Si Core

May 25, 2010 - We report electronic transport properties of doped Ge-core/Si-shell and Si-core/Ge-shell nanowires (NWs) from first-principles. We obta...
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Single-Impurity Scattering and Carrier Mobility in Doped Ge/Si Core-Shell Nanowires Hyungjun Lee and Hyoung Joon Choi* Department of Physics and IPAP, Yonsei University, Seoul 120-749, Republic of Korea ABSTRACT We report electronic transport properties of doped Ge-core/Si-shell and Si-core/Ge-shell nanowires (NWs) from firstprinciples. We obtain single-impurity scattering properties of electrons and holes using density-functional methods for quantum conductance and then estimate charge-carrier mobilities considering multiple impurity scatterings. It is found that holes in the Gecore/Si-shell NW with B-doped Si and electrons in the Si-core/Ge-shell NW with P-doped Ge have higher mobilities than holes and electrons in other chemical and doping configurations. These results reflect asymmetric radial confinements of charge carriers in the core-shell NWs and show that Si-core/Ge-shell NWs with electron donors in the shell are as promising for nanoelectronic devices as Ge-core/Si-shell NWs with electron acceptors in the shell. KEYWORDS Core-shell nanowires, radial confinement, impurity scattering, carrier mobility

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the dopants as in the study of doped Si NWs,16 firstprinciples calculations are required not only for the electronic structures but also for the transport properties. Moreover, if p- or n-type doping can be site-controlled, different combinations of core-shell compositions6 may enlarge the functionality of the core-shell NWs even further, which can also be explored by first-principles calculations. In this Letter, we report electronic transport properties of doped Ge/Si core-shell NWs obtained by first-principles pseudopotential density-functional methods.17-19 We consider both Ge-core/Si-shell and Si-core/Ge-shell NWs and calculate transmission spectra through a single B or P impurity in the core or the shell. Then charge-carrier mobilities are estimated as functions of dopant and carrier concentrations for different chemical and doping configurations. As a result, we show that, while the highest hole mobility occurs in the Ge-core/Si-shell NW with the B impurities at the shell, the highest electron mobility, comparable to or even greater than the hole case, can be achieved in the Sicore/Ge-shell NW with the P impurities at the shell. This suggests that doped Si-core/Ge-shell NWs can be as suitable for nanoelectronic devices as doped Ge-core/Si-shell NWs. Our present calculations are based on ab initio pseudopotential density-functional methods for electronic structures17,18 and for electronic transports,19 which employ norm-conserving pseudopotentials for constituent atoms, the local density approximation for the exchange-correlation energy, and pseudoatomic orbitals for expansion of electronic wave functions. A large enough supercell is used to simulate an isolated NW, and the electronic density in the NW is obtained by integrating wave functions in the onedimensional (1D) Brillouin zone. Atomic positions and unitcell lengths along the NWs are optimized by minimizing the

i and Ge semiconducting nanowires (NWs) have attracted wide interest as building blocks for nanometer-scale electronic devices due to their intriguing electronic properties depending on their size, shape, and chemical compositions1 and due to their compatibility with Si-based technology.2 Among them, the Ge-core/Si-shell NW, which is a radial heterostructure of Si and Ge NWs, shows higher mobility and transconductance than the homostructural Si or Ge NW3,4 and has been applied successfully to field-effect transistors,5-7 semiconductor-superconductor hybrid nanostructures,8 double quantum dots,9 and threedimensional integrated circuits.10 The Ge-core/Si-shell NW is reported to have hole carriers accumulated in the core region even without any intended doping,5 possibly due to dangling-bond defects or Au impurities from catalysts.11 Since electronic transport properties of undoped core-shell NWs are mainly determined by their electronic structures, first-principles calculations have been extensively performed to characterize the radial quantum confinement and surface effects in undoped Ge/Si core-shell NWs12-14 while their transport properties have been analyzed, so far, by a semiclassical ballistic transport model.15 For more versatile functionality suited for wider device applications and optimizations, doping of various elements may be intended in Ge/Si core-shell NWs. If dopants are introduced, electrons or holes released from the dopants will become mobile charge carriers in the NWs while the dopants themselves will produce charge-carrier scatterings that limit the carrier mobilities. Since theoretical description of these scatterings requires realistic wave functions of electrons near

* To whom correspondence should be addressed. E-mail: [email protected]. Received for review: 03/30/2010 Published on Web: 05/25/2010 © 2010 American Chemical Society

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DOI: 10.1021/nl101109p | Nano Lett. 2010, 10, 2207–2210

FIGURE 1. Undoped core/shell [110] nanowires. (a) Cross-sectional and (b) side views of a Ge-core/Si-shell NW in which blue and red balls are Ge and Si atoms, respectively, while light gray balls are H atoms. In (b), six unitcells are shown along the NW. The same bonding structure applies to a Si-core/Ge-shell NW with slightly different bond-lengths. (c,d) Electronic band structures and projected densities of states (PDOS) per eV·unitcell in the Ge-core/Sishell and Si-core/Ge-shell NWs, respectively.

FIGURE 2. Ge/Si core-shell NWs with an impurity. (a) Side view of a doped NW composed of a doped scattering region, where the electronic potential is perturbed by the impurity placed in the marked doped layer, and two infinitely long undoped perfect regions. (b,c) Cross-sectional views of the doped layer with an impurity, marked in gray and yellow, in the shell and the core, respectively.

total energy of electrons and ions until all residual atomic forces are smaller than 0.05 eV/Å. Figure 1a,b shows our atomic model of an undoped Gecore/Si-shell (and topologically equivalent Si-core/Ge-shell) NW along the [110] direction, which is reportedly a preferred growth direction in the case of synthesized Si NWs with small diameters (10 nm or less)20 and also in the case of synthesized Ge-core/Si-shell NWs with the diameter of 18 nm.7 For simplicity, an octagonal cross-sectional shape is assumed, which has (001), (-111), and (-110) surfaces on its sides. Hydrogen atoms are placed on the surfaces in canted dihydride forms,21 passivating dangling bonds and suppressing surface states that may appear inside the energy gap. A primitive unit cell contains 104 atoms: 30 atoms in the core, 46 atoms in the shell, and 28 H atoms. When geometries are optimized, the undoped Ge-core/Si-shell NW has the primitive unit-cell length of 0.389 nm, the whole diameters (dwhole) of 2.254 nm, which is the average of the height and the width of the NW measured between H atoms, and the ratio of the core diameter to the whole diameter (dcore/dwhole) of 0.62. For the undoped Si-core/Ge-shell NW, the primitive unit-cell length is 0.392 nm, dwhole is 2.256 nm, and dcore/dwhole is 0.59. Radial confinements of charge carriers occur differently in the core-shell NWs depending on the core and shell compositions (Figure 1c,d), consistently with previous theoretical results.13 In the Ge-core/Si-shell NW, which has the energy gap (Egap) of 0.85 eV, the projected density of states (PDOS) (Figure 1c) shows that electronic states near the valence band maximum (VBM) are radially confined mainly in the Ge core region but those near the conduction band © 2010 American Chemical Society

minimum (CBM) are spread out in both of the core and shell regions. Meanwhile, in the Si-core/Ge-shell NW, where Egap ) 0.66 eV, the radial confinement is substantial near CBM; electronic states near CBM are radially confined mainly in the Si region, as shown in PDOS in Figure 1d. We quantify radial confinement energies by comparing the PDOS onto the core and shell regions. We regard that electronic states at an energy are confined in a region if the PDOS onto the region is more than 75% of the whole. Then, in the Ge-core/Si-shell NW, the confinement energy is 0.23 eV near VBM while it is negligible near CBM (Figure 1c). In the Si-core/Ge-shell NW, the confinement energy is 0.47 eV near CBM while it is rather small (0.08 eV) near VBM (Figure 1d). Schematic valence- and conduction-band alignments considering these confinement energies are depicted with color-shades in PDOS plots in Figure 1c,d. To study scattering properties of electrons and holes by a single impurity, we consider an infinitely long core-shell NW with a single B or P impurity in it. One can divide this doped NW into three regions: a scattering region, which is a region near the single impurity, and two perfect regions, which are infinitely long undoped NWs, as shown in Figure 2a. Cross-sectional locations of the single impurity (Figure 2b,c) are chosen in accordance with the reported stable dopant positions in Si NWs;22 then atomic positions and the unit-cell length near the impurity are optimized by minimizing the total energy, using a supercell of three unit cells with the single impurity in it. Relaxed supercell lengths are almost independent of dopant positions and they are 1.165 and 1.170 nm for B- and P-doped Ge-core/Si-shell NWs, respectively, and 1.173 and 1.177 nm for B- and P-doped Si-core/ 2208

DOI: 10.1021/nl101109p | Nano Lett. 2010, 10, 2207-–2210

vn(k), where λn(k) ) λn(En(k)) and pvn(k) ) (∂En(k))/(∂k). Since the 1D electrical conductivity σ1D of the NW is the 1D current density, which is the amount of current per NW, divided by the electric-field strength along the NW, we have

σ1D ) e2

n

∂fFD(E) ∂E

]

E)En(k)

(1)

where fFD(E) is the Fermi-Dirac distribution function that depends on the chemical potential and temperature. Finally, the 1D electron and hole mobilities, µe and µh, are given by µe ) σe/ene and µh ) σh/enh, respectively, where σe and σh are respective contributions of electrons and holes to σ1D, and ne and nh are the numbers of electrons and holes per unit length, respectively. We note that σ1D is proportional to 1/Ni because τn(k) ∝ λn(k) ∝ 1/Ni, and that the integral in eq 1 requires τn(k) over an energy interval of 200 meV centering at the chemical potential for a numerically converged value of σ1D at temperature of 300 K. We obtain ne and nh as functions of the chemical potential from the densities of states of undoped core-shell NWs and fFD(E). In our Ge-core/Si-shell and Si-core/Ge-shell NWs, the intrinsic chemical potentials without doping are -2.2 and 8.6 meV from the middle of the energy gap, respectively, at temperature of 300 K. Although ne and nh are affected by Ni in doped NWs, we assume that a gate is present so that its voltage can change the chemical potential, and thereby ne and nh independently of Ni. Then the mobilities µe and µh are proportional to 1/Ni because σ1D ∝ 1/Ni but ne and nh are made independent of Ni by the gate voltage. Figure 4 shows carrier concentrations, 1D electrical conductivities, and mobilities of holes and electrons as functions of the chemical potential. We assume that the impurity density Ni is 2.75 × 106 cm-1, corresponding to one B or P impurity per 3.64 nm, the typical length of the scattering region in Figure 2a. We consider carrier concentrations of nh e 9.2 × 106 cm-1 and ne e 2.4 × 107 cm-1 for Ge-core/Si-shell NWs, and nh e 1.7 × 107 cm-1 and ne e 1.1 × 107 cm-1 for Si-core/Ge-shell NWs. These concentrations can be achieved by varying the chemical potential up to 0.6 eV above or below the middle of the energy gap. When the chemical potential is right at VBM, the obtained hole mobility is 940 cm2/V·s in the Ge-core/Si-shell NW with B-doped Si, and it decreases if B is placed in the Ge region instead of the Si region (Figure 4a). When the chemical potential is right at CBM, the electron mobility is 870 cm2/ V·s in the Si-core/Ge-shell NW with P-doped Ge, and it decreases if P is placed in the Si region instead of the Ge region (Figure 4b). In contrast, the electron mobility at CBM in P-doped Ge-core/Si-shell NWs and the hole mobility at VBM in B-doped Si-core/Ge-shell NW are rather insensitive to locations of impurities and they are smaller than the above optimal values. These results show that, when Si-Ge band

FIGURE 3. Transmission spectra T(E) through single impurities in (a) Ge-core/Si-shell and (b) Si-core/Ge-shell NWs. Each colored line shows T(E) through a B or P impurity in the Si or Ge region as a function of the electron energy E. The middle of the energy gap in each NW is set to zero. Thin black lines, drawn for comparison, are T(E) in the case of no impurity.

Ge-shell NWs, respectively. With these lengths near the impurity, the length of the scattering region in Figure 2a is set at 3.498 and 3.505 nm for B- and P-doped Ge-core/Sishell NWs, respectively, and 3.524 and 3.529 nm for B- and P-doped Si-core/Ge-shell NWs, respectively. The scattering region in Figure 2a is the only region in which we consider perturbation in the electronic potential due to the impurity. Figure 3 shows the transmission spectra T(E), as functions of the electron energy E, through a single impurity in Ge/Si core-shell NWs obtained by the first-principles scatteringstate method for quantum conductance.19 With multiple bands, we have T(E) ) ∑nTn(E), where Tn(E) is the transmission of an electron in the nth band. We find that in B-doped Ge-core/Si-shell NWs (upper two panels in Figure 3a), T(E) near VBM is sensitive to the position of the B impurity, but it is almost insensitive near CBM. In the case of P-doped Gecore/Si-shell NWs (lower two panels in Figure 3a), T(E) near CBM is sensitive to the position of the P impurity and, unexpectedly, it is also sensitive near VBM. In the cases of doped Si-core/Ge-shell NWs (Figure 3b), the positions of the B and P impurities make significant differences in T(E) only in the valence and conduction bands, respectively. Hole or electron mobilities can be estimated from Tn(E). When a nanowire has Ni number of impurities per length along the wire, the probability Pn(x) for an electron or a hole in the nth band to travel a distance x before being backscattered is given by Pn(x) ) (Tn(E))Nix ) eNix ln(Tn(E)) because Tn(E) is equivalent to the forward scattering probability in each impurity scattering. As Pn(x) ) e- x/λn defines a mean-free path λn, we have λn ) -(Ni ln Tn(E))-1. Then, the lifetime τn(k) of the electron or the hole of wavevector k is τn(k) ) λn(k)/ © 2010 American Chemical Society

[

τ (k)vn(k)2 ∑ ∫ dk π n

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eV below VBM (top panel in Figure 3b). Especially, in the Sicore/Ge-shell NW with P-doped Ge, T(E) is almost one in an energy range of about 0.2 eV near CBM (bottom panel in Figure 3b), resulting in the highest mobility among the NWs considered. In summary, we have studied effects of single impurities on electronic transport properties in doped Ge/Si core-shell [110] NWs and obtained electron and hole mobilities as functions of carrier concentrations for different atomic kinds of dopants, doping sites, and core-shell compositions. We found that holes in the Ge-core/Si-shell NW with the B-doped shell and electrons in the Si-core/Ge-shell NW with the P-doped shell have high mobilities among the considered doped core-shell NWs. Our results suggest that doped Sicore/Ge-shell NWs are as much qualified for nanoelectronic devices as doped Ge-core/Si-shell NWs. Acknowledgment. This work was supported by the NRF of Korea (Grant R01-2007-000-20922-0 and Grant 20090081204). H.L. acknowledges support from the Seoul Science Scholarship. Computational resources have been provided by KISTI Supercomputing Center (Project No. KSC2008-S02-0004).

FIGURE 4. One-dimensional (1D) carrier concentrations, conductivities, and mobilities versus the chemical potential µ in (a) Ge-core/ Si-shell NWs and (b) Si-core/Ge-shell NWs at the electron temperature of 300 K. The chemical potential, which is measured from the middle of the energy gap, is assumed to be changeable by an applied gate voltage; it is negative (positive) when hole (electron) carriers are the majority. Vertical dashed lines indicate values of the chemical potential when it is at the valence band maximum or at the conduction band minimum.

REFERENCES AND NOTES (1) (2) (3)

offsets are significant, carriers confined in one region may propagate without strong impurity scattering if impurities are placed in the other region, and that the carrier mobilities are rather small and independent of impurity positions if the confinement is weak. In the considered range of the chemical potential µ, the maximal hole mobility in the Ge-core/Si-shell NW with B-doped Si is 4960 cm2/V·s when nh ) 6.1 × 106 cm-1 corresponding to µ of 0.13 eV below VBM. In the Si-core/ Ge-shell NW with B-doped Si, the maximum hole mobility is also as large as 5060 cm2/V·s when nh ) 1.2 × 107 cm-1 corresponding to µ of 0.15 eV below VBM, even though the confinement energy at VBM is rather small (0.08 eV). For the electron mobility, the largest value occurs in Si-core/Geshell NW with P-doped Ge, reaching 9570 cm2/V·s when ne ) 7.0 × 106 cm-1 corresponding to µ of 0.26 eV above CBM. In contrast, in P-doped Ge-core/Si-shell NWs, where radial confinement is insignificant near CBM, electron mobilities are rather small regardless of dopant positions. The obtained large carrier mobilities are due to almost perfect transmission, that is, T(E) equal to the number of bands at the energies near VBM or CBM. In Figure 3a, for the Ge-core/Si-shell NW with B-doped Si, T(E) is almost one at about 0.1 eV below VBM (top panel in Figure 3a), implying that the holes at the energy are hardly scattered by the impurity. Similarly, in the Si-core/Ge-shell NW with B-doped Si, T(E) is almost two, the number of bands, at about 0.15

© 2010 American Chemical Society

(4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)

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DOI: 10.1021/nl101109p | Nano Lett. 2010, 10, 2207-–2210