Phonon-limited mobility for electrons and holes in highly-strained silicon

npj | computational materials Article Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences https://doi.org/10.1038/s41524-024-01425-0 Phonon-limited mobility for electrons and holes in highly-strained silicon Check for updates 1234567890():,; 1234567890():,; Nicolas Roisin 1 , Guillaume Brunin2,3, Gian-Marco Rignanese Jean-Pierre Raskin1 & Samuel Poncé 2,4 2,4 , Denis Flandre1, Strain engineering is a widely used technique for enhancing the mobility of charge carriers in semiconductors, but its effect is not fully understood. In this work, we perform first-principles calculations to explore the variations of the mobility of electrons and holes in silicon upon deformation by uniaxial strain up to 2% in the [100] crystal direction. We compute the π11 and π12 electron piezoresistances based on the low-strain change of resistivity with temperature in the range 200 K to 400 K, in excellent agreement with experiment. We also predict them for holes which were only measured at room temperature. Remarkably, for electrons in the transverse direction, we predict a minimum room-temperature mobility about 1200 cm2 V−1 s−1 at 0.3% uniaxial tensile strain while we observe a monotonous increase of the longitudinal transport, reaching a value of 2200 cm2 V−1 s−1 at high strain. We confirm these findings experimentally using four-point bending measurements, establishing the reliability of our first-principles calculations. For holes, we find that the transport is almost unaffected by strain up to 0.3% uniaxial tensile strain and then rises significantly, more than doubling at 2% strain. Our findings open new perspectives to boost the mobility by applying a stress in the [100] direction. This is particularly interesting for holes for which shear strain was thought for a long time to be the only way to enhance the mobility. Semiconductors are the core of the electronics industry. In particular, silicon is the most used material due to its electronic performances and the mature CMOS technology, leading to reduced manufacturing cost1. However, the relatively low electron (resp. hole) mobility of silicon compared to other semiconductors such as GaN2,3 and InGaAs4,5 (resp. Ge6) limits its use in highspeed devices, essential for optical7,8 and microwave9,10 communications. Elastic strain is a proven technique to boost the performances of semiconductors, in particular to increase the mobility of their charge carriers at low additional cost11. Mechanical stress can be applied using different types of microfabrication schemes such as substrate-induced12,13, on-chip actuator14,15 or pressure-induced membrane16,17 methods. The deformation of the crystal structure has, in fact, several possible benefits. It can be used for tuning the band gap in a range favorable for optoelectronic applications18,19, for varying the vibrational energies that can be exploited in metrology such as Raman spectroscopy20, and for modifying the carrier mobilities which can be used for strain sensing21, or to improve the conductivity of electronic devices22. The type of deformation applied to the crystal (uniaxial, biaxial, or hydrostatic) and the direction of the applied stress are critical for tuning the semiconductor properties23,24. The mobility enhancement in strained devices is a well-known mechanism that has already been exploited to develop semiconductor strain gauges or high-mobility silicon transistors25,26. Different theoretical methods have been elaborated to predict the improvement or degradation of the mobility, starting from an empirical linear model27 valid at low strain, and later based on semi-empirical analysis using the calculated band structure of strained silicon24. The band structure and its strain dependence can be theoretically computed using semiempirical approaches based on fitting parameters such as the k ⋅ p method28,29, empirical pseudopotentials30, and the tight-binding method31,32, or with ab initio computations based on densityfunctional theory (DFT) that do not require fitting parameters33. Previous mobility models were based on the theoretical knowledge of the band structure but they relied on empirical parameters for the electronphonon scattering34,35 that is important in the transport mechanism in silicon. Only recently, first-principles calculations of the Boltzmann transport equation (BTE) have become available within EPW36,37, PERTURBO38, PHOEBE39 or ABINIT40,41, which made it possible to predict the mobility in semiconductors fully-theoretically42. These new 1 Institute of Information and Communication Technologies, Electronics and Applied Mathematics, Université catholique de Louvain, Louvain-la-Neuve, Belgium. European Theoretical Spectroscopy Facility, Institute of Condensed Matter and Nanosciences, Université catholique de Louvain, Louvain-la-Neuve, Belgium. 3 Matgenix, A6K Advanced Engineering Center, Charleroi, Belgium. 4WEL Research Institute, Wavre, Belgium. e-mail: 2 npj Computational Materials | (2024)10:242 1 Article https://doi.org/10.1038/s41524-024-01425-0 techniques can now be used to compute the mobility variation in a deformed crystal from first principles, without the empirical bias. In this work, the mobility variation for holes and electrons is computed from first principles for silicon tensely strained (up to 2%) in the [100] direction, and measured for n-doped sample, up to 1%, in the same crystallographic direction. The presented method is general and can be used for any strain type and direction for any semiconductor for which the mobilities are limited by the electron-phonon scattering. We find excellent agreement for the electron and hole mobility variations with strain and temperature compared to experimental data. In particular, resorting to first-principles calculations is found to be particularly crucial for the holes for which the proximity of the valence bands and the important modification of their shapes under strain conditions limit the accuracy that can be achieved when adopting an analytic approach. In addition, the calculations allow the study of intrinsic phenomena that cannot be achieved experimentally due to the challenge of obtaining low-doped samples43. a. kz X001 [100 ] stres s ky kx npj Computational Materials | (2024)10:242 X100 Poisson effect b. ΔEc Eg,Δ2 Eg,Δ4 Results Impact on the electronic structure A direct impact of the crystal deformation under uniaxial tensile strain is the modification of the band structure, particularly the variations of the band edges, as illustrated in Fig. 1. In silicon, the six electron valleys are located along the high-symmetry path from Γ towards X while the top of the valence bands is located at Γ. The spin-orbit (SO) valence band is slightly separated (~47 meV) from the heavy-hole (HH) and light-hole (LH) valence bands due to the spin-orbit coupling (SOC). Under uniaxial stress in the [100] direction, the degeneracy of the conduction band mi (...truncated)