A first-principles study of GaAs nanowires
In this paper we present a detailed analysis of the atomic and electronic structure of GaAs nanowires using first-principles pseudopotential calculations. We consider six different types of nanowires with different diameters all grown along  direction and reveal interesting trends between cohesive energy and nanowire type with varying diameters. Generally, the average cohesive energy of nanowires with wurtzite stacking is higher than those with zinc blende stacking for small diameters. We found that most of the bare nanowires considered here are semiconducting and continue to be semiconducting upon the passivation of surface dangling bonds with hydrogen atom. However, the surface states associated with the surface atoms having two dangling bonds in zinc blende stacking occur in the band gap and can decrease the band gap to change the nanowire from semiconducting to metallic state. These nanowires become semiconducting upon hydrogen passivation. Even if the band gap of some nanowires decreases with increasing diameter and hence reveals the quantum confinement effect, generally the band gap variation is rather complex and depends on the type and geometry, diameter, type of relaxation and also whether the dangling bonds of surface atoms are saturated with hydrogen.
pacs:61.46.Km, 62.23.Hj, 73.22.-f
In a search to discover semiconductor materials and metallic interconnect for new generation miniaturized electronic devices, nanostructures have been a focus of attention. Electronic devices, such as transistors based on carbon nanotubesreview (), attracted interest in nanowires. Rodlike Si nanowires have been fabricatedsinw1 () with a diameter 1.3-7 nm.sinw2 () It has been shown that such Si nanowires can display metallic, semiconducting and half-metallic properties depending on their functionalization.sinw3 () Being an alternative to silicon based microelectronics GaAs is one of the most important materials used in semiconductor physics. Due to the high electron mobility, GaAs always carried a potential of being used in high speed electronic devices. GaAs/AlGaAs heterostructuresesaki () have served as a media for the two dimensional electron gas studies.
Similar to bulk crystals researcher have envisioned GaAs naowires to be a potential alternative for Si nanowires. Recent advances in fabrication technology made it possible to grow GaAs nanowires. They are grown by metal catalysts in vapor-liquid-solid (VLS) mechanism.harmand () Generally, GaAs nanowires are grown along  direction in zincblende () structure, whereas nanowires with wurtzite () structure with diameter as small as 10 nm are also observed.mariager () Several models were developed to predict the transition radius from wurtzite to zinc blende structure.glas (); dubrovskii () Actually, there is no sharp transition but instead there are many different stacking configurations with very similar energies and more sophisticated models are needed to predict the ground state configuration. Together with the crystal structure, surface facet structure is also an important parameter affecting the structural and electronic properties. Nanowires reported so far are either grown in structure with  or  facet orientation or in structure with  or  facet orientation.leitsmann (); hiruma (); ohlsson () One of the unusual structures is so called A-wire, which has been grown in defect free triangular shape on (111)A surface of GaAs.wacaser () More complex structure of nanowires grown on GaAs(111)B surface have been also reported.lisa ()
There are many experimental and theoretical works on growth and structure of GaAs nanowires, but not much work is done to predict the electronic properties of these wires. Theoretical studies so far have focused on the electronic properties of superlattices composed of GaAs nanowires. For example, the electronic structure of InAs/GaAs nanowire superlattices with radius R=10 nm was examined using a semiempirical tight-binding model.niquet () Another atomistic tight-binding calculation was carried out to reveal the electronic structure of freestanding GaAs/AlGaAs nanowire superlattices oriented along the  crystallographic direction.persson () Also a first-principles investigation has been performed on the hexagon-shaped, /-oriented III-V semiconductor nanowires, which was concentrated on the surface effects on the structure and stability of these nanowires.leitsmann ()
The purpose of this paper is to provide a detailed analysis of GaAs nanowires, which is necessary for further experimental and theoretical studies. To this end we present a systematic, first-principles investigation on structural and electronic properties of GaAs nanowires grown along  direction. Six different types of GaAs nanowires are distinguished depending on the shape of their cross sections and the crystallographic orientation of their side surfaces. Their optimized atomic structure and cohesive energies are calculated revealing interesting trends between atomic structure and cohesive energy. Based on the calculations of electronic structure and isosurface charge density of specific states we analyzed the character of states at the band edges and variation of band gap with diameter. The effects of hydrogen saturation of dangling bonds of surface atoms on the atomic and electronic structure are examined. We found that most of bare GaAs nanowires are semiconducting and remain semiconducting even before the passivation of surface dangling bonds. Only one type is metallic due to the states localized at the surface.
We have performed first-principles plane wave calculationspayne (); vasp () within Density Functional Theory (DFT)kohn () using ultrasoft pseudopotentials.vasp (); vander () The pseudopotentials having three electrons for Ga (4 4), five electrons for As (4 4) and one electron for H (1) were used. A plane-wave basis set with kinetic energy up to 250 eV was used. Cutoff energies used were at least 30 higher than maximum values suggested.vasp () The exchange correlation potential is approximated by generalized gradient approximation (GGA) using PW91 functional.gga () For partial occupancies we use the Methfessel-Paxton smearing method.methfessel () The adopted smearing width is 0.1 eV for the atomic relaxation and 0.01 for the accurate band structure analysis and density of state calculations. All structures have been treated within a supercell geometry using the periodic boundary conditions. Vacuum spacing was arranged so that the minimum distance between two atoms in adjacent unit cells were larger than 10 , provided that atoms have negligible interaction at that far distances. We chose bare as a test structure and increased the vacuum spacings to 16 . This resulted in an energy difference around 0.2 meV/atom. In the self-consistent potential and total energy calculations the Brillouin zone (BZ) is sampled in the k-space within Monkhorst-Pack schememonk () by (1x1x9) mesh points for and (1x1x7) mesh points nanowires. Increasing the k-space sampling for bare structure from (1x1x9) to (1x1x15) resulted in a total energy difference around 1 meV. All atomic positions and lattice constant are optimized by using the conjugate gradient method where total energy and atomic forces are minimized. The criterion of convergence for energy is chosen as 10 eV between two ionic steps, and the maximum force allowed on each atom is 0.05 eV/. We have reduced the maximum force criterion down to 0.0025 eV/ in our test structure, bare . This had no considerable effect, since the change in energy and band gap was around 0.3 meV/atom and 0.006 eV, respectively. Clearly, the criterion for the maximum allowed force 0.05 eV/ is appropriate for systems including large number of atoms.
Iii Structures and Cohesive Energies
GaAs nanowires studied here are cut from ideal bulk structure along  direction. Nanowires having wurtzite () and zinc blende () stacking have four and six atomic layers in the unit cell, respectively. Except A-wire, they have hexagonal cross section. A-wires, by themselves have stacking and display a triangular cross-section with three () planar side surfaces. Atomic structures of cross-section of all nanowires are shown before and after relaxation (structure optimization) in Fig. 1. Here we consider the largest members of all types of GaAs nanowires. Upon relaxation the surfaces of ideal wires undergo a reconstruction, while inner parts preserve the bulk configuration. In spite of the fact that the indices of their planar side surface are the same, and are still different. structure have six identical surfaces, whereas structure have three surfaces same as and three surfaces with hanging Ga-As atom pairs before relaxation.
At the surfaces of some ideal structures in Fig. 1 the atoms can have the coordination number smaller than four. Upon relaxation the coordination numbers may undergo a change. Two adjacent surface atoms having low coordination number can form new bonds, whereby these atoms increase their coordination number and the nanowire, in turn, lowers its energy (i.e it becomes more energetic). It turns out that, the coordination number of surface atoms is crucial for the value of the cohesive energy per atom pair. For example, nanowire with has three surfaces each having an atom pair with coordination number two, while other three surfaces have surface atoms with coordination number three. nanowires having 25, 60 and 85 atom pairs in the unit cell have surface atoms with coordination number two only at the corners of the hexagonal cross-section. On the other hand, nanowires having 42 and 114 atom pairs in the unit cell and all members of nanowires have surface atoms with coordination number three. In light of these arguments one expects to have larger cohesive energy per atom pair than the rest of nanowires, which, in turn, should be larger than that of nanowires. These arguments are actually confirmed in Fig. 2, where we present the trends of cohesive energies per atom pair for all nanowires considered here.
Ideal A-wires have surface atoms with coordination number of two on each surface, but apart from that, it has a triangular cross-section, which makes the surface to volume ratio even higher compared to that of other types. Consequently, A-wires have the lowest cohesive energy per atom pair as seen in Fig. 2.
Interestingly, structures have relatively larger cohesive energies than structures, with structure being the largest. The bulk structure however, is energetically more favorable than that of structure by nearly 20 meV per Ga-As pair, also confirmed by our calculations. As radii increase, the cohesive energy per Ga-As atom pair values should converge to the bulk value. It is energetically easier to form a surface than to form a surface. In other words surface energy of structure is larger. That is why, structures become energetically more favorable as the surface to volume ration increases, namely as the radius of the nanowire decreases. So there should be some point where cohesive energy of and structures cross each other. The radius at this point can be interpreted as the critical radius for transition from to structure or vice versa. Here it should be noted that, energetically favorable does not mean that these structures will start to grow in experiments. One should also include effects of formation path, like nucleation growth. Actually, when we are around the critical radius we can see some hybrid stacking patterns like ABAC, which is defined as 4H structure.dubrovskii ()
Table 1 gives the calculated values for the structure and cohesive energies of nanowires after the relaxation. Cohesive energies per Ga-As atom pair increase with increasing diameter, approaching the bulk values, but we don’t see the critical radius because it is expected to be an order of magnitude larger than that of our nanowires.dubrovskii () Surprisingly, the lattice constant decrease as the diameter of nanowires in structure increase, while for structures the reverse situation occurs. The ratio of number of surface atoms to the total number of atom pairs give a measure of surface to volume ratio, which is decreasing with increasing diameter. Note that the number of surface atoms having coordination number of two is .
Iv Electronic Structure
Most of the relaxed GaAs nanowires presented in Fig. 1 are semiconducting even without hydrogen saturation. As we will see, in some cases these bare GaAs nanowires don’t even have the surface states at the band edges. This situation is in contrast with Si nanowires.sinw1 (); sinw2 (); sinw3 (); sinw4 () Si nanowires as cut from the bulk crystal and subsequently relaxed are found to be metallic. Their metallicity occurs due to the partial filling of the dangling bonds surface states. Upon passivation of the dangling bonds with hydrogen atoms the surface states are discarded from the band gap and eventually Si nanowire becomes semiconductor. In what follows, we will examine the electronic structure of bare GaAs nanowires and reveal the effects of geometry and the passivation of dangling bonds with hydrogen.bandgap ()
Figure 3 presents results of the band structure and charge density analysis performed for structure. Ideal structure of these wires have Ga-As atom pairs hanging on three surfaces, while other three surfaces have the same profile as type. Upon relaxation these hanging pairs tend to bend towards each other and lower energy by making unusual Ga-Ga and As-As bonds. For example, structure having 64 atom pairs per unit cell () have four hanging pairs before relaxation. After relaxation first two and last two of them bend to each other and form a stable structure. and nanowires have one hanging pair after relaxation. As a result, all surface atoms of structure have coordination number of three, while and have six surface atoms with coordination number of two (See Table 1). Isosurface charge densities show that three states at the top of the valence band edge of and structures have bulklike character, while conduction band edge states concentrate at the surface. It is hard to define the nature of states in structure because it has low diameter. To calculate the band structure of the relevant surface we cut a slab from the bulk wurtzite structure parallel to surfaces so that the resulting structure have 11 atomic layers. This slab has two dimensional periodicity in the surface and a vacuum region between adjacent surfaces, so that they don’t interact. The band structure of this slab plotted along k, parallel to -direction, provide us with information about the band structure of nanowires if they were grown thick enough to have a reasonable bulk region in order to reduce the corner effects. One expects the band gap of the infinite slab structure to be lower than that of the nanowires. Comparison of the bands of with those of surface confirm their similarity and expected size effect.
Figure 4 shows the band structure and charge density analysis for structures. Here all surface atoms of and structures have coordination number of three, while the rest of considered structures have surface atoms with coordination number of two at the corners. Note that the valence band edge of and structures mimic that of infinite slab structure, while other structures fail to do that. We see that the band gap of is larger than that of structure in agreement with the quantum confinement effect. On the other hand, however, structure have the lowest band gap. This is caused by large surface to volume ratio, which make surface effects pronounced. Isosurface charge densities show that the valence band edge have bulklike character, while conduction band edge states concentrate on the surface.
Band structure and charge density analysis of structure is illustrated in Fig. 5. This structure has the largest cohesive energy per atom value compared to other types of nanowires in Fig. 1. In this nanowire all surface atoms make three bonds and surface states derived therefrom do not occur in the band gap. Consequently, the band structure of the related infinite slab is very similar to that of nanowire. Charge density of the states at both band edges are spread throughout the nanowire cross-section showing the bulk character. Since band gap is not diminished by surface states bands, one can clearly see the quantum confinement effect in these nanowires.
Passivation of dangling bonds of semiconducting nanowires by hydrogen atoms, generally, results in significant changes in the electronic structure. These changes depend on whether the passivation is done before or after the relaxation of bare nanowires. We find the latter case more suitable for the simulation of the experimental procedure.sinw3 () Figure 5 includes information about the effect of hydrogen passivation of all surface dangling bonds on the band structure of nanowires. In contrast to Si nanowires (where surface states of bare structure were carried out from forbidden region to the band continua upon passivation with hydrogen)sinw3 (); ref1 (); ref2 (), the band edge states of GaAs nanowires remain in their place after the hydrogen saturation. This conclusion is corroborated by the analysis of isosurface charge density of states at the center of BZ located at both edges of band gap. We found that the character and charge distribution of these states do not undergo a change after passivation of surface dangling bonds with hydrogen atoms. Also the similarity in the profile of band edges before and after hydrogen saturation is found to be striking. Hydrogen atoms mostly affect the surface states, which, in structure, are found in the valence band continua. That is why, the effect of hydrogen saturation is not reflected on the band edges in the way it was in silicon nanowires having surface states at the band edges. Here the increase of the band gap occur not because the edge states are cleared out, but because the atomic structure of the nanowires become more bulklike. This effect is illustrated in Fig. 6, where we analyzed the interatomic distance distribution of structure before and after the hydrogen saturation. Plots given here are done by making a histogram of interatomic distances and then smearing it out. The first plot indicates the interatomic distance distribution of the core region of bare nanowire. We get the same result also in the core region of hydrogen saturated structure and the peaks match the first, second, third and fourth nearest neighbor distances of bulk GaAs in structure. The crucial difference between bare and hydrogen saturated structures is reflected in the shell part of the nanowires. One can clearly see that hydrogen saturated structure have more bulklike character.
Furthermore we performed atom projected density of states analysis by calculating the localized density of states on the surface atoms, as well as on the core atoms before and after passivation with hydrogen atoms. As seen in Fig. 6 (d) and (e), surface as well as core atoms have comparable contributions to the state distribution at both edges of band gap. These results further corroborate the fact that GaAs nanowires like having surface atoms with coordination number of three do not have dangling bond surface states in the band gap. For nanowires with large the passivation with hydrogen have negligible effects on the band gap. Note that the increase in the band gap with decreasing diameter holds also for hydrogen saturated structures.
In Fig. 7 we present the band structures and isosurface charge densities of selected states at -point of A-wires. Note that, we cannot have an infinite slab corresponding to the large A-wires because these nanowires have triangular cross-section. All relaxed A-wires have surface atoms with coordination number of two. The band gap have no obvious trend with varying diameter and is determined by surface states especially falling in the conduction band edge. For small the edges of band gap are determined by surface states with low dispersion. As the diameter increases the valence band edge starts to show bulk character, while states at the conduction band edge remain to display surface character. Even for nanowire the flat band states at the edge of the conduction band are located on one of the planar side surface.
The effect of surface states on the band gap is dramatic for the bare GaAs nanowires having stacking. The values of band gap as large as 1.5 eV occurring in -type nanowires reduce to the values as small as 0.2 eV in -type nanowires. The calculated band gap can even be closed in -type nanowires. It appears that many of the dangling bond surface states, in particular those associated with the surface atoms having coordination number of two, are placed in the band gap. Therefore one expect dramatic changes in the electronic properties after the passivation of surface dangling bonds with hydrogen atoms. The electronic structures of bare -type nanowires are illustrated in Fig. 8. Here all surface atoms of and nanowires have coordination number of three after relaxation, but certain surface atoms of structure cannot find a pair to form a bond and stay with coordination number of two. As a result the band gap of does not have a value between that of and , as one would expect. To get more information about the nature of the states at both edges of the band gap we have saturated all dangling bonds of and structures by hydrogen. After the hydrogen saturation the band edge profile does not stay the same, as it was in the case of structures. This means that the band gap of bare nanowire has increased upon hydrogen passivation of dangling bonds, since the band gap is cleared from the surface states. Apparently, the band edge states of the bare nanowire originate from the surface, while in structure they have bulklike behavior. The different behavior of triply coordinated surface atoms in and structure is attributed to their structural orientation. Surface atoms of structure make atomic rings perpendicular to the wire axis. Dangling bonds of these atomic rings form several minibands (with low dispersion along Z) in the band gap, with charge density localized at the surface. Surface atoms of structure, however, form atomic chains along the wire axis. The dangling bonds of these atomic chains form surface states with high dispersion, which fall in the band continua. The effect of hydrogen saturation is investigated in the case of a slab in Fig. 8. Here the first and third valence band edge states of bare slab structure are labeled as A and B, respectively. These dispersive states have bulklike character. Upon the passivation band A rises to A’ forming the conduction band edge, and band B rises to B’ forming the valence band edge of the passivated structure. Surface states having low dispersion are lowered to valence band continua.
The effects of the dangling bond surface states are even more dramatic in -type bare GaAs nanowires in Fig. 9, where the band gaps are closed and the nanowires become metallic. Therefore, -type is the only type we have considered to have metallic character. It is interesting that the corresponding infinite slab structure have a finite band gap. structures would have a band gap which is larger than that of the slab structure if they would had no partly filled corner states crossing the Fermi level. Isosurface charge density plots support the idea of band edge states being originated from the dangling bonds at the corners of hexagonal cross-section. We further tested these arguments by passivating the dangling bonds of with hydrogen. Upon hydrogen passivation all flat surface state bands in the range of energy from -0.5 eV to 0.5 eV disappeared and a band gap of 0.9 eV opened. The resulting band profiles near the edges of conduction and valence bands are similar to those of the slab slab surface given at the right hand side of Fig. 9.
V Discussions and Conclusions
We have performed first-principles DFT calculations to reveal the atomic and electronic structure of six different types of bare and hydrogen saturated GaAs nanowires. Nanowires considered have a diameter less than 3 nm, and at this sizes -type have the highest cohesive energy per atom pair, while A-wires have the lowest. In general, nanowires have higher cohesive energy than nanowires, but the difference in cohesive energies decrease with increasing diameter.
We found that all bare GaAs nanowires are semiconducting, except -type. In the latter structure, dangling bond states of atoms having coordination number of two cross the Fermi level and hence the structure becomes metallic. For bare GaAs nanowires in structure with surface atoms all having coordination number of three, the dangling bond states associated with these surface atoms do not appear in the band gap, but rather in the band continua. Therefore, quantum (size) confinement effect is apparent in their band gap variation with radius. For these -type nanowires the band gap may increase upon the passivation of dangling bonds with hydrogen, since bonding of surface atoms become more bulklike.
Surface states of GaAs nanowires in structure fall in the band gap, even if all surface atoms have coordination number of three. In structure surface atoms form atomic chains along the wire axis, while in structure they form non-interacting atomic rings perpendicular to the wire axis. Hydrogen saturation dramatically increases the band gap of structures by clearing the surface states from the forbidden region. Generally, increasing diameter results in more bulk like valence band edge, however conduction band edge behave more surface like.
For reasons explained in detail, the band gap variation of GaAs nanowires is rather complex and depends on their type and geometry, diameter, relaxation and also whether the dangling bonds of surface atoms are passivated with hydrogen. We believe that present results are valuable for further research on GaAs and other III-V compound nanowires dealing with their doping, forming heterostructure and multiple quantum well structure and their fuctionalization to get new electronic and magnetic properties.gudiksen (); bjork ()
Acknowledgements.Part of the computations have been carried out by using UYBHM at Istanbul Technical University through a grant (2-024-2007).
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