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Text: o  corresponds to the frequency of the interlayer breathing mode in 2 layer BP , that is 75.0 cm -1 in our calculation . The curve of equation ( 1 ) shown as the red line in Figure    3(b ) fits both the LDA and experimentally measured frequency well . The good fitting suggests that the interlayer interactions are dominated by the interactions between nearest - neighboring layers . Similar to few layer graphene and MoS 2 , 8,9 adsorbates or substrate effects seem to have negligible influence on the frequencies of the interlayer breathing mode . In Fig . 3(b ) , where  is the mass per unit area of the few layer BP , c is the speed of light in cm / s , and k z is the out of plane interlayer force constants . The thus derived out - of - plane force constant k z ( 14.1×10 19 N / m 3 ) is related with the elastic modulus of few - layer BP by C 33 = k z l with l being the distance between the centers of mass for adjacent puckered layers . As a result , the calculated elastic modulus of C 33 is around 73.9 GPa , consistent with calculated values for the bulk . 26 The interlayer out - of - plane force constant k z is significantly larger than those derived in the same way for other materials ( k z is 6.11×10 19 , 9.6×10 19 and 9.3×10 19 N / m 3 for Bi 2 Se 3 , 27 graphite 28 and MoS 2 , 8 respectively ) . Similarly , if we fit the linear chain model to our experimental data , we obtain k z of 12.3×10 19 N / m 3 , larger than k z obtained by fitting the experimental frequencies of other materials . 8,27,29 The larger force constants contribute to a larger o  factor in equation ( 1 ) , and thus a larger thickness dependence for the A 3 g frequency . Specifically , the A 3 g frequency changes 54 cm -1 from 2 - layer to 8layer BP , two times more than that in the transition metal dichalcogenides such as few layer MoS 2 ( 31 cm -1 from 2 - layer to 8 - layer ) and WSe 2 ( 20 cm -1 from 2 - layer to 8 - layer ) . 8 The above analysis can also be applied to the shear modes B 3 3 g and B 2 2 g , and we can get a shear elastic modulus of 8.9 GPa and 20.4 GPa along the armchair and zigzag directions , respectively . The difference in the shear elastic moduli along the two in - plane directions reflects the strong anisotropic elastic properties of BP . 13 The larger force constant in BP suggests that the interlayer bonding may be different from that in the other well - studied layered materials like MoS 2 and Graphite .<|eot_id|><|start_header_id|>assistant<|end_header_id|>

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