NANO LETTERS
Spatial Carrier Confinement in Core-Shell and Multishell Nanowire Heterostructures
2008 Vol. 8, No. 10 3341-3344
A. Nduwimana, R. N. Musin, A. M. Smith, and Xiao-Qian Wang* Department of Physics and Center for Functional Nanoscale Materials, Clark Atlanta UniVersity, Atlanta, Georgia 30314 Received June 19, 2008; Revised Manuscript Received August 8, 2008
ABSTRACT We have derived an analytical effective-mass model and employed first-principles density functional theory to study the spatial confinement of carriers in core-shell and multishell structured semiconductor nanowires. The band offset effect is analyzed based on the subband charge density distributions, which is strongly dependent upon the strain relaxation. First-principles calculation results for spatially confined Si/Ge and GaN/GaP nanowires indicate accumulation of a Ge-core hole gas and a GaN-core electron gas, respectively, in agreement with experimental observations.
The current technological pursuit of electronic nanodevices based on two-dimensional (2D) quantum wells, onedimensional (1D) quantum wires, and zero-dimensional (0D) quantum dots is rapidly approaching systems where carrier transport is entirely controlled by quantum effects.1-5 Semiconductor nanowires (NWs) exhibit unique electrical and optical properties in connection with the quantum confinement effect and are increasingly used in electronic devices including field-effect transistors, sensors, detectors, light-emitting diodes, and solar cells.6 Recent experimental work on growing core-shell and core-multishell nanowire heterostructures provides the capability of controlling the size, composition, and band gap engineering. The band offset at the radial heterojunction generates an effective potential barrier for electrons and holes moving through the interface, which serves as an additional confinement potential for the carriers and yields reduction of the dopant scattering over single component NWs.7-10 A wealth of quantum phenomena associated with the quantum confinement remains to be explored. The band alignments can be classified as either type-I (both electrons and holes are confined in one material) or type-II, with distinctive electronic structures. Although the band offset induced quantum effects are well studied for the bulk and 2D systems, the physics behind the 1D NWs is not fully understood. The band offset in NWs is expected to be considerably different from that in bulk heterojunctions and to be strongly modified by the quantum confinement effect. * Corresponding author,
[email protected]. 10.1021/nl8017725 CCC: $40.75 Published on Web 08/28/2008
2008 American Chemical Society
The present study builds on our previous work on the structural and electronic properties of Ge/Si core-shell structured NWs, which demonstrated the important role played by strain relaxation for indirect-to-direct band gap transitions,11 scaling behavior of band structure,12 and nontrivial size dependence of the band offset.13 We derive an effective-mass model for core-shell and multishell cylindrical NW heterostructures and compare the predictions with first-principles calculations of near-band gap charge distributions for Si/Ge and GaN/GaP NWs. The analysis sheds considerable light on the nature of spatial carrier confinement, which is of great importance to developing novel nanoscale devices. The effect of band offset on the band structure of a cylindrical core-shell structure can be studied by introducing a potential barrier, ∆, in a 2D “particle-in-a-box” model as shown in the insets of Figure 1. Along the NW axis, one has a continuous spectrum of plane waves, and the quantization comes from the radial confinement. 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 free motion in the zdirection and a radial equation
(
)
∂2ψ 1 ∂ψ 2m* l2 + + (E - V(r)) - 2 ψ ) 0 2 2 r ∂r ∂r p r
(1)
where m* is the effective mass and l is the angular momentum. From the requirement that the wave function stays single-valued as φ is changed from φ ( 2πn, one gets the quantization condition for the azimuthal quantum number l ) 0, (1, (2,.... For V(r) ) ∆ or 0 depending on the core-shell structure,
κ[κJl(λRc)Yl′(κRc) - λJl′(λRc)Yl(κRc)]Jl′(κRs) + κYl′(κRs)[λJl′(λRc)Jl(κRc) - κJl(λRc)Jl′(κRc)][Yl(λRs)Jl(λR) Jl(λRs)Yl(λR)] ) λ[κJl(λRc)Yl′(κRc) λJl′(λRc)Yl(κRc)]Jl′(κRs) + λYl(κRs)[λJl′(λRc)Jl(κRc) κJl(λRc)Jl′(κRc)][Yl′(λRs)Jl(λR) - Jl′(λRs)Yl(λR)] (3)
Figure 1. Radial charge distribution for the 1p state in the central well (left) and the 1s state in the central barrier (right) effectivemass model, for Rc/R ) 0.5. and E ) ∆/5 (black), E ) ∆ (green), and E ) 5∆ (red), respectively. The model involves three parameters (see insets), ∆, Rc, and R, corresponding to the potential barrier, the core radius, and the radius of the nanowire, respectively.
eq 1 is solved by Bessel functions with a length scale κ ) (2m*(E - ∆))1/2/p or λ ) (2m*E)1/2/p. In the central well case, we have ψ(r)∝Jl(λr) at the core region, while ψ(r) ) c1Jl(κr) + c2Yl(κr) at the shell region, where Jl and Yl are the Bessel functions of the first and second kinds, respectively. Employing boundary conditions such that the wave function as well as its derivative are continuous at the interface (Rc) and vanish at the edge (R), the resulting secular equation for solving Enl is of the form κJ′l(κRc)[Jl(λR)Yl(λRc) - Yl(λR)Jl(λRc)] ) λJl(κRc)[Jl(λR)Y′l(λRc) - Yl(λR)J′l(λRc)] (2)
for the central well case. For the central barrier case, the corresponding secular equation can be obtained by switching λ with κ (λ T κ) in eq 2. The energy and length units are p2/2m*R2 and R, respectively. The analytical effective-mass model describes a set of parabolic subbands with corresponding value Enl at the zone center. Following the conventional notation we denote l ) 0 as the s-state (1-fold degenerate) and l ) (1, (2, (3,..., as the p, d, f,..., states (2-fold degenerate). The limiting cases of eq 2 can be readily extracted. For ∆ ) 0, the low-lying states are all unconfined in the order of 1s, 1p, 1d, 2s, 1f,..., while for ∆ ) ∞ all the charges are confined in the tube (width R - Rc) for central barrier case. The energies can be calculated through a simplified equation Yl(λR)Jl(λRc) ) Yl(λRc)Jl(λR), and the low-lying states are in the order of, e.g., 1s, 1p, 1d, 1f,..., for Rc/R ) 0.5. For general values of ∆, the classification of confined (E < ∆) and unconfined (E > ∆) states can be obtained through numerical solution of eq 2. The calculated charge densities for the two prototype cases (the 1p state with central well and the 1s state with central barrier) are shown in Figure 1. The effect of the band offset is clearly observable in that the charge densities are compressed toward to the center with central well and spreading out toward the shell region with central barrier. The effective-mass model for core-multishell structures can be constructed similarly. For a multishell NW with inner shell radius Rs, the resultant equation is of the form 3342
for the central well model and κ T λ for the central barrier case. While it is gratifying that the results from the analytical effective-mass model provide qualitative interpretation of novel spatial carrier confinement in agreement with experimental observations (e.g., on the hole injection into the Ge region for Ge/Si core-shell and multishell NWs),3 it is worth noting that the band offsets for realistic heterostructured NWs are not known a priori. To analyze the experimental results in NWs, researchers typically use the band offsets from bulk heterojunctions or superlattices, which are not accurate in NWs because of the special geometry at reduced scales.13 In order to provide a more accurate description of the strong spatial confinement of the carriers, we have performed first-principles calculations for Si/Ge and GaN/GaP core-shell and multishell NWs along the [111] direction. There is a ∼4% mismatch of Si and Ge bulk lattice constant and ∼20% mismatch for GaN and GaP. As such, Ge/Si and GaP/GaN constitute limiting cases of strain in the interface of heterostructures. The representative structures of the NWs are shown in Figure 2. Summarized in Table 1 are the diameters and the numbers of atoms in one unit cell for the wires we have studied. In accordance with experimental results, all of the NWs under consideration have approximately a cylindrical shape. In order to study the intrinsic properties, the NW surfaces are passivated by hydrogen or pseudohydrogen atoms. Our calculations are carried out using density functional theory within the local density approximation (LDA) parametrized with the Ceperley-Alder exchange-correlation
Figure 2. Top view of representative ball-and-stick models of Si/ Ge (top panel) and GaN/GaP (bottom panel) core-shell (left) and multishell (right) NWs. Black, orange, purple, blue, and green spheres represent Si (Ge), Ge (Si), Ga, N (P), and P (N) atoms, respectively. Light spheres on the boundary are passivating hydrogen or pseudo-hydrogen atoms. Nano Lett., Vol. 8, No. 10, 2008
Table 1. Calculated Diameters for Core-Shell and Multishell NWs Investigated in This Worka type
Ncore
Nshell
Nmultishell
d (nm)
Si/Ge 38 204 0 2.77 Si/Ge/Si 38 84 120 2.73 Ge/Si 38 204 0 2.70 Ge/Si/Ge 38 84 120 2.74 GaN/GaP 38 72 0 1.75 GaN/GaP 62 156 0 2.17 GaN/GaP 110 192 0 2.27 GaN/GaP/GaN 38 132 132 2.77 GaP/GaN 110 192 0 3.03 GaP/GaN/GaP 38 132 132 2.86 a The numbers of Si/Ge, GaN/GaP atoms in the core and shell regions are shown.
potential.14 Periodic-boundary conditions are employed in the xy plane with a supercell large enough to eliminate the interaction between neighboring wires. The energy cutoff is in the range of 219-300 eV. The Monkhorst-Pack k-point mesh of 1 × 1 × 4 is found to provide sufficient accuracy in the Brillouin-zone integration. The structures are obtained by fully minimizing the forces and stress. Similar techniques have been used in recent studies of semiconductor NWs.11-15 Our relaxed structures for core-shell and multishell NWs show that the lattice constant along axial direction agrees with predictions from Vegard’s law.11 This is also true for optimized diameters of Ge/Si NWs. However, as seen in Table 1, the diameter of the last four GaN/GaP NWs (each has 392 atoms) is in the range of 2.27-3.03 nm, which has strong deviation from the interpolated value of ∼2.6 nm and indicates strong strain relaxation in the heterostructures. The confinement effect is readily observable by examining the spatial distribution of discrete states near the gap. First we focus on the states in Si/Ge core-shell and multishell NWs. The real-space charge density distributions, integrated along the axial direction within one unit cell, of representative states are shown in Figure 3. The four NWs have approximately the same size. For Ge-core, the state at the top of the valence bands in Figure 3 has a charge distribution mainly in the core region. In contrast, for Si-core the state has a charge distribution mainly in the shell region. This type-II core-shell confinement feature can be easily concluded for these two states from the charge distribution. Moving down in energy from the valence band maximum (VBM), for Si-core, the next two states are of 1p and 1d character (2-fold degenerate), in agreement with effectivemass model predictions. For Ge-core, the 2s state crosses over 1p and 1d states in energy order for core-shell and multishell NWs, respectively. The crossover can be attributed to the fact that the band offset induced potential well prefers core-type (1s and 2s) over shell-type (e.g., 1p and 1d) states. The charge distribution of the 2s state extends into the shell region, and its energy relative to VBM can be regarded as a confinement energy that serves as an upper bound to the band offset.13 For NWs with the same core-size, the effective-mass model results show that the crossover of 2s state with 1p or 1d state is easier for multishell than core-shell, in agreement with first-principles results shown above. This implies the reduction of confinement energy and less charge separation Nano Lett., Vol. 8, No. 10, 2008
Figure 3. Charge distributions for the three near-band gap valence states of the Si/Ge core-shell and multishell: the heavy hole (VBM), the light hole (VB2), and the third valence band (VB3). Shown from top to bottom panels are those for Si/Ge core-shell, Si/Ge/Si multishell, Ge/Si core-shell, and Ge/Si/Ge multishell, respectively.
Figure 4. Charge distributions of a GaN/GaP core-shell NW of diameter 2.17 nm after integrating along the axis. Twelve states are shown (six for conduction bands at the Γ point and six for valance bands) together with the symmetry classification of each state.
in multishell. As seen in Figure 3, the charge density distribution of VBM heavy hole state shows increased charge in the Si region for multishell structures. The corresponding confinement energy is indeed reduced from both firstprinciples and effective-mass model calculation results. We plot a representative case in Figure 4 for a [111] GaN/ GaP wire containing 62 GaN atoms in the core and 156 GaP atoms in the shell regions. The valence bands have charge mainly in the shell region, while the conduction bands have charge mainly in the core. The 1px and 1py states are nearly degenerate (so as 1dx and 1dy). The asymmetry of the charge distribution arises from the fact that the x and y directions are not exactly equivalent. This is also the case for NWs along other directions (such as [110] NWs), where carriers have different effective masses along x and y, leading to different confinement extents in these two directions. The first state that loses this core confinement feature is the 2s state, as shown in Figure 4. 3343
Figure 5. Charge distributions in the cross sections of core-shell and multishell NWs after integrating along the axis. Listed from top to bottom are those for GaN/GaP (110/192) core-shell, GaN/ GaP/GaN (38/132/132) multishell, GaP/GaN (110/192) core-shell, and GaP/GaN/GaP (38/132/132) multishell, respectively.
Because the core and shell regions are expected to induce different confinement effects, the confinement energies for the valence and conduction states do not have to be the same in core-shell NWs. Of particular interest is the NWs with a core confinement, in which the conduction states have a significant confinement energy of the order of 1.7 eV, while the result for valence states is much smaller, about 0.4 eV. In the latter case, all the shell-confined states in the valence bands are close in energy, and the 2s state moves up largely due to the difference in effective masses. We find that the band offset for conduction states with a GaN core is consistently larger than other cases. The three GaN/GaP core-shell NWs listed in Table 1 have very close Rc/R ratio. The optimized diameters for the three structures follow a linear N relationship, indicating similar strain relaxation effects. Interestingly, the conduction band charge distributions for the three NWs have close resemblance (see the bottom panel of Figure 4 and top panel of Figure 5 for NWs of d ) 2.17 and 2.27 nm, respectively). The corresponding confinement energy decreases with the increase of the NW diameter with a roughly R-2 fashion. This is in conformity with the effect-mass model prediction in that with the same Rc/R ratio and ∆ ∝ R-2, the shape of charge density remains invariant. Shown in Figure 5 are the charge distributions of GaN/ GaP core-shell and multishell NWs. It is worth noting that the conduction states of GaN/GaP/GaN multishell and valence states of GaP/GaN/GaP multishell show coaxial charge confinement, with the former more pronounced. This
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is to be contrasted to the case of Ge/Si/Ge multishell, where no coaxial distribution is observed. This demonstrates that the strain relaxation plays an important role in charge separations. In conclusion, we have presented detailed analysis of the charge density distributions and the clarification of the quantum confined states based on analytical effective-mass model predictions and first-principles calculations. The charge separation in Ge-core valence states and GaN-core conduction states is evident, which are in good agreement with experimental observations of accumulation of a Gecore hole gas1 and GaN-core electron gas16 in core-shell structured NWs. Furthermore, our calculation results reveal a coaxial electron charge distribution in GaN/GaP/GaN multishell NWs. We remark, before closing, that it is straightforward to use this approach to novel semiconductor NWs, and the investigation of the relevant band offset effect will provide an invaluable tool for developing future nanodevices. Acknowledgment. This work is supported in part by the National Science Foundation (Grant Nos. DMR-02-05328 and HRD-0630456), and Army Research Office (Grant No. W911NF-06-1-0442). References (1) Lu, W.; Xiang, J.; Timko, B. P.; Wu, Y.; Lieber, C. M. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 10046. (2) Duan, X.; Huang, Y.; Cui, Y; Wang, J.; Lieber, C. M. Nature 2001, 409, 66. (3) Lauhon, L. J.; Gudlksen, M. S.; Wang, D.; Lieber, C. M. Nature 2002, 420, 57. (4) Cui, Y.; Zhong, Z.; Wang, D.; Wang, W. U.; Lieber, C. M. Nano Lett. 2003, 3, 149. (5) Greytak, A. B.; Lauhon, L. J.; Gudiksen, M. S.; Lieber, C. M. Appl. Phys. Lett. 2004, 84, 4176. (6) Zhang, Y.; Wang, L.-W.; Mascarenhas, A. Nano Lett. 2007, 7, 1264. (7) Gudiksen, M. S.; Lauhon, L. J.; Wang, J.; Smith, D. C.; Lieber, C. M. Nature 2000, 415, 617. (8) Noborisaka, J.; Motohisa, J.; Hara, S.; Fukui, T. Appl. Phys. Lett. 2005, 87, 093109. (9) Bjo¨rk, M. T.; Ohlsson, B. J.; Sass, T.; Persson, A. I.; Thelander, C.; Magnusson, M. H.; Deppert, K.; Wallenberg, L. R.; Samuelson, L. Appl. Phys. Lett. 2002, 80, 1058. (10) Kim, B.-K.; Kim, J.-J.; Lee, J.-O.; Kong, K.-J.; Seo, H. J.; Lee, C. J. Phys. ReV. B 2005, 71, 155313. (11) Musin, R. N.; Wang, X.-Q. Phys. ReV. B 2005, 71, 155318. (12) Musin, R. N.; Wang, X.-Q. Phys. ReV. B 2006, 74, 165308. (13) Yang, L.; Musin, R. N.; Wang, X.-Q.; Chou, M. Y. Phys. ReV. B 2008, 77, 195325. (14) Kresse, G.; Furthmuller, J. Phys. ReV. B 1996, 54, 11169. Comput. Mater. Sci. 6, 15 (1996). (15) Zhao, X. Y.; Wei, C. M.; Yang, L.; Chou, M. Y. Phys. ReV. Lett. 2004, 92, 236805. (16) Li, Y.; Xiang, J.; Qian, F.; Gradecak, S.; Wu, Y.; Yan, H.; Blom, D.; Lieber, C. M. Nano Lett. 2006, 6, 1468.
NL8017725
Nano Lett., Vol. 8, No. 10, 2008
