9702
J. Phys. Chem. C 2010, 114, 9702–9705
Core and Shell States of Silicon Nanowires under Strain Alexis Nduwimana†,‡ and Xiao-Qian Wang*,† Department of Physics and Center for Functional Nanoscale Materials, Clark Atlanta UniVersity, Atlanta, Georgia 30314, and Georgia Perimeter College, CoVington, Georgia 30014 ReceiVed: March 19, 2010; ReVised Manuscript ReceiVed: April 28, 2010
The electronic structure characteristics of silicon nanowires under tensile and compressive strain are studied using first-principles density functional theory. The unique wirelike structure leads to distinct hole distributions in the core and shell regions, which can be characterized by the electronic band structures of the light-hole and heavy-hole states. The onset transition pressure for silicon nanowires is shown to be lower than the value of bulk counterpart, in conformity with experimental observations. These results demonstrate that the impact of strain on the electronic characteristics is important for nanodevice applications. Introduction Semiconducting nanowires are basic building blocks for the function and integration of nanoscale devices such as nanoswitches, single electron transistors, optoelectronic units, sensors, and solar cells.1-7 Silicon nanowires (SiNWs) are prototype semiconducting wires and have attracted a great deal of attention. To tailor their physical properties, it is important to determine the phase stability and understand the effect of quantum confinements. Tensile and compressive strain can have a profound influence on the properties of materials at the nanoscale. Strain-induced changes in structures alter the intrinsic interatomic distances and modify the energy levels. For semiconductors, the associated deformation significantly changes the electronic and optical properties. The impact of tensile and compressive strain on SiNWs is fundamentally important since the strain can yield nanostructures with novel properties, and nanomaterials respond to the strain differently from their bulk counterparts. At the nanoscale, the increase of the surface-volume ratio exerts great influence on its overall mechanical properties, and the quantum confinement effect plays an important role in determining physical properties. Experimental studies1-3 demonstrated pivotal effects of strain on the electronic transport that may find significant applications in nanomechanics and nanoelectronics. Moreover, recent experimental studies of phase transition8 and bulk modulus of SiNWs under hydrostatic conditions revealed reduced compressibility and lower onset transition pressure as compared to the corresponding values for bulk silicon.9,10 Despite its significance, the microscopic mechanism for how the phase transformation and compressibility depends on their size and surface conditions is poorly understood. While the structural and electronic properties of SiNWs at ambient conditions have been well studied, much less attention has been given to the study of properties of SiNW under tensile and compressive strain. Understanding the strain-induced changes in SiNWs is thus a problem of considerable interest.11-16 Here we present a comprehensive investigation of phase transitions and electronic properties of SiNW under tensile and compressive strain. Our study reveals the characteristic core and shell states * To whom correspondence should be addressed. E-mail:
[email protected]. † Clark Atlanta University. ‡ Georgia Perimeter College.
Figure 1. Ball-and-stick presentation of the cross section view of the SiNWs along [110] (left) and [111] (right) directions.
associated with SiNWs and sheds light on the potential application of SiNW-based nanodevices. Results and Discussion A fundamental question that needs to addressed is how to distinguish the surface and bulk contributions. Ideally, one can separate the nanowire into two regions: a core region that mimics the bulk behavior and a shell region that represents the characteristics of the NW surface.13-16 To assess the feasibility of this scenario, we have examined SiNWs along the [111] and [110] orientations as prototype of SiNW systems (see Figure 1). To explore the effect of strain, we applied uniaxial strains on the NW and optimized the resultant atomic structure. A careful examination of the structures through optimizing the length of the unit cell and positions of all atoms indicates that changing the length of the unit cell along the wire axis is efficient in the construction of structures under tensile and compressive strain. The calculated band structures of near band gap states of a [111] SiNW in a range of (2.5% are illustrated in Figure 2. For the valence bands, the lowest band corresponds to the singly degenerate light-hole (LH) state. The next double degenerate (at Γ point) band is characterized as the heavy-hole (HH) states. The splitting between the LH and HH states in nanowires can be attributed to the quantum confinement in radial plane.17 It is readily observable from Figure 2 that the light holes are predominantly confined in the core region and the heavy holes are mainly confined in the proximity of the shell region. As seen from Figure 2, the electronic band gap is direct at ambient conditions, yet indirect under compressive strain. Under compression, the conduction band minimum (CBM) shifts toward the band edge along with a downward shift for
10.1021/jp102514b 2010 American Chemical Society Published on Web 05/12/2010
Silicon Nanowires under Strain
J. Phys. Chem. C, Vol. 114, No. 21, 2010 9703
Figure 2. Calculated band structure of a SiNW along the [111] direction with diameter 1.9 Å at at 2.5% compression (-2.5%), at ambient condition (0%), and at 2.5% tensile expansion (+2.5%). Inset: the corresponding charge distribution of light-hole (LH) and heavyhole (HH) states.
the HH band. In contrast, under tensile strain, the gap remains direct, and the HH states moves up in energy. The energy split between the HH and the LH states, which is absent in the bulk, is sensitive to the strain. The energy split increases with applied compressive force and decreases under tensile strain. This is manifested by a negative deformation potential between -4.8 and -1.6 eV. The deformation potential is defined as aB ) ∂∆/∂ ln V, where ∆ is the split between the LH and HH energies at the band center and V is the unit cell volume. The effect of tensile and compressive strain on the band structure of a cylindrical core-shell structure can be studied by introducing a pressure-induced potential in a 2D “particlein-a-box” model.18,19 The quantization comes from the radial confinement and the one-electron equation in cylindrical coordinates (r, φ, z) can be solved by assuming the wave function of the form Φ(r,φ,z) ) ψ(r)ei(lφ+kz), which separates into the free motion in the z-direction and a radial equation
(
)
∂2ψ 2m* 1 ∂ψ l2 + (E V(r)) ψ)0 + r ∂r ∂r2 p2 r2
(1)
where m* is the effective mass and l is the angular momentum. One of the sensible approximations for pressure-induced potential is of the form V(r) ) γr/R, where γ is a parameter characterizing the strains (γ > 0 for compressive strain and γ < 0 for tensile stress) and R is the radius of the nanowire. In the absence of strain (γ ) 0), eq 1 is equivalent to the Bessel function with a length scale λ ) (2m*E)1/2/p, and the charge density can be obtained analytically.19 The charge distributions for the lowest two states (1s and 1p) are in accordance with those for the LH and HH states, as shown in the insets of Figure 1, which are of core and shell character, respectively.20,21 For small tensile or compressive strains, the solution of eq 1 can be obtained through a perturbation calculation. The correction to the energy levels is 0.425γ, 0.553γ, and 0.623γ for the three lowest states (corresponding to LH, HH, and the next 2-fold degenerate state of 1d character), respectively. The results obtained from the model calculation indicate that the energy split between the LH and HH states increases with compressive strain and decreases with tensile stress, which correlates well with the first-principles results. Furthermore, the model provides a simple picture for first-principles calculation results in that there is a paucity of modification for the charge density of LH under tensile or compressive strain while there exists level switching for HH states under 2.5% tensile strain. The former indicates that a perturbation treatment is valid, and thus there is no corrections to the charge density. The latter can be attributed to the fact that taking into account the
Figure 3. Composition dependence of the energy split between the valence band maximum (light holes) and next low-lying band (heavy holes) for the Ge-core/Si-shell and Si-core/Ge-shell structured nanowires oriented along the [111] direction.
difference in effective mass for near-gap valence states, the HH (1p character) and the 1d state can switch order for tensile stress (γ < 0). The two types of strains work in a concerted fashion, resulting in a spatial separation of LHs and HHs.19 The strain-induced potential is qualitatively similar to a step potential induced by band offsets of heterostructures.17,19,21 Consequently, the spatial separation of light and heavy holes are reminiscent of that of Ge/Si core-shell NWs.5,6,19,21 Specifically, for Ge-core Si-shell NWs, there is an enhanced spatial separation of heavy and light holes, while the converse is true for Si-core/Ge-shell NWs. In fact, there is a one-to-one correspondence between the qualitative behavior of the energy split of LH and HH bands, along with the switch of 1p and 1d levels. Illustrated in Figure 3 is the band structure of Si, Ge, Ge/Si, and Si/Ge NWs. As can be seen from Figure 3, the shifting of LH and HH bands is in accordance with the behavior of SiNWs under tensile and compressive strain in that Ge-core Si-shell NWs act as compressive strain, whereas Si-core/Ge-shell NWs behave as SiNWs under tensile stress. The charge separation of light and heavy holes can be characterized by the energy split ∆E.19,22 Shown in Figure 4 are the size and composite dependence of the energy split. The energy split is correlated to the strain and provides useful information about the response of charge separation under strain. There is a noticeable difference between band structures of the Si-core and Ge-core nanowire heterostructures. The level spacing for the valence bands of Ge-core nanowires (including pure Ge) is larger compared to that of Si-core cases. The converse is true for conduction bands. A larger level spacing corresponds to larger band offset due to stronger confinement effect. For NWs of different sizes, the overall character of the band structure is qualitatively similar, apart from the fact that the level spacing decreases with the increase of the diameter. We are now in a position to discuss the effect of charge separation on the electronic transport. A few theoretical studies attempted an interpretation based on the changes due to the effective mass.23 The predicted drastic change of the effective mass was based on first-principle calculation of an unpassivated metallic SiNW. The charge transport is qualitatively different than the p-doped semiconductor SiNWs. To clarify this point,
9704
J. Phys. Chem. C, Vol. 114, No. 21, 2010
Nduwimana and Wang
Figure 5. Energy of SiNWs along the [111] (top panel) and [110] (bottom panel) directions, along with cross-section views of the corresponding high-pressure and ambient condition structures.
TABLE 1: Calculated Silicon-Silicon Bond Length (d0), Silicon-Hydrogen Bond Length (d1), Unit Cell Volume per Atom V0, Transition Pressure Pt, Bulk Modulus B, and the Deformation Potential a for the SiNWs V0 (Å3) [110] [111]
Figure 4. Composition dependence of the energy gap between the valence band maximum (light holes) and next low-lying band (heavy holes) for the Ge-core/Si-shell and Si-core/Ge-shell structured nanowires oriented along the [111] direction.
we have carefully analyzed the change of effective mass due to compressive and tensile strains. Particular attention was paid to the LH and HH switch that can result in a drastic change of the effective mass. The estimated compressive strain for a switching between HH and LH states is larger than 10%, and the maximal change of effective mass for properly passivated NWs is less than 60%. On the basis of the analysis, it becomes evident that the change of effective mass is a less important factor as compared to the charge separation and the resultant change in the relaxation process, analogous to the mechanism of increased hole mobility in Ge-core/Si-shell NWs.17 Further increasing the strain can be realized through applying an external pressure, which leads to structural phase transitions. Figure 5 displays the energy change of SiNWs under pressure. Near ambient conditions, the shape of energy change is parabolic and the bulk modulus at ambient conditions can be extracted accordingly. A high-pressure phase is formed around 20% compression. Our calculation results indicate a distinct behavior between the [111] and [110] oriented NWs. For the NWs in the [110] direction, the hexagonal conformations in ambient conditions change to rectangles as seen in the inset of Figure 5. The structural change is attributed to a core expansion and a shell shrinkage, along with the anisotropy of the [110] wires. By contrast, for SiNWs along the [111] direction, the predominant changes of the structure are confined near the surface, where the corresponding shell region expands and reconstructs. Summarized in Table 1 are structural properties of SiNWs for ambient and high-pressure phases. There is an increase in the bond length of Si-Si and Si-H as compared to bulk for [111] NWs, while the converse is true for the [110] NWs. In
16.2 18.1
d0 (Å)
d1 (Å)
1.98-2.33 1.23-1.47 2.32-2.43 1.51-1.52
Pt (GPa) B (GPa) a (eV) 10.2 6.7
130 120
-1.6 -4.8
connection to the core-shell behavior discussed above, the onset transition temperature Pt is weakly dependent on the diameter for [110] NWs. For [111] NWs, the thinner the NW, the smaller the onset transition pressure. Nevertheless, SiNWs along both directions have lower transition pressure than the bulk value of ∼11.5 GPa,9 which is consistent with the experimental result (8.5-9.9 GPa for [111] SiNWs). The extracted value of bulk modulus of 120 GPa for SiNW along [111] is also in good agreement with experimental results of 123 GPa.9 It is worth mentioning that our predicted structure of the high-pressure phase of SiNWs along [111] is in conformity with experimental assessment of a primitive hexagonal phase.24 We depict in Figure 6 the band structures of thin SiNWs in the [111] and [110] directions under the ambient conditions and for the high-pressure phase. For the [111] SiNW, both phases are semiconducting with a gap of 2.1 and 0.3 eV for the ambient and high-pressure phases, respectively. While the gap along [111] direction is direct at the Γ point, the high-pressure phase has an indirect gap between the band center and the edge of the Brillouin zone. For the higher pressure phase, the HH states are pushed up and the conduction states are lowered, while a smaller gap survives. This is to be compared to the Si/Ge core-shell in which the shell expands along with the associated lifting of the HH states. In the case of the [110] direction, a phase change from a semiconducting state with a gap of 1.4 eV to a metallic state is observed. The valence bands near the Fermi energy, especially LH, are elevated during compression, thereby closing the gap. The conduction bands are also lowered as in the case of [111] NWs. This is similar to the Ge/Si core-shell NWs in which the core expands, with the lifting of core-type states. Computational Section On the basis of various experimental and theoretical studies, the SiNWs were constructed from diamond structure with a roughly cylindrical shape along with hydrogen passivation on
Silicon Nanowires under Strain
J. Phys. Chem. C, Vol. 114, No. 21, 2010 9705 mass of hole propagation changes during strain, but not drastic in order to be accounted for the dominant mechanism for explaining the experimentally observed electronic transport. In contrast, the increased level split between core and shell states leads to substantial enhancement of the hole mobility, which is an intrinsic feature of thin SiNWs under strain and in analogous to the phenomenon in Ge-core, Si-shell heterostructured NWs. These findings through the first-principles density-functional calculations are important for guiding the syntheses of SiNWs in a controlled manner and for tailoring their physical properties for different exploitations. Furthermore, we have studied the structural and electronic properties of the high-pressure phases of SiNWs. The conclusion of a reduction of the transition pressure for [111] SiNWs is in good agreement with the corresponding experimental results. We remark, before closing, that it is straightforward to employ this approach to novel semiconductor NWs, and the investigation of the relevant strain effect will provide an invaluable tool for developing NW-based nanodevices. Acknowledgment. We thank R. N. Musin and M. Y. Chou for stimulating discussions. The work was supported by the National Science Foundation (Grant DMR-0934142) and Army Research Office (Grant W911NF-06-1-0442).
Figure 6. Calculated band structure of SiNWs along the [111] (top panel) and [110] (bottom panel) directions. Left panel is for the ambient condition phase, while the right panel is for the high-pressure phase. The valence band minimum (VBM) is set to 0 eV for semiconductor phases.
the surface. The Vienna ab initio simulation package (VASP) was used to solve the Kohn-Sham equation.25,26 The local density approximation to the exchange-correlation functional was employed along with the ultrasoft pseudopotential for Si for the description of electron-ion interaction.19 For a given size of the SiNWs, the geometry optimization was performed using the conjugate-gradient algorithm of the total energy minimization. The convergence of electronic properties, with an accuracy of 10-4 eV for the total energy and 0.01 eV/Å for the ionic forces, was achieved using a Monkhorst-Pack 1 × 1 × 6 k-points and a kinetic energy cutoff of 219 eV. A supercell with 5 Å of separation on neighboring wires was chosen to eliminate interactions of more than 1 meV per H atom. The kinetic energy cutoff and the supercell separation were tested carefully at various pressures to ensure the convergence of results. The pressure was computed as the stress on the NW cross section. As the lattice constant is reduced, the pressure increases to a maximum, followed by a declining trend before another steady increase. The transition pressure corresponds to the first point of inflection as the pressure curve moves toward the first maximum. The bulk modulus was obtained through fitting to Birch-Murnaghan equation of state.27 The volume was extracted from the area of the NW cross section and the lattice constant, and the error bar was estimated to be within 5%. The deformation potential as well as pressure-dependent properties can be extracted readily. Conclusions In summary, closer scrutiny of electronic structure characteristics of SiNW under compressive and tensile strain reveals that the light and heavy hole states are of features of core and shell confinement, respectively. The modification of the effective
References and Notes (1) Duan, X. F.; Huang, Y.; Cui, Y.; Wang, J. F.; Lieber, C. M. Nature 2001, 409, 66. (2) Lauhon, L. J.; Gudiksen, M. S.; Wang, C. L.; Lieber, C. M. Nature 2002, 420, 57. (3) Cui, Y.; Zhong, Z.; Wang, D.; Wang, W.; Lieber, C. M. Nano Lett. 2003, 3, 149. (4) Greytak, A. B.; Lauhon, L. J.; Gudiksen, M. S.; Lieber, C. M. Appl. Phys. Lett. 2004, 84, 4176. (5) Lu, W.; Xiang, J.; Timko, B. P.; Wu, Y.; Lieber, C. M. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 10046. (6) Xiang, J.; Lu, W.; Hu, Y.; Wu, Y.; Yan, H.; Lieber, C. M. Nature 2006, 441, 489. (7) Wu, Z.; Neaton, J. B.; Grossman, J. C. Nano Lett. 2009, 9, 2418. (8) Seong, H.; Jeon, E.; Kim, M.; Oh, H.; Lee, J.; Kim, J.; Choi, H. Nano Lett. 2008, 8, 3656. (9) Wang, Y.; Zhang, J.; Wu, J.; Coffer, J. L.; Lin, Z.; Sinogeikin, S. V.; Yang, W.; Zhao, Y. Nano Lett. 2008, 8, 2891. (10) Smith, A. M.; Mohs, A. M.; Nie, S. Nature Nanotechnol. 2009, 4, 56. (11) Ma, L.; Wang, J.; Zhao, J.; Wang, G. Chem. Phys. Lett. 2008, 452, 183. (12) Leu, P. W.; Svizhenko, A.; Choet, K. Phys. ReV. B 2008, 77, 235305. (13) Hong, K.-H.; Kim, J.; Lee, S.-H.; Shin, J. K. Nano Lett. 2008, 8, 1335. (14) Huang, L.; Lu, N.; Chou, M. Y.; Wang, C.-Z.; Ho, K.-M. J. Phys. Chem. C 2008, 112, 15680. (15) Sajjad, R. N.; Alam, K. J. Appl. Phys. 2009, 105, 044307. (16) Maegawa, T.; Yamauchi, T.; Hara, T.; Tsuchiya, H.; Ogawa, M. IEEE Trans. Electron DeVices 2009, 56, 553. (17) Park, J.-S.; Ryu, B.; Moon, C.-Y.; Chang, K. J. Nano Lett. 2010, 10, 116. (18) Zhao, X.; Wei, C. M.; Yang, L.; Chou, M. Y. Phys. ReV. Lett. 2004, 92, 236805. (19) Nduwimana, A.; Musin, R. N.; Smith, A. M.; Wang, X.-Q. Nano Lett. 2008, 8, 3341. (20) Yang, L.; Musin, R. N.; Wang, X. Q.; Chou, M. Y. Phys. ReV. B 2008, 77, 195325. (21) Musin, R. N.; Wang, X.-Q. Phys. ReV. B 2005, 71, 155318. (22) Tombler, T. W.; Zhou, C.; Alexseyev, L.; Kong, J.; Dai, H. J.; Liu, L.; Jayanthi, C. S.; Tang, M. J.; Wu, S. Y. Nature 2000, 405, 769. (23) Cao, J. X.; Gong, X. G.; Wu, R. Q. Phys. ReV. B 2007, 75, 233302. (24) Hu, Z. J.; Merkle, D. L.; Menoni, S. C.; Spain, L. I. Phys. ReV. B 1986, 34, 4679. (25) Kresse, G.; Furthmu¨ller, J. Phys. ReV. B 1996, 54, 11169. (26) Kresse, G.; Furthmu¨ller, J. Comput. Mater. Sci. 1996, 6, 15. (27) Durandurdu, M. J. Phys.: Condens. Matter 2008, 20, 325232.
JP102514B
