screening，kohnanomaly，friedeloscillation，andrkkyinteractioninbilayergraphene-外文文獻(xiàn)(參考版)
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【正文】 Phys. Rev. Lett. 83, 164 (1999).[13] C. Bena, Phys. Rev. Lett. 100, 076601 (2020).[14] . Peierls, Quantum Theory of Solids (Clarendon Press,Oxford, 1995).[15] S. Das Sarma and Wuyan Lai, Phys. Rev. B 32, 1401(1985).[16] . Vozmediano, . LopezSancho, T. Stauber,and F. Guinea, Phys. Rev. B 72, 155121 (2020).[17] L. Brey, . Fertig, and S. Das Sarma, Phys. Rev. Lett.99, 116802 (2020).PRL 101, 156802 (2020)PHYSICAL REVIEW LETTERSweek ending10 OCTOBER 20201568024。 S. Adam et al., Proc. Natl.Acad. Sci. . 104, 18392 (2020).[10] W. Kohn, Phys. Rev. Lett. 2, 393 (1959)。 V.Cheianov and V. Fal’ko, Phys. Rev. Lett. 97, 226801(2020)。 K. Novoselov et al., Nature Phys. 2, 177(2020)。 M.Koshino and T. Ando, Phys. Rev. B 73, 245403 (2020)。 J. Nilsson, . Castro Neto, . Peres, andF. Guinea, Phys. Rev. B 73, 214418 (2020)。 2kF) with the BLG screening having astrong cusp at q 188。kFr222。2kFr222。r222。r222。 formally scales as 1=r3, itsmagnitude does not converge], which means that intrinsicSLG is susceptible to ferromagic ordering in the presence of magic impurities [16,17] due to the divergentRKKY coupling.In doped (or gated) BLG, the oscillatory term in RKKYinteraction is restored due to the singularity of polarizability at q 188。N0localized magic moments are not correlated by the longrange interaction and there is no magic moment. InSLG, the Fourier transform of polarizability [Eq. (8)] diverges [even though C5240。r222。q222。q222。r222。r222。188。, where J is the exchange coupling01234q/k01Π(q)/NF0SLGBLG01234q/k01Π(q)/NF0SLGBLG01234q/k012Π(q)/NF0SLGBLG2D(a) (b)(c)FIG. 1 (color online). Calculated (a) intraband, (b) interband,and (c) total static polarizability of bilayer graphene. For parison, the single layer polarizabilities are shown. In (c), wealso show the regular 2D static polarizability (dashed line).PRL 101, 156802 (2020)PHYSICAL REVIEW LETTERSweek ending10 OCTOBER 20201568023constant. The RKKY interaction between two localizedmoments via the conduction electrons may then be writtenin the following form: HRKKY240。C14240。s240。JS240。r222。r222。q222。 2kFcusp in the polarizability, rather similar to the 1D Peierls instability [14] since the q 188。q222。, that the screened phonon dispersion will exhibit amuch stronger Kohn anomaly in the BLG than in the SLGwith the 2DEG being intermediate. This arises from thestronger singularity at q 188。 2kF. It is obvious from Fig. 1(c), andfrom the discussion above based on our analytical resultsfor C15240。 2kFhasother interesting consequences related to Kohn anomaly[10] and RKKY interaction, which we discuss below.The strong cusp in the BLG polarizability atq 188。2kFr222。r222。=2C138, which is similar to the2DEG except for the additional constant C (C 188。1254。???5pC0log189。2。240。2sin240。2kF254。r222。 at q 188。!1 for bothBLG and 2DEG.The strong cusp in BLG C5240。 for SLG and C15240。gC25e2=240。q !1222。q222。q222。=dq / 1=???????????????????q2C04k2Fq. This behavior isexactly the same as that of the regular 2DEG, which alsohas a cusp at q 188。 .,as q ! 2kF, dC5240。 2kF. Even though the BLG polarizability is continuous at q 188。 2kFas existsin SLG and 2DEG.A qualitative difference between SLG and BLG polarizability functions is at q 188。 increases as q2=2k2F. This behavior es from the overlapfactor Fss0 in Eq. (3). For SLG, intraband (interband)polarizability decreases (increases) linearly as q increases,and these two effects exactly cancel out up to q 188。 decreases as 1C0q2=2k2F, and C5inter240。. For smallPRL 101, 156802 (2020)PHYSICAL REVIEW LETTERSweek ending10 OCTOBER 20201568022q, C5intra240。=dC138N240。C0df240。188。q 188。188。N0andC5inter240。0222。C138: (16)Equation (16) with Eq. (15) is the basic result obtained inthis Letter, giving the doped BLG polarizabilityanalytically.In Fig. 1, we show the calculated static polarizabilityas a function of the wave vector. Figur
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