Erratum: Caroli formula in near-field heat transfer between parallel graphene sheets [PHYSICAL REVIEW B 96, 155437 (2017)]
EErratum: Caroli formula in near-field heat transfer betweenparallel graphene sheets [PHYSICAL REVIEW B 96, 155437 (2017)]
Jia-Huei Jiang and Jian-Sheng Wang (Dated: February 26, 2021)
This erratum aims to correct 1) the wrong expressions, 2) some typographical errors, and 3) some erroneous pointsmade in discussion of the disparity of heat flux ratios between our full RPA model and the local conductivity model,appearing in the original paper [1], which might confuse or mislead readers. Collectively, they don’t change the keyphysics we sought to present. In the following, whenever we say the/ our original paper, it refers to Ref. [1].First, we would like to correct wrong expressions:1. In Eq.(A1), the electronic annihilation and creation operators were given wrongly in the paper. They should beas follows: c ( l ) k = (cid:16) c ( l ) A k , c ( l ) B k (cid:17) T ; c k † ( l ) = (cid:0) c A k † ( l ) , c B k † ( l ) (cid:1) . This does not affect our calculations, presentationof key physics, nor conclusions for this was only an accidental misimplementation during the drafting phase ofthe paper.2. The statement Π r → (cid:80) j, j (cid:48) Π rjj (cid:48) above Eq. (D1) should be restated as: Π r → (cid:80) j, j (cid:48) Π rjj (cid:48) . The division by 4is to give the sense of averaging over four components of Π rjj (cid:48) , when A and B sublattices are indistinguishable.This statement is not used elsewhere in the paper and, therefore, does not affect our calculations, presentationof key physics, nor conclusions.Second, the typographical errors are corrected as below:1. The expressions for the f ( k ) function appearing in Eqs.(A1) and (7) have the wrong signs for k x in the secondand third terms. They should be: f ( k ) = e − i k x a + e i k x a / i √ k y a / + e i k x a / − i √ k y a / .2. The expression for q z should have lim ˜ c →∞ , instead of lim ˜ c → .3. In Eq. (B3), the denominators on the right-hand side should be multiplied by a to make dimensions match forthe Laplacian.4. In Eq. (D3), the z (cid:48) and z are finally taken to be d − at the end ( z (cid:48) = z = d − ).Third, we re-clarify some points in discussing the disparity of heat flux ratios between our full RPA model and thelocal conductivity model:1. We inappropriately adopted the analytical expressions for the plasmon branches ( ω L and ω H [2]) valid only forhighly doped sheets deviated by mild temperature difference (see Eq. (E1) with µ (cid:29) k B T ). The inappropriate-ness can be obviously seen in the parameters of doping as light as 0 . eV and temperatures as 300 K and 1000 K used in our original paper. To remedy this, we derive a more general form: ω L/H = (cid:34) Z avg (cid:32) ∓ (cid:115) − ( Z Z avg )( Z Z avg )(1 − e − qd ) (cid:33) (cid:35) / , q > , (E1)where the ω L corresponds to the minus branch and ω H the positive; Z avg = ( Z + Z ) / Z l = ( e k B T l /(cid:15) π ¯ h ) ln[2 cosh( µ l / k B T l )] q [3]; the η (= 0 . eV ) was neglected because it is very smallcompared to the energy spans we have chosen. One can readily verify that this formula goes back to the form ω L/H = ω s √ ∓ e − qd , ω s = (cid:112) Z avg , when Z = Z = Z avg [2]. One can also show that ω L = (cid:112) ( Z Z /Z avg ) qd as qd →
0. This way of estimating plasmon branches consider the Drude part of conductivity only, which is fairwhen the frequency (or q ) is small and is dominant in wider frequency span when µ is larger than k B T [3].The purpose for drawing the ω L and ω H lines using Eq. (E1) is for approximating the orientations of the twobranches on the ω − q plane. The approximation can tell the initial orientations of the plasmon branches in thelow frequency regime; it is our observation that the orientations in the higher frequency regime then develop onthe basis of these initial orientations without making drastic change.We replace the cyan and green lines in the FIGs. 8 - 9 of our original paper with Eq. (E1) and give the updatedplots as follows ( the figure indexing follows the same figure indexing as in the original paper): a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b (a) (b) FIG. 8:
The updated FIGs. 8 (a) and (b) with the cyan and green dashed lines based on Eq. (E1) replotted; curves standingfor Eq. (E1) in the µ (cid:29) k B T limit are not plotted because the results closely overlap with the cyan and green dashed lines,respectively; the white dashed line marks the ω = v F q border; other contents are the same as the original paper. (a) (b)(c) (d) FIG. 9:
The updated FIGs. 9 (a)-(d) with the cyan and green dashed lines based on Eq. (E1) replotted; the red and orangedashed lines stand for Eq. (E1) in the µ (cid:29) k B T limit ( these lines are not plotted for (c) and (d) because the results closelyoverlap with the cyan and green dashed lines, respectively); the yellow and pink dashed lines were mistakenly and respectivelytaken as the red and orange dashed lines, in the original paper; the white dashed line marks the ω = v F q border; othercontents are the same as the original paper.
2. The values of ”critical distances” d c s are nearly the same as previously reported [1] so we don’t change them inlight of Eq. (E1).3. We report misplacement of the cyan ω L lines drawn on FIGs. 9a and 9b in the original paper. The correct linescan be obtained from Eq. E1 and with the µ (cid:29) k B T limit applied (see the red, orange, yellow, and pink dashedlines FIGs. 9a and 9b in this erratum). Also, we report misplacement of figures: FIGs. 8a and 8b in the originalpaper were mistakenly interchanged due to the ultra-similarity (the local-conductivity curves should bend moreslightly inwards toward the ω = v F q border and appear less blurry at the high-frequency ends).4. Our previous statement in the original paper: “The full RPA calculation of ours has rescued the extinction ofacoustic plasmon mode under local conductivity approximation by constraining the mode to stay within borderof ω = v F q line.” is not well-phrased and confusing, for the full RPA calculation doesn’t need to rescue itsacoustic plasmon mode from invalidating the basic assumption ( ω > v F q ) that the local conductivity modeltook. The better statement is: “The full RPA calculation of ours seems to have the acoustic mode line staywithin the border of ω = v F q line.” [1] J.-H. Jiang and J.-S. Wang, Caroli formalism in near-field heat transfer between parallel graphene sheets, Phys. Rev. B ,155437 (2017).[2] H. Iizuka and S. Fan, Analytical treatment of near-field electromagnetic heat transfer at the nanoscale, Phys. Rev. B ,144307 (2015).[3] O. Ilic, M. Jablan, J. D. Joannopoulos, I. Celanovic, H. Buljan, and M. Soljaˇci´c, Near-field thermal radiation transfercontrolled by plasmons in graphene, Phys. Rev. B85