In an interesting recent study [Allen et al., J. Appl. Phys. 118, 074302 (2015)] (see also their Erratum [Allen et al., J. Appl. Phys. 118, 159902 (2015)]), Allen and co-workers measured the mean square amplitudes of graphene lattice vibrations between 100 and 1300 K and used a simplified theoretical approximation for the acoustic phonon modes to evaluate the maximum phonon wavelengths supported by the lattice. By fitting their data using the smallest wave-vector as the fitting parameter, they found this to be significantly smaller than the physical size of the graphene crystallites.
In addition to amending the simplified analysis of Allen et al.,1,2 in this comment, we suggest that using an accurate model of the density of states (DOS) for the out-of-plane modes in graphene, for example, as calculated by Tewary and Yang (Fig. 1 in Ref. 3; reproduced as our Fig. 1(a)), could help shed light on this discrepancy. The complete treatment of this problem, including the evaluation of the phonon DOS through calculating the eigenvalues of the dynamical matrix at a large number of points in the Brillouin zone, is beyond the scope of this comment. Instead, we limit here our treatment to available literature data in order to gauge the direction and magnitude of this correction for the out-of-plane acoustic ZA mode.
We have evaluated the ZA contribution by simply integrating the out-of-plane phonon DOS to the highest ZA energy, since in graphene this is equal to the lowest ZO energy. This energy is approximately 85 meV,4 corresponding to ωmax = 1.29 × 1014 s−1 (Fig. 1(a)). This way we can make a direct comparison to the model used by Allen et al, which is by definition only valid for the acoustic modes. Note that the ωmax is not the same as the “Debye frequency” ωmZ for out-of-plane phonons, as defined by Tewary and Yang,3 because the optical mode ZO is not included here. (Note that in the model by Allen et al., the integral is carried out until ωmZ, but only one phonon mode is assumed.)
The mean square displacement can be calculated as a thermal average of the squared modulus of the second quantization displacement operator5
where g(ω) is the phonon DOS (in the current case, the DOS for the out-of-plane acoustic phonons). (This seems to be missing in the derivation of Allen et al. given in their supplementary information. The “density of states σ” mentioned there is only the normalization term for the DOS; see further.)
In principle, a DOS calculated with any approximation can be used here. For example, following Allen et al., we can use the dispersion ω(k) ∼ αk2, where α = 6.2 × 10−7 m2/s is obtained6 through fitting the ZA dispersion near Γ. From this, it is straightforward to derive an approximate 2D density of states as a function of frequency as
where kD = 1.65 × 10−10 m−1 is the wave vector corresponding to the highest out-of-plane phonon frequency ωmZ in graphene, and the prefactors ensure the correct normalization of g (Fig. 1(a)). As described above, since the treatment is limited to ZA, instead of ωmZ, we use ωmax in the following. Inserting this into (2) and the resulting density into (1) leads to the mean-square displacement
Using the identity coth(x/2) = + 1/(exp(x) − 1), we see that this corresponds to, but differs from, the model of Allen et al. (Equation (7) in Ref. 1) due to the density of states term in (1). Fitting their data (Fig. 7 in Ref. 1) using (3), we obtain a minimum frequency of 8.3 × 1011 s−1. Inverting the approximate quadratic dispersion relation, we can express the wavelength as a function of the frequency as
This gives λmin = 5.4 × 10−9 m, which is close to 2.5 × 10−9 m, the value originally obtained by Allen et al.
However, we can equally well insert into (1) the better-motivated out-of-plane phonon DOS published by Tewary and Yang3 (Fig. 1(a)), which yields ωmin = 5.9 × 108 s−1. If we assume that the dispersion relation is still valid (as it should be for low frequencies in the case of acoustic phonons), this corresponds to a wavelength of λmin = 6.4 × 10−8 m, which is about 25 times larger than that obtained by Allen et al., while the fit to the data remains equally good (Fig. 1(b)). While still small, this suggests that this surprising finding by Allen et al. may be partly due to the limitations within the used approximations.
Based on this quick analysis, it appears to us that a more complete theoretical treatment of the problem, including both acoustic and optical phonon modes as well as the in-plane and out-of-plane vibrations, would be worth its own separate study.
We acknowledge funding from the Austrian Science Fund (FWF) project P 28322-N36 and the Wiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF) project MA14-009.