Dear all,
1) I am trying to calculate the elastic constants (C2D) of a 2D material by the fitting of a quadratic polynomial to a curve of energy difference (Eequ - Estr) versus strain. As suggested in several studies (E.g. Article First-principles calculations of elastic constants for epsil...
), if I fit my curve to y = ax2 + bx + c, my elastic constant along x should be 2a/V0 (through the double differentiation of y w.r.to x, V0 is the volume at equilibrium). Is this correct? In some articles, I found the plots and calculated values are not in agreement although they mentioned a similar approach. And simply, V0 = a1 * a2 in 2D. Is not it correct?2) Also, I am trying to calculate the effective mass (m*) using a commonly known formula: m* = ℏ2/(∂2E / ∂k2). Here if I fit my conduction band minima (CBM) [or valence band maxima (VBM)] curve to y = ax2 + bx + c, and take the double differentiation of y w.r.to x, it gives rise to ∂2E / ∂k2. Is it correct? And how can I calculate m* in the unit Kg (or as the product of me), my E is in eV?
Please let me know if there are any journals/discussions related to the above queries.
Best,
Abhiyan Pandit
University of Arkansas
Dear Abhiyan,
The elastic modulus of two-dimensional sheets is obtained from the equation below:
(Eε − E0) = 1/2 S0 C2D ε2,
where Eε is the total energy of the sheet under strain, E0 is the total energy of the pristine sheet, and the S0 is the surface of the unit (super) cell. Through the fitting of the curve of energy difference (Eε − E0) versus strain (ε), the factor 1/2 S0 C2D is achieved. Having the value of this factor, it is not difficult to calculate the elastic modulus (C2D). It should be mentioned that the unit of S0 is m2. Also, in the 2D sheet, the volume makes no sense. For more detail refer to the article below.
[1] Wang Y, Ding Y. Electronic structure and carrier mobilities of arsenene and antimonene nanoribbons: a first-principle study. Nanoscale research letters. 2015 Dec 1;10(1):254.
The unit conversion of the effective mass.
Note the unit of energy in your calculation is eV while in my explanation it's in J.
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1 eV = 1.60217×10−19 J
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Hi Mohammad,
Thank you very much for your answers. Regarding the calculation of the elastic constants, I think mine [C2D = 2a/V0, where a is the constant obtained by the fitting of a quadratic polynomial to a curve of energy difference (Eequ-Estr) versus strain percentage (%), V0 in 2D is S0] and your [C2D = 2 ((Eε− E0)/ε2) / S0] expressions look similar. I just want to make sure, whether you use a value (Eε− E0)/ε2 for only a certain squared-strain (ε2) or you obtain by the fitting of a quadratic polynomial to a curve of energy difference (Eequ-Estr) versus strain percentage (%) or else?
Best,
Abhiyan Pandit
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