External Electric Field Modulated Electronic and Structural Properties

Key Laboratory of Automobile Materials, Ministry of Educations, and Department of Materials Science and Engineering, Jilin University, Changchun 13002...
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External Electric Field Modulated Electronic and Structural Properties of 〈111〉 Si Nanowires R. Q. Zhang, W. T. Zheng, and Q. Jiang* Key Laboratory of Automobile Materials (Jilin UniVersity), Ministry of Educations, and Department of Materials Science and Engineering, Jilin UniVersity, Changchun 130022, China ReceiVed: October 25, 2008; ReVised Manuscript ReceiVed: February 26, 2009

The effects of external electric field F on band gap Eg(D, F) of silicon nanowires (SiNWs) in a diameter range of D ) 0.45-1.79 nm are quantitatively calculated using density functional theory. The results show that Eg(D, F) decreases with increasing F due to the rapid drop of the conduction band maximum of SiNWs. As F increases, Eg(D, F) f 0 except at D ) 0.45 nm. Further increasing of F results in breakdown of the geometry structure of SiNWs. In addition, the bond lengths and angles of SiNWs are also functions of F. These findings imply that Eg(D, F) functions of SiNWs can be modulated by manipulating D and F. 1. Introduction Silicon nanowires (SiNWs) have attracted intense interest in recent years because of their interface compatibility with existing silicon-based technology and satisfaction for the need of high integration in very-large-scale integration circuits.1 They can function both as devices and as wires that access them.2 SiNWs have been synthesized recently by various techniques,3-9 which are promising in applications such as field effect transistors,10-14 photodetectors,15 sensors,16-18 field emission device,19 photovoltaic cells,20 and logic gates.21,22 With the device miniaturization,1 the size of SiNWs will further decrease, where the quantum confinement effect plays a key role.10 Since the Sibased high-performance electronic and optoelectronic devices have been required to obtain increasingly strict and well-defined performance, a detailed control at the atomic level is needed, which could be realized with the help of computer simulations. The electronic properties of SiNWs, such as band gap Eg(D), are size-dependent23-27 where D denotes the diameter of SiNWs. While the drop of D is understandable due to the requirement of device miniaturization, strain ε is a useful and economical technique to modulate the Eg(D, ε) function.28-31 However, the effect of ε on Eg(D, ε) of SiNWs is inconspicuous when D is only several nanometers,32 which suggests that other techniques adjusting the Eg(D, ε) function of SiNWs are necessary. An external electric field F can affect electronic properties of semiconductor materials.33 Thus, Eg(D, F) of SiNWs could be a function of F. With the scanning tunneling microscope (STM), by combining a strong F formed between the STM tip and the surface, Si atoms can be transferred from the surface to the tip and redeposited on a predetermined surface site.34 The distance of the charge layer to surface of SiNWs is significant to ensure its sensor sensitivity.17 In fact, F also affects the quantum conduction of Cu nanowires,35 the energies of both the electron and the hole ground state of the colloidal CdSe quantum rods,36 and a dynamic gap between electron and hole bands in the spectrum of graphene quasiparticles.37 Since F is easy to control in actual applications, F should be a powerful tool to modulate properties of SiNWs. * Corresponding [email protected].

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In this contribution, we report Eg(D, F) functions (D ) 0.45-1.79 nm) along the 〈111〉 axis of SiNWs using density functional theory (DFT). The F field induced modifications of the electronic structures, introduced and illustrated here, suggest new research avenues on Eg(D, F) functions of SiNWs. 2. Calculation Methodology In accordance with experimental observations7,38 and Wulff construction,39 the SiNWs have the low-free-energy facets of (111), (110), and (100). The structures of four 〈111〉 directional SiNWs used are shown in Figure 1. The SiNWs were placed in unit cells where the interwire distance is larger than 10 Å, which effectively prevents the interaction effect from neighboring cells. The unit cell of SiNWs consists of six atomic planes. The number of Si and H atoms in each unit cell is presented in Figure 1. The cross section of 〈111〉 SiNWs is a regular hexagon. D of the SiNWs is defined as the distance between two opposite vertexes of Si atoms in the hexagon, which is given in Table 1. The starting lattice constant a ) 0.5470 nm is obtained from bulk Si, which is similar to the experimental value of a ) 0.5429 nm.40 The dangling bonds of the surface Si atoms are terminated by H at an initial bond length of 0.154 nm. The axial lattice constant of SiNWs ac before and after optimization, which are regarded as the lengths of the repeating unit of the SiNWs along the 〈111〉 direction, are also shown in Table 1. The relaxation along the SiNW axis is more important because the band structure is determined from this direction. For all SiNWs, the relative change of ac from that of the bulk Si due to the relaxation is smaller than 0.43%. ac increases with decreasing D due to the contribution of surface effect, which is consistent with previously calculated results.29 The D-dependent relaxation brings out structural elongations. The electronic properties of 〈111〉 SiNWs were calculated using DFT,41,42 which is implemented in the DMol3 model.43,44 The generalized gradient approximation (GGA) functional with the PW91 method45 was employed as the exchange-correlation functional. All-electron relativitistic (AER),46 which includes all core electrons explicitly and introduces some relativistic effects into the core, was used for core treatment. In addition, double numerical plus polarization (DNP)43 was chosen as the basis set with orbital cutoff of 4.6 Å. We use smearing techniques47 to achieve self-consistent field convergence with

10.1021/jp809455w CCC: $40.75  2009 American Chemical Society Published on Web 05/27/2009

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Figure 1. Top and side views of the ball and stick structures of hydrogen terminated 〈111〉 SiNWs where the stoichiometry of each SiNW unit cell is given. The big and small balls denote Si and H atoms, respectively. The numbers 1-6 show different Si layers in the unit cell. A, B, and C are row symbols of Si atoms. The arrow shows the direction of F.

TABLE 1: D Values of Four SiNWs and ac before and after Optimization with a Percentage Change δ

stoichiometry Si14H18 Si38H30 Si74H42 Si122H54

D (nm) 0.45 0.89 1.34 1.79

ac (Å) before optimization

ac (Å) after optimization

δ (%)

9.475

9.516 9.489 9.486 9.482

0.43 0.15 0.12 0.07

a smearing value of 0.005 Ha (1 Ha ) 27.2114 eV). The structure of SiNWs was then relaxed using the state-of-the-art delocalized internal coordinate optimization scheme. The convergence tolerance of energy of 1.0 × 10-5 Ha, maximum force of 0.002 Ha/Å, and maximum displacement of 0.005 Å in the geometry optimization were taken. F was directly added along the 〈111〉 direction with F ) 0.00, 0.25, 0.50, 0.75, and 1.00 V/Å, respectively, and later geometry relaxation was carried out until the forces on each atom were smaller than 0.002 Ha/Å. It is well-known that DFT underestimates Eg(D, 0) but is normally proportional to experimental values and GW calculations (a combination of the Green function G and the screened Coulomb interaction W),48 while GW calculations give values of Eg(D, 0) close to experimental results.24,25,27 To confirm the validity of our simulation technique, our simulated Eg(D, 0) values are compared with other DFT and GW calculations,24,25,27-29 which are shown in Figure 2. For bulk Si, Eg(∞, 0) ) 0.79 eV, which is lower than the experimental result of 1.17 eV and GW calculations of 1.08 eV27 but is larger than other DFT results ranging from 0.58 to 0.63 eV.10,24,26,27,29 The corresponding GW correction, or the difference between GW and our DFT result, is 0.29 eV. As shown in Figure 2, for the 〈111〉 SiNW at D ) 0.89 nm, GW correction increases to 1.66 eV ()4.17-2.51 eV), being larger than the bulk one.27 Compared with another GW result,25 the GW corrections are 2.10, 1.01, and 0.50 eV for three 〈111〉 SiNWs with D ) 0.45, 0.89, and 1.34 nm, respectively. Thus, the size dependence of Eg(D, 0) using GW calculations is stronger than that using DFT.24,25,27 Note that the GW result itself shows also a big difference to determine Eg(D, 0). Comparing our results with other DFT simulations shown in the overlapping part of D values, good correspondence of Eg(D, 0) values is found within the error range. As D drops, especially when D < 1 nm, Eg(D, 0) increases evidently. Our

Figure 2. Eg(D, 0) function of SiNWs along the 〈111〉 direction. The filled circles are our calculated results for Si14H18, Si38H30, Si74H42, and Si122H54. The unfilled squares,28 unfilled circles,24 filled triangles,29 down triangles,27 filled stars,25 and unfilled stars27 are cited data for SiNWs. The inset is the calculated Eg(D, 0) (filled circles) functions of SiNWs along the 〈110〉 direction compared with GW calculation (unfilled up27 and down triangles24) and experimental results along 〈110〉 (plus50) and 〈112〉 (filled squares50) directions (since we have only one experimental result along the 〈110〉 direction, several additional ones along the 〈112〉 direction are shown). In the figure, due to different definitions for diameter of the same SiNWs in references, we have unified the diameter in all cited data using our definition.

results have extended present results to a wider size range for the Eg(D, 0) function than the present data. In order to check our DFT results more carefully, because there are little experimental results for 〈111〉 SiNWs, we give Eg(D, 0) functions for 〈110〉 SiNWs in the inset of Figure 2 where experimental49 and GW calculations24,27 results are also shown. Similar to the case of 〈111〉 SiNWs, our results are again in agreement with other DFT results for 〈110〉 SiNWs24,27 with the same tendency of the size dependence of Eg(D, 0) although the absolute values of DFT results are lower than experimental results and GW calculations. The above results present only the Eg(D, 0) function, while the Eg(D, F * 0) function of SiNWs is absent. It is known that the Eg(D, F * 0) function for silica nanowires has been simulated using DFT calculations.50,51 Thus, DFT should be a proper method to simulate the Eg(D, F * 0) function of SiNWs when GW calculations are with difficulty realized now. In

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Figure 3. Eg(D, F) functions of SiNWs for (a) Si14H18, (b) Si38H30, (c) Si74H42, and (d) Si122H54. The straight lines are drawn as a guide to the eyes.

Zhang et al.

Figure 6. Density of states (DOS) of Si14H18 NW under F ) 0 (thick curve) and F ) 0.75 V/Å (thin curve).

TABLE 2: Layer Distance Function L(F) in Å for SiNWs, F in V/Å, and ∆, the Percent Difference between F ) 1 and F )0 L L1-2 L2-3 L3-4 L4-5 L5-6 L6-1

Figure 4. Band structures of Si14H18 NW under different F values. The valence band maximum has been shifted to zero.

Figure 5. Orbital energy levels E for Si14H18 NW under (a) F ) 0, (b) F ) 0.25, (c) F ) 0.5, (d) F ) 0.75, and (e) F ) 1 V/Å. Fermi level Ef shown as the dashed line is fixed at the zero of energy axes. Ten energy levels above and below Ef are given.

addition, we here focus on the relative change of the Eg(D, F * 0) function as a function of F using the DFT technique, which should provide qualitative results for SiNWs as the case under strain field.28 3. Results and Discussion The Eg(D, F) function is shown in Figure 3. First, as D decreases, Eg(D, F) increases as shown in Figure 2. At a constant

F ) 0.00 F ) 0.25 F ) 0.50 F ) 0.75 F ) 1.00 2.354 0.817 2.367 0.817 2.354 0.811

2.360 0.758 2.369 0.864 2.353 0.811

2.371 0.692 2.377 0.905 2.355 0.816

2.390 0.613 2.387 0.937 2.362 0.826

2.423 0.521 2.395 0.955 2.375 0.846

∆ (%) 2.93 -36.23 1.18 16.89 0.89 4.32

D, Eg(D, F) decreases with increasing F. The tendency is consistent with that of the Eg(D, F) of SiO2.50,51 Meanwhile, the effect of F on Eg(D, F) becomes weaker with smaller D. For instance, when F changes from 0 to 0.25 V/Å, the variations of Eg(D, F) are 0.36%, 0.56%, 3.37%, and 3.69% for Si14H18, Si38H30, Si74H42, and Si122H54 nanowires, respectively. As F increases from 0.25 to 0.5 V/Å, the effect of F on Eg(D, F) becomes evident. The largest variation of Eg(D, F) is 45.23% for Si122H54 nanowire. The drop tendencies of Eg(D, F) with F at any D are similar although a constant relationship between D and F is absent. As shown in Figure 3, Eg(D, F) f 0 when F is large enough. The corresponding F value is referred to as the “breakdown threshold”. Eg(1.79,0.75) ) Eg(1.34, 1) ) 0 and Eg(0.89, 1) ≈ 0. However, Eg(0.45,F) > 0 up to the structural breakdown at F ) 1.372 V/Å. Since the geometric structure at the threshold remains, the threshold is classified as a soft breakdown.51 When F increases from 0 to 0.5 V/Å, the variation of Eg(1.79, F) is 47.3%. Note that if ε ) 5% is applied on the same SiNW, the variation of Eg is only 2.9%.29 Thus, F is more effective than ε to modulate Eg. In addition, as shown in Figure 3, F could compensate the increase of the Eg(D, F) function with decreasing D. Eg(D, F) ) CBM - VBM, where CBM shows the conduction band minimum while VBM denotes the valence band maximum. CBM and VBM functions of a Si14H18 nanowire (NW) are plotted in Figure 4. With increasing F, the energetic levels of both CBM and VBM drop, while this change becomes evident when F > 0.25 V/Å for CBM and when F > 0.5 V/Å for VBM. Since the former drops more rapidly than the latter, Eg(D, F) reduces. Simultaneously, the bands themselves are more dispersive, or the distribution of band energies is wider. This phenomenon could be induced by the bond length change under F since the bandwidth is reversely proportional to the bond length. When F varies from 0 to 0.75 V/Å, the distributions of

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TABLE 3: Bond Angles θ of Each Si Atom in the Si14H18 NW under F ) 0 and F ) 1.371 V/Åa θ (deg) layer

F (V/Å)

1

0 0.75 1.371 0 0.75 1.371 0 0.75 1.371 0 0.75 1.371 0 0.75 1.371 0 0.75 1.371

2 3 4 5 6

a

A 110.2 111.0 113.1

108.5 113.6 117.6 108.6 105.0 105.6

110.1 109.8 111.3

110.1 111.0 113.4 110.2 103.8 95.1 108.9 113.4 119.2 108.9 105.1 105.4 110.2 114.6 114.2 110.1 110.0 111.7

B 108.8 106.9 104.7

110.1 111.0 113.6

110.3 105.0 97.7 110.2 113.7 111.8

108.5 113.6 117.6 108.6 105.0 105.6

108.6 110.1 109.0

110.1 110.0 111.6

110.1 111.0 113.4 110.2 103.8 94.7 108.6 113.6 118.1 108.5 105.0 106.5 110.2 114.5 113.7 110.1 110.0 111.0

C 109.1 107.0 104.4

110.1 111.0 113.1

110.2 105.0 98.0 110.2 113.5 113.8

108.9 113.4 119.2 108.9 105.1 105.4

109.1 110.1 109.3

110.1 110.0 111.9

110.1 111.0 112.5 110.2 103.8 95.5 108.6 113.6 118.1 108.5 105.0 106.5 110.2 114.6 113.9 110.2 109.8 110.8

108.6 106.9 105.1

110.2 105.0 97.2 110.3 113.7 113.2

108.8 110.1 109.2

A, B, and C are column symbols of Si atoms shown in Figure 1a.

the bond length are from 2.354-2.369 nm to 2.349-2.392 nm, which lead to larger bandwidth. The above result can be supported by calculating the orbital energy levels of SiNWs. Such calculated orbital energy levels of Si14H18 NW under different F values are presented in Figure 5. In the released state of equilibrium without F, the total orbital distribution is shown in Figure 5a. It is found that as F increases, the distribution of orbital energy is broader, which explains why the band structures are more dispersive, both for VBM and CBM. These results confirm also the observed Eg(D, F) functions shown in Figure 3. The plots of density of states (DOS) of SiNWs as a function of F are given in Figure 6. Compared with DOS without F, the blue shift is present in almost all valence band energy levels while the red shift occurs in all conduction band energy levels under F, which results in a decrease of Eg(D, F). Because of the smearing technique used in plotting DOS, the density state of VBM is not located in the Fermi level. The DOS of SiNWs splits into more peaks due to the Stark effect52 with a change in polarization behaviors and reduction of optical transition probability,53 which is consistent with Figure 4. The F field, being similar to a mechanical field, should deform atomic structures of SiNWs with atomic movements.54 We study the geometry information by determining the layer distance L and the bond angle θ. Let Li-j denote a distance of two neighboring layers i and j in the unit cell of Si14H18 NW shown in Figure 1a; the simulated Li-j values under different F values are shown in Table 2. As F increases, L1-2, L3-4, L4-5, L5-6, and L6-1 increase, while L2-3 has an opposite trend. The relative change ∆ from F ) 0 to F ) 1 V/Å is larger for L2-3 and L4-5, while it is only several percent for the others, particularly ∆5-6 ) 0.89%. L2-3 and L4-5 have the largest contractive and elongated deformations of -36.23% and 16.89%, respectively. Since the total length of SiNWs is fixed in our simulation, which is also the actual case when SiNWs are industrially applied, F results in inhomogeneity of the length distribution. The deformation amount of SiNWs increases with increasing F. θ values under F ) 0 and F ) 0.75 V/Å are listed in Table 3. Under F ) 0, θ has two distribution ranges of θ1 ) 108.5°-109.1° and θ2 ) 110.1°-110.3°. The average value of 109.7° is only a little different from the bulk θ ) 109.5°. When

Figure 7. Bond length lij(F) functions of Si14H18 NW, where the subscript ij denotes the bond length between layers i and j in the unit cell. i, j ) 1-6, whose definition is shown in Figure 1a.

F ) 0.75 V/Å, θ has more dispersive distribution with θ1 ) 105.0°-113.6° and θ2 ) 103.8°-114.6°. Although the size ranges of θ1 and θ2 become larger, both in low- angle and highangle sides, the average value of 109.5° is almost the same as 109.7° without F. When F increases from 0 to 0.75 V/Å, the bond angles of layer 2 atoms decrease while those of layer 5 atoms increase, and the variations are -5.81% and 3.90%, respectively. We can see that θ2 values in layers 2 and 5 have the largest variations among all layers. Compared with Table 2, these changes are proportional to that of L2-3 and L4-5. As F increases, L and θ vary, which certainly brings out the change of the bond length l, which is present in Figure 7. It is observed that l12, l23, l34, and l61 increase and l45 decreases when F increases. Meanwhile, the variation tendencies of l12, l34, and l61 are consistent with that of L1-2, L3-4, and L6-1. However, l23 and l45 have opposite variation directions of L2-3 and L4-5, respectively. It should be related to the largest variations of θ in layers 2 and 5, as in the discussion above. Figure 7 shows also that lij changes little when F increases from 0 to 0.25 V/Å, which is in agreement with the cases of Li-j and further Eg(D, F). Only when F > 0.25 V/Å are more variations present for Li-j, θ, and lij, and thus for Eg(D, F).

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Figure 8. Slice images of the electronic deformation density of Si14H18 NW under F ) 0 and F ) 1.371 V/Å. The blue region shows the electron accumulation, while the red region shows the electron loss.

It is interesting that l56 decreases when F varies from 0 to 0.25 V/Å and increases when F does from 0.25 to 1 V/Å while L5-6 increases with F. This can be illustrated by different variations of θ in layer 6. Table 3 also shows θ values under F ) 1.371 V/Å, which is the largest field applied before the geometry structure of Si14H18 NW breaks down. The detailed discussion about the field is given later. Here, as shown in Table 3, the θ(0 < F < 1.371) function in layer 6 is not monotonous where a maximum or minimum must be present, which leads to different trends of l56 in the range of 0 < F < 0.25 and 0.25 < F < 1 V/Å. When F reaches 1.372 V/Å, the atomic structure of SiNWs breaks down and there is no longer a stable geometry structure due to the existence of nonconvergence of energy. To understand the limiting case of the geometry structure, we study the changes of L and θ values under F ) 1.371 V/Å. At F ) 1.371 V/Å, θ has more dispersive distribution with θ1 ) 104.4°-119.2° and θ2 ) 94.7°-114.2°, while the average value of 108.9° is only a little lower than that of 109.7° without F. Atoms in layers 2 and 3 have the biggest variations of bond angles. θ1 increases 9.46% while θ2 decreases 14.06%, which are related with the largest variation of L2-3. In this case, L2-3 drops to 0.304 Å with a percentage change of -62.79%. Other L values increase as F increases. The electronic densities under F ) 0 and F ) 1.371 V/Å are shown in Figure 8. The slice image shows that the electron density deformation of the biggest variation is between layers 1 and 2. The bond length increases 10.2% and reaches 2.595 Å. Since the elastic limit of SiNWs with D < 10 nm is about 0.1 ( 0.02,55 the breakdown of the structure of atoms indeed corresponds to the elastic limit. As shown in Figure 8a, the Si-Si bond is covalent because the preferential electrons mainly accumulate between two atoms rather than around a particular atom. However, with F applied, the electrons accumulate near the left Si atom, while another Si atom has more positive charge. At last, the covalent bond disappears, and therefore the geometric structures of SiNWs break down. Note that the above calculation is carried out using DFT while GW calculation for Eg(D, 0) of SiNWs has shown better agreement with experimental results. For the simulation of Eg(D, 0) of SiNWs, the better accuracy of GW comes in a large part from the increase of CBM, while VBM has a similar size of DFT calculation.27,48 As stated above, with increasing F, both CBM and VBM drop while CBM decreases more rapidly than VBM, which brought the decreasing of Eg(D, F) in our DFT calculations. Taking the D ) 0.45 nm SiNW for instance, CBM decreases 7.8%, 17.1%, and 30.0% with F ) 0.25, 0.50, and 0.75 V/Å, respectively, while VBM drops only 1.6%, 2.7%, and 3.1%. Since GW could better simulate CBM values, GW could be expected to supply more accurate simulation results than DFT, which should be our future work to determine the Eg(D, F) function.

In summary, Eg(D, F) functions of 〈111〉 SiNWs are calculated using the DFT technique. The results show that Eg(D, F) decreases with increasing F and D due to the rapid drop of CBM, respectively. As F increases, Eg(D, F) f 0, where the electronic density, bond lengths, and bond angles change. Both variables of D and F could modify the Eg(D, F) function in a wider range, which supplies a new research avenue on the Eg(D, F) function of SiNWs. Acknowledgment. The authors acknowledge support by National Key Basic Research and Development Program (Grant No. 2004CB619301) and NNSFC (Grant No. 50525204). References and Notes (1) Peercy, P. Nature 2000, 406, 1023. (2) Zhang, P. P.; Mayer, T. S.; Jackson, T. N. ACS Nano 2007, 1, 6. (3) Morales, A. M.; Lieber, C. M. Science 1998, 279, 208. (4) Zhang, Y. F.; Tang, Y. H.; Yu, D. P.; Lee, C. S.; Bello, I.; Lee, S. T. Appl. Phys. Lett. 1998, 72, 1835. (5) Holmes, J. D.; Johnston, K. P.; Doty, R. C.; Korgel, B. A. Science 2000, 287, 1471. (6) Wu, Y.; Yang, P. J. Am. Chem. Soc. 2001, 123, 3165. (7) Wu, Y.; Cui, Y.; Huynh, L.; Barrelet, C. J.; Bell, D. C.; Lieber, C. M. Nano Lett. 2004, 4, 433. (8) Wang, Y. F.; Lew, K.-K.; Ho, T.-T.; Pan, L.; Novak, S. W.; Dichey, E. C.; Redwing, J. M.; Mayer, T. S. Nano Lett. 2005, 5, 2139. (9) Shan, Y. H.; Fonash, S. J. ACS Nano 2008, 2, 429. (10) Kobayashi, M.; Hiramoto, T. J. Appl. Phys. 2008, 103, 053709. (11) Cui, Y.; Zhong, Z.; Wang, D.; Wang, W. U.; Lieber, C. M. Nano Lett. 2003, 3, 149. (12) Cui, Y.; Lieber, C. M. Science 2001, 291, 851–853. (13) Koo, S.-M.; Li, Q.; Edelstein, M. D.; Richter, C. A.; Vogel, E. M. Nano Lett. 2005, 5, 2519. (14) Goldberger, J.; Hochbaum, A. I.; Fan, R.; Yang, P. Nano Lett. 2006, 6, 973. (15) Yang, C.; Barrelet, C. J.; Capasso, F.; Lieber, C. M. Nano Lett. 2006, 6, 2929. (16) Cui, Y.; Wei, Q. Q.; Park, H. K.; Lieber, C. M. Science 2001, 293, 1289. (17) Cattani-Scholz, A.; Pedone, D.; Dubey, M.; Neppl, S.; Nickel, B.; Feulner, P.; Schwartz, J.; Abstreiter, G.; Tornow, M. ACS Nano 2008, 2, 1653. (18) Zhang, G.-J.; Zhang, G.; Chua, J. H.; Chee, R.-E.; Wong, E. H.; Agarwal, A.; Buddharaju, K. D.; Singh, N.; Gao, Z. Q.; Balasubramanian, N. Nano Lett. 2008, 8, 1066, and references therein. (19) She, J. C.; Deng, S. Z.; Xu, N. S.; Yao, R. H.; Chen, J. Appl. Phys. Lett. 2006, 88, 013112. (20) Peng, K. Q.; Wang, X.; Lee, S.-T. Appl. Phys. Lett. 2008, 92, 163103. (21) Huang, Y.; Duan, X.; Cui, Y.; Lauhon, L. J.; Kim, K.-H.; Lieber, C. M. Science 2001, 294, 1313. (22) Sheriff, B. A.; Wang, D. W.; Heath, J. R.; Kurtin, J. N. ACS Nano 2008, 2, 1789. (23) Delley, B.; Steigmeier, E. F. Appl. Phys. Lett. 1995, 67, 2370. (24) Zhao, X. Y.; Wei, C. M.; Yang, L.; Chou, M. Y. Phys. ReV. Lett. 2004, 92, 236805. (25) Bruno, M.; Palummo, M.; Marini, A.; Sole, R. D.; Ossicini, S. Phys. ReV. Lett. 2007, 98, 036807. (26) Ng, M. F.; Zhou, L. P.; Yang, S. W.; Sim, L. Y.; Tan, V. B. C.; Wu, P. Phys. ReV. B 2007, 76, 155435. (27) Yan, J. A.; Yang, L.; Chou, M. Y. Phys. ReV. B 2007, 76, 115319. (28) Hong, K. H.; Kim, J.; Lee, S.-H.; Shin, J. Nano Lett. 2008, 8, 1335. (29) Leu, P. W.; Svizhenko, A.; Cho, K. Phys. ReV. B 2008, 77, 235305. (30) Lyons, D. M.; Ryan, K. M.; Morris, M. A.; Holmes, J. D. Nano Lett. 2002, 2, 811. (31) Audoit, G.; Mhuircheartaigh, E´. N.; Lipson, S. M.; Morris, M. A.; Blau, W. J.; Holmes, J. D. J. Mater. Chem. 2005, 15, 4809. (32) Peng, G. W.; Feng, Y. P. Appl. Phys. Lett. 2007, 91, 083116. (33) Novoselow, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Science 2004, 306, 666. (34) Lyo, I.-W.; Avouris, P. Science 1991, 253, 173. (35) He, C.; Zhang, P.; Zhu, Y. F.; Jiang, Q. J. Phys. Chem. C 2008, 112, 9045. (36) Li, X. Z.; Xia, J. B. Phys. ReV. B 2003, 68, 165316. (37) Fistul, M. V.; Efetov, K. B. Phys. ReV. Lett. 2007, 98, 256803.

Electric Field Effect on Band Gap of Si Nanowires (38) Li, C. P.; Lee, C. S.; Ma, X. L.; Wang, N.; Zhang, R. Q.; Lee, S. T. AdV. Mater. 2003, 15, 607. (39) Hong, S.; Chou, M. Y. Phys. ReV. B 1998, 57, 6262. (40) Landolt-Bo¨rnstein, Numerical Data and Functional Relationships in Science and Technology; Madelung, O., et al., Eds.; New Series, Group X, Vol. 17a; Springer: Berlin, 1982. (41) Hohenberg, P.; Kohn, W. Phys. ReV. 1964, 136, B864. (42) Kohn, W.; Sham, L. J. Phys. ReV. 1965, 140, A1133. (43) Delley, B. J. Chem. Phys. 1990, 92, 508. (44) Delley, B. J. Chem. Phys. 2000, 113, 7756. (45) Perdew, J. P.; Wang, Y. Phys. ReV. B 1992, 45, 13244. (46) Koelling, D. D.; hartreermon, B. N. J. Phys. C: Solid State Phys. 1977, 10, 3107. (47) Delley, B. In Modern Density Functional Theory: A Tool for Chemistry; Seminario, J. M., Politzer, P., Eds.; Theoretical and Computational Chemistry Vol. 2; Elsevier Science: Amsterdam, 1995. (48) Hybertsen, M. S.; Louie, S. G. Phys. ReV. B 1986, 34, 5390.

J. Phys. Chem. C, Vol. 113, No. 24, 2009 10389 (49) Ma, D. D. D.; Lee, C. S.; Au, F. C. K.; Tong, S. Y.; Lee, S. T. Science 2003, 299, 1874. (50) Zhang, D. J.; Guo, G. L.; Liu, C. B.; Zhang, R. Q. J. Phys. Chem. B 2006, 110, 23633. (51) Sun, Q.; Wang, Q.; Kawazoe, Y.; Jena, P. Nanotechnology 2004, 15, 260. (52) Miller, D. A. B.; Chemla, D. S.; Damen, T. C.; Gossard, A. C.; Weigmann, W.; Wood, T. H.; Burrus, C. A. Phys. ReV. Lett. 1984, 53, 2173. (53) Weman, H.; Martinet, E.; Dupertuis, M. A.; Rudra, A.; Leifer, K.; Kapon, E. Appl. Phys. Lett. 1999, 74, 2334. (54) Sun, C. Q. Prog. Mater. Sci. 2009, 54, 179. (55) Kizuka, T.; Takatani, Y.; Asaka, K.; Yoshizaki, R. Phys. ReV. B 2005, 72, 035333.

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