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We previously reported the resistivity of graphene with vacancies induced by ion irradiation in ultra-high vacuum (UHV; ref. 13). Here we present a detailed study of the gate voltage (Vg) and temperature (T) dependence of the resistivity ρ(Vg,T) in similar graphene with vacancies over a wider temperature range 300 mK<T<290 K. Apart from weak-localization (WL) corrections17,18, we find that ρ(Vg,T) is explained by a temperature-independent contribution ρc(Vg) due to non-magnetic disorder plus a temperature-dependent contribution ρK(Vg,T), not present in as-prepared graphene13, which follows the universal temperature dependence expected for Kondo scattering from a localized 1/2-spin with a single scaling parameter TK.

Graphene with vacancies is prepared as described in ref. 13. After irradiation, the devices were annealed overnight at 490 K in UHV, and then exposed to air during transfer to a 3He sample-in-vacuum cryostat. Figure 1a shows σ(Vg) measured at 17 K for a graphene device (sample Q6) before irradiation, immediately after irradiation, and measured at 300 mK after annealing and transfer to the 3He cryostat.Vg is applied to the Si substrate to tune the carrier density n=cgVg/e, where cg=1.15×10−8 F cm−2 is the gate capacitance, and e the elementary charge. The mobility of the device is approximately 4000, 300, and 2000 cm2 V−1 s−1, respectively, for these three measurements; the conductivity and mobility recover significantly after annealing and air exposure, consistent with our previous study13. From the post-annealing mobility we estimate that this device has a defect density, nimp, of approximately 3×1011 cm−2, although greater understanding of the effects of annealing and ambient exposure on vacancies in graphene is needed. See Supplementary Information for the calculation of defect density and also Raman spectra of the device before and after irradiation. Slight asymmetry between electron and hole conduction in the σ(Vg) curve is also observed in the irradiated sample, which could indicate a non-zero on-site energy for the defects in graphene19.

Figure 1: Gate voltage dependent conductivity σ(Vg) and magnetoresistance of the graphene sample.
figure 1

a, σ(Vg) of the graphene sample Q6 before (black solid line) and after (red dashed line) irradiation with 500 eV He+ at a temperature T=17 K, and after annealing at 490 K overnight in ultra-high vacuum and exposure to ambient before cooling to T=300 mK (blue short-dashed line). Magnetic field B=0 for all data. The gate voltage of minimum conductivity V g,min=−8 V, 5 V, 5.3 V for pristine, irradiated and annealed sample, respectively. b, Magnetoresistance of irradiated and annealed graphene sample for B=0–8 T at various Vg. c, Normalized detailed magnetoresistance of irradiated and annealed graphene sample from −1.2 to 1.2 T at Vg−V g,min≈−65 V.

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Figure 1b shows the perpendicular magnetic field dependence of the resistivity ρ(B) of the irradiated sample Q6 at T=300 mK at several different gate voltages. Negative magnetoresistance is observed at small B, indicating the dominance of weak localization arising from intervalley scattering due to lattice defects17,18. Figure 1c shows a detail of the magnetoresistance at small B, at 300 mK and at Vg−V g,min=−65 V (see Supplementary Information for the gate voltage and temperature dependent phase coherence length, which is extracted from analyzing the WL magnetoresistance). Shubnikov–de Haas (SdH) oscillations appear at high B field. To measure the resistivity without WL and SdH corrections, the WL contribution is suppressed by application of B=1 T in further measurements. For |Vg−V g,min|<5 V, SdH corrections may affect the data slightly at 1 T. However, as shown below, the ρ(T) behaviour for |Vg−V g,min|>5 V and |Vg−V g,min|<5 V show no qualitative differences.

Figure 2a shows the temperature-dependent resistivity ρ(T) of the irradiated graphene measured at several different gate voltages at B=1 T. Positive slopes, dρ/dT>0, are seen in ρ(T) from room temperature to about 200 K for Vg not too near V g,min, indicating phonon contributions20; between ∼10 and ∼100 K, we find dρ/dT<0 and the resistivity increases logarithmically with decreasing temperature at all Vg. At low temperature the resistivity at all Vg saturates (dρ/dT→0), indicating that there is no disorder-induced metal to insulator transition (MIT; ref. 16) or opening of a bandgap.

Figure 2: Universal Kondo behaviour of graphene with defects.
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a, Temperature-dependent resistivity ρ(Vg) of graphene sample Q6 under a perpendicular magnetic field of 1 T, at 12 different gate voltages, with temperature changing from 300 mK to ∼290 K. b, The normalized Kondo part of the resistivity (ρ−ρc1)/ρK,0 versus T/TK(Vg), where TK(Vg) is the Kondo temperature at respective gate voltage (see Fig. 4). The red line is the expected universal Kondo behaviour from numerical renormalization group calculations21.

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In metallic systems where localized magnetic moments couple anti-ferromagnetically to the conduction electrons, spin-flip scattering gives rise to an anomalous component of the resistivity ρK(T) which is characterized by a Kondo temperature TK (refs 6, 21). For T≈TK, ρK(T) is approximately logarithmic in T (similar behaviour is observed in situ in UHV before ambient exposure; see Supplementary Information). In principle, interaction effects in the presence of disorder could also lead to logarithmic ρ(T) even at high magnetic field (the Altshuler–Aronov effect); however this would also lead to similar corrections to the Hall resistivity, which are not observed (see Supplementary Information). For T≪TK, the conduction electrons screen the spins of the local moments and the resistivity saturates, with a negative correction proportional to T2 (ref. 22). To compare the observed data in graphene with vacancies to theories of the Kondo effect, we model the temperature-dependent resistivity in the low temperature regime and the intermediate temperature regime (region of maximum logarithmic slope, roughly between 10 and 100 K), respectively, as

where ρK,0 is the Kondo resistivity at zero temperature, ρc1 and ρc2 the non-temperature-dependent part of the resistivity, presumably from impurity scattering that does not involve the spin degree of freedom23,24. The numerical factors in equations (1) and (2) are from the theory of the spin-1/2 Kondo effect21. As ρ(Vg,T=0) is known, there are three degrees of freedom in these equations at each Vg : ρc1, ρc2 and TK; if ρK(T) follows the universal Kondo form then ρc1=ρc2. We keep ρc1 and ρc2 as independent parameters to test the internal consistency of the model. Least square fits to the equations (1) and (2) are carried out on ρ(Vg,T) in the low and intermediate temperature ranges respectively (see Supplementary Information for details).

Using the extracted parameters, we can scale the ρ(T) curves at different Vg and compare them to the universal Kondo behaviour21,25. Figure 2b shows the normalized Kondo resistivity (ρ−ρc1)/ρK,0 versus T/TK and the universal Kondo behaviour from numerical renormalization group calculations (NRG; ref. 21). From Fig. 2b one can find that: (1) all the experimental curves collapse to a single functional form for 300 mK<T<∼3TK and (2) the functional form matches well the universal Kondo behaviour from NRG calculations. At higher temperature (T>200 K), phonon contributions become important20 and the observed positive deviations from the NRG calculations are expected. However, at the lowest gate voltages, the deviation is negative, possibly due to thermal activation of carriers.

Now we discuss the gate voltage dependence of the extracted parameters, ρc1, ρc2, ρK,0 and TK. Figure 3a shows ρc1 and ρc2 versus Vg, which peak around the actual minimum conductivity gate voltage V g,min≈5.3 V. We find that ρc1 and ρc2 are practically identical, which indicates that the logarithmic divergence and T2 saturation of the resistivity indeed arise from the same effect (the Kondo effect). From now on we use ρc1 for the non-Kondo resistivity and label it as ρc. Figure 3b shows the non-Kondo conductivity Gc=1/ρc as a function of Vg, which has a similar gate voltage dependence as an as-prepared graphene sample. That is linear G(Vg) at high Vg, and a minimum G of a few e2/h where h is Planck’s constant. It is worth noting that the minimum non-Kondo conductivity Gc,min=6.9e2/h is the same as the minimum conductivity of the pristine sample before irradiation (see Fig. 1a).

Figure 3: The non-Kondo and Kondo part of the resistivity versus Vg.
figure 3

a, Comparison between the non-Kondo resistivity obtained from fitting ρ(Vg,T) to equation (1) in the low temperature regime (ρc1) and to equation (2) in the intermediate temperature regime (ρc2) at different Vg. b, non-Kondo conductivity Gc=1/ρc as a function of Vg. c, The zero temperature Kondo resistivity ρK,0 and d, Kondo conductivity GK,0=1/ρK,0 as a function of Vg. The red solid line in d is a power law fit to GK,0,(Vg). The blue dashed line is the expectation for unitary scatterers of concentration 3×1011 cm−2 (ref. 15).

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Figure 3c shows the Kondo resistivity ρK,0=ρ(T=0)−ρc versus Vg, which also peaks around the V g,min, and decreases rapidly with increasing |Vg−V g,min|, and Fig. 3d shows GK,0=1/ρK,0 versus Vg. In the low-temperature limit (saturated resistivity), we expect GK,0≈(πe2/h)(n/nimp) (ref. 15). However, GK,0 is 3–10 times larger than expected for nimp=3×1011 cm−2, and varies more rapidly; the red solid line is a power law fit to GK,0(Vg) that yields GK,0∼A+B Vgα, with α=2.1±0.1 for electron conduction and α=2.2±0.2 for hole conduction.

Figure 4 shows the gate voltage dependence of the Kondo temperature TK. TK is of order 50 K, which indicates strong coupling between the localized magnetic moment and the conduction electrons. Moreover, TK is tunable by gate voltage, with a minimum of about 30 K near V g,min and maxima close to 90 K at Vg−V g,min≈−20 V (hole conduction) and close to 70 K at Vg−V g,min≈25 V (electron conduction). At higher gate voltages (|Vg−V g,min|>25 V), TK decreases slightly with gate voltage, although the experimental error becomes large at large Vg, as the Kondo resistivity become very small. TK versus Vg−V g,min for a second sample (sample L2, see Supplementary Information for details) shows similar magnitude and similar variation with gate voltage. The higher base measurement temperature (1.7 K for sample L2 as compare to 0.3 K for sample discussed in main text) results in larger uncertainty in TK, and might result in a systematically smaller TK obtained from the fitting. It is also possible that TK depends on the initial disorder in the sample, as electron–hole puddling may play a role in determining TK.

Figure 4: The gate-tunable Kondo Temperature.
figure 4

The Kondo temperature TK of two graphene samples with vacancies as a function of gate voltage Vg as determined from fits to equations (1) and (2). The sample Q6 (black squares) has been discussed in the main text. Details on sample L2 (blue circles) can be found in Supplementary Information. The error bars represent the ± one standard deviation of TK, calculated using error propagation from the standard deviation of raw fitting parameters (see Supplementary Information).

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In a conventional (non-relativistic) Fermi liquid, we can roughly estimate the Kondo temperature:

where kB is Boltzmann’s constant, D is the electronic bandwidth (∼10 eV for π electrons in graphene), J>0 the antiferromagnetic coupling constant, and is the density of states of graphene at the Fermi energy. Equation (3) predicts TK vanishes exponentially as EF→0. This is in contrast to the observed variation of only ∼3× in TK. The Kondo effect in a relativistic electron system is expected to differ qualitatively from the conventional non-relativistic Kondo effect14,26,27. In particular, for systems with ρ(EF)∝|EF|r, the case of r=1 (expected for graphene) is critical, and TK vanishes as EF→0 either linearly or as a power law, depending on the sign of EF. For r>1,TK remains finite, and for r<1,TK is exponentially suppressed, as EF→0. A recent theoretical study27 of (AB sublattice symmetric) Co on graphene found large TK, of order 100 K, and strong asymmetry in TK about EF=0, which we do not observe. However, the defects studied here are expected to break sublattice symmetry and locally open a gap at the Dirac point, with a sharp resonant midgap state leading to resonant impurity scattering as EF→0(refs 13, 16). In this case, it has been pointed out that the dimensionless parameter which determines the strength of the Kondo effect, J ρ(EF) in equation (3), is proportional to the sine of the scattering phase shift for the defect, which for the large scalar potentials induced by defects in the graphene plane is finite and of order unity16 as EF→0, leading to a robust Kondo effect. For a resonant state located at exactly the Dirac point we would also expect a strong particle–hole asymmetry as the defect state is filled and emptied for positive or negative EF. However, we suppose that a state located modestly below the Dirac point could be filled at all observable EF (for Vg<50 V,EF<240 meV) and still produce a strongly resonant scattering around EF=0 (ref. 19). It is also probable that in disordered graphene with potential fluctuations28, the resistivity probes only the few defects with the largest TK; this is consistent with GK,0 larger than expected in the low-temperature limit (Fig. 3d). A complete theory of the Kondo effect in graphene will probably require both a microscopic understanding of the defect and its interaction with conduction electrons, as well as an effective medium theory of Kondo scattering in the presence of potential variations in disordered graphene.

The high TK in graphene with its small density of states is a unique consequence of defect scattering in a Dirac system16, and we anticipate that new graphene Kondo systems may be realized by a variety of physical or chemical modifications to the graphene lattice. Defect engineering thus provides a powerful route to introduce and control magnetism in carbon nanostructures, such as graphene and carbon nanotubes, without the presence of transition metal elements. The observation of Kondo scattering from defects in graphene may also explain the anomalous short spin lifetimes observed in graphene spin valves in the currently explored temperature range29; as a small native concentration of defects could be present in these (and perhaps all) graphene devices30.