Ab initio study of electronic structures of InAs and GaSb nanowires along various crystallographic orientations

Computational Materials Science 50 (2010) 780–789

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Ab initio study of electronic structures of InAs and GaSb nanowires along various crystallographic orientations Wei-Feng Sun, Mei-Cheng Li ⇑, Lian-Cheng Zhao Department of Information Material Science and Technology, School of Material Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China

a r t i c l e

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Article history: Received 10 June 2010 Received in revised form 5 October 2010 Accepted 7 October 2010 Available online 3 November 2010 Keywords: Density functional theory Semiconductor nanowire Electronic structure Effective mass

a b s t r a c t InAs and GaSb nanowires oriented along different crystallographic axes—the [0 0 1], [1 0 1] and [1 1 1] directions of zinc-blende structure—have been studied utilizing a first-principles derived nonlocal screened atomic pseudopotential theory, to investigate the band structure, polarization ratio and effective masses of these semiconductor nanowires and their dependences on the wire lateral size and axis orientation. The band energy dispersion over entire Brillouin zone and orbital energy are determined and found to exhibit different characteristics for three types of wires. There is an explicit dispersion hump in the conduction bands of [0 0 1] nanowires with two larger diameters and [1 0 1] nanowires with the smallest diameter considered. Moreover, the [1 1 1] nanowires are shown to exhibit very different orbital energy for the maximum valence state at the zone-boundary point, compared with [0 0 1] and [1 0 1] nanowires. These differences present significant and detailed insight for experimental determination of the band structure in InAs and GaSb nanowires. Furthermore, we study the polarization ratio of these nanowires for different orientations. Our calculation results indicate that, for the same lateral size, the [1 1 1] nanowires give extraordinarily higher polarization ratio compared to nanowires along the other two directions, and at the same time have larger band-edge photoluminescence transition intensity. Consequently, the [1 1 1] nanowires are predicted to be better suitable for optoelectronic applications. We also significantly find that polarization ratio and transition intensity displays different varying trend of dependence on lateral size of nanowires. Specially, the calculated polarization ratio is shown to increase with the decreasing size, which is opposite to the behavior displayed by the optical transition intensity. The predicted polarization ratios of [1 0 1] and [1 1 1] nanowires for 10.6 Å diameter approach the limit of 100%. In addition, the electron and hole masses for InAs and GaSb nanowires with different crystallographic axes have been calculated. For the [1 0 1] and [1 1 1] oriented nanowires, the hole masses are predicted to be around 0.1–0.2 m0, which are notably smaller than the values (0.5 m0) along the same direction for their bulk counterparts. Thus, we demonstrates an inspired possibility of obtaining a high hole mobility in nanowires that is not available in bulk. The small hole mobility is interpreted as to be associated with the strong electronic band mixing in nanowires. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Semiconductor nanowires (NWs) are substantially important for the advanced development of the electronic and optoelectronic applications such as integrated nanocircuits [1–4] and low-dimensional photodetectors [5,6], as well as semiconductor nanowire lasers [7–9], which necessitate the understanding their electronic and optical properties. The high conductivity along wire axis which could be obtained by feasible doping in semiconductor NWs makes carriers transmit easily, and one the other hand, the tunable quantum confinement in lateral directions provide possible enhanced efficiency. As a result, though the theoretical research and ⇑ Corresponding author. Tel.: +86 15846592798; fax: +86 0451 86418745. E-mail address: [email protected] (M.-C. Li). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.10.011

understanding of NWs are much less than those of nanodots [10–16], the NWs are more suitable for these applications than the three dimensionally confined nanoparticles. Several theoretical [17–20] and experimental studies [21–25] of InAs and GaSb NWs have been reported. dos Santos and Piquini [17] performed first-principles calculations of mechanical, electronic and structural properties of [1 1 1] zinc-blende InAs NWs. Pistol and Pryor [18] reported the calculated band structures of strained segmented InAs, GaSb, and several other semiconductor NWs for the [0 0 1] and [1 1 1] crystallographic axis directions, using strain-dependent k  p theory. Persson and Xu [19] presented a theoretical study of the electronic structure of free-standing InAs NWs along [0 0 1] crystallographic orientation, based on the atomic tight-binding approach. The structural stability and electronic properties of different shapes of GaSb NWs have been studied by

W.-F. Sun et al. / Computational Materials Science 50 (2010) 780–789

ab initio approach within the generalized gradient approximation [20]. The electronic transport in InAs NWs and InAs/InP NW heterostructures studied by Thelander et al. [21] showed strong confinement effects and carrier depletion for wires with diameters less than 30 nm. Experimental studies on the diameter dependence of the transport coefficients [22] and electron mobility [23] in InAs NWs have been performed. Jeppsson et al. [24,25] reported growth-optimization study and the successful growth of [1 1 1] GaSb NWs by MOVPE, with the structural property characterization by transmission electron microscopy, X-ray diffraction and single nanowire photoluminescence. The crystallographic orientation of the wire axis for NWs is a more possibly useful degree of freedom in effectively modifying the material properties, which, however, does not exist in nanoparticles. For an equal diameter, the NWs formed along different axes, along the [1 1 1] or [1 0 1] crystallographic directions of zincblende structure for example, could exhibit distinct features of zone-center electronic state, optoelectronic transition wavelength and intensity, etc. Thus, it is deduced that only some wires along a certain axis orientation can preferably dominate properties than the others. The polarization ratio determining the photon absorption or emission in NWs is an essential property, which often provides considerable higher resolution of contrast and sensitivity [26,27]. Therefore, more preferably concerned than the direct transition intensity (optical absorption or luminescence), polarization ratio is a regular quantity employed in photodetectors and optical communication. Recently, highly polarized luminescence has been reported to be as high as 90% of polarization ratio in free-standing semiconductor NWs [28,29]. The observed extremely high polarity was interpreted as the dielectric confinement, in which the environment with small dielectric susceptibility surrounding NWs allows the only axis-polarized electric field of excitation laser to penetrate effectively into the wires. As to our knowledge, there has hitherto no report for the polarization spectroscopy of the InAs and GaSb NWs. It is noted that the dielectric confinement model does not explicitly depend on the electronic structures of wires, and essentially represents an extrinsic mechanism, but not an intrinsic one. The other intrinsic mechanism causing a high polarization of optoelectronic transition is correlated with the electronic states of each wire. Especially, the optoelectronic transition polarizes when the dipole matrix elements between band-edge states depend significantly on the polarization direction of the excitation electric field. The analytical technique and theory of polarization-dependent optical transition matrix elements including band mixing between heavy hole and light hole states were originally suggested and demonstrated for V-shaped semiconductor quantum wires [30–33], and further investigated for free-standing NWs [34–37]. The second mechanism can give substantial insight into the microscopic electronic properties of semiconductor NWs. In previous studies, the tight-binding or k  p theory was employed to calculate the polarization ratio for free-standing NWs [36,37] only along the [1 1 1] wires axis. In present paper, we concern on how the crystallographic axis of NWs will affect the optical polarization ratio. The results of this exploration could demonstrate whether it is possible to improve the polarization ratio by modifying the nanowire orientation. Because the energy separation and coupling between heavy hole and light hole subbands vary with the different wire orientations, the polarization ratio of NWs depends on crystallographic direction of wire axis. The effective mass of electron or hole that determines the carrier mobility and speed of device performance is a special property for semiconductor NWs. Particularly, with a high interest of relevance, one desires to know whether it is possible to obtain a considerably smaller effective mass (electron or hole) in NWs than that in the corresponding bulk, assuming that carriers transmit

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along the same crystallographic direction in both systems. If this is validated to be available, it will lead to a faster operation speed of nanodevice that is not possible in bulk. Previous studies reveal that this is definitely feasible in strained quantum wells, due to the heavy hole and light hole coupling that depends on the strain conditions in system [31,32]. However, it is assumed there is no strain in ideal free-standing NWs, and besides, electronic structures of NWs are generally different from those of two-dimensional quantum wells. This indicates why obtaining a small effective mass in NWs is particularly significant now. In this report, we also investigate the carrier effective mass (along wire axis) in NWs for different crystallographic orientations. For most III–V bulk materials, it is known that the effective mass of hole along [0 0 1] direction is obviously smaller than along [1 0 1] and [1 1 1] directions [38]. We particularly wonder whether this trend can apply to NWs, i.e., whether we can reasonably believe that the [0 0 1] wires remain to have a much smaller mass and are thus more suitable for achieving fast device performance than the wires along two other directions. Finally, for better understanding of effective mass in NWs, one may like know whether it is possible to explore a general rule that presides over the effective mass in these NWs. This procurable rule may be of high value for analysis and design of carrier mobility in semiconductor nanostructures. In present paper, we endeavor to represent the above issues that could be significant for a more comprehensive and detailed understanding of semiconductor NWs, by performing pseudopotential calculations for InAs and GaSb (with near lattice-constants of bulk in same crystallographic structure) NWs of different lateral sizes grown along different crystallographic axes. Our results predictably indicate that varying crystallographic axis is indeed able to considerably change optoelectronic transition intensity, polarization ratio and carrier effective mass in NWs, and thus can provide another possible approach to engineer these properties in addition to modifying size or dimensionality. Furthermore, we find that the [1 1 1] oriented NWs are capable of giving both larger band-edge optical intensity and higher polarization ratio when their lateral sizes become small, as compared to the NWs along other directions. Based on our study results, effective mass along a given mobility direction in NWs can be remarkably smaller than that in bulks, which demonstrates the feasibility and provides a useful theoretical support to experimental design and fulfillment of faster device performance by nanowire applications. Moreover, our results explain that the obtained strong polarity in InAs and GaSb NWs can be due to the intrinsic electronic structures. Polarization ratio and transition intensity in NWs are further shown to vary with wire diameter in an opposite trend. We also find that the reduced effective mass in strained quantum wells [39,40] and in free-standing NWs result from different origins. It is unexpected from our results that the varying trend of effective mass in bulks cannot be directly applied to the NWs.

2. Theoretical methodology The first-principles density functional theory (DFT) [41] is preferably used to accurately describe the electronic structure of semiconductor wires, taking into account the charge redistribution that can occur both inside and at the surfaces of nanostructures. Furthermore, the dynamical-mean-field theory within non-crossing approximation is used to study the electronic structure of strongly correlated electron systems [42]. The quasi-particle first-principles calculations computed within GW approximation of many-body theory have been used to study the excitons, optical properties, and hot-electron lifetimes under electron–phonon interaction in many-electron systems [43,44]. However, the calculations directly employing first-principles approaches are generally time consum-

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ing, which limits the first-principles theory to be applied for dots or wires with large number of atoms in one supercell. Moreover, it is necessary to consider the spin–orbit interaction for some narrowgap semiconductor such as InAs or InSb. In the majority of nanostructure experiments, the material surfaces have been passivated by capping organic materials, and these passivated nanoparticles or NWs are technologically more useful due to their higher quality. For example, the influence of surface passivation on the transport properties of InAs NWs, determined by Hang et al. [45], indicated that electron mobility in passivated devices is superior to the unpassivated ones. The passivation chemically saturates the dangling bonds on the surfaces, and remodels the defect states by forming bonding and anti-bonding states. Alternative approaches take advantage of this fact. Therefore, in the completely passivated nanodots or NWs, many properties are measured to be mainly ‘‘bulk-like”, which means that they originate from the inner space of the nanostructures and rely little on surface conditions. These bulk-like properties are intrinsic and of our predominant interest because they can be effectively controlled by properly varying size. On the other hand, the interesting surface effects in semiconductor nanostructures vary from sample to sample. Alternative approaches for ideally passivated nanostructures to determine their electronic structure include linear combination of atomic orbits (LCAO) or tight-binding [46,47], empirical pseudopotentials [48,49] and multi-band k  p theory [15,50]. Here we calculate the electronic structure and associated properties of NWs using first-principles theory derived nonlocal screened atomic pseudopotentials (SAPs) [51–53]. Compared with bare atomic pseudopotentials, SAP takes into account the self-consistent charge screening effects by forming chemical bonds in solids, and are acquired and parameterized straightforwardly from first-principles density functional calculations of bulk semiconductors with different crystal structures and different cell volumes [52]. Thus, these DFT derived pseudopotentials obviously remain the problems of first-principles theory in local density or general gradient approximations (LDA or GGA) and underestimate the energy band-gap. To obtain the corrected LDA and GGA band-gaps, these DFT derived atomic pseudopotentials are accordingly rectified to reproduce the experimental or quasiparticle transition gap. It has been demonstrated that a small modification of the screened atomic potential around the nuclear core is sufficient to correct the band-gap, while maintaining wave functions nearly unchanged [51]. Furthermore, the SAPs at the same time result in reliable electron and hole effective masses that are in excellent agreement with experimental values. Compared with tight-binding and empirical pseudopotential methods, nonlocal screened pseudopotentials are shown to be able to reproduce accurately the single-particle DFT wave functions [53], and thus preferred to be used in determining transition matrix as well as other optical properties. In comparison with k  p theory which is reliable near the zone-center, the SAP method can generates accurate band dispersion over the entire Brillouin zone. The SAP deriving details and demonstration of the potential reliability were presented in Refs. [51–53]. Using a similar folded Hamiltonian technique [51] including spin–orbit coupling, electronic structure (orbital energies as well as wave functions of electron states) in NWs is calculated by regularly solving the Schrödinger equation for quantum systems with a large number of atoms:

" 

r2 2

# þ VðrÞ Wi ðrÞ ¼ Ei Wi ðrÞ

Wave functions Wi(r) are expanded in term of plane wave basis set. The potential V(r) is built by superposing the SAP pseudopotentials of all atoms in the nanostructures. Momentum (optical transition)

matrix elements are determined by integrating the realistic wave functions derived from the Schrödinger equation. Nanowire surfaces are passivated by pseudo-hydrogen atoms [52]. 3. Results and discussion Since the NWs is naturally more preferable and easier to be grown along the crystallographic axes of high symmetry, we investigate InAs and GaSb (with almost same bulk lattice-constants) NWs grown with their axes respectively along the [0 0 1], [1 0 1] and [1 1 1] crystallographic directions of the zinc-blende structure to exemplify NWs with all possible axis orientations. It is expected that these studies can generalize some conclusions that will also be valid and valuable for the NWs in other orientations. The investigated [0 0 1], [1 0 1] and [1 1 1] NWs have different cross-section shapes of square, rectangle, hexagon respectively. Here, we define the nanowire diameters as those of the thinnest cylinders that contain the NWs excluding the saturation layer. In our calculations, for all the three types of NWs with different axis orientations, the diameter D of lateral size is set to be four values of 10.6, 21.2, 31.7 and 42.3 Å to study the size effects on nanowire properties. 3.1. Band structures of different NWs Fig. 1 show the calculated band structures of [0 0 1] axis-oriented InAs and GaSb NWs with different lateral sizes. It is indicated from the results of [0 0 1] axis-oriented NWs (Fig. 1) that the InAs and GaSb NWs with three larger diameters have a direct band-gap, thus compatible for optoelectronic applications, whereas the minimum of the lowest conduction band for the smallest wire diameter (D = 10.6 Å) is located away from Brillouin zone-center, making an indirect band-gap. More profound, we reveal that, for [0 0 1] InAs and GaSb NWs with the lateral size in the range of 21.2–42.3 Å, the top valence bands exhibit rather parabolic over Brillouin zone, whereas the conduction bands display a pronounced hump for D = 31.7 and 42.3 Å diameters. The hump of the lowest conduction band is calculated to locate at khump  0.3 (2p/a) for both D = 31.7 and 42.3 Å NWs, where a = 6.058 and 6.096 Å are the lattice-constants of bulk InAs and GaSb respectively. Besides, the humps of high-leveled bands occur at khump  0.3 and 0.2 (2p/a) respectively for D = 31.7 and 42.3 Å NWs. The cusp point in band structure generates singularity in density of states and can be used to study electronic states coupling due to band crossing, and thereby is of our high interest. By comparing the projected wave function projection, we find that the electron states at the cusp point are similar to the lowest conduction band states of bulk InAs and GaSb at the k points near the midway between C and X. The electronic energy level at the cusp point (khump) is calculated to be 1.0 and 0.8 eV above the conduction band minimum (CBM) at C for D = 31.7 and 42.3 Å diameters respectively, similarly in InAs and GaSb NWs as shown in Fig. 1. It is also demonstrated from band structures of two largest NWs investigated (D = 31.7 and 42.3 Å) in Fig. 1 that the considerably separated conduction subbands near C can be applied for infrared detection with wavelength modified by varying lateral size. The calculated energy difference between two lowest subbands at C is 1.1 and 1.8 eV for the D = 42.3 Å InAs and GaSb NWs respectively. The lowest conduction state energy at the zone-boundary X point is found to show a sizable dependence on the wire diameter, varying from 4.0 to 5.5 eV and from 4.1 to 5.0 eV for InAs and GaSb NWs respectively as the lateral size increases from D = 10.6 Å to D = 42.3 Å. It is noticed that this 1.5 (1.9) eV energy shift due to different quantum confinement for the lowest conduction state at X is substantial and comparable with that of 1.45 (0.95) eV at C for InAs (GaSb) nanowire within the size range con-

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Wave Vector k (Γ →X) Fig. 1. Band structures of the [0 0 1] oriented InAs (up panels) and GaSb (low panels) NWs with different lateral sizes. The panels from left to right represents the NWs with diameters D = 10.6, 21.2, 31.7 and 42.3 Å respectively. The wave vector is in the unit of 2p/a, where a = 6.058 and 6.096 Å are the lattice-constants of bulk InAs and GaSb.

sidered. Size effect on the electronic states at the zone-boundary is seldom reported and scantily understood, though many studies concern on the band energy at zone-center. The indirect band-gap or lower conduction band energy at X can be explained by the characteristics of bulk InAs and GaSb band structures. The energy dispersion of the lowest conduction band along the X–W direction is precipitous for bulk InAs and GaSb, resulting in large energy shift of the conduction states at X in NWs. Compared with the conduction band states, our calculations indicate that, except for the D = 21.2 Å size (only 0.5 eV higher), the highest valence state energies at X of [0 0 1] InAs and GaSb NWs are almost fixed to be about 7.5 and 7.0 eV respectively, depending much less on wire diameter as shown in Fig. 1. Figs. 2 and 3 show the calculated band structures of differently sized NWs with the axes along [1 0 1] and [1 1 1] crystallographic directions respectively. The results of [1 1 1] InAs NWs agree well with the values from the other first-principles study [17]. In a notable contrast with the [0 0 1] NWs, Except for smallest diameter, the conduction bands of the [1 0 1] NWs do not show a hump, and exp hibit instead a band crossing at k  0.3 (2p/ 2a) only for D = 21.2 Å NWs, as shown in Fig. 2. Furthermore, the conduction band structures of all the investigated [1 1 1] NWs exhibit completely no hump character as shown in Fig. 3. This clear axis dependence of nanowire band structure is due to the band folding effect which is different along the different crystallographic directions. Moreover, for [1 0 1] NWs, the calculated highest valence state energies at zone-boundary K for the D = 10.6 Å InAs and D = 42.3 Å GaSb NWs are shown to be 7.8 and 8.2 eV respectively, which are appreciably different from those for the same size [0 0 1] InAs and GaSb NWs. Besides, the other [1 0 1] NWs show almost the same value 7.0 eV of the highest valence state energy at K. Especially as shown in Fig. 3, the top valence states at the zone-

boundary L point for [1 1 1] InAs (GaSb) NWs are determined to be slowly increasing from 7.6 (6.8) to 6.8 (6.1) eV of orbital energy as the diameter increasing from 10.6 to 42.3 Å, showing much difference with the [0 0 1] and [1 0 1] NWs. Therefore, our results indicate the notable difference in the highest valence orbital energy at the zone-boundary for NWs along different axis orientations, which could be examined and confirmed by photon luminescence spectroscopy in experiments [38]. 3.2. Size dependence of band-edge orbital energies Fig. 4 depict the energy values of the conduction band minimum (CBM) and valence band maximum (VBM) at C for all the investigated InAs and GaSb NWs. Compared with bulk values, the differences between the obtained single-particle band-gaps of NWs in this study and the band-gaps of their bulk counterparts generally agree well with other theoretical calculations [17–19]. For instance, the difference between the determined band-gap of the D = 10.6 Å [1 1 1] InAs nanowire and the band-gap of bulk InAs is 2.59 eV here, almost equivalent to 2.31 eV from the other DFT calculation [17]. It is noticed from the results in Fig. 4 that the VBM energies of the [1 0 1] and [1 1 1] oriented NWs are very close for both small and large lateral sizes. Interestingly, we find the curves of VBM energies (shown in Fig. 4) can be analytically fitted to be E = Ebulk + M/Dn (Ebulk denotes bulk energy) with universal n  2.63 and 2.89 with ±6% fluctuation for InAs and GaSb NWs respectively for different crystallographic axis orientations, despite the M parameters are different. The analytically fitting parameters M and n for different oriented NWs are given in the Table 1. Similarly, the fitting exponent parameters n  1.16 and 1.31 of the respective InAs and GaSb nanowire CBM energies also exhibit the same value fluc-

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tuation ±6% of little orientation dependency. In addition, to investigate how the n value depends on nanowire dimensionality (along axis) instead of the wire axis orientation, we have further performed calculations for InAs and GaSb spherical dots using the same method of pseudopotentials and passivation as the NWs. The fitting parameters for nanodots are also given in the Table 1. The n values of InAs (GaSb) dot for the VBM and CBM energy are determined to be 1.61 (1.97) and 0.81 (0.99) respectively, demonstrating the significant variation of the parameter n with nanowire dimensionality along axis. 3.3. Polarization ratio in NWs We have calculated the optical transition matrix elements P  Px + Py + Pz at the zone-center C point for InAs and GaSb NWs

with differently oriented axes and different lateral sizes. The constituent component Pi (i = x, y, or z) is determined as ||2 and found to be dependent on the wire diameter as well as axis crystallographic orientation. In our calculations, we define the z axis to be along the nanowire axis. The Fig. 5 depicts the calculated zone-center optical transition matrix elements P and each component Pi of differently oriented InAs and GaSb NWs with different wire diameters. It is noticed that the obtained band-edge transition matrix P declines with the decreased lateral diameter for each type of NWs. This behavior of results is inconsistent with the normally supposed expectation that the enhanced quantum confinement of electron and hole in NWs could result in larger magnitude of transition matrix, as wire diameter becomes smaller. Furthermore, it is indicated from Fig. 5 that the value of P does not depend sensitively on the wire axis orientation for large size NWs. In fact, for the same diameter D = 42.3 Å, the P magnitudes of the [0 0 1], [1 0 1] and [1 1 1] oriented InAs (GaSb) NWs are 0.66 (0.81), 0.70 (0.86) and 0.74 (0.86) respectively, showing much little variance. However, when the size becomes smaller, the P values of the [0 0 1] and [1 0 1] InAs (GaSb) NWs decline more sensitively than [1 1 1] NWs, by respectively decreasing to 0.33 (0.47) and 0.32 (0.63) for D = 10.6 Å diameter. In contrast, the reduction of the transition matrix magnitude is the least for the [1 1 1] crystallographic orientation, and consequently, the D = 10.6 Å [1 1 1] InAs and GaSb NWs maintain the large P values of 0.53 and 0.65 respectively. Therefore, base on our theoretical calculations, it is proposed that the [1 1 1] oriented NWs will be more suitable for the optoelectronic applications requiring a sharp band-edge absorption and particularly small size. The more interesting polarization ratio is defined as R = (Pz  Px  Py)/(Pz + Px + Py). Since the D = 10.6 Å [0 0 1] InAs and GaSb NWs have indirect band-gaps, they are not considered in term of polarization ratio here. The calculated Px and Py are considerably smaller than Pz for all direct-gap NWs (shown in Fig. 5), resulting in remarkably high polarization ratios in InAs and GaSb NWs. The photon polarization ratios of all the direct-gap NWs are revealed to be enhanced as the lateral size decreases, and especially approaches to the extreme limit of 100% for the D = 10.6 Å. However, in notable difference, the total transition magnitude P is varying in an opposite trend and becomes weaker with the size decreased. Therefore, we can conclude that size dependences of polarization ratio and transition matrix in semiconductor quantum wires are very different. Such extraordinarily high polarization ratios in small NWs confirm that polarization ratio is more preferable to be utilized for nanowire optoelectronic applications than the optical intensity. Moreover, by comparing the transition matrix and polarization ratio of NWs with different crystallographic axes, we find interestingly that the [1 1 1] NWs produce both stronger optical intensity and higher polarization ratio for the same given wire diameter, predicting better performance applications of NWs in this crystallographic axis. Based on our calculation results, for the highly polarized photoluminescence observed in semiconductor NWs [6], the orbital-in-

Table 1 The fitting parameters M and n obtained from the analyses of the single-particle energies with our pseudopotential calculations. The M and n values for InAs and GaSb spherical dots are also given for comparison. NWs

Parameters

InAs

CBM VBM

GaSb

CBM VBM

[0 0 1]

[1 0 1]

[1 1 1]

Nanodot

M n M n

32.47 1.23 250.7 2.63

27.48 1.12 294.4 2.55

28.59 1.15 411.0 2.70

18.31 0.81 144.4 1.61

M n M n

28.89 1.38 374.6 2.76

18.69 1.25 268.3 2.86

19.45 1.29 483.7 3.07

12.45 0.99 98.2 1.97

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Fig. 5. Zone-center optical transition matrix element P, constituent components Pi (i = x, y, z) and polarization ratio R of (a) InAs and (b) GaSb NWs with the axes along the [0 0 1] (up panels), [1 0 1] (middle panels) and [1 1 1] (low panels) crystallographic directions, as a function of wire diameter. P and Pi are described by the left vertical coordinate axis, and polarization ratio R is described by the right vertical coordinate axis.

duced origin [35–37] is confirmed rather than the dielectric confinement. However, we consider the presence of vacuum environment that is not related to the dielectric confinement in our calculations. The incident electric field of the excitation photons and the dipole matrix are two independent quantities in determining the polarization ratio and absorption intensity [38]. The dielectric confinement is represented by its effects on the electric field penetrating into the NWs, while the dipole matrix is determined by the electronic wave functions. Thus the high polarization ratio calculated using the dipole matrices results from the intrinsic electronic states in NWs, as shown in Fig. 5. Our theoretical high polarization ratios of 85% are comparable and similar to the experimental values of around 90% in other free-standing NWs [28,29]. The observed high polarity of photoluminescence in practical experiments is possible to result from both the intrinsic electron properties and the dielectric confinement. To further distinguish these two mechanisms, it is necessary to differentiate the certain

observed characteristic. The size dependence of the polarization ratio can indicate the existence of such difference. Since the confinement broadens band-gaps and thus decreases the dielectric constant in the NWs, the dielectric contrast between the nanowire and the environment is to reduce as the NWs become smaller. Consequently, the polarization ratio deriving from the dielectric confinement and the intrinsic electronic states will respectively decline and increase with the decreasing wire diameter, as shown in Fig. 5. It is notably pointed out that the electron–hole Coulomb interaction is not considered in our calculations of transition rate and polarization ratio. Therefore, our approach is preferably used and valid for the strong confinement region with the electron states well separated in energy. Since the electron–hole binding increase due to stronger confinement is much smaller than the increase in the single particle energy separation, the Coulomb interaction as a perturbation does not contribute significantly to the mixing of single-particle wave functions in the strong confinement region. Therefore, the calculation of the transition rate using

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single-particle wave functions is a reasonable approximation. For the large size nanowire, the more sophisticated approach such as solving the Bethe–Salpeter equation [43] is required in the calculation considering the important wave function mixing from the electron–hole interaction. 3.4. Electron and hole effective masses The band-edge effective mass is a critical quantity in determining carrier mobility. Although the bulk carrier mobility in semiconductors has been widely and elaborately studied, both the experimental and theoretical investigations for carrier mobility in NWs remain scanty and are of high interest. Moreover, while the optical band-gap in nanostructures has been indicated to considerably depend on size, the size dependence of the effective mass is however less perceived and understood. Therefore, here we concern on the dependence of the carrier mobility on the axis orientation and lateral size in InAs and GaSb NWs. Since the necessary spin–orbit (SO) coupling need to be included in the calculation of effective mass for some semiconductors, we have performed two calculations with and without SO coupling included respectively. In the calculation including SO, the average orbital energy over SO-splitting states is utilized to determine the effective mass. Our results of the two calculations show similar values of effective masses with and without SO coupling. Fig. 6 shows the obtained results from the calculation without spin–orbit coupling included, of the (band-edge) electron and hole effective masses at zone-center in InAs (a) and GaSb (b) NWs. The effective mass values are represented here in the unit of m0 (free electron mass). It is found from the Fig. 6a and b that the electron effective mass increase substantially and decrease slowly as the

wire diameter becomes smaller for InAs and GaSb NWs respectively, demonstrating the size dependency of effective mass in semiconductor NWs. Our results also indicate that the electron mass is slightly larger for the [0 0 1] NWs than that for the NWs along the other two axis orientations, which is inconsistent with varying trend in their bulk counterparts (along corresponding directions). However, the predicted hole effective masses of the NWs along three considered orientations exhibit a strikingly different manner of dependence on the nanowire diameter. For the [0 0 1] orientation, the calculated hole mass depends considerably on the lateral size, varying from mh ¼ 0:90 (0.77) for the D = 10.6 Å to mh ¼ 0:22 (0.19) for the D = 42.3 Å by a factor of 4 in InAs (GaSb) NWs. Furthermore, the hole masses of the [1 0 1] and [1 1 1] InAs (GaSb) NWs are shown to be little dependent on the wire diameter, changing slightly from 0.11 (0.13) to 0.19 (0.22) and from 0.1 (0.12) to 0.21 (0.25) respectively with the decreasing diameter in the considered range. This particularly predicted size dependency of the hole effective mass is attractively useful for obtaining high carrier mobility in nanowire applications when the small size is needed. On the other hand, the determined hole masses (0.1–0.2) in our realistic calculations for the [1 0 1] and [1 1 1] NWs are remarkably smaller than the hole effective masses along the same directions in bulk InAs and GaSb. Using the same screened pseudopotential as used in nanowire calculations, the hole masses in bulk InAs (GaSb) are calculated to be 0.84 (0.99) and 0.51 (0.61) along [1 0 1] and [1 1 1] directions respectively. Thus, our calculations demonstrate that the carrier masses in semiconductor NWs can be drastically decreased, and consequently the carrier mobility can be significantly enhanced compared with bulk. To illustrate quantitatively improvement of carrier transport by the reduction of the hole mass mh in NWs, by assuming that mobility in semicon-

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Fig. 6. The band-edge electron (up panels) and hole (low panels) effective masses at zone-center of (a) InAs and (b) GaSb NWs with different crystallographic axes, as a function of wire diameter. Because of indirect band-gaps for the D = 10.6 Å [0 0 1] InAs and GaSb NWs, the calculated band-edge electron masses of respective 4.6 and 3.5 (much higher) are not shown.

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ductor is proportional to jmh j5=2 [38], we estimate that the mobility lh is able to increase by 200 times from eight fold decrease in mh when going from bulk to the [1 0 1] NWs with transport along the same [1 0 1] direction in both cases. Therefore, our study indicates that engineering crystallographic orientation is definitely an effective route to obtain novel and advanced material properties in semiconductor NWs. More especially, the coupling between bulk electronic sates occurring in forming the nanowire sates is found to be responsible for the small effective mass in the [1 0 1] and [1 1 1] NWs. To explain this, we expand the wire wave function Wwire with wave vector q in terms of bulk wave functions nbulk nk for q band n and wave vector k:

Wwire ¼ q

X

F nk vbulk nk

ð1Þ

nk

where Fnk denotes the expansion coefficient. Omitting the surface effects which was demonstrated to be a good approximation for bulk-like states in nanostructures [54], we can obtain the relationship between the orbital energies of NWs and bulk as P 2 bulk Ewire q nk jF nk j Enk . Since the coefficients Fnk do not change for two closed q’s used in determining the nanowire effective mass of wire whereas the bulk k points contributing the wire states are varying with q accordingly, we can obtain the nanowire effective mass to the first-order perturbation by:

ð1=m Þwire ¼

X 1 nk

h

2

jF nk j2 r2k Ebulk nk

ð2Þ

In fact, the nanowire effective mass depends on the curvature of the bulk dispersion at the k points that contribute to the nanowire states. Since the calculation of Fnk is computationally demanding, we only consider the [0 0 1] and [1 1 1] NWs of D = 31.7 Å as a paradigm. Using this approach, we determine the hole masses of D = 31.7 Å [0 0 1] and [1 1 1] InAs (GaSb) NWs to be 0.37 (0.31) and 0.12 (0.15) compared with the obtained values of 0.34 (0.29) and 0.09 (0.11) in our direct calculations, which are much smaller than the bulk effective masses along the corresponding directions. It needs to be pointed out that, as represented in the right side of Eq. (1), there is a dominant contribution from some specific bulk k points relating with the wire diameter D by 1/D [37]. The transition matrix element between the conduction band minimum and valence band maximum states decreases as these bulk k points moving away from Brillouin zone-center, and thus decline as the nanowire lateral size becomes smaller, being consistent with the results as shown in Fig. 5. It is hypothetically considered that the small hole effective masses in NWs will approach the bulk values as the wire diameter continues to increase. However, our results from direct calculations show in Fig. 6 that the mh hole masses in the [1 0 1] and [1 1 1] semiconductor NWs do not appear to approach to the values of their bulk counterparts. This demonstrates the possible existence of an electronic phase transition, occurring as the wire diameter becomes larger. It has been shown in Fig. 3 for the D = 10.6 Å [1 1 1] GaSb nanowire that the energy dispersions of the first and the second (or the third) valence bands are steep (small effective mass) and flat (large effective mass) respectively, with band crossing at p k  0.35 (2p/ 3a). The energy separation between the first and second valence bands is significantly reduced as the wire diameter becomes larger, due to their different size dependence. Furthermore, the point of band crossing between these two bands shifts towards p zone-center (e.g., to k  0.18 (2p/ 3a) for the D = 42.3 Å GaSb nanowire) with the increasing wire diameter. As the wire diameter continues to increase further, it is expected that the second band will eventually becomes the highest valence band with its effective mass approaching the bulk value, due to the faster up-shift of the second band than the first band. On the other hand, for the valence

bands belonging to the same irreducible representation (symmetry), band anti-crossing occurs at the k point that also shifts towards zone-center with the increasing wire diameter, and will vanish into zone-center as the wire diameter further increase to a certain large value, also resulting in alternation of valence bands, as shown in Fig. 3 for InAs NWs. Though the computationally demanding calculations for the very large NWs have not been performed here, our results of the smaller NWs substantially indicate the very possible occurrence of the electronic phase transition. In conventional quantum well systems, the exchange between the heavy hole and light hole band when varying the well wideness (layer thickness) is mainly caused by lattice-mismatch (between well and barrier materials) induced strain [39,40]. However, in the free-standing NWs without strain as investigated here, the band level exchange is only due to the size effect causing different energy shift from quantum confinements for different bands. 4. Conclusions In summary, using first-principles DFT derived nonlocal screened atomic pseudopotentials, we have performed systematic calculations on InAs and GaSb NWs with different crystallographic axes and for different lateral sizes. We have calculated the band structure, optical transition intensity, optical polarization ratio and effective masses, as well as their dependence on the crystallographic axis orientation and wire diameter. The determined conduction bands of the [0 0 1] NWs with two larger sizes and the [1 0 1] NWs with the smallest size, but no [1 1 1] NWs, exhibit a hump in their dispersions. The conduction states of [0 0 1] NWs clearly yield subbands when wire diameter varying from 10.6 to 42.3 Å. For all three types of wires studied, the calculated band structures exhibit significantly dependency of the orbital energies at zone-boundary and energy dispersions of higher valence bands on the crystallographic axis orientation. The fitting parameter n exponent is revealed to be universally independent of the nanowire orientation, and determined to be 2.36 (2.89) for the VBM and 1.06 (1.31) for the CBM at C in InAs (GaSb) NWs, which are considerably different from those for InAs and GaSb dots, demonstrating the dimensional dependency of n parameter. The predicted transition matrix element and polarization ratio decreases and increases respectively with the decreasing wire diameter. Furthermore, the magnitude of transition matrix declines most intensively and slightest for the [0 0 1] and [1 1 1] NWs respectively as wire diameter becomes smaller, indicating the best compatibility of the [1 1 1] axis orientation for optoelectronic application by small size NWs. The polarization ratio is found extremely high as 85% for both the [1 0 1] and [1 1 1] NWs, varying with wire diameter. This result reveals another possible mechanism rather than the dielectric confinement model to explain the highly polarized photoluminescence observed in experiments. The obtained electron effective masses depend substantially on the wire diameter. On the other hand, the calculated hole masses only for [0 0 1] NWs depends significantly on the wire size, varying from 0.90 to 0.22 and from 0.77 to 0.19 for InAs and GaSb NWs respectively, whereas the hole masses for the [1 0 1] and [1 1 1] NWs are found to have little size dependency. Our calculations demonstrate that, due to the strong band mixing of NWs, it is feasible to acquire much smaller hole effective mass (along wire axis) in NWs than that in bulk for the same carrier transport direction, and consequently obtain much higher mobility that is not available in bulk materials. Acknowledgements This work has been partially supported by the NSFC (Under Grant Numbers: 50502014 and 50972032), 863 project

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