NANO LETTERS
Orientational Dependence of Charge Transport in Disordered Silicon Nanowires
2008 Vol. 8, No. 12 4146-4150
Martin P. Persson,† Aure´lien Lherbier,†,‡ Yann-Michel Niquet,*,† François Triozon,§ and Stephan Roche| Commissariat à l’Energie Atomique, INAC/SP2M/L_Sim, INAC/SP2M/GT, and LETI-MINATEC, 17 rue des Martyrs, 38054 Grenoble cedex 9, France, and Laboratoire des Technologies de la Microe´lectronique (LTM), UMR 5129 CNRS, 17 rue des Martyrs, 38054 Grenoble cedex 9, France Received April 21, 2008; Revised Manuscript Received September 26, 2008
ABSTRACT We report on a theoretical study of surface roughness effects on charge transport in silicon nanowires with three different crystalline orientations, [100], [110] and [111]. Using an atomistic tight-binding model, key transport features such as mean-free paths, charge mobilities, and conductance scaling are investigated with the complementary Kubo-Greenwood and Landauer-Bu¨ttiker approaches. The anisotropy of the band structure of bulk silicon results in a strong orientation dependence of the transport properties of the nanowires. The best orientations for electron and hole transport are found to be the [110] and [111] directions, respectively.
Chemically grown semiconducting nanowires have become the subject of intense study due to their outstanding electronic and optical properties.1 The progress made in nanowire growth techniques has indeed allowed the fabrication of highquality nanowires and the demonstration of p-n junction diodes,2 logic gates,3 field-effect transistors (FETs),4,5 and nanoscale sensing devices.7 Among semiconducting materials, silicon nanowires (SiNWs) are of particular interest as they stand as potential building blocks for nanoscale electronics.8 As a matter of fact, compared to classical planar device technology, SiNWs can better accommodate “all-around” gates, which improve field effect efficiency and device performances.5,9,10 Moreover, SiNWs with different orientations have been fabricated, using both epitaxial5,6 (bottom up) and lithographic10 (top down) techniques, allowing for further device optimization. Indeed, the transport properties of low-dimensional SiNWs are expected to be sensitive to the crystal orientation, due to the anisotropy of the bulk band structure, but to date this issue remains poorly addressed. Additionally, at variance with many other materials, structurally stable and electrically active SiNWs can be manufactured with diameters d < 5 nm, which opens new opportunities for the exploration of * Corresponding author,
[email protected]. † Commissariat à l’Energie Atomique, INAC/SP2M/L_Sim. ‡ Laboratoire des Technologies de la Microe ´ lectronique (LTM), UMR 5129 CNRS. § Commissariat à l’Energie Atomique, LETI-MINATEC. | Commissariat à l’Energie Atomique, INAC/SPSMS/GT. 10.1021/nl801128f CCC: $40.75 Published on Web 11/08/2008
2008 American Chemical Society
quantum transport phenomena in low dimensional systems.10,11 However, the impact of structural imperfections such as surface roughness and defects becomes increasingly important with decreasing wire diameter due to the large surface to volume ratios. In the case of lithographic SiNW-FETs, surface roughness disorder (SRD) is already believed to be a key limiting factor for ballistic conduction.12,13 It is thus important to assess the impact of SRD on the transport properties of small SiNWs and to determine which nanowire orientation is best suited to the engineering of highly performant SiNW-based field effect transistors. In this Letter, we report on a comparative study of the transport properties of rough SiNWs oriented along the [100], [110], and [111] crystal directions. The electronic structure of the SiNWs is described by an accurate third nearest neighbor sp3 tight-binding (TB) Hamiltonian, previously validated by ab initio calculations and comparison with experimental data.14 In contrast to fully ab initio approaches, the tight-binding method indeed allows for quantum transport calculations on long and disordered nanowires. Following a prior work,15 two complementary approaches are used to explore the transport regimes in SiNWs; namely an order N Kubo-Greenwood method, which gives a straightforward access to the intrinsic elastic mean free paths and charge mobilities,16 and a Landauer-Bu¨ttiker approach,17,18 which is particularly well suited to the quasi-ballistic regime, where contact effects start to prevail over intrinsic phenomena. We
Figure 1. Top panel: Cross sections of [100], [110], and [111] oriented nanowires with radius R0 ) 1 nm. Middle and bottom panels: Conduction and valence band structures and DoS (in eV-1 nm-1 spin-1) for clean SiNWs (R0 ) 1 nm) oriented along the [100] (a and b), [110] (c and d), and [111] (e and f) directions.
Table 1. Basic Band Structure Parameters for SiNWs with Radius R0 ) 1 nm Oriented along the [100], [110], and [111] Directions: Electron and Hole Effective Masses along the Nanowire, Degeneracy of the Conduction Band (CB) Valley and Highest Valence Subband (VB)19 hole mass electron mass VB degeneracy 19 CB valley degeneracy
[100]
[110]
[111]
0.97m0 0.30m0 2 4
0.18m0 0.14m0 1 2
0.16m0 0.96m0 1 6
first briefly review the band structure of SiNWs as a function of their orientation and then discuss their transport properties. Recently, the electronic structure of small-diameter SiNWs has been computed with tight-binding models and shown to be sensitive to the wire orientation.14 As an illustration, the band structures and densities of states (DoS) of clean Si nanowires with radius R0 ) 1 nm, and oriented along the [100], [110], and [111] directions, are plotted in Figure 1 (using the same sp3 TB model as in ref 14). The Van Hove singularities typical of 1D subbands are clearly visible in the DoS. Some key features (electron and hole effective masses, degeneracy of the conduction band valley and highest valence subband) are also collected in Table 1. The orientation dependence of the transport properties of small SiNWs can be understood from a comparison of their band structures. Bulk silicon has six conduction band minima located around (0.8ΓX. Each of these valleys has a heavy longitudinal effective mass (ml* ) 0.92m0) and a light transverse effective mass (mt* ) 0.19m0) along and perpendicular to the ΓX direction, respectively. These six minima fold differently into the first Brillouin zone of the SiNWs. In [100] oriented SiNWs, four valleys fold at the Γ point, while the other two fold at a higher energy around k ) (0.4π/l (where l is the length of the unit cell of the nanowire). In [110] oriented SiNWs, only two minima fold at the Γ point, while the other four fold at k = ( 0.8π/l. The effective mass of the electrons at Γ is close to mt* in both [100] and [110] oriented nanowires. The 2- and 4-fold Nano Lett., Vol. 8, No. 12, 2008
degeneracy expected between the lowest conduction subbands is however lifted by the intervalley couplings in small SiNWs (especially in the [110] orientation), as shown in Figure 1, panels a and c. Finally, in [111]-oriented SiNWs, the six bulk valleys fold (in principle) at k =(0.4π/l with an intermediate mass m* = 0.4 m0. However, in small nanowires, they fold closer to the Γ point and show a much larger effective mass, as evidenced in Table 1. As for the holes, in small [110]- and [111]-oriented SiNWs, the highest valence subband is rather dispersive (light hole mass) and well separated from the other bands. It is, at variance, 2-fold degenerate,19 close to the next subband, and much flatter in [100]-oriented SiNWs. We can therefore anticipate that the [110] direction will be the best orientation for electron transport (since the 6-fold valley degeneracy is completely lifted), while the [110] and [111] directions will be the best orientations for hole transport. These first observations will be confirmed by our detailed transport analysis. The effects of chemical disorder on the electronic and transport properties of small SiNWs have recently been investigated with ab initio approaches.20 Such first principles calculations, however, can hardly give access to the main transport length scales given computational limitations. On the other hand, the effects of surface roughness have also been analyzed using phenomenological models with a simplified description of the band structure and/or geometry of the nanowires,21 impeding a quantitative understanding of charge transport in real materials. A computationally less demanding but quantitatively correct description can be achieved using a tight-binding model.15,22 Here we describe surface roughness disorder at the atomic scale, removing any silicon atom outside a given surface profile and saturating the dangling bonds with hydrogen atoms. This surface profile is defined as a random fluctuation of the radius of the nanowire around its average value R0, characterized by a exponential Lorentzian autocorrelation function.13,15,17 It reads in cylindrical coordinates: 4147
Figure 2. Top panel: A typical example of surface roughness profile. Middle panel: Conductance (as a function of energy) in clean and rough SiNWs (R0 ) 1 nm) oriented along the [100] (a), [110] (b), and [111] (c) directions. The thin blue line is the quantized conductance in clean (disorder-free) SiNWs, while the red dashed and black solid lines are the averaged (ten samples) conductance in rough SiNWs with lengths L = 20 nm and L = 160 nm, respectively. Bottom panel: Mean free paths of the electrons and holes in the rough SiNWs.
R(θ, z) ) R0 +
∑
ankeinθei
(2π⁄L)kz
(1a)
(n,k)*(0,0)
where
( [( ) ( ) ] )
ank ) eiφnk 1 +
2πk 2 n + L R0
2
Lr2
-3⁄4
(1b)
φnk ∈ [0, 2π[ is a random number, Lr is the correlation length of the SRD, and L is the length of the nanowire. In the following, we set R0 ) 1 nm, Lr ) 2.17 nm, and renormalize the ank so that 〈(R - R0)2〉1/2 ) 1 Å. An example of a SRD profile is shown in Figure 2. The transport properties of the nanowires have been investigated with the Landauer-Bu¨ttiker17,18 and KuboGreenwood methods.15,16 In the former approach, the nanowire is coupled to ideal semi-infinite leads with radius R ) 1.2 nm. The total transmission probability and conductance are computed from the SiNW Green function,17 which is recursively evaluated with a standard decimation technique.23 The results are averaged over 10 roughness configurations for each nanowire orientation. The Kubo-Greenwood method, on the other hand, is used to extract the intrinsic transport length scales (elastic mean free path le and charge mobility µ) from the saturation of the quantum diffusivity. Computational details can be found in refs 15 and 16. Figure 2 shows the conductance G(E) as a function of energy, for clean SiNWs with different orientations, as well as for rough SiNWs with two typical lengths L ) 20 nm and L ) 160 nm. In clean SiNWs, G(E) does not depend on length and is, as expected, an integer multiple of the conductance quantum G0 ) 2e2/h. Indeed, the onset of transport through a new subband gives rise to an additional plateau in G(E), which results in a staircase pattern. In 4148
particular, the almost 2- and 4-fold degeneracies of the conduction band minimum in [110]- and [100]-oriented SiNWs are clearly visible in Figure 2, panels a and b. The 6-fold degeneracy of the conduction band minimum in [111]oriented SiNWs is hardly noticeable however, as the valleys around k ) (0.2π/l are very shallow in such small wires (see Figure 1, panels e and c). Quantized conductance in the first subband has been reported experimentally in highly pure core-shell semiconducting nanowires.24 In the presence of disorder, the conductance becomes length-dependent, owing to the increasing contribution of backscattering and quantum interference effects. The decay of the conductance can be characterized by the mean free path le(E), whose features clearly depend on the crystalline orientation of the nanowires (see bottom panel of Figure 2). As for the holes, the most favorable orientation for long ballistic transport in the first subband is the [111] direction with le ∼ 75 nm. This is notably evidenced by the robustness of the first plateau of G(E) for lengths L from 20 to 160 nm (see Figure 2) and is consistent with the rather light hole mass in these nanowires (see Table 1). However, le(E) markedly decays at the onset of the second subband because of the reduced wave packet velocity and increase of the scattering probability. The mean free path of the holes in [110]-oriented SiNWs peaks around le ) 20 nm in the first subband, which is also supported by the persistence of a conductance plateau for 20 nm long nanowires. The mean free path of the electrons reaches le = 40 nm in [110]-oriented nanowires and le = 50 nm in [111] oriented nanowires. The backscattering of the electrons and holes is much more efficient in [100]-oriented nanowires. Indeed, conductance Nano Lett., Vol. 8, No. 12, 2008
Table 2. Room Temperature Mobility of the Electron and Holes (Carrier Concentration n or p ) 1018 cm-3), for [110]- and [111]-Oriented SiNWs with Radius R0 ) 0.5, 1, and 1.5 nm [110] [111]
Figure 3. Room-temperature electron (a) and hole (b) mobilities in rough SiNWs (R0 ) 1 nm) oriented along the [100], [110], and [111] directions, as a function of the carrier density.
quantization is lost in the first subbands even in 20 nm long SiNWs, owing to very short electron and hole mean free paths. This can be related to the much heavier hole mass and to the enhancement of interband (or intervalley) scattering within the 4-fold degenerate conduction band minimum and denser valence subbands. This translates, in particular, into an almost complete suppression of the conductance around the conduction band edge in the 160 nm long nanowires. As a matter of fact, G(E) is found to decay exponentially in long enough nanowires, pinpointing the emergence of a strong Anderson localization regime, as expected from the conventional scaling theory of localization.25 This is especially pronounced close to the conduction and valence band edges, where the mean free path is the shortest and the carriers are easily trapped by the protrusions along the nanowire.15 A key transport quantity for assessing device performances is the charge carrier mobility µ. By definition, µ(E) ) σ(E)/ [n(E)e], where σ(E) is the conductivity, n(E) is the carrier density, and e is the elementary charge.16,17 Figure 3 shows the room temperature mobility as a function of the electron density n and hole density p, computed from the intrinsic Kubo conductivity for the three nanowire orientations. As expected, the mobility is almost constant for Fermi energies in the band gap, i.e., in the “low” carrier density limit. The [110]-oriented nanowires exhibit the largest electron mobilities, µe = 315 cm2 V-1 s-1 up to n = 1019 cm-3. The mobility further increases at larger n, as the Fermi level moves deeper in the conduction band where the mean free paths are longer, and peaks around µemax ) 1040 cm2 V-1 s-1 at n ) 2.3 × 1020 cm-3. In contrast, the [100]- and [111]-oriented SiNWs show poor electron mobilities. The almost 4- and 6-fold valley degeneracies indeed give rise to intervalley scattering and to a large increase of the DoS around the conduction band edge (especially in [111]-oriented SiNWs where the electrons are quite heavy). This tends to pin the Fermi energy near the bottom of the conduction band where the mean free paths are the shortest and explains why the mobility peak evidenced in [110]-oriented SiNWs is shifted to much higher carrier densities (only a shoulder being visible in Figure 3b). As for the holes, the [111]-oriented SiNWs exhibit the Nano Lett., Vol. 8, No. 12, 2008
R0 (nm)
µh (cm2 V-1 s-1)
µe (cm2 V-1 s-1)
0.5 1.0 1.5 0.5 1.0 1.5
78 264 2275 81 480 4179
102 315 1760 34 49 467
highest mobility, µh = 480 cm2 V-1 s-1 for pj1019 cm-3 and up to µhmax ) 1340 cm2 V-1 s-1 at p ) 1.5 × 1020 cm-3. The mobility is actually enhanced by the large separation between the highest two valence subbands, which prevents interband scattering and the occupation of lower lying states with shorter mean free paths. The hole mobility is also significant in [110]-oriented SiNWs (around half the [111] value at low carrier densities), while it is much smaller in [100]-oriented nanowires. This is consistent with the trends evidenced in Figure 2 and with the degeneracies of the highest valence subband (see Table 1). It is important to discuss how the effects of surface roughness, which appear significant in small SiNWs, scale with the nanowire diameter. We have therefore computed the mobilities in [110]- and [111]-oriented SiNWs with three different radii R0 ) 0.5, 1.0, and 1.5 nm. They are reported in Table 2. Although surface roughness is efficiently limiting transport in small SiNWs, its impact rapidly decreases with increasing nanowire diameter. At R0 ) 1.5 nm the calculated mobility is already in the order of the most performant undoped SiNWs reported so far,4 suggesting that other scattering mechanisms, such as chemical impurities and electron-phonon coupling,26 might be predominant. The trends seen in Figure 3 hold true also at R0 ) 1.5 nm, with a [111] electron mobility that is significantly lower than the [110] electron mobility. Even though we have focused on surface roughness, we expect that the main trends evidenced in this work also hold for other kinds of disorders (trapped surface charges or defects), since they follow from the anisotropy of the band structure of silicon more than from the detailed scattering properties of the disorder. A strong orientational dependence of the mobility can, therefore, be expected in the quantum regime where only one or a few subbands are occupied. In this respect, the splitting between the 2-fold degenerate conduction band valley at Γ and the heavier, 4-fold degenerate valley at k ) (0.8π/l in [110]-oriented SiNWs remains greater than kT at room temperature (25 meV) up to R0 = 3 nm.14 Likewise, the splitting between the highest two valence subbands in [111]-oriented SiNWs is also greater than kT up to R0 = 3 nm. Quantum effects will likely average out beyond this radius at room temperature. However, strain engineering might further be used to split the electron and hole subbands and alter their effective masses, as shown for example in ref 27. In conclusion, the effects of surface roughness on the transport properties of SiNWs oriented along the [100], [110], and [111] directions have been analyzed with the Landauer4149
Bu¨ttiker and a real space Kubo-Greenwood methods. The backscattering strength is strongly dependent on the nanowire orientation in small SiNWs, which can be traced back to the anisotropy of the band structure of silicon. This dependence is enhanced in the quantum regime where only a few electron or hole subbands are occupied. The electrons are less sensitive to surface roughness in [110]-oriented nanowires, while the holes show best transport properties along [111]-oriented SiNWs. These conclusions likely hold for other kinds of disorder (e.g., trapped charges), as they result from the anisotropy of silicon more than from the details of the scattering mechanism. Acknowledgment. This work was supported by the EU Project No. 015783 NODE, by the French National Research Agency (ANR) project “PREEANS” and “QUANTAMONDE”, and by the CEA ChimTronique program. The calculations were performed at the CCRT supercomputing center.
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NL801128F
Nano Lett., Vol. 8, No. 12, 2008
