Supporting Information: Strain Induced Optimization of Nanoelectromechanical Energy Harvesting and Nanopiezotronic Response in MoS 2 Monolayer Nanosheet Nityasagar Jena, Dimple, Shounak Dhananjay Behere, and Abir De Sarkar*, Institute of Nano Science and Technology (An Autonomous Institute Supported by Department of Science and Technology, Govt. of India), Habitat Centre, Phase-X, Sector-64, Mohali, Punjab-160 062, India National Institute of Technology, Sikkim, Barfung Block, Ravangla Sub-Division, South Sikkim- 737 139, India *Address correspondence to Abir De Sarkar: abir@inst.ac.in, abirdesarkar@gmail.com Supplementary Notes: Note: 1 The polarization dependent linear response properties have been calculated using density functional perturbation theory (DFPT), as implemented in VASP, which calculates piezoelectric stress coefficients, = ( / ) E,T = -( / ) ε,t and the same in reduced Voigt notation, = ( / ) E,T = -( / ) ε,t, where, i = {x, y, z} ; j = {xx, yy, zz, xy, yz, zx} or, in index representation, i = {1, 2, 3} ; j = {1, 2, 3, 4, 5, 6}. Due to the point group symmetry in pristine ML-MoS 2 nanosheet, the non-vanishing piezoelectric stress tensor components occur at, and with the same magnitude. Therefore, for the sake of completeness, the variation in all these non-vanishing piezoelectric stress tensors, with different strain modes have been studied and plotted in Figure S1. is found to be most sensitive to strain as it shows the largest variation with strain. As discussed in the main text, ( =- ) uniquely defines the piezoelectric strength of this system. So, in order to quantify the variation in piezoelectric strength with strain, we have focused on the changes in tensor component (where = - ) for all modes of strain studied in this work. Likewise, for symmetry reasons, the piezoelectric strain coefficient, which uniquely defines the piezoelectric strength of ML-MoS 2 is related to the piezoelectric stress tensor S1
component, as, where and are in-plane elastic stiffness tensors and discussed in Note 3. Since piezoelectric strain coefficient, directly relates to piezoelectric stress coefficient, the increase in the magnitude of or ( since ) corresponds to an increase in and hence, the piezoelectric strength in ML-MoS 2. Significant enhancement in piezoelectric stress coefficients for uniaxial tensile strain along zig-zag (zz) direction and shear (S 1 ) mode of strain has been observed. While for biaxial strain, all the non-vanishing tensor components are found to be of same magnitude for any magnitude of strain, as the strain is uniformly applied along both the lattice directions. Figure S1. Piezoelectric stress tensor coefficients, to strain. (a)-(d) Variation in in-plane S2
piezoelectric stress coefficients,, and (non-vanishing tensor components) to different strain modes. Note: 2 The Born-effective charges on ionic species have been correlated with piezoelectric strain coefficient to account for its variation with strain. For simplicity, only the uniaxial strain along the zig-zag (zz) direction has been presented here. As can be seen in Figure S2(a) and (b), piezoelectric strain coefficient, reaches high in magnitude when the effective dynamical charges on Mo- and S-atoms decreases along the zig-zag direction and increases along the arm-chair direction. Figure S2. Correspondence between piezoelectric strain coefficient, and in-plane effective dynamical charges, on ionic species. (a) Correlation between piezoelectric strain coefficient, and in-plane effective dynamical charges, on ionic species, Mo & S along the zig-zag lattice direction for uniaxial strain applied along the zig-zag (zz) direction. (b) The same along armchair (ac) direction. Note: 3 In-plane elastic stiffness constants, and have been calculated to obtain the Young s S3
moduli and Poisson s ratios and in turn, to determine the mechanical stiffness and elastic deformation limit under different strain modes. It also helps to compute the piezoelectric strain tensor, which relates to piezoelectric stress coefficient, via the relation,. Figure S3. In-plane elastic stiffness coefficients to strain. (a)-(d) Variation in in-plane elastic stiffness constants, and to different strain modes. zz: zigzag direction, ac: armchair direction. Note: 4 The variation in charge carrier mobilities with lattice deformation has been calculated using S4
acoustic phonon limited deformation potential and effective mass approximation. Effective mass of carriers (electron and holes) has been computed at the band edges (electrons: at conduction band minimum, CBM, and holes: at valence band maximum, VBM) in response to uniaxial strain, as shown in Figure S4. The drastic drop in carrier effective mass at certain levels of strain represents band gap switching from direct to indirect and the same effect has been observed in carrier mobilities as well, which has been discussed in the main text. The positive and negative numbers along y-axis represent electron and hole effective masses respectively. Figure S4. Effective mass of charge carriers at band edges (CBM, VBM). (a)-(b)variation in effective mass of charge carriers at VBM (valence band maximum) and CBM (conduction band minimum) with different modes and magnitude of uniaxial strain applied along ZZ and AC directions, respectively. (Note: positive and negative numbers along the x-axis represent tensile and compressive strain, whereas positive and negative numbers along the y-axis correspond to effective mass of electrons ( ) and holes ( ), respectively in units of rest mass of electron, ) Note: 5 The Figure S5 below has been included here for completeness. Biaxial strain subjects both the lattice directions, namely, zig-zag and arm-chair, to the same magnitude of strain and therefore, S5
the effective dynamical charges on the ionic species along both the lattice directions, zig-zag and arm-chair, come out to be equal. Figure S5. Variation in Born effective charges, on each Mo- and S-ions along the zz (i.e., zigzag) and ac (i.e., arm-chair) directions with biaxial tensile and compressive strain. S6