Transport anomaly at the ordering transition for adatoms on graphene

Impurities in metals experience a long-range RudermanKittel-Kasuya-Yosida (RKKY) interaction due to polarization of the electron Fermi sea (Friedel oscillations).1 For surface adsorbents such an interaction may result in their structural ordering, repeating the pattern of the Friedel oscillations of electron density.2 In particular, a dilute ensemble of adatoms on graphene may undergo a partial ordering transition.3–6 Unlike other materials, the RKKY interaction between adatoms on graphene exists even at zero carrier density, with a characteristic long-range 1/r3 dependence, and it exhibits Friedel oscillations which are commensurate with the underlying honeycomb lattice. For adatoms residing above the centers of the honeycombs, the intervalley scattering of the electrons by adatoms leads to Friedel oscillations that resemble a √ 3 ×√3 charge-density wave superlattice. Positions of each individual adatom relative to such superlattice can be characterized by one of three vectors, u = (cos 2πm 3 , sin 2πm 3 ) with m = −1,0,1. The transition of an ensemble of adatoms into a “Kekulé mosaic” ordered state,3 characterized by the order parameter u, falls in the symmetry class of three-state Potts models.7 In this paper we analyze how partial “Kekulé” ordering of adatoms on graphene affects its resistivity ρ in the regime of low coverage, nia 1 (ni is the concentration of adatoms, a is the lattice constant). The behavior of the temperaturedependent resistivity correction δρ(T ) = ρ(T ) − ρ(∞) is sketched in Fig. 1. At T Tc (region I) the temperature dependence of the resistivity is dominated by a nonvanishing order parameter u causing the amplified intervalley mixing and opening a gap ∝ u ∼ (Tc − T ) in the corners of the Brillouin zone. As the temperature increases from T = 0 to T = Tc the resistivity correction monotonically decreases as (Tc − T )2β . At T > Tc, critical fluctuations of the order parameter, which are characterized by the correlation length ξ ∝ |T − Tc|−ν and precede the formation of the ordered phase, lead to a nonmonotonic feature in δρ(T ). At high temperature, T Tc (region III), the constructive interference of electron waves, scattered by adatoms within ordered clusters of size ξ , enhances resistivity. The effect, which becomes stronger upon approaching Tc, is similar to the critical opalescence8 in materials undergoing a structural phase transition or the resistivity anomaly in bulk metals with magnetic impurities undergoing a ferromagnetic transition.9 This enhancement saturates when ξ becomes comparable to the electron wavelength λF (i.e., λF ≈ ξ ). In region II of temperatures T → Tc + 0, where ξ λF , scattering of electrons is affected only by the gradient of the fluctuating order parameter u. The resistivity is thus reduced and a cusp-shape minimum at T = Tc should be expected. In the following we assume that the electron concentration ne = 4π/λF is not high (i.e., ne ni), but the electron Fermi wavelength is shorter than its mean-free path, λF l. This assumption also implies that, in the ordered phase, kBT , εF . The electrons are described by a four-component Dirac-like spinor = [ψK,A,ψK,B,ψK ′,B,ψK ′,A] with the components corresponding to different valleys (K,K ′) and sublattices (A,B).10 In the absence of adatoms, quasiparticles are characterized by the linear spectrum |εp| = h̄vp and plane-wave states (for εp > 0):