The Green-kubo method has been broadly used to calculate the thermal conductivity of 2D materials such as single layer graphene (SLG), however, there is a "V" term in the Green-Kubo method. What is the correct volume for the 2D materials, should it be the surface area of the SLG? or the surface area times the layer height, which is one atomic diameter height?
I also noted that at finite temperatures, the surface of the 2D materials should have some wrinkles, it is not "flat", then the surface area should be the real surface area with wrinkles or the projected one perpendicular to the plane direction?
For 2D systems, V represents the surface area of the system, not volume. Expressions of this type invariably involve 1/V in conjunction with a summation over a set of wave vectors (the set conform the box boundary condition). In d dimensions (here d=2), one has
limV→∞ (1/V) Σk f(k) = (1/2π)d ∫ ddk f(k),
where the integration on the right-hand side is over the continuum limit of the discrete set on the left-hand side over which the summation is carried out. Note that confining the system in a hypercube of side-length L, one has V = Ld.
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