Chen et al. reply —

We reported1 that irradiation-induced vacancies in graphene have local moments that couple to conduction electrons through the Kondo effect, with a resistivity increasing logarithmically with decreasing temperature and saturating at low temperature. The local moment of graphene vacancies has since been verified in susceptibility measurements2; however, that experiment was agnostic on the Kondo effect, which would be difficult to probe magnetically as vacancies significantly outnumbered carriers. Weber et al. comment that our resistivity data might be confounded by Altshuler–Aronov corrections3, recently measured in graphene on SiO2 (ref. 4) and SiC (refs 5, 6, 7). The Altshuler–Aronov effect gives graphene's resistivity as:

where ρxx,0 is the uncorrected longitudinal resistivity, A ≤ 1 a constant, e the elemental charge, T temperature, μ the charge carrier mobility, B the magnetic field, and τtr the transport momentum relaxation time. Here we show that equation (1) cannot explain the magnitude of the logarithmic resistivity and magnetoresistance, or the saturation of the resistivity at low temperature for our devices.

Figure 1a shows fits of equation (1) to our ρxx(B) for 1 T < B < 6 T at T = 300 mK, for four different gate voltages (Vg); the global fit parameter is A = 0.32 (A is systematically smaller, and magnetoresistance overestimated, for small Vg − Vg,min). Our devices are etched Hall bars with an aspect ratio exceeding three; geometric errors in ρxx are at most a few per cent. A = 0.32 is reasonably consistent with other experiments; Kozikov et al.4 analysed ρxx(T) and found 0.35 < A < 1.05 for graphene on SiO2, and Jobst et al.7 found that 0.35 < A < 0.92 for graphene on SiC. The Kondo effect itself also produces negative magnetoresistance, so we consider A = 0.32 to be an upper bound. Figure 1b shows ρxx(T) for the same four Vg and the Altshuler–Aronov correction from equation (1) with A = 0.32. The Altshuler–Aronov effect accounts for at most 12–16% of the logarithmic divergence of the resistivity in our data. Likewise, had we had fit ρxx(T), we would find 1.9 < A < 2.5, and would drastically overestimate the observed magnetoresistance. A > 1 is unphysical and a priori rules out the Altshuler–Aronov effect as the sole source of logarithmic ρxx(T).

Figure 1: Temperature and magnetic-field dependent resistivity of graphene with defects.
figure 1

a, Magnetoresistance of graphene sample Q6 with defects at a temperature T = 300 mK and at gate voltages Vg − Vg,min = −15.3 V (blue stars), −25.3 V (purple triangles), −35.3 V (red circles) and −45.3 V (green crosses); Vg,min = 5.3 V is the gate voltage of minimum conductivity. b, Temperature-dependent resistivity of graphene under 1 T of transverse magnetic field and at the same four Vg values in a. In a and b the solid lines are equation (1) with A = 0.32, chosen to fit the data in a. c, Experimentally determined Kondo temperatures (TK) for sample Q6 (red squares) and L2 (blue circles). The solid lines are the expected Tsat from the Altshuler–Aronov effect for samples Q6 (red) and L2 (blue). Data taken from ref. 1.

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Now we discuss the saturation of ρxx(T). The Thouless length LT sets the scale for Altshuler–Aronov corrections, which become T independent for LT > lsample (lsample is the shortest sample dimension)8. Assuming9

the saturation temperature (Tsat) is

where vF is the Fermi velocity and EF is the Fermi energy. We reported data for two samples: Q6 (μ = 2,000 cm2 Vs−1 and lsample = 2.0 μm) and L2 (μ = 1,100 cm2 Vs−1 and lsample = 6.5 μm). Figure 1c plots the experimentally determined Kondo temperature and the Tsat predicted by Altshuler–Aronov theory as a function of gate voltage Vg ∝ EF2. Altshuler–Aronov theory predicts these two samples should show Tsat differing by a factor of 35 and strongly Vg dependent; neither is observed. Notably, saturation of ρxx(T) is not observed down to T = 0.2 K in exfoliated graphene on SiO2 (ref. 4) or 1.5 K in graphene with lsample = 5 μm on SiC (ref. 5). Thus the observed resistivity saturation is novel, and strong evidence for the Kondo effect in graphene with lattice defects. Other saturation mechanisms (such as coherence length lФ = lsample as in weak localization) are also inconsistent with the data.