PHYSICAL REVIEW B 86, 199904(E) (2012)

Erratum: Theory of interfacial plasmon-phonon scattering in supported graphene [Phys. Rev. B 86, 165422 (2012)] Zhun-Yong Ong and Massimo V. Fischetti (Received 12 November 2012; published 29 November 2012) DOI: 10.1103/PhysRevB.86.199904

PACS number(s): 79.60.Jv, 73.50.Dn, 74.78.Fk, 72.80.Vp, 99.10.Cd

In this paper, we used the long-wavelength approximation for the polarizability

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FIG. 1. (Color online) The remote-phonon-limited mobility for SiO2 , HfO2 , h-BN, and Al2 O3 is computed using (a) the old approximation in Eq. (1) and (b) the new approximation in Eq. (3). The new approximation yields a mobility that plateaus at high n in SiO2 , h-BN, and Al2 O3 .

n. This agrees with the findings from Refs. 2 and 3, where the inelastic part of the resistivity was determined to be inversely proportional to n. We compute μRP from T = 50 to T = 500 K for n = 1012 and n = 5 × 1012 cm−2 for SiO2 in the old [Eq. (1)] and new [Eq. (3)] screening approximation. Figure 2(a) plots 1/μRP from 50 to 500 K for n = 1012 cm−2 . In Fig. 1, we see that the old and new approximations for μRP do not differ significantly in the low n limit. Similarly, at n = 1012 cm−2 , 1/μRP exhibits a temperature dependence which depends weakly on whether we employ Eq. (1) (old results) or Eq. (3) (new results). In Fig. 2(b), we show that 1/μRP is smaller, −2

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Equation (3) implies that the polarizability and hence the dynamic screening remain finite in the Q → ∞ limit, in contrast to Eq. (1). We use the expression in Eq. (3) to reevaluate the electron-IPP (interfacial plasmon-phonon) coupling coefficient MQ and scattering rates RP , the detailed expressions for which are given in this paper. We recompute μRP , as in this paper. To see the effect of the new approximation, we compute μRP at room temperature (300 K) at different carrier densities from n = 0.5 to 5.0 × 1012 cm−2 for the substrate dielectrics SiO2 , HfO2 , h-BN, and Al2 O3 . The results are shown Figs. 1(a) and 1(b) and compared to the “old” results presented in this paper. Using the long-wavelength approximation for the polarizability, given by Eq. (1), we found in this paper that μRP increases rapidly with n, especially at high n [see Fig. 1(a)], where the short-wavelength polarization is important. The new approximation in Eq. (3) yields a μRP that saturates at high n [see Fig. 1(b)] in SiO2 , h-BN, and Al2 O3 . We find this particular property appealing because it means that the remote-phonon-limited conductivity should scale linearly with

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where gs (gv ) is the spin (valley) degeneracy, L2 is the area, s indexes the band (+1 for conduction and −1 for valence), fsK is the Fermi-Dirac occupation of the (s,K) state, EsK is the electron energy, and Fss  (K,K + Q) is the overlap integral between the (s,K) and (s  ,K + Q) states. The Dirac-conical approximation for the electron energy dispersion is EsK = s¯hvF |K|. This implies that when ω < vF Q, the approximation for the polarizability in Eq. (1) overestimates the polarizability, the consequence of which is that it overestimates the dynamic screening effect at high carrier densities resulting in an excessively high remote-phonon-limited mobility μRP . Given that the approximation in Eq. (1) is only applicable when ω > vF Q, i.e., in the high-frequency limit, a more accurate approximation for the polarizability would be1 ⎧ 2 ⎨ Q 2EF2 , ω > vF Q ω (3) (Q,ω) = π¯h 2E F ⎩ − 2 2 , ω < vF Q. π¯h v

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Q2 EF , (1) (Q,ω) = πh ¯ 2 ω2 where Q is the wave vector, EF is the Fermi energy (in the Dirac-conical approximation), and ω is the frequency. The Lindhard formula for the polarizability is1 fsK − fs  K+Q g s gv  (Q,ω) = 2 L Kss  EsK − Es  K+Q + h ¯ω × Fss  (K,K + Q),

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FIG. 2. (Color online) Temperature dependence of the inverse mobility 1/μRP in the old and new approximation for (a) n = 1012 cm−2 and (b) n = 5 × 1012 cm−2 . The results from the old and new approximations are plotted with solid and hollow symbols, respectively.

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FIG. 3. (Color online) The remote-phonon-limited mobility computed with different screening models: no screening (triangles), the old approximation (squares), the new approximation (circles), and the FG approximation (diamonds), for (a) Al2 O3 and (b) SiO2 .

but the extent of the decrease is different for the old and new approximation. In the old approximation, μRP increases by almost two orders of magnitude as the carrier density is increased from n = 1012 cm−2 to n = 5 × 1012 cm−2 , as a consequence of dynamic screening. This implies that the temperature dependence of the total mobility would become negligible at high carrier densities. In the new approximation, this increase is significantly smaller as a result of the weaker dynamic screening. Nonetheless, it still implies that the temperature dependence of the total mobility should weaken as n increases as reported in this paper, albeit by a smaller extent. We also compare four different screening models: (a) no screening, (b) the old screening approximation [given by Eq. (1)], (c) the new screening approximation [given by Eq. (3)], and (d) the phenomenological static screening

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E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007). J. Chen, C. Jang, S. Xiao, M. Ishigami, and M. Fuhrer, Nature Nanotechnol. 3, 206 (2008).

model used by Fratini and Guinea4 (referred to here as the FG approximation). The μRP for each screening model is computed for Al2 O3 [see Fig. 3(a)] and SiO2 [see Fig. 3(b)]. In Al2 O3 , μRP without screening scales approximately as n−0.5 ; μRP in the FG approximation decreases with increasing n. In the old approximation, μRP increases rapidly with n. In contrast, with the new approximation, μRP increases significantly with n at low density but quickly saturates when n > 1.5 × 1012 cm−2 . In SiO2 , the same trends for μRP are present, although the rate of change in μRP with respect to n is slightly smaller. The steep gradient of μRP at low n in the old and new approximations is due to dynamic screening, which we noted in this paper, and remains a distinguishing feature of our model. The main effect of using the revised expression for the polarizability [Eq. (3) as presented here in place of Eq. (43) in the paper] on the findings and conclusion of our paper are: (1) The remote-phonon-limited mobility μRP for HfO2 is not comparable to the μRP for h-BN at high carrier densities. (2) High-frequency phonons are still screened more weakly by the graphene plasmons than low-frequency modes are, but the contribution of the former to μRP is less significant. Therefore, the results in Fig. 1(b) presented here replace those in Fig. 6 in this paper. We no longer claim that electron-IPP scattering in HfO2 is negligible at high carrier densities. Our new results agree qualitatively with the conclusion in the original text that the temperature dependence of mobility is reduced at higher carrier densities. We would like to acknowledge helpful technical discussions with Antonio Castro-Neto [National University of Singapore (NUS)] and Dave Ferry (Arizona State University). Z.Y.O. would also like to acknowledge the hospitality of the Center for Computational Science and Engineering at NUS where part of this work was carried out.

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K. Zou, X. Hong, D. Keefer, and J. Zhu, Phys. Rev. Lett. 105, 126601 (2010). 4 S. Fratini and F. Guinea, Phys. Rev. B 77, 195415 (2008).

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