Advances in Condensed Matter Physics

Volume 2015 (2015), Article ID 453125, 8 pages

http://dx.doi.org/10.1155/2015/453125

*Ab Initio* Study of Strain Effects on the Quasiparticle Bands and Effective Masses in Silicon

^{1}Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany^{2}Department Physik, Universität Paderborn, 33095 Paderborn, Germany

Received 1 December 2014; Accepted 16 January 2015

Academic Editor: Da-Ren Hang

Copyright © 2015 Mohammed Bouhassoune and Arno Schindlmayr. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Using *ab initio* computational methods, we study the structural and electronic properties of
strained silicon, which has emerged as a promising technology to improve the performance of silicon-based
metal-oxide-semiconductor field-effect transistors. In particular, higher electron mobilities
are observed in *n*-doped samples with monoclinic strain along the [110] direction, and experimental
evidence relates this to changes in the effective mass as well as the scattering rates. To assess the
relative importance of these two factors, we combine density-functional theory in the local-density
approximation with the approximation for the electronic self-energy and investigate the effect
of uniaxial and biaxial strains along the [110] direction on the structural and electronic properties of
Si. Longitudinal and transverse components of the electron effective mass as a function of the strain
are derived from fits to the quasiparticle band structure and a diagonalization of the full effective-mass
tensor. The changes in the effective masses and the energy splitting of the conduction-band
valleys for uniaxial and biaxial strains as well as their impact on the electron mobility are analyzed.
The self-energy corrections within lead to band gaps in excellent agreement with experimental
measurements and slightly larger effective masses than in the local-density approximation.

#### 1. Introduction

Silicon retains its place as the most prominent material used in technological applications such as metal-oxide-semiconductor field-effect transistor (MOSFET) based devices. This success stems from the sustained increase in performance that could be achieved, over many years, by continuous geometric downscaling. Nowadays, however, as devices approach their physical and geometrical limits, other solutions must be found to uphold the trend of Moore’s law. Strain engineering has become one of the promising scaling vectors in this context, because it can lead to higher carrier mobility and faster switching times than in conventional devices based on unstrained silicon while remaining fully compatible with existing manufacturing technology [1, 2]. Strained silicon is already used in mass-scale industrial production since the 90 nm node, together with new dielectric materials and other boost factors. The most established method to obtain strained layers of silicon is to grow a silicon-germanium film on top of a standard wafer, which then serves as an atomic template for the subsequent epitaxial growth of silicon layers [3]. These adopt the larger lateral lattice constant of the silicon-germanium substrate and therefore exhibit biaxial tensile strain. With this growth process, around 1% strain in the Si layer can be achieved, which enhances the electron mobility by a factor of 1.8–2.0 [2, 4–6]. The enhancement of the hole mobility is negligible [6], on the other hand, as this would require significantly higher tensile strain, which can not easily be realized by epitaxy. Different techniques have been pursued in this situation until uniaxial strain was introduced into the MOSFET channel [7–10], which promises to enhance the electron and hole mobility even at smaller strain than 1% [10, 11]. In order to implement uniaxial strain, stress liner techniques are commonly used, where capping layers, usually of silicon nitride, are grown on top of the transistor to produce compressive or tensile strain depending on the deposition conditions. Heteroepitaxial strain techniques, where epitaxial growth of SiGe and SiC in etched recesses near the source and drain generates uniaxial strain due to the lattice mismatch with the silicon channel, may also be employed. In this way, a hole mobility enhancement by a factor of 2 has been observed for -type MOSFETs with 2 GPa uniaxial stress [12, 13].

Although early theoretical studies [14–18] already provided a qualitative understanding of strain effects on the electronic structure of Si, actual progress was so far mainly driven by experiments that measure the impact of a particular strain configuration on the charge carrier mobility, for instance, through changes in the resistivity. Several parameters, in fact, independently influence the carrier mobility in MOSFETs: first, the effective masses are inversely proportional to the curvature of the energy bands. If strain modifies the energy bands, this will lead to changes in the effective masses. Second, the scattering rates of the charge carriers can be reduced by a partial lifting of band-edge degeneracies in strained samples with reduced symmetry. Indeed, for certain strain configurations, the number of available scattering channels for electrons residing in the lowest conduction-band valleys naturally decreases. In addition, surface roughness and other process-induced factors may influence the mobility in actual devices. Therefore, it is difficult to assess the quantitative role of the individual contributing factors on the basis of experimental measurements alone. To answer this question, we conducted a combined theoretical and experimental study of biaxial tensile strain in the (001) plane of -type Si [19], corresponding to a tetragonal distortion of the unit cell. Our measurements indicate no change in the electron effective mass, obtained from the temperature dependence of the Shubnikov-de Haas oscillation amplitudes, which is consistent with our* ab initio* band-structure calculations that show no warping of the lowest conduction band up to 1% applied strain. The observed enhancement of the mobility is hence attributed mainly to the partial splitting of the originally sixfold degenerate conduction-band valleys, which reduces the scattering rate. In the case of biaxial tensile strain in the (111) plane, corresponding to a trigonal distortion of the unit cell, the six conduction-band valleys remain degenerate, so that the scattering channels are unaffected. As all components of the effective mass tensor furthermore increase with strain [20], the trigonal distortion may even reduce the electron mobility. On the other hand, for monoclinic deformations along the direction, experimental evidence suggests that both the effective mass of the charge carriers and the scattering rate contribute to the observed high electron mobility in -doped samples. It was also reported that uniaxial strain induces a much larger mobility enhancement for both electrons and holes [21–23] than wafer-based biaxial strain [24].

Theoretical simulations of the carrier mobility require an accurate knowledge of the electronic band structure under strain, but the vast majority of band-structure calculations dealing with strained Si rely on empirical approaches that require input parameters from experiments or, like density-functional theory (DFT) within the local-density approximation (LDA), exhibit systematic errors. To circumvent these problems, here we instead employ* ab initio* computational methods that combine DFT-LDA and the approximation for the electronic self-energy in order to obtain highly accurate quantitative results for the band structure, the variation of the effective masses, and the valley splitting as functions of the applied strain. We focus on uniaxial and biaxial strains, because this monoclinic deformation is the most promising for achieving high charge carrier mobilities.

This paper is organized as follows. In Section 2 we give an overview of our computational method and discuss the structural parameters of silicon under a monoclinic deformation along the direction. In Section 3 we then present our results for the electronic structure and compare the effects of tensile biaxial and uniaxial strains on the electron mobility by analyzing the changes in the effective masses and the splitting of the conduction-band valleys. We finally summarize our conclusions in Section 4.

#### 2. Computational Method

We start by performing a series of* ab initio* calculations using density-functional theory [25] within the local-density approximation [26] to determine the structural properties of silicon under uniaxial and biaxial strains in the direction. While the ground-state total energy for a given atomic configuration is accurately described by the DFT-LDA approach, the Kohn-Sham eigenvalue dispersion differs systematically from the true quasiparticle band structure. In particular, band gaps are severely underestimated, but the band curvature and hence the effective masses also show deviations. For this reason, we employ many-body perturbation theory and the approximation for the electronic self-energy [27], which yields quasiparticle band structures in excellent agreement with experimental photoemission data. The derived effective masses also agree well with experimentally measured values, not only for solids like silicon [19], but also for very different materials, such as organic semiconductors [28, 29].

Our calculations use nonlocal norm-conserving pseudopotentials and a plane-wave basis set with a cutoff energy of 20 Ry. We choose the parametrization of Perdew and Zunger [30] for the exchange-correlation functional. The structural optimization within DFT-LDA is performed with a -point set in the full Brillouin zone, whereas a sampling of points suffices for the quasiparticle shifts in the approximation. The self-energy is constructed with 100 unoccupied bands in the standard perturbative way from the Kohn-Sham orbitals, and we evaluate the dynamic screening function in the random-phase approximation without recourse to plasmon-pole models [31, 32].

Crystal deformations are described by the strain tensor , which transforms the primitive lattice vectors , , and according to , where denotes the identity matrix. For monoclinic deformations along the direction, the strain tensor takes the following form: where is the strain component in the (110) plane and is the out-of-plane strain parameter along the direction. Under equilibrium conditions silicon crystallizes in the diamond structure with two atoms per unit cell and 24 symmorphic point symmetries. The monoclinic distortion reduces the number of symmetry operations to four. As a consequence, the relative positions of the two atoms inside the unit cell are no longer fixed by symmetry constraints, and internal displacements will occur. In our calculations we take this internal relaxation fully into account and determine the optimal sublattice spacing by minimizing the total energy separately for each configuration. The internal displacement parameter derived from our results depends linearly on the strain, but the deviation from the limiting value at zero strain is small for experimentally achievable deformations.

The elastic moduli as well as related quantities can be determined from the change in total energy as a function of the applied strain. A contour plot of the total energy for monoclinic deformations is shown in Figure 1. To simulate uniaxial (orange dots) and biaxial (red dots) strains, we fix one of the two strain parameters and determine the other by a constrained energy minimization: respectively. For small deformations the relation between the two components of the strain is linear and is given by the Poisson ratio . The values for biaxial and uniaxial strains obtained here directly from first principles are consistent with those derived from the elastic constants calculated in [20] and with experiments.