low energy range of the spectrum, a small optical peak confirms that the transition is allowed. It is worth to stress that this situation is similar to the paradigmatic case of GaAs, whose absorption spectrum also grows smoothly around 1.4 eV, notwithstanding that a clear photoluminescence (PL) peak is experimentally visible.
8 grid is needed. To move one step further in the direction of a realistic model of the hexagonal SiGe alloy, these calculations were performed after fully relaxing the atomic positions and the cell vectors until the forces on the atoms and the stress were lower than 0.025 eV/Å and 0.5 kbar, respectively. In order to correctly estimate the electronic bandgap value, we performed a one-shot perturbative GW calculation on top of the ground state DFT simulation. In this way, the fundamental direct electronic bandgap at Γ (0.37 eV at the DFT level) becomes 0.52 eV after the GW correction. This value provides a reasonable estimate of the optical bandgap (which in turn gives the energy of the emitted light) if excitonic and local-field effects can be neglected. This is indeed the case here, as we have proven by solving the Bethe-Salpeter Equation (BSE),40 which accounts for such effects, and obtaining minor corrections to the GW spectrum (see the inset of Figure 4). C
DOI: 10.1021/acs.nanolett.7b01441 Nano Lett. XXXX, XXX, XXX−XXX
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in the calculation of the retarded polarization function only conduction states up to some tens of millielectronvolts above the conduction band minimum. In this way, we obtain a clear PL peak, which we show in Figure 5. This simplified approach
Figure 4. Low-energy absorption spectrum of Si0.6Ge0.4 calculated within the RPA and including the GW correction of the electronic bandgap. The blue arrow indicates the onset of the optical absorption associated with the direct transition at the Γ-point. Inset: comparison between the absorption spectrum within the GW approach and solving the Bethe-Salpeter equation on top of the GW calculation.
Figure 5. Calculated photoluminescence spectrum of Si0.6Ge0.4 exhibiting a sharp feature at the emission energy that corresponds to the direct transition at Γ. The spectrum of pure hexagonal Si (x = 0.0), completely flat in the selected energy window, is shown for comparison.
Indeed, the crucial quantity to be calculated, if interested in the PL spectrum (or, similarly, in the electroluminiscence response), is the lesser polarization response function, rather than the retarded one, which instead provides a good description of absorption spectra.41−43 At small light-pump intensity and in a direct-gap pure material (neglecting important red-shift due to Stokes shift), one can safely assume that a PL peak occurs at the same energy of the optical bandgap. In other words, we can consider that electrons will be excited from the valence to the conduction band at energies all across the absorption spectrum and then will thermalize toward the conduction band minimum at Γ. It is clear that when this happens in bulk cubic Si, emission is extremely inefficient, because the bandgap is indirect and a phonon is required to satisfy momentum conservation, thus nonradiative recombination is the dominant process. Using the nonequilibrium Green’s function formalism, the lesser polarization function can be constructed from the eigenvectors and eigenvalues of the BSE, as discussed in ref 43. If many-body effects are small, like in the present case, BSE eigenvectors and eigenvalues change only slightly from the independent-quasiparticle ones. In this case, the lesser polarization response can be well approximated as the product of the imaginary part of the retarded response function and a Bose− Einstein distribution centered at the total chemical potential μ of the system41 P
