Silicon and Germanium Nanostructures for Photovoltaic Applications: Ab-Initio Results
© The Author(s) 2010
Received: 16 June 2010
Accepted: 1 July 2010
Published: 18 July 2010
Actually, most of the electric energy is being produced by fossil fuels and great is the search for viable alternatives. The most appealing and promising technology is photovoltaics. It will become truly mainstream when its cost will be comparable to other energy sources. One way is to significantly enhance device efficiencies, for example by increasing the number of band gaps in multijunction solar cells or by favoring charge separation in the devices. This can be done by using cells based on nanostructured semiconductors. In this paper, we will present ab-initio results of the structural, electronic and optical properties of (1) silicon and germanium nanoparticles embedded in wide band gap materials and (2) mixed silicon-germanium nanowires. We show that theory can help in understanding the microscopic processes important for devices performances. In particular, we calculated for embedded Si and Ge nanoparticles the dependence of the absorption threshold on size and oxidation, the role of crystallinity and, in some cases, the recombination rates, and we demonstrated that in the case of mixed nanowires, those with a clear interface between Si and Ge show not only a reduced quantum confinement effect but display also a natural geometrical separation between electron and hole.
Photovoltaic (PV) energy is experiencing a large interest mainly due to the request for renewable energy sources. It will become mainstream when its costs will be comparable to other sources. At the moment it is too expensive for competitive production. For this reason an intense research activity is of fundamental importance to develop efficient PV devices ensuring a low cost and a low environmental impact. Until now three generations of solar cells have been envisaged . Currently, PV production is 90% first-generation and is based on Si wafers. First generation refers to high quality and hence low-defect single crystal devices and is slowly approaching the limiting efficiencies of about 31%  of single-band gap devices. These devices are reliable and durable, but half of the cost is the Si wafer. The second generation of cells make use of cheap semiconductor thin films deposited on substrates to produce low-cost devices of lower efficiency. These thin-film cells account for around 5–6% of the market. For these second-generation devices, the cost of the substrate represents the cost limit and higher efficiency will be needed to maintain the cost-reduction trend . Third-generation cells use, instead, new technologies to produce high-efficiency devices [4, 5]. They are photo-electrochemical cells based on dye-sensitized nanocrystalline wide bandgap semiconductors  or multiple energy threshold devices based on nanocrystalline silicon for the widening of the absorbed solar spectrum, due to the quantum confinement (QC) effect that enlarges the energy gap of the nanostructures, and for the use of excess thermal generation to enhance voltages or carrier collection . Moreover recently also silicon and germanium nanowires have been used and envisaged for PV applications [8–13].
Besides the intense experimental work, devoted to the improvement of the nanostructures growth and characterization techniques and to the realization of the nanodevices, an increasing number of theoretical works, based on empirical and on ab-initio approaches, is now available in the literature (see for example Refs. [14–16]). The importance of the theoretical efforts lies not only in the interpretation of experimental results but also in the possibility to predict structural, electronic, optical, and transport properties aimed at the realization of more efficient devices. Important progresses in the description of the electronic properties of Si and Ge nanostructures have been reported, but an exhaustive understanding is still lacking. This is due, on one side to the not obvious transferability of the empirical parameters to low- dimensional systems and on the other side to the deficiency of the ab-initio Density Functional Theory (DFT) approach in the correct evaluation of the excitation energies. In fact, due to their reduced dimensionality, the inclusion of many-body (MB) effects in the theoretical description, in the so-called many-body perturbation theory (MBPT), is mandatory for a proper interpretation of the excited state properties. In particular, the quasiparticle structure is a key for the calculation of the electronic gap and to the understanding of charge transport as the inclusion of excitonic effects is really important for a description of the optical properties. In this paper, we apply DFT and MBPT to the calculation of the structural, electronic and optical properties of two classes of systems: pure and alloyed Si/Ge nanocrystals (NCs) embedded in wide band gap SiO2 matrices, and free-standing SiGe mixed nanowires (NWs). These systems have been chosen for their application in photovoltaics, and therefore our results will be discussed with respect to their potentiality. The paper is organized as follows: in section “Ab-initio Methods: DFT and MBPT”, we sketched the theoretical methods used in our computations, section “Embedded Si and Ge Nanocrystals” is devoted to the presentation of the results related to the embedded Si and Ge NCs, whereas section “Si/Ge Mixed Nanowires” discusses the outcomes for the mixed SiGe NWs, finally some conclusion is outlined in section “Conclusions”.
Ab-initio Methods: DFT and MBPT
DFT [17, 18] is a single-particle ab-initio approach successfully used to calculate the ground-state equilibrium geometry and electronic properties of materials, from bulk to systems of reduced dimensionality like surfaces, nanowires, nanocrystals, nanoparticles.
However, the mean-field description of the MB effects, taken into account in this method, by the so-called exchange-correlation (XC) term, is not enough to describe excited state properties. Even the time-dependent development of this approach, the TDDFT [19, 20], formally appropriate to calculate the optical excitations and the dielectric response of materials, presents problems due to the limited knowledge of the exact form of the XC functional [21, 22]. For these reasons, excited state calculations based on MBPT, performed on top of DFT ones, have become state-of-the-art to obtain a correct description of electronic and optical transition energies. The DFT simulations of our nanostructures are performed using the Quantum Espresso package , with a plane-wave (PW) basis set to expand the wavefunctions (WF) and norm-conserving pseudopotentials to describe the electron-ion interaction. The local density approximation (LDA) is used for the XC potential. A repeated cell approach allows to simulate NCs and NWs. A full geometry optimization is performed and, after the equilibrium geometry is reached, a final calculation is made to obtain not only the occupied but also a very high number of unoccupied Kohn-Sham (KS) eigenvalues and eigenvectors (ε n k , ψ n,k ) [24, 25]. In fact, although they cannot be formally identified as the correct quasi-particle (QP) energies and eigenfunctions, they are the starting point to perform MB calculations.
where β n k is the linear coefficient (changed of sign) in the energy expansion of the SE around the KS energies. In eq. 2, Σ x represents the exchange part and Σ c is the correlation part. To determine Σ c , a plasmon pole approximation for the inverse dielectric matrix, is assumed [28, 29].
This formula relies on the fact that, although in an inhomogeneous material the macroscopic field varies with frequency ω and has a Fourier component of vanishing wave vector, the microscopic field varies with the same frequency but with different wave vectors q + G. These microscopic fluctuations induced by the external perturbation are at the origin of the local-field effects (LF) and reflect the spatial anisotropy of the material. In particular for NWs, like other one-dimensional nanostructures [30, 31] it has been demonstrated [32–34] that the classical depolarization is accounted for only if LF are included and it is responsible of the suppression of the low energy absorbtion peaks in the ⊥ direction, rendering an isolated wire almost transparent in the visible region. A similar anisotropic behavior has been observed in the optical absorption of carbon nanotubes , in the photoluminescence spectra of porous Si  and in the optical gain in Si elongated nanodots .
Embedded Si and Ge Nanocrystals
In this section we present ab-initio results for Si and Ge NCs, pure and alloyed, that are embedded in a SiO2 matrix. The role of crystallinity (symmetry) is investigated by considering both the crystalline (betacristobalite (BC)) and the amorphous phases of the SiO2, while size and interface effects emerge from the comparison between NCs of different diameters. A mixed half-Si/half-Ge NC is additionally introduced in order to explore the effects of alloying. The BC SiO2 is well known to give rise to one of the simplest NC/SiO2 interface because of its diamond-like structure . The crystalline embedded structures have been obtained from a BC cubic matrix by removing all the oxygens included in a cutoff-sphere, whose radius determines the size of the NC. By centering the cutoff-sphere on one Si atom or in an interstitial position it is possible to obtain structures with different symmetries. The pure Ge-NCs and the Si/Ge alloyed NCs are obtained from such structures by replacing all or part of the NC Si-atoms with Ge-atoms. In such initial NC, before the relaxation, the atoms show a bond length of 3.1 Å, larger with respect to that of the Si (Ge) bulk structure, 2.35 Å (2.45 Å). No defects (dangling bonds) are present, and all the O atoms at the NC/SiO2 interface are single bonded with the Si (Ge) atoms of the NC.
To model NCs of increasing size, we enlarge the hosting matrix so that the separation between the NC replicas is still around 1 nm, enough to correctly describe the stress localized around each NC [39–41] and to avoid the overlapping of states belonging to the NC, due to the application of periodic boundary conditions .
Structural characteristics of the embedded crystalline (upper set) and amorphous (lower set) NCs: number of NC atoms, number of core atoms (not bonded with oxygens), symmetry (cutoff sphere centered or not on one silicon), number of oxygens bonded to the NC, number of bridge-bonds (see the text), average diameter d, supercell volume V s
V s (nm3)
Together with the crystalline structure, the complementary case of an amorphous silica (a-SiO2) has been considered. The glass model has been generated using classical molecular dynamics (MD) simulations of quenching from a melt, as described in Ref. . The amorphous a-Si10 and a-Si17 embedded NCs and their corresponding Ge-based counterparts have been obtained starting from the Si64O128 glass (supercell volume V s = 2.76 nm3), while for the a-Si32 and a-Ge32 NCs the larger Si216O432 glass have been used (supercell volume V s = 9.13 nm3). The structural characteristics of the embedded amorphous NCs are reported in Table 1 (lower set). We find that the number of bridge bonds (Si–O–Si or Ge–O–Ge, where Si or Ge are atoms belonging to the NC) increases with the dimension of the NC (three for the largest case and none for the smallest NC) in nice agreement with other structures obtained by different methods [50, 51]. For each structure we calculated the eigenvalues and eigenfunctions using DFT-LDA and in some cases MBPT [23, 24, 52]. An energy cutoff of 60 Ry on the PW basis has been considered.
Pure Si Nanocrystals
We resume here results previously obtained for pure Si NCs embedded in SiO2 matrices [24, 53–57]. These results provide not only a good starting point for the comparison between pure Si and pure Ge NCs (see III B) and with alloyed NCs (see III C), but also allow to discuss our results in view of the theoretical methods used (MBPT vs DFT-LDA) and with respect to the technological applications.
DFT-LDA HOMO-LUMO gap values (in eV) for the crystalline and amorphous silica, and for the embedded Si nanocrystals
Many-body effects on the energy gap values (in eV) for the crystalline and amorphous embedded Si10 dots
First, we note that the HOMO-LUMO gap for the crystalline cases seems to increase with the NC size, in opposition to the behavior expected assuming the validity of the QC effect. As discussed in Ref.  such deviation from the QC rule can be explained by considering the oxidation degree at the NC/SiO2 interface: for small NC diameters the gap is almost completely determined by the average number of oxygens per interface atom, while QC plays a minor role. Besides, also other effects such as strain, defects, bond types, and so on, contribute to the determination of the fundamental gap, making the system response largely dependent on its specific configuration. Moreover, looking at Table 3 we note that for the Si10 and a-Si10 embedded NCs the SE (calculated through the GW method) and the e-h Coulomb corrections (calculated through the Bethe–Salpeter equation) more or less exactly cancel out each other (with a total correction to the gap of the order of 0.2 eV) when the LF effects are neglected. Besides we note the presence of large exciton binding energies, of the order of 1.5 eV, similarly to other highly confined Si and Ge systems [32, 58–60]. Furthermore, some our recent calculations (still unpublished) show that the LF effects actually blue-shifts the absorption spectrum of the smallest systems (d < 1 nm), with corrections of the order of few tenths of eV. Instead, for larger NCs no blue-shift is observed. Therefore, while such corrections should be taken into account for a rigorous calculation, we expect that the LF effects will have the same influence on Si and Ge NCs of the same size and geometry, allowing in principle a straightforward comparison between the responses of the two compounds. Besides, Table 3 and previous MB calculations on Si-NCs show absorption results very close to those calculated with DFT-LDA in RPA [24, 55, 61, 62]. In fact these results show that the energy position of the absorption onset is practically not modified by the inclusion of MB effects. The arguments remarked above justify the choice of DFT-LDA for the results discussed in sections. “Comparison Between Pure Si and Ge Nanocrystals” and “Alloyed Si/Ge Nanocrystals”, assuring a good compromise between results accuracy and computational effort.
Concerning the applications we demonstrated  that the emission rates follow a trend with the emission energy that is nearly linear for the hydrogenated NCs and nearly cubic for the NCs passivated with OH groups or embedded in SiO2. Moreover, the hydrogenic passivation produces higher optical yields with respect to the hydroxilic one, as also evidenced experimentally. Besides, for the hydroxided NCs the emission is favored for systems with a high O/Si ratio. In particular the analysis of the results for the embedded NCs reveals a clear picture in which the smallest, highly oxidized, crystalline NCs, belong to the class of the most optically-active Si/SiO2 structures, attaining impressive rates of less than 1 ns, in nice agreement with experimental observations. From the other side, a reduction of five orders of magnitude (10 ms) of the emission rate is achievable by a proper modification of the structural parameters, favoring the conditions for charge-separation processes, thus photovoltaic applications . In the case of strongly interacting systems (i.e. when the separation between the NCs lowers under a certain limit), the overlap of the NCs WF becomes relevant, promoting the tunneling process. Therefore, while for the single Si/SiO2 heterostructure the e-h pair is confined on the NC, in the case of two (or more) interacting NCs a charge migration from one NC to the neighbor can occur . Evidence of an interaction mechanism operating between NCs has been frequently reported [64–66], sometimes indicated as an active process for optical emission , and sometimes even exploited as a probing technique . This interaction has been widely interpreted in terms of a kind of excitonic hopping or migration between NCs, although only more recently the mechanisms for carrier transfer among Si-NCs have been more clearly elucidated [69, 70]. Roughly speaking, the possibility of charge migration reduces the QC effect, possibly leading to the formation of minibands with indirect gaps . It should be noted that, contrary to photonics applications, for PV purposes the indirect nature of the energy bandgap in Si-NCs is advantageous, since the photogenerated e-h pair has a longer lifetime with respect to direct bandgap materials. Therefore, the NC–NC interaction can be considered as an additional parameter (tunable by the NC density) that concurs to the characterization of the system behavior: while the NC-size primarily determines the absorption/emission energy, the interaction level affects the absorption/emission rates. This picture opens to the possibility of creating from one side (high rates) extremely efficient Si-based emitters , and from the other side (low rates) PV devices capable to harvest the full solar energy with high yields. While the role of the NC size has been extensively investigated by many works, as theoretically like as experimentally, the study of the effects of NC–NC interplay is still at an early stage, due to the difficulties encountered.
Comparison Between Pure Si and Ge Nanocrystals
The analysis of Fig. 1 reveals that Ge-NCs present reduced gaps with respect to their Si counterparts. This could be reasonably associated with the reduced band-gap value of bulk-Ge with respect to bulk-Si. The Ge10 case is an outlier to this rule, showing a gap slightly larger than the Si10 NC. This exception can be justified by considering that such NCs represent a limit case in which all the NC atoms are localized at the interface.
It is noteworthy that the DOS profile arising from conduction states is similar for Si- and Ge-based structures of the same size, while the DOS profile arising from valence states differs for the two species. In particular, in the case of Ge-NCs the energy region around the valence band edge tends to be densely occupied, while for Si-NCs only few discrete levels appear in that region.
This result can motivate the employment of Ge together with Si for the production of semiconductor-based NCs, in order to improve the possibility of tuning the opto-electronic response by selecting, in addition to the structural configuration, also the composition of the NC. Another opportunity comes from the exploitation of alloyed Si/Ge-NCs, that could provide additional control over the final response as discussed in the next section.
Alloyed Si/Ge Nanocrystals
In this section, we consider the case of the Ge16Si16 NC, that has been built starting from the pure Si32 NC by replacing half of the NC with Ge atoms and then by totally relaxing the resulting alloyed. The crystallinity of the system permits a net specularity in the geometry of the two halves of the compound. This choice eliminates any complication that may arise from differences in the structural configuration of the two halves, eventually overbalancing the response of one species with respect to the other.
Si/Ge Mixed Nanowires
The scientific and technological importance of SiGe NWs is related to the peculiar physical properties that they present and that make them more suitable for PV with respect to the corresponding pure Si and Ge NWs. In fact it has been demonstrated, both experimentally and theoretically [12, 72–74], that the electronic and optical properties of SiGe NWs can be strongly modified by changing the size of the system (like in the pure nanowires [75–77]), but also by changing the relative composition of Si and Ge atoms and the geometry of Si/Ge interface [12, 78, 79]. This additional degree of control on the electronic structure makes these type of wires a possible route for PV because it offers a very wide range of possibilities to modulate the electronic structure of the material in order to obtain the desired properties. As discussed in section “Introduction”, in order to improve the efficiency in PV it is necessary or to maximize the absorption spectrum, or to obtain inside the material, after the absorption of light, a strong separation of electron and hole, or to improve the rapidity of transfer electrons and holes to metallic electrodes. Here, we show how a particular type of SiGe NWs, called Abrupt SiGe NWs and characterized by a clear planar Si/Ge interface, can satisfy (more than the corresponding pure NWs) the requirements of a material for solar cell.
The free-standing NWs considered here are oriented along the  direction (that guarantees thermodynamic stability ) and have an approximately cylindrical shape; the diameter range is from 0.8 to 1.6 nm and all the surface atoms have been passivated with H atoms in order to eliminate the intra-gap states. For the details of the construction of the geometry of NWs we refer to Ref.[12, 78]. We have analyzed pure Si, pure Ge and Abrupt SiGe NWs. This particular type of SiGe NWs is characterized by the presence of a planar Si/Ge interface along the shortest dimension of the transverse cross-section of the wire [12, 78]. The compositional range for Abrupt SiGe NWs is 0 ≤ x ≤ 1, where x is the relative composition of one type of atom with respect to the total number of atoms in the unit cell. An energy cutoff of 30 Ry, a Monkhorst-Pack grid of 16 × 1 × 1 points and 10 Å of vacuum between NWs replicas have been evaluated enough to ensure the convergence of all the calculated properties. In order to obtain the geometry of minimum energy of our structures, we have performed total energy minimization of the positions of atoms in the plane normal to the growth direction; while to take into account the effect of the strain in the direction of growth, we have used the Vegard’s law for semiconductor bulk alloys , which very recently has been demonstrated also valid for nanoalloys and which states that the relaxed lattice parameter of a binary system is a linear function of the composition of the system. After the evaluation of DFT-LDA ground state properties, in order to calculate the optical properties of the wires, in particular the excitonic wave function localization, we have solved the Bethe–Salpeter equation (BSE) in the basis set of quasi-electron and quasi-holes states, as described in section “Ab-initio Methods: DFT and MBPT”.
DFT-LDA electronic gaps (in eV) as function of Ge composition x Ge for Abrupt SiGe NWs with d = 1.6 nm
Composition x Ge
DFT-LDA E g
In this paper, we have presented ab-initio computational methods for determining the structural, electronic and optical properties of Si and Ge nanostructures. We have concentrated our interest to those nanostructures that play a role in PV applications. In particular, we presented one-particle and many-body results for Si and Ge nanocrystals embedded in oxide matrices and for mixed SiGe nanowires. The discussed results shed light on the importance of many-body effects in systems of reduced dimensionality. In particular, we showed for embedded Si and Ge nanoparticles how the absorption threshold depends on size and oxidation and we have calculated the exciton binding energies. Besides, we have elucidated the role of crystallinity and through the calculation of recombination rates and absorption properties we have highlighted the best conditions for technological applications. In the case of Si/Ge embedded alloyed nanocrystals, we have shown the dependence of the absorption spectra on the alloying and the presence of a different localization for HOMO and LUMO. Regarding the SiGe nanowires, we demonstrated that those which show a clear interface between Si and Ge originate not only a reduced quantum confinement effect but display also a direct band gap and a natural separation between electron and hole, a property directly related to PV potentiality.
The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement n. 211956 and n. 245977, by MIUR-PRIN 2007, Ministero Affari Esteri, Direzione Generale per la Promozione e la Cooperazione Culturale and Fondazione Cassa di Risparmio di Modena. The authors acknowledge also CINECA CPU time granted by CNR-INFM.
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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