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Carrier Localization in Nanocrystalline Silicon Luigi Bagolini,† Alessandro Mattoni,‡ Reuben T. Collins,† and Mark T. Lusk*,† †

Department of Physics, Colorado School of Mines, Golden, Colorado 80401, United States Istituto Officina dei Materiali del CNR (CNR-IOM) UOS Cagliari, Cittadella Universitaria, I-09042, Monserrato (Ca), Italy

‡

S Supporting Information *

ABSTRACT: The localization of electronic energy orbitals is computationally examined for silicon condensed matter composed of crystallites encapsulated within continuous random networks. Density functional theory is used to elucidate the relationship between the orbital character, energy, and crystallite size for diameters up to 4 nm. The difference in long-range order alone is sufficient to induce hole localization within crystalline regions provided they exceed a critical size (1.7 nm), and the confinement power of the matrix is found to be the same as that associated with planar boundaries: 0.68. The spatial distribution of confined valence edge states can vary from nearly cylindrical to narrow ribbons. Conduction edge electrons, on the other hand, tend to be localized within satellite states at the interface between regions of local and extended order due to the presence of a shallow (56 meV) energy well surrounding the crystallites.

These two fieldstransport through aSi and quantum confined nanostructuresare now beginning to merge. The encapsulation of crystalline silicon nanocrystals (SiNCs) within hydrogenated amorphous silicon (a-Si:H) protects them from oxidation and takes advantage of the efficient photon absorption of the amorphous silicon in solar applications. Carrier transport is tremendously complex, though, because structural localization and quantum confinement are both active, and strain effects couple disordered and ordered regions in this nanocrystalline condensed matter.21 It may well be that new wave-like and/or hopping paradigms are associated with such intimately coupled, covalently bonded semiconductor matter. A broad range of microstructures, differing by amorphous-to-crystal ratio and by average grain size/shape, could be created via direct encapsulation of dots or by thermally activating their growth through amorphous recrystallization.22,23 The electronic structure of quantum confined nanocrystalline matter is a natural starting place for exploring order/disorder transitions and transport within each regime. Beyond such fundamental physics issues, a technological grasp of nanocrystalline optoelectronics would allow optical property control beyond just quantum dot size and could be used to construct multijunction cells such as that shown in Figure 1. Such research is in its infancy, but an important first step in this direction was a recent semiempirical tight binding (TB) study of quantum confined crystalline wires encapsulated within a-Si:H.24 The analysis suggested that valence band edge states could be confined to the NCs with conduction band edge states located within the matrix. A red shift in the NC optical gap was predicted

T

he interplay of structural disorder, correlation, and charge transport within amorphous silicon (aSi) has been studied for more than half a century,1,2 and it is arguably the canonical material system for exploring disorder-induced quantum phase transitions.3−5 Because hydrogenation passivates dangling bond defects and precipitated silicon nanocrystals stabilize aSi against light induced degradation,6,7 considerable effort has also been made to understand charge transport through mixed amorphous−crystalline nc-Si:H condensed matter. Here networks of precipitated crystallite needles offer carrier percolation pathways and may allow for hot electron harvesting.8 While this has resulted, for instance, in impressive solar cell devices with conversion efficiencies as high as 23.0%,9,10 the relationship between structure and transport is not yet understood within this more complex structural setting.11 In parallel with investigations of transport in aSi, efforts to understand the optoelectronic properties of nanostructured crystalline silicon12−14 have culminated in a recent, intense focus on quantum confined silicon nanostructures. This is justified by the breadth of technological opportunities associated with the ability to dial in optoelectronic character and by the prospect of achieving efficient multiple exciton generation15 and hot carrier collection.8 Such confinement also plays a critical role in siliconbased light-emitting diodes, promising candidates for the next generation of flat displays with advantages such as full-color emission, complementary metal-oxide-semiconductor compatibility, system feasibility, and low cost of fabrication.16,17 Moreover, quantum confined Si quantum dots (QDs) are increasingly important in medical applications since dots with optimized fluorescence profiles may be encapsulated within phospholipids for use as biomarkers to selectively adhere to cancer cells.18,19 Finally, quantum confinement must certainly be taken into account for emerging sub-20 nm silicon on insulator technology.20 © 2014 American Chemical Society

Received: March 11, 2014 Revised: June 5, 2014 Published: June 6, 2014 13417

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confinement power is quantified and compared to a special, one-dimensional setting. Intriguingly, the spatial character of the localized hole states is observed to be slaved to interfacial defects. A pair of defects on nearly opposing sides of the crystalline region results in a narrow, ribbon-like orbital. On the other hand, relatively defect-free interfaces result in nearly cylindrical hole states. The dimensionality of such quantum confined orbitals will be one and two, respectively, so interface defects play a direct role in influencing the optical properties of the composite material.

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METHODOLOGY There are two approaches to the computational construction of amorphous materials, and both are still the subject of methodological scrutiny: (1) classical molecular dynamics simulations carried out at finite temperature in which Si and H are free to make and break bonds; and (2) the so-called Wooten, Winer, Weaire (WWW) bond switching algorithm.29 The reliability of the first method rests on the accuracy of the classical potentials used in the dynamics and has no structural constraints leading to highly defected structures. These generate a finite density of states in the band gap region30 and result in an absorption coefficient far from that measured experimentally. The WWW algorithm takes a different, more empirical, approach by producing a continuous random network (CRN) in which no dangling bonds are present. When sufficiently relaxed,31,32 the band gap regions generate densities of state that are in line with experimental measurements. It is this second procedure that we adopted to create aSi-H:NC structures using a code developed by the authors. The resulting structures were found to have structural statistics consistent with experiment for distributions of bond length, bond angle, and dihedral angle.32 Hydrogenation was carried out by removing silicon atoms responsible for the most localized band tail states. This hydrogenation procedure is detailed elsewhere,31 but a few remarks are in order because the approach was tailored to the current setting. The number of atoms whose orbitals contribute to a given energy level is the fundamental quantity in this procedure and is used to determine the degree to which a given state is localized. Removal of the most localized states, followed by hydrogen passivation of the dangling bonds, eliminates the corresponding energy levels. Candidate states for removal are those composed of less than 15% of the totality of atomic orbitals. The number of atoms removed per energy level defines the hydrogen content of the structure that, in the present work, is set to be between 8% and 10% in order to match typical experimental values.33,34 As discussed in the Supporting Information, it is even possible to identify the eigenstate that will become the band edge after hydrogenation. The hydrogenated structures were relaxed using an in-house molecular dynamics (MD) code implemented at zero temperature with an environmentally dependent potential.35 This MD setting was previously found to be accurate in the analysis of hydrogenated nanocrystalline silicon.27 Density functional theory (DFT), detailed below, was then used to complete the relaxation process. This relaxation sequence was employed for computational expediency and found to be much more efficient than relaxation using DFT alone. Electronic structure calculations were carried out using DFT as implemented in the Siesta suite.36 Exchange and correlation effects accounted within the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE)37 which accurately describes functionalized and encapsulated Si quantum dot structures,38−40 the bonds in a-Si:H systems, and well-

Figure 1. Quantum confined silicon quantum dots encapsulated within a-Si:H. The size of quantum dots (light blue) can be used to control the effective optical gap of each layer in a multijunction solar cell, and the matrix (pixellated gray) causes a red-shift in these gaps.

that was consistent with photoluminescence measurements on laser annealed samples. The crucial role of hydrogenation24 in achieving quantum confinement was also elucidated.25,26 Specifically, hydrogen atoms tend to segregate within the amorphous phase27 where they passivate dangling bonds, restoring a clean gap to the amorphous phase that enables quantum confinement. Despite the merits of such TB analyses, the approximate nature of the approach did not allow for the electronic structure to be fully characterized, leaving many questions unansweredin particular, the nature and the spatial distribution of the charge carriers. In the present work we perform a quantitative investigation of the electronic structure of hydrogenated nanocrystalline silicon using a suite of computational methods that produce highly reliable Kohn−Sham (KS) orbitals. Such KS orbitals are physically reasonable approximations, for systems of the size proposed here, to the true wave functions,28 and thus will be used to elucidate the nature of quantum confinement in relation to the spatial distribution of band edge states. We find that band edge holes are confined to regions exhibiting long-range order, provided that they are sufficiently large. The corresponding band edge electrons, on the other hand, tend to be localized at the interface separating regions of local and long-range order. This has important implications for relaxed excitonic states as well as recombination and carrier transport rates. The associated 13418

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Figure 2. Isosurface of the HOMO energy levels for an encapsulated 2.1 nm diameter nanowire before (left) and after (right) 8% hydrogenation (magenta); 248 atoms are present in the crystalline phase periodic cross section and the isosurface level is set at 0.0085 e−;/Å3.

Figure 3. Isosurfaces of the HOMO energy level for a sequence of inclusions of increasing size. Isosurface levels are set at 0.085 e−/Å3.

due to periodic boundary conditions: 1.09 × 5.49 × 5.49 nm3 (2 × 5 × 5 cSi unit cells).

reproduced proton−proton separations.41 The basis set for Si was extended to include 3d and 4f orbitals and a second radial function was used for all channels, i.e., double-ζ basis set. The simulation supercells for most calculations were chosen to have dimensions 1.09 × 4.39 × 4.39 nm3 (2 × 4 × 4 crystalline silicon (cSi) conventional unit cells); for the largest nanocrystals, though, a bigger supercell was employed to avoid spurious effects

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RESULTS Eight specimens of crystalline wires, with diameters ranging from 1.6 nm up to 4.0 nm, were computationally created and encapsulated within a-Si:H using the approach described above. After hydrogenation and subsequent relaxation, the nanocrystal13419

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Figure 4. HOMO, LUMO, and gap (difference) energies as a function of dot size. Yellow circles (left panel) identify the states localized on the a−c interface. δE is the difference between bulk a-Si:H LUMO and that of the composite NC structure as discussed in the text.

(LUMO). As discussed in detail below, its spatial footprint is at the interface between the aSi and ncSi for all configurations considered, and this has important consequences for the nature of electron conduction. At this point, though, it is the trend in energy levels with NC size that is considered. In Figure 4 the HOMO (EH), LUMO (EL), and gap (HOMO − LUMO) energies are plotted as a function of dot diameter, showing the effect of the finite barrier supplied by the encapsulating aSi:H matrix. The cSi conduction energy levels were scissor-shifted to give a crystalline gap of 1.12 eV. The conduction levels of the aSi:H and of the mixed phases were then scissor-shifted by the same amount. The HOMO and LUMO energy level data were then fitted to obtain the following confinement relations, which are also plotted in the figure.

line specimens were subjected to an ab initio electronic structure analysis. The top valence band level exhibits the most interesting features from the perspective of both spatial distribution of the KS orbitals and influence on quantum confinement.42,43 This will be referred to as the highest occupied molecular orbital (HOMO) because it is generated at the wavenumber Gammapoint and represents the lowest energy hole state. As noted above, hydrogenation serves to passivate dangling bonds and has a significant effect on the spatial distribution of the HOMO (Figure 2). The valence edge hole localizes on a defect within the a-Si, but hydrogenation frees the hole, allowing it to take up a ribbon-like distribution within the dot. The comparison of the localized highest valence band KS orbitals of nc-Si:H with those of bulk cSi (available in the Supporting Information), reveals a clear resemblance in both shape and symmetry between such energy levels. As a consequence the HOMO of the mixed phase system may be considered to have a local crystalline character, allowing the particles in such orbitals to (1) be confined in the crystalline region and (2) possess physical properties like those of a tetrahedrally symmetric environment. These ribbons tend to connect localized defect pockets on the NC interface, presumably in response to local strain, and their width varies with that of the defect footprint. Since there is no experimental characterization of their shape as yet, the existence of such ribbons is a computational prediction with important consequences. For instance, the quantum confinement character of associated states will be influenced by this spatial distribution, with a two- and one-dimensional confinement characters in the extremes. A second important topological feature of such hole states is the existence of a critical dot diameter, D*, for these valence band edge states to be associated with the crystallite as shown in Figure 3. For the samples with diameters smaller than 1.6 nm, the HOMO level is localized in the amorphous matrix, while it is always confined to the nanowires for diameters greater than 1.7 nm. Even at hydrogenation levels up to 12%, it is not possible to confine the HOMO to the ordered region for the small crystallites. Reassuringly, semiempirical TB calculations performed on similar systems predicted a minimum size for confinement of 1 nm.24 This value is larger than that predicted by DFT and is attributable to the fact that the TB methodology only accounts for very localized interactions. We next turn attention to the lowest level of the conduction band, associated with low energy electron transport, which will be referred to here as the lowest unoccupied molecular orbital

E H = −4.04 −

0.44 D0.68

(1)

E L = −3.09 +

4.24 D5.52

(2)

The large exponent in the denominator of eq 2 reflects a weak dependence of the LUMO level on NC size. As a consequence, the confinement power of the HOMO/LUMO gap is determined by the exponent of the HOMO level in eq 1. These fits confirm our qualitative observations and can be compared directly with 1-D confinement. In a specialized 1-D setting, the quantum confinement power of cSi slabs with aSi:H was estimated to be 0.68,40 the same as the fit for the current 2-D setting. This is most likely due to the ribbon-like spatial character of the hole states confined to the nanocrystal that mimics onedimensional confinement. Such a small dependence of the confinement power with the dimensionality suggests that a similar trend can be expected within 3-D settings as well. The same NCs were also considered within a stand-alone (SA) setting by replacing the a-Si with hydrogen termination. An inverse power law fit gave an exponent of 1.37 for SA slabs40 which is typical of confined crystals.44−47 This is in contrast to a confinement power of 2.0 that might be expected by idealizing the system potential to that of an infinite square well. The difference is understood to be due to the nonparabolicity of the bands and quantum tunneling effects.48 Encapsulation of the NCs therefore has a dramatic effect on their confinement power (1.37 → 0.68) resulting in a remarkable red-shift in their optical response. Band offset considerations help to explain how the matrix modifies the confinement character of NCs. In the limit of bulk 13420

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Figure 5. LUMO isosurfaces are shown for the states highlighted with yellow in Figure 4: a, D = 2.04 nm; b, 2.5 nm; c, D = 3.24 nm; and d, D = 4.0 nm. Isosurface level is 0.007 e−/Å3. Panels (a) and (b) refer to the samples formed by 2 × 4 × 4 cSi unit cells, while panels (c) and (d) are associated with larger domains composed of 2 × 5 × 5 cSi unit cells.

The result of this shallow potential well is that conduction edge electrons will occupy the interfacial region at the order/ disorder boundary in what might be referred to as satellite states. This is shown in Figure 5 where the interface states are those highlighted with yellow in Figure 4. Interestingly, the arch of the interfacial LUMOs extends with increasing NC size, and at the same time its width decreases. The result, for the widest inclusion, is an eigenstate with a shape that exactly follows the line of the interface; the flattening of the LUMO curve in Figure 4 coincides with the rise of such a kind of state. The above alignment analysis is valid under the assumption that no dipole layers form at the interface due to mismatches of the sp3 hybrid orbitals.31 This is a reasonable assumption, though, because of local ordering in the aSi.

material properties, the valence band edge of a-Si:H is 0.28 eV lower than that of cSi, while its conduction band edge is 0.11 eV lower than that of cSi. The alignment between the separate amorphous and crystalline bulk phases is in agreement with those in the literature.49 A Type-II interface alignment50 therefore exists at the interface between regions of localized and extended order. Holes will lower their energy going from a-Si:H to cSi and the top valence band energy will be thus confined in the crystalline region. The opposite will occur for electrons in the conduction band and the LUMO energy level will be pushed out of the crystalline region. The first panel of Figure 4 also indicates that the LUMO of the composite NC structure is lower than that of pure a-Si:H by δE = 56 meV. This is due to the existence of a small potential well located at the order/disorder boundary, where conduction edge electrons can be trapped as interface states. This is consistent with the K·P theory for heterostructures in which a small well between different phases is predicted due to mild differences between the structural parameters between the materials.51 In our case, the fourfold coordination of atoms, also at the boundary between the ordered and the disordered phases, prevents strong reconstruction of chemical bonds that would otherwise have encouraged the formation of deep defect states similar to those of the band tail states that exist in the absence of hydrogenation.

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CONCLUSIONS

The electronic structure of realistic models of nanocrystalline silicon have been elucidated by hierarchically combining Monte Carlo, molecular dynamics, and density functional theory calculations. The methodology amounts to a computationally efficient protocol for the generation of accurately relaxed, hydrogenated structures with distinct interfaces between regions of extended and localized orderreferred to here as order/ disorder boundaries. Consistent with previous semiempirical 13421

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Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

calculations, quantum confinement is quantified and attributed to the localization of positively charged carriers within the crystal grains. The analysis goes further, though, characterizing the spatial distribution of band edge states in the mixed amorphous/ crystalline system. The electronic gap of the mixed-phase material can be tuned by varying its diameter and, surprisingly, the confinement power is found to be essentially the same as that associated with quantum confined planar geometriesi.e., 0.68. This is attributed to the 1-D ribbon-like nature of the confined holes that, in turn, results from strains generated at the order/disorder interface. This implies that an analogous effect may result in very similar quantum confinement character for 3-D settings. Nanocrystals with diameters less than 1.7 nm are not capable of confining valence band edge states. For crystallites slightly larger than this, the valence band edge state resides within the crystal and tends to have a ribbon-like shape. This is attributed to localized strain effects. As the nanocrystal size is increased, this confined state tends to become more symmetrically distributed, presumably because of the reduction in influence of interfacial strain effects with decreasing surface-to-volume ratio. The conduction band edge state is also influenced by the nanocrystal and tends to be trapped in a potential well of 56 meV that surrounds the order/disorder boundary. As a result, the band edge state follows the curvature of the interface but otherwise has an amorphous character. This is in contrast to the confined holes which exhibit Kohn−Sham orbitals reminiscent of bulk crystalline wave functions. Quantum confinement, working in conjunction with an order/ disorder interface, therefore results in both band edge states being associated with the nanocrystal. The implication for transport is that both electron and hole hopping may well be observed to move from nanocrystal to nanocrystal with holes contained within the crystal and electrons in satellite states localized to the surrounding amorphous layer. It may even be that colocated holes and electrons exhibit an indirect excitonic character that influences carrier dynamics.

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ASSOCIATED CONTENT

S Supporting Information *

Additional figures: Results of a convergence study using three types of atomic cells; X-point gap and actual indirect gap; HOMO before and after hydrogenation; and bulk isosurfaces, with explantions. This material is available free of charge via the Internet at http://pubs.acs.org/.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.M. acknowledges financial support by Istituto Italiano di Tecnologia (IIT-SEED POLYPHEMO and “Platform Computation”), Regione Autonoma della Sardegna (L.R. 7/2007 CRP 18013 and CRP-24978). This material is based upon work supported by the U.S. Department of Energy and the National Science Foundation under Award Numbers DE-EE0005326, DMR-0820518, and CNS-0722415. The work benefited from interactions with P. C. Taylor. This report was prepared as an account of work sponsored by an agency of the United States 13422

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