Hole Mobility in Low-Doped Silicon Nanowires - American Chemical

Article pubs.acs.org/JPCC

Hole Mobility in Low-Doped Silicon Nanowires: An Atomistic Approach D. Sharma*,† and G. Fagas‡ †

Department of Physics, University of Namur, 61 rue de Bruxelles, B-5000 Namur, Belgium Tyndall National Institute, University College Cork, Cork, Ireland

‡

ABSTRACT: We performed first-principle calculations by the Density Functional Theory in order to study the scattering properties of silicon nanowires with two different crystalline orientations, [100] and [110], and different types of dopants. The nanowire axis orientation is found to have a strong influence on transmission, which can alter the normal behavior of dopants. Boron, a p-type dopant, can act indeed as a strong scatterer in the conduction band of [100]-oriented silicon nanowires. Using Boltzmann transport theory, we calculated the charge mobility of boron-doped silicon nanowires with different diameters. Although the scattering strength is shown to be strongly dependent on dopant locations and wire cross-sectional size, the hole mobilities are rather insensitive. It was found that the doping density and the nanowire width strongly influence the hole mobility. It was also found that in small diameter nanowires scattering by neutral impurities can be quite significant and act as a limiting factor to the mobility. At low doping densities, the hole mobility increases monotonically with the width of the nanowire. distributions in SiNWs.12 Nevertheless, the carrier mobility in SiNWs is expected to be affected by different scattering mechanisms, including acoustic phonon scattering,13 neutral dopant scattering, ionized impurity scattering,14 and surface roughness scattering.15 Electron−phonon scattering depends on the nanowire and device dimensions, as well as the nanowire axis.15,16 It has been confirmed that when quantum confinement was not significant in SiNWs of diameter above 30−50 nm, scattering from ionised dopants or phonons limited the carrier mobility.16 In ultrascaled devices, the interplay between the various scattering mechanisms is expected to depend on orientation and cross-section and surface properties. The effect of surface confinement on the low-field mobility of [110] p-type Si nanochannels has been reported.17 Using firstprinciples and the Boltzmann transport method, it was found that [110] channels show a significant enhancement in hole mobility compared to the [100] channel.17 In some recent theoretical work, the current flow in a SiNWs transistor has been shown to improve compared to nanobelts transistor.25 By suppressing the quantum mechanical coupling between heavy and light holes in p-type SiNWs, a doubling of the current is observed in devices with a channel width of 5 nm.18 It has also been shown that SiNWs oriented along the [110] direction support more efficient current processes19 and longer scattering lengths.19 Previously, transistor behavior in junctionless nanowire-based gate-all-around (GAA) setups with just a 3 nm gate length was predicted.20,21 The excellent device characteristics at such ultrascaled dimensions have been recently confirmed.22

1. INTRODUCTION One dimensional semiconductor nanostructures, particularly silicon nanowires (SiNWs), are prominent nanomaterial candidates for many applications including nanoelectronics such as scaled field-effect transistors,1 ballistic field-effect transistors (FETs),2 sensitive nanosensors,3 solar cells,4 or biodevices5 due to their unique electronic properties, their abundance, and their high absorption of light. Chemical doping is one route to tailor the optical, electronic properties, as well as reactivity of SiNWs. Significant control of the semiconducting behavior of SiNWs can be achieved via doping to tune their electronic and transport properties. The main reason for doping a material is to increase the carrier density. Unintentional doping is also present due to the fabrication process of electronic devices and the prohibitive cost of producing wafers with very low dopant impurity concentrations.6 As discussed in the refs 7 and 8, several applications of SiNWs require being able to tune the doping profile; these include solar cells, photodiodes, and FETs for sensors and logic gates. Typically boron and phosphorus are used as p- and n-type dopants, respectively, but Ga and As have also been used. Common methods to incorporate dopants are during the vapor−liquid− solid (VLS) growth.9 The effect of doping in the electronic and transport properties of silicon nanowires has been discussed both theoretically10,11 and experimentally. Different behaviors can be seen depending on the radial distribution of the dopants along the nanowire cross-section. For example, if the dopant is located in the body of the nanowire it can cause back scattering.10 This is absent if the dopant is located near the surface while it couples to dangling bonds at the surface.10,11 A recent experiment reveals different surface segregation behaviors of B and P dopants, which resulted in varying radial © 2015 American Chemical Society

Received: February 9, 2015 Revised: April 23, 2015 Published: May 4, 2015 11934

DOI: 10.1021/acs.jpcc.5b01342 J. Phys. Chem. C 2015, 119, 11934−11940

Article

The Journal of Physical Chemistry C

lations were performed using the OpenMX code with the GGA-PBE functional and norm conserving pseudopotentials. The structures were geometry optimized with a force threshold of 0.01 eV/Å. A Monkhorst−Pack k-point sampling on a 15 × 1 × 1 grid in the Brillouin zone and an optimized double-ζ polarized basis set were used. The mobility μ of a semiconductor material can be obtained from the Boltzmann transport equation within the relaxation time approximation

A factor to consider in electronic devices based on SiNWs are sample-to-sample fluctuations since it is expected that, as the width of the nanowire decreases, the enhanced impurity scattering can depend on the exact number of dopants and their location along the radial direction. Moreover, charge carrier mobilities are limited by other scattering mechanisms like surface roughness and phonon scattering, which have not been systematically studied. Experimental data have just started to emerge and are difficult to interpret. In early measurements, the mobility of [100]- and [110]-oriented SiNWs has been estimated by using a capacitance−voltage method.23 The authors found that both electron and hole mobilities decrease monotonically with nanowire width and that the electron mobilities of nanowires along the [100]- and [110]-axis are comparable. This contradicts more recent studies. In ref 24, it is shown that the [110]-orientation yields enhanced mobility, in agreement with theoretical results.23 In this paper we are motivated by the results of ref 24, where quantum confinement has been studied in [110] channels made of silicon nanowires. The authors performed detailed electrical characterization in SiNW FETs with a sub-5 nm channel width and nominally doped with boron (doping density 2 × 1015 cm−3). They found that the mobility is enhanced due to quantum confinement compared to a nanobelt control sample; hole mobilities in the range of 400−1200 cm2/(Vs) are reported. In the present paper, the hole mobility in borondoped [110]-oriented Si nanowires is calculated for different diameters (1.15−4.47 nm) for various low dopant concentrations and place the substitutional dopant impurity at different positions in the cross-section to estimate the mobility variations due to neutral impurity scattering. Finally, it is shown that hole mobility of Si nanowires nominally doped with boron decreases monotonically with decreasing width at fixed doping density and it increases with decreasing increasing dopant concentration. Significant mobility variations are identified, which can explain experimental observations. The structure of the paper is as follows. Section 2 provides the detailed description of calculating mobility from the relaxation time approximation of Boltzmann transport equation followed by details on the preliminary calculations regarding the electronic structure as well as the scattering properties of single boron impurities as a function of nanowire diameter and dopant position in sections 3.1, 3.2, and 3.3. These calculations are based on first-principles using OpenMX and TiMeS as discussed in previous papers.7,8 In section 3.4, the results on the hole mobility are discussed. Here, the estimations are obtained by considering scattering from independent impurities and applying the relaxation time approximation within the Boltzmann transport method.

μ=

e τm m*

(1.1)

where e is the electron charge and m* is the effective mass calculated from the band structure of the undoped SiNWs.26 The average momentum relaxation time ⟨τm⟩ = ⟨τm(ν)⟩ is ⎛ ∂f ⎞

∞

τm

=

+∞

∫−∞ τm(v)·vx⎜⎝− ∂v0 ⎟⎠d3v/∫−∞ x

f0 (E)d3v

(1.2)

where the Fermi−Dirac distribution f 0(E) = {1/[1 + exp((E − Ef)/(kBT))]} and the group velocity vx is calculated from the first derivative of the bands, as described in section 3.3. For a one-dimensional system ⟨τm⟩, the integral of eq 1.2 can be expressed as a function of energy via the mean free path λ τm

=

m* KBT

∫0

∞

λ(E) ·f0 ·(1 − f0 )dE /

∫0

+∞

f0

1 dE vx (1.3)

Substituting eq 1.3 in eq 1.1 gives the mobility as μ=

e

∫ KT 0

∞

B

λ(E) ·f0 ·(1 − f0 )dE /

∫0

+∞

f0

1 dE vx

(1.4)

To estimate the mean free path, we use the implicit assumption of scattering from independent impurities. As in refs 7 and 27, this is calculated via λ(E) = ((Ts(E))/(Nch − Ts(E))) × ld, where Ts is the transmission value along the wire with the impurity, Nch is the number of modes, as defined in Section II B in ref 8, and ld is the mean distance between dopants, which for a fixed dopant concentration decreases with increasing width, as shown in Table 2. Here, low-doped SiNWs with nominal doping density ∼1015 cm−3 is assumed. Table 1. Band Gap and Hole Effective Mass of [110]Oriented Si Nanowires and Increasing Width

2. METHODOLOGY The geometry of the pristine nanowire and electronic structure calculations have been discussed in detail in refs 7 and 8. We focus on hydrogenated silicon nanowires grown along the [110] and [100] crystallographic directions, with a lattice constant of 3.84 Å, along with the wire axis and diameter equal to 1.15 and 1.27 nm, respectively. We doped the SiNWs with boron, arsenic, phosphorus, and gallium with similar doping concentration, while for calculating the hole mobility, we doped hydrogenated SiNWs of diameters of 1.15−4.47 nm with different doping concentrations by boron. The calculation of mobility from electronic structure has been discussed as follows. Structural relaxations and electronic structure calcu-

SiNW diameter (nm)

band gap (eV)

effective mass (me) calcd here

1.15 2.24 3.34 4.47

2.24 1.74 1.36 1.25

0.14 0.17 0.18 0.19

3. RESULTS AND DISCUSSION 3.1. Impact of Type of Dopant on Transmission. In this section we discuss the impact of the type of dopants on transmission in [110]- oriented SiNWs. When a dopant is introduced into the silicon lattice it causes scattering of charge carriers. This is due to the new chemical environment (potential) introduced by the dopant which modifies the periodicity of the lattice. We doped [110]-SiNW with four different types of single impurity, which are boron, gallium, 11935


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The Journal of Physical Chemistry C Table 2. Estimates of the Distance ld between Dopants for Varying (Low-)Dopant Density and Nanowire Widtha

W (nm)

ld (μm) at n 1 × 1015 cm−3

ld (μm) at n 2 × 1015 cm−3

ld (μm) at n 3 × 1015 cm−3

ld (μm) at n 4 × 1015 cm−3

ld (μm) at n 2 × 1016 cm−3

1.15 2.22 3.34 4.47

964 258 114 64

482 130 56 32

320 86 38 22

240 64 28 16

48 12 5.6 3.2

3.2. Impact of Nanowire Crossection on Transmission. For completeness, in Figure 2 we present the transmission of

This is calculated from π(w/2)2 × ld × n = 1, where W is the nanowire diameter and n is the doping concentration. a

phosphorus, and arsenic. The transmission has been calculated from TiMeS.8 In Figure 1a we have plotted the transmission for SiNW with a single impurity for energies below the valence band edge and

Figure 2. Transmission function across [100]-oriented SiNWs with 1.27 nm diameter and with single substitutional boron impurity as indicated. All calculations are performed using optimized double-ζ basis sets. Results are shown for the valence subbands in (a) and conduction subbands in (b). Energies are referenced to band extrema.

[100] oriented SiNW with a diameter of 1.27 nm across a single boron impurity substituting a Si atom in the center. In this case, boron acts as a weak scatterer in the valence band, whereas strong scattering is observed in the conduction band. This indicates that for small diameter nanowires the scattering properties of dopant impurities depends strongly on the wireaxis orientation. As mentioned earlier, quantum confinement determines the exact alignment of the electronic states of the boron dopant with the valence and conduction bands of the nanowire. Since quantum confinement is a function of diameter and wire orientation the scattering properties are also expected to change with these parameters. 3.3. Electronic Structure and Transmission Properties. In Figure 3.1a−d, the band structure of ideal SiNWs surfacepassivated with hydrogen and varying width (diameter ranges from 1.15 to 4.47 nm) is plotted. The cross-section of the nanowires is displayed in the insets. The effective mass is calculated from the band structure by taking the second

Figure 1. Transmission function across [110]-oriented SiNWs with 1.15 nm diameter and doped with single substitutional dopants as indicated. All calculations are performed using optimized double-ζ basis sets. Results are shown for the first valence band in (a) and conduction band in (b). Energies are referenced to band extrema.

for the first subband. Evidently, boron and gallium atoms act as strong scatterers. This is attributed to their electronic states being closely aligned to the valence band edge. However, the exact dependence is expected to depend on orientation and diameter. Holes injected within the first valence subband would show weak scattering in the presence of P and As dopants; for example, this configuration could arise in n+-p-n+ transistor junctions. In contrast, strong (weak) scattering is observed for P and As (B and Ga) in the first conduction subband. The results are shown in Figure 1b. 11936


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Figure 3.1. Band structures of [110]-oriented Si nanowires with increasing width (diameter: 1.15, 2.22, 3.34, and 4.47 nm). The cross sections are given as insets. The calculated band gap of bulk Si is 0.96 eV.

derivative of the first valence band at the Γ point7. The results are tabulated along with the band gap. The band gap increases with decreasing nanowire width as expected from the stronger quantum confinement and the effective mass increases slightly with the width. These results agree with ref 30. In ref 10 it is shown that a dopant substituting a Si atom in the “bulk” body of a nanowire induces stronger back scattering than a surface dopant interacting with a dangling bond, in which case ballistic transport occurs. Here, all dangling bonds are saturated and only the possibility of dopant substitution in various positions is considered. The different distribution of boron substitutional dopants in the cross-section of nanowires with increasing width is shown in Figure 3.2. For the smallest nanowire with width (d = 1.15 nm), one position in the silicon body and one at the surface is considered. For the nanowire with a diameter of 2.22 nm, six different positions are considered: three sites for “bulk” doping and three dopant positions at the surface. Similarly, seven and eight different positions for the 3.34 and 4.47 nm nanowires, respectively, are considered. The structures are optimized using a supercell of 19.2 Å × 25 Å × 25 Å with the nanowire axis oriented along the x-direction and an optimized double-ζ polarized basis set as before. However, the approximate single-ζ basis set to perform transport calculations within TiMeS is used. Based on the results of ref 7, this is sufficient for a first qualitative discussion of the effect of dopant position and nanowire diameter on the hole mobilities. The transmission of holes across the considered SiNWs with a single boron impurity is shown in Figure 3.3. For the same dopant position, the scattering properties are different with varying nanowire widths. For example, backscattering is

Figure 3.2. Cross sections of the [110]-oriented silicon nanowires with boron substitutional impurities. The various positions of the boron atoms are indicated. The diameter of the nanowires of radius (a) 1.15, (b) 2.22, (c) 3.34, and (d) 4.47 nm.

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case, it can act as a limiting factor to the mobility, which plays an important role in the device performance, such as drive current and speed. Figure 3.4 shows the hole mobilities as a function of diameter and for various dopant positions. Evidently, although mobility

Figure 3.3. Transmission at different dopant location of boron doped SiNWs with width vs eigenenergies of valence band edge: (a) 1.15, (b) 2.2, (c) 3.34, and (d) 4.47 nm. The transmission is normalized to one number of normalized channel.

observed with the transmission dropping to zero when the boron atom is positioned exactly at the center (bulk) and the diameter is 1.15 nm, but for larger nanowires, there is no such strong scattering for this dopant position. For SiNWs with width 2.22 and 4.47 nm backscattering is seen for both bulk and surface locations of the dopant atom. However, at nanowire with 3.34 nm width back scattering is seen mostly in the bulk site, except at position 6 where the dopant is located near the bulk. This shows that dopants located at the surface can also act as strong scatterers. These results show that the scattering behavior is strongly dependent on nanowire width along with dopant location, which may explain some of the variations seen in electrical characterization experiments. 3.4. Hole Mobility. As seen in the previous section, when a dopant is introduced into the silicon lattice it causes scattering of charge carriers. In bulk semiconductors, there are two sources of scattering from dopant impurities, namely, neutral and ionized impurity scattering. The typical ionization energy in bulk silicon for various dopants is ∼50 meV, which implies that, at room temperature, ionized impurity scattering is the dominant mechanism. However, this may not be the case for highly doped silicon nanowires where the impurity bands strongly overlap with the charge carrier channels.20 Also, quantum confinement and the dielectric mismatch with the surrounding material can significantly increase the ionization energy in low-doped silicon nanowires, thereby making dopants inactive.28,29 As demonstrated, neutral impurity scattering can be quite significant in small diameter nanowires, and in this

Figure 3.4. Hole mobility of lightly doped [110]-oriented SiNWs with diameter as indicated. Mobility values for fixed diameter correspond to scattering from boron atoms located at various nonequivalent sites (both at surface and bulk). Doping density is (a) 2 × 1015 and (b) 2 × 1016 cm−3.

variations are observed depending on the dopant position a much weaker dependence than expected is exhibited. This is attributed to the cutoff introduced in the integral of eq 1.4 by the distribution function, shown in Figure 3.5(a), which suppresses any prominent features observed in the scattering properties much below the top of the valence band (e.g., the mean free path dependence shown in Figure 3.5(b)). At such low-dopant densities, the most significant dependence is observed with respect to increasing nanowire diameter and decreasing doping concentration. In both cases it scales linearly with increasing diameter. Comparing the calculated values with those of the experiment at doping density 2 × 1015 cm−3 and nanowire width 4.47 nm,20 the results show good agreement. They may also explain the observed variations between different channel lengths and nanowire diameters. Using the distance ld between dopants as a parameter that captures the statistical variations between samples the hole mobility assuming “bulk” boron dopants (position 1 in Figure 3.2) is plotted in Figure 3.6. It is found that for nanowires with similar width and nominally doped at ∼1015 cm−3 the hole mobility shows similar variation as in the 11938


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according to Mathiensen’s rule, our results show that scattering at neutral dopants can be the most important contribution to the mobility at these diameters for this type of lightly doped nanowires.

4. CONCLUSIONS In summary, the impact of dopant position and nanowire width on the transmission and mobility of SiNWs nominally doped with substitutional boron is evaluated. The scattering properties of single dopant impurities strongly depend on both the nanowire size and dopant location. Backscattering and weak scattering may be observed when an impurity is placed either in the bulk or near the surface. However, despite strong backscattering for many of the dopant positions the hole mobility limited by neutral impurity scattering increases monotonically with increasing nanowire width at fixed doping density. The opposite behavior is expected with increasing dopant concentration. Most importantly, it is found that the mobility can show significant variations in nominally doped nanowires. For example, at 4.47 nm width and n ≈ 1015 cm−3, the hole mobility is in the range of 340−1360 cm2/(Vs), which is in agreement with experimental observations. Finally, motivated by recent experiments the hole mobility in [110]-oriented SiNWs of different widths (diameters in the range 1.15−4.47 nm) and nominally doped with boron (doping density) is calculated. To this end, we combined first-principles results on the band analysis and scattering from single impurities with the Boltzmann transport approach within the relaxation time approximation. In addition to our previous finding on the importance of the wire axis on the scattering properties of single dopants, we have shown that there is interplay with dopant position and wire diameter regarding the effect of neutral dopant impurities in transport properties. Unlike earlier calculations, it has identified examples where backscattering and weak scattering may be observed when an impurity is placed either in the bulk or near the surface. Interestingly, for such low-doping concentrations, the cutoff imposed by the Fermi−Dirac distribution function implies a weak dependence of the mobility from this effect. This has as an effect that the hole mobility limited by neutral impurity scattering is determined predominantly by the nanowire width and the doping density. The hole mobility decreases monotonically with decreasing nanowire width at fixed doping density and increasing dopant concentration. Most importantly, it is shown that the mobility can show significant variations, which can explain experimental observations.

Figure 3.5. (a) Statistical distribution function for holes and (b) mean free path of boron doped SiNW with diameter 1.15 nm. Energies have been referenced to the top of the valence (sub)band.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research was funded by Science Foundation Ireland (SFI) under the Principal Investigator Grant No. 06/IN.1/I857. Partial support was provided through the European Union 7th Framework ICT-FET-Proactive program, SiNAPS project under Contract No. 257856. We also acknowledge computing resources provided by SFI to the Tyndall National Institute and by the SFI and Higher Education Authority Funded Irish Centre for High End Computing.

Figure 3.6. Hole mobility of [110]-oriented SiNWs with varying width and doping distance between dopants as indicated in Table 2. Boron substitutional dopants are located in the bulk of the SiNW (referred as location 1 in Figure 3.2).

experiment, namely, in the range of 340−1360 cm2/(Vs) for the nanowire with 4.47 nm width. Although to estimate the total mobility the contribution from phonon scattering, surface roughness and ionized impurities should be taken into account 11939


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(21) Ansari, L.; Feldman, B.; Fagas, G.; Colinge, J.-P.; Greer, J. C. Subthreshold Behavior of Junctionless Silicon Nanowire Transistors from Atomic Scale Simulations. Solid-State Electron. 2012, 71, 58−62. (22) Migita, S.; Morita, Y.; Masahara, M.; Ota, H. Electrical Performances of Junctionless-FETs at the Scaling Limit (LCH = 3 nm). Electron Devices Meeting. IEDM 2012, IEEE International, 2012; pp 191−194. (23) Packan, P.; Cea, S.; Deshpande, H.; Ghani, T.; Giles, M.; Golonzka, O.; Hattendorf, M.; Kotlyar, R.; Kuhn, K.; Murthy, A.; Ranade, P.; Shifren, L.; Weber, C.; Zawadzki, K. High Performance HiK + Metal Gate Strain Enhanced Transistors on (110) Silicon. Electron Devices Meeting. IEDM 2008, IEEE International, 2008; p 1. (24) Buin, A. K.; Verma, A.; Svizhenko, A.; Anantram, M. P. Significant Enhancement of Hole Mobility in [110] Silicon Nanowires Compared to Electrons and Bulk Silicon. Nano Lett. 2008, 8, 760− 765. (25) Trivedi, K.; Yuk, H.; Floresca, H. C.; Kim, M. J.; Hu, W. Quantum Confinement Induced Performance Enhancement in Sub-5nm Lithographic Si Nanowire Transistors. Nano Lett. 2011, 11, 1412− 1417. (26) Seeger. K. Semiconductor Physics: An Introduction, 9th ed.; Springer-Verlag Berlin Heidelberg: New York, U.S.A., 2004. (27) Fagas, G.; Greer, J. C. Ballistic Conductance in Oxidized Si Nanowires. Nano Lett. 2009, 9, 1856−1860. (28) Rurali, R.; Aradi, B.; Frauenheim, T.; Gali, Á . Donor Levels in Si Nanowires Determined by Hybrid-Functional Calculations. Phys. Rev. B 2009, 79 (115303), 1−7. (29) Bjork, M. T.; Schmid, H.; Knoch, J.; Riel, H.; Riess, W. Donor Deactivation in Silicon Nanostructures. Nat. Nano. 2009, 4, 103−107. (30) Harris, C.; O’Reilly, E. P. Nature of the Band Gap of Silicon and Germanium Nanowires. Phys. E 2006, 32, 341−345.

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Hole Mobility in Low-Doped Silicon Nanowires - American Chemical

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