Letter pubs.acs.org/NanoLett
Effects of Strain on the Carrier Mobility in Silicon Nanowires Yann-Michel Niquet,*,† Christophe Delerue,*,‡ and Christophe Krzeminski‡ †
L_Sim, SP2M, UMR-E CEA/UJF-Grenoble 1, Institut Nanosciences et Cryogénie (INAC), Grenoble, France Department ISEN, Institut d'Electronique de Microélectronique et de Nanotechnologie (IEMN), UMR CNRS 8520, Lille, France
‡
S Supporting Information *
ABSTRACT: We investigate electron and hole mobilities in strained silicon nanowires (Si NWs) within an atomistic tight-binding framework. We show that the carrier mobilities in Si NWs are very responsive to strain and can be enhanced or reduced by a factor >2 (up to 5×) for moderate strains in the ±2% range. The effects of strain on the transport properties are, however, very dependent on the orientation of the nanowires. Stretched ⟨100⟩ Si NWs are found to be the best compromise for the transport of both electrons and holes in ≈10 nm diameter Si NWs. Our results demonstrate that strain engineering can be used as a very efficient booster for NW technologies and that due care must be given to process-induced strains in NW devices to achieve reproducible performances. KEYWORDS: Transport, Silicon, Nanowires, Strains, Phonons, Modeling
M
processing steps (e.g., thermal stress) and can be stretched on purpose in flexible electronics. In a 1D geometry, axial stress in the GPa range translates into forces in the μN range, making NWs very sensitive to strains and offering unprecedented opportunities to engineer strains in semiconductor devices. In this context, quantitative predictions of the strain-induced variations of the mobility in Si NWs are highly desirable but still lacking. The problem is indeed very complex, especially for the electron−phonon scattering mechanism. In ultimate NWs, the carriers are coupled to many phonon modes,17,19,20,22 and strains do not only alter the electron band structure but also the phonons and the electron−phonon interaction. In this letter, we present fully atomistic calculations of the phonon-limited mobility of electrons and holes in strained Si NWs with different orientations and diameters d up to 8 nm. Combining tight-binding (TB) and valence-force field (VFF) models, we describe the effects of strains and confinement on both electrons and phonons. Intervalleys and spin−orbit couplings are included in the calculation. We consider uniaxial strains along the NW axis, as well as oxidation-induced strains. We demonstrate that, depending on the NW orientation, the mobility can be enhanced or reduced by a factor >2 for moderate strain, and we discuss the physics behind these trends. We conclude that strain engineering is a very efficient booster of the mobility in ultimate NWs and that ⟨100⟩ Si NWs under tensile strain provide the best compromise for the transport of both electrons and holes around d = 10 nm. We focus on phonon scattering because it is a strongly limiting
ultigate or gate-all-around silicon nanowire (Si NW) transistors are promising candidates for the next generation of electronic devices because they provide excellent gate control and are less sensitive to short channel effects.1 They are also considered as attractive building blocks for flexible electronics.2−5 Scaling down NW technologies to ultimate dimensions, transistors with diameters between 3 and 10 nm have been recently fabricated and characterized.6−10 The performances of these devices are, however, strongly dependent on the size and orientation of the NWs, due to the interplay between transport and quantum confinement.11−22 In general, the carrier mobilities in sub-10-nm Si NWs are reduced with respect to the bulk, in particular because lateral confinement enhances scattering. Therefore, innovative strategies are needed to boost the transport properties of ultimate Si NWs and meet the requirements of the International Technology Roadmap for Semiconductors (ITRS).23 A promising solution is to use mechanical strain to engineer the band structure and reduce the transport effective masses. Strain engineering is actually widely used in advanced logic devices and has contributed to extend two-dimensional (2D) Si complementary metal−oxide−semiconductor (CMOS) technologies to at least the 32 nm node.24,25 Recent computational studies of the band structure of Si NWs25−31 have suggested that strain could also improve the performance of onedimensional (1D) NW devices but did not provide quantitative data about carrier mobilities. From the experimental point of view, the enhancement of the mobility (or the transconductance) under strain has been reported in Si NWs with diameters between 10 and 150 nm4,32−34 and even in quantumconfined sub-10-nm NWs.35,36 As a matter of fact, Si NWs can withstand large nonintentional strains due to oxidation or other © 2012 American Chemical Society
Received: March 21, 2012 Revised: May 30, 2012 Published: June 13, 2012 3545
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mechanism in Si NWs, and because it is intrinsic to the material, at variance with surface roughness and impurity scattering which are much more dependent on the device geometry and environment. However, the conclusions drawn in this letter also hold for other scattering mechanisms, as evidenced in the Supporting Information. The electronic structure of the Si NWs is computed with the TB model of ref 37, which reproduces the main band gaps, effective masses, and deformation potentials38 of silicon. The dangling bonds at the surface of the Si NWs are saturated with hydrogen atoms. The phonons are computed with the VFF model of ref 39, including anharmonic third- and fourth-order terms (which actually give harmonic contributions to the dynamical matrix in strained Si NWs). The electron−phonon scattering rates are calculated from the derivatives of the TB Hamiltonian with respect to the atomic positions using the Fermi golden rule, for all electronic bands within at least 200 meV of the band edges, and for all phonon modes. The linearized Boltzmann transport equation is then solved exactly for the low-field mobility. Brillouin zone integrations are performed on a regular grid of 1024 k-points for ⟨110⟩ Si NWs, 724 k-points for ⟨100⟩ Si NWs, and 420 k-points for ⟨111⟩ Si NWs. Details can be found in ref 19. Within such an atomistic description, we are able to reproduce the nonlinearities of the band structure of silicon under strain, to catch the interplay between confinement and strains, and to track the effects of strains on the electron−phonon interaction, without making any extra assumptions about the electron−phonon coupling Hamiltonian as, for example, in effective mass descriptions.22 The mobilities discussed in this paper have been computed at room temperature and low carrier densities. To begin with, we investigate the transport properties of uniaxially strained Si NWs. Si NWs can indeed be elongated by at least 2% in, for example, wafer bending experiments. Axial (though nonuniform) compressive strains can also be achieved in NWs etched in a strained Si layer. In the following, the unit cell of the Si NW is stretched along the nanowire axis by an amount ε∥, and then the atoms within the unit cell are relaxed with the same VFF model as for phonons. As expected from elasticity theory (see the Supporting Information), the resulting strain ε⊥ perpendicular to the nanowire is fairly homogeneous and proportional to ε∥ (but opposite in sign). For convenience, we define the uniaxial component of the strain εuni = ε∥ − ε⊥. We will later show that the mobility mostly depends on εuni rather than on ε∥ and ε⊥ separately. The electron and hole mobilities are plotted as a function of the uniaxial strain εuni in stretched Si NWs with diameter d = 8 nm in Figure 1 (εuni = 1.278ε∥ for ⟨100⟩ Si NWs, εuni = 1.212ε∥ for ⟨110⟩ Si NWs, and εuni = 1.180ε∥ for ⟨111⟩ Si NWs). The mobility is strongly dependent on the strain, whatever the carriers and NW orientation, and can vary by up to a factor >2 (up to 5×) in the investigated range ε∥ ∈ [−2%, 2.5%]. We now discuss the physics behind these trends, starting with electrons. The transport properties of Si NWs are highly anisotropic.12−14,21,22 In unstrained nanowires, the ⟨100⟩ Si NWs show the best electron mobilities, closely followed by the ⟨110⟩ Si NWs, while the ⟨111⟩ Si NWs lag well behind. The calculated mobilities are, nonetheless, much lower than in bulk (1400 cm2/V·s22), whatever the orientation, due to the enhancement of the electron−phonon interaction in Si NWs.11,15,19,21,22,40,41 As discussed in refs 14 and 22, this hierarchy mostly results from band structure effects: In [100] Si
Figure 1. Phonon-limited mobility of (a) electrons and (b) holes as a function of the uniaxial strain εuni = ε∥ − ε⊥ in ⟨100⟩, ⟨110⟩, and ⟨111⟩ Si NWs with the diameter d = 8 nm. The open symbols are data for Si NWs compressed by a SiO2 shell with fixed (plain symbols) and free (crossed symbols) ends (see text). The horizontal dash−dotted lines are the bulk mobilities computed with the TB model.
NWs, the confinement splits the 6-fold degenerate valleys of bulk Si into 4-fold degenerate Δy,z valleys at Γ with low transport mass m* ≃ m*t = 0.19m0, and 2-fold degenerate Δx valleys 25 meV above (near k = ± 0.3π/a) with mass m* ≃ m*t = 0.92m0. The situation is somewhat less favorable in [110] Si NWs at this diameter, with only 2-fold degenerate Δz valleys at Γ with mass m* ≃ m*t, and 4-fold degenerate Δx,y valleys 15 meV above (near k = ± 0.85(2)1/2π/a) with mass m* ≃ 0.55m0. This leads to lower average carrier velocities and stronger scattering. Last, the six conduction band valleys remain almost degenerate in ⟨111⟩ Si NWs, with an intermediate mass m* ≃ 0.43m0; this is however the worst scenario for the mobility which is expected to scale as m*−3/2 in the single subband limit.21 The differences between the various orientations diminish with increasing NW diameter as the transport becomes strongly multisubband and is not, therefore, mostly conditioned by the longitudinal transport mass any more. The above features can be either strengthened or reversed by the strains. In ⟨100⟩ and ⟨110⟩ Si NWs, a positive (tensile) strain lowers the light valleys at Γ with respect to the heavier, off-center valleys (see Figures 2−4). This increases the proportion of fast electrons in the light valleys and suppresses intervalley scattering, which enhances the mobility. The electron mobility near ε∥ ∼ 1% is (incidentally) comparable 3546
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Figure 4. (a) Splitting between the 2-fold degenerate Δz valleys at Γ and the 4-fold degenerate Δx,y valleys near k = ± 0.85(21/2π)/(a) as a function of the longitudinal strain ε∥ in a stretched [110] Si NW with the diameter d = 8 nm. The dashed line is the result of the deformation potential theory within the two-band k·p model (see the Supporting Information). (b) Transport effective mass of the Δz and Δx,y valleys as a function of ε∥. The inset depicts the behavior of the Δz valleys under shear strain.
Figure 2. Conduction band structure of a stretched ⟨110⟩ Si NW with the diameter d = 8 nm, at two longitudinal strains (a) ε∥ = −1% and (b) ε∥ = +1%. a = 5.431 Å is the lattice parameter of silicon.
valleys when ε∥ ≲ −0.25%. The resulting increase of the average electron mass and the strong enhancement of the scattering from the light to the heavy valleys come, of course, with a drastic reduction of the mobility. Simple models reproducing the behavior of the different valleys under strain are discussed in the Supporting Information. The strain does not have sizable effects on the transport mass of the different valleys in ⟨100⟩ Si NWs. It has, however, significant effects on the transport mass of the Δz valleys at Γ in [110] Si NWs, as shown in Figure 4b. The nonlinear behavior of the energy and mass of the Δz valleys can be ascribed to the effects42−44 of the shear conduction band deformation potential Ξ′u, which can only be accounted for in atomistic or two-band k·p models of silicon (see the Supporting Information). This deformation potential controls the opening of the gap between the two conduction bands (actually the two opposite valleys) crossing at X, Y, and Z points in bulk silicon. Namely, the shear component εxy of an uniaxial [110] strain opens the gap at Z which increases the longitudinal confinement mass m*t (see inset of Figure 4b). It also breaks the symmetry between the
Figure 3. Splitting between the 4-fold degenerate Δy,z valleys at Γ and the 2-fold degenerate Δx valleys near k = ± 0.3 π/a as a function of the longitudinal strain ε∥ in a stretched [100] Si NW with the diameter d = 8 nm. The dashed line is the result of the deformation potential theory (see the Supporting Information).
to the bulk mobility. Conversely, a compressive strain lowers the heavy off-center valleys, which become the ground-state 3547
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transverse mass axes of the Δz valleys, which decreases the transport effective mass for positive strains (Figure 4b) but increases the transverse confinement mass (see refs 42 and 43 and Figure 2 of ref 45). As a consequence, the splittings between the subbands of the Δz valleys get smaller at large positive strain (clearly visible in Figure 2b), which enhances intersubband scattering and results in a loss of mobility when ε∥ > 1%, despite very low transport masses. The importance of Ξ′u on the band structure and electron−phonon coupling in strained Si NWs has been overlooked in many previous works, some based on TB models which do not actually reproduce this deformation potential.27 The shear strains are also responsible for the behavior of the mobility in ⟨111⟩ Si NWs. The opening of the gap at X, Y, and Z by shear strains indeed flattens the conduction band and degrades the mobility for both positive and negative ε∥, even though the ⟨111⟩ Si NWs were already the worst candidates for n-type channel devices in the unstrained limit. In the case of holes, the mobility in unstrained Si NWs is much larger in the ⟨110⟩ and ⟨111⟩ orientations than in the ⟨100⟩ orientation. Indeed, as discussed in refs 14 and 22, the confinement tends to promote light hole bands in ⟨110⟩ and ⟨111⟩ Si NWs. These light hole bands have a main |3/2, ±1/2⟩ character (mixed with |1/2, ±1/2⟩) and are predominantly formed by bonding combinations of p and dyz, dxz, and dxy orbitals oriented along the NW axis.46 They are therefore very dispersive along that axis but are much less confined than the bonding |3/2, ±3/2⟩ combinations of p- and d-orbitals oriented perpendicular to the nanowire. The mass of the highest valence subband is as low as ≃0.13m0 in ⟨110⟩ Si NWs and ≃0.18m0 in ⟨111⟩ Si NWs, while the splitting with the next hole subband is still >10 meV at d = 8 nm. Like for electrons, this increases the average velocity of the carriers, reduces intersubband scattering, and therefore enhances the mobility. The hole mobility in unstrained ⟨111⟩ Si NWs with diameter d = 8 nm is even greater than the bulk mobility (678 cm2/V·s within this TB model22). On the contrary, the topmost valence bands in ⟨100⟩ Si NWs are almost degenerate and show heavy masses (m* ≃ m0), which is detrimental to the mobility. Indeed, at variance with ⟨110⟩ and ⟨111⟩ Si NWs, quantum confinement hardly splits |3/2, ±3/2⟩ from |3/2, ±1/2⟩ holes at k = 0. The highest |3/2, ±3/2⟩ and |3/2, ±1/2⟩ hole bands are therefore almost degenerate in square and cylindrical ⟨100⟩ Si NWs, and both show heavy masses because they are repelled by lower-lying subbands. In a k·p framework, this would be related to the small γ2 (little splitting at k = 0) but large γ3 (strong mixing at k ≠ 0) of silicon. In the tight-binding picture, this results from the very anisotropic coupling between the p- and d-orbitals in light covalent materials: this coupling is indeed much stronger along the bonds, none of which has a privileged orientation with respect to the ⟨100⟩ axes. Compressive strain strengthens the light hole character of the highest valence subbands whatever the NW orientation, pushing heavy hole subbands down and eventually ruling them out from transport. This explains the huge increase of the mobility visible in Figure 1b (the mobility can be up to 5× larger than in bulk!). The hole mass is little affected by strains in ⟨110⟩ and ⟨111⟩ Si NWs (Figure 6) but decreases steadily in the ⟨100⟩ orientation as the repulsion between the |3/2, ±1/2⟩ bands going up and the |3/2, ±3/2⟩ bands going down diminishes. On the contrary, tensile strain brings back up heavy hole subbands, as illustrated in Figures 5 and 6, which degrades
Figure 5. Valence band structure of a stretched ⟨110⟩ Si NW with the diameter d = 8 nm, at two longitudinal strains (a) ε∥ = −1% and (b) ε∥ = +1%.
Figure 6. Transport effective mass of the highest valence subband as a function of the longitudinal strain ε∥ in stretched ⟨100⟩, ⟨110⟩, and ⟨111⟩ Si NWs with the diameter d = 8 nm.
the mobility. The hole mass abruptly jumps at the crossing between the heavy and the light hole subbands in ⟨110⟩ and ⟨111⟩ Si NWs and then decreases. In ⟨100⟩ Si NWs, tensile strains also lift the near degeneracy between the heavy and the 3548
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Table 1. Strains (%) and Phonon-Limited Mobility (cm2/ V·s) of Electrons and Holes in ⟨100⟩, ⟨110⟩, and ⟨111⟩ Si NWs with Diameter d = 8 nma
light hole bands, which improves the hole mass and mobility for the same reasons as for compressive strains. The enhancement of the mobility is not, however, as strong for ε∥ > 0 as for ε∥ < 0, because the highest lying bands are mostly “heavy” |3/2, ±3/ 2⟩ holes in the former case (m* = 0.287 in bulk), while they are mostly “light” |3/2, ±1/2⟩ holes in the latter case (m* = 0.202 in bulk). This makes, nonetheless, stretched ⟨100⟩ Si NWs good candidates for efficient electron and hole transport. From the experimental point of view, Hashemi et al.35,36 have investigated the performances of n-type ⟨110⟩ Si NW transistors with diameters down to 8 nm and uniaxial stress up to ∼2 GPa, that is, ε∥ ∼ 1% and εuni ∼ 1.25% (see the Supporting Information). The measured 1.8−2× enhancement of the transconductance is in excellent agreement with our predictions (Figure 1a). We have also investigated the mobility in smaller Si NWs with diameter d = 4 nm and reached the same conclusions as for the 8 nm NWs (see the Supporting Information). We have, moreover, found the same quantitative trends for the impuritylimited mobility in a quantum Green's functions framework, which shows that the conclusions of the present work are neither restricted to electron−phonon scattering nor an artifact of Boltzmann’s transport equation. The dependence of the electron and hole mobility on the strain is strong in sub-10-nm Si NWs whatever their orientation. Strain engineering therefore provides real opportunities to enhance the electrical performances of Si NWs. We now discuss the case of oxidized Si NWs. The oxidation of Si NWs indeed builds up strain in the Si core. The radial stress at the Si/SiO2 interface has been estimated using a kinetic model for wet oxidation,47 allowing for plastic relaxation in the oxide.48 Starting from a 20 nm diameter Si NW, a 30 min long annealing in oxidizing ambient leads to a 8 nm diameter Si core embedded in a 6.5 nm thick SiO2 shell. This results in a radial compressive stress σ⊥ = −3 GPa at the Si/SiO2 interface, compatible with the stress levels measured using contact resistance atomic force microscopy on oxidized Si NWs.49 The strain in the Si core was next evaluated in two opposite limits (see ref 50 and Supporting Information): (i) The Si/SiO2 interface is not coherent so that the Si NW can extend within the SiO2 gangue (σ∥ = 0 ⇒ ε∥ > 0), and (ii) the Si NW is fixed at both ends, so that its length cannot vary (ε∥ = 0). Depending on the device geometry and coherence at the Si/SiO2 interface, the actual strains shall lie between these two limits. The strain, electron, and hole mobilities computed in oxidized Si NWs with the diameter d = 8 nm and different orientations are reported in Table 1, for both free and fixed ends. We recover the same qualitative trends as for stretched Si NWs: radial compression (axial tension) enhances the mobility of electrons (except in the poor ⟨111⟩ orientation) but degrades the mobility of holes (except in the ⟨100⟩ orientation). More importantly, the data of Table 1 compare quantitatively with Figure 1 (see open symbols). Indeed, all mobilities lie on the same “universal” curve when plotted as a function of the uniaxial component of the strain εuni = ε∥ − ε⊥. The hydrostatic component is not, actually, expected to have significant effect on the mobility since it does not break any symmetry. The data of Figure 1 can therefore be extrapolated to a large variety of experimental situations. In this particular case, the electron mobility in ⟨100⟩ Si NWs reaches the same limit as in stretched nanowires (≃ 1350 cm2/ V·s) whatever the boundary conditions (free or fixed ends). The electron mobility in ⟨110⟩ Si NWs is close to the
SiO2 unstrained ⟨100⟩
⟨110⟩
⟨111⟩
ε⊥ ε∥ μph μph ε⊥ ε∥ μph μph ε⊥ ε∥ μph μph
electrons holes
949 201
electrons holes
745 500
electrons holes
520 750
free ends
fixed ends
−1.66 1.28 1355 754 −1.40 0.75 1125 373 −1.31 0.58 263 131
−1.30 0 1304 450 −1.24 0 1390 271 −1.20 0 382 131
a
The NWs are either unstrained (first column) or embedded in SiO2 with free ends (second column) or fixed ends (i.e., fixed length, third column). The radial stress at the Si/SiO2 interface is σ⊥ = −3 GPa.
maximum of Figure 1a for fixed ends but beyond the optimum for free ends. As for holes, the mobility in ⟨110⟩ and ⟨111⟩ Si NWs is consistent with the large strain data of Figure 1b, while the mobility in ⟨100⟩ Si NWs is far from saturation and is therefore significantly more dependent on the boundary conditions. It is enhanced by a factor ∼2 for fixed ends and by a factor ∼4 for free ends. These calculations show that the mobility in sub-10-nm Si NWs can be very dependent on thermal or oxidation stress. The control of process-induced strain is therefore essential to achieve efficient and reliable devices. To conclude, we have computed the transport properties of strained Si NWs. The mobility in sub-10-nm Si NWs is very responsive to strain and can show variations by a factor >2 for moderate strain in the ±2% range. We have compared the case of uniaxially stretched Si NWs and the case of oxidized Si NWs radially compressed by a SiO2 shell. We have found that the mobility follows the same quantitative trends as a function of the uniaxial component of the strain εuni = ε∥ − ε⊥. The tensile strain εuni > 0 improves the mobility of electrons in ⟨100⟩ and ⟨110⟩ Si NWs, while compressive strain degrades the transport properties. The ⟨111⟩ Si NWs respond poorly to strains whatever their sign but are anyhow the worst choice for electron transport. As for holes, compressive strains εuni < 0 improve the mobility in ⟨110⟩ and ⟨111⟩ Si NWs, a behavior however opposite to electrons. Tensile as well as compressive strain, on the other hand, enhance the mobility of holes in ⟨100⟩ Si NWs, making stretched ⟨100⟩ Si NWs with diameters near 10 nm suitable for both electron and hole transport. These results show that strain engineering is an efficient way to improve the transport properties of Si NWs and that due care must be given to the process-induced strains in ultimate Si NW devices to achieve reproducible transport characteristics.
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ASSOCIATED CONTENT
S Supporting Information *
Simple models for the strains and band structure of uniaxially stretched Si NWs, for the strains in oxidized Si NWs, and additional data for 4 nm diameter Si NWs and other scattering mechanisms. This material is available free of charge via the Internet at http://pubs.acs.org. 3549
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected];
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the French National Research Agency (ANR) project Quasanova (contract ANR-10-NANO011-02). The calculations were run at the GENCI-CCRT supercomputing center.
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dx.doi.org/10.1021/nl3010995 | Nano Lett. 2012, 12, 3545−3550
