Abstract
Graphene is a model system for the study of electrons confined to a strictly twodimensional layer^{1} and a large number of electronic phenomena have been demonstrated in graphene, from the fractional^{2,3} quantum Hall effect to superconductivity^{4}. However, the coupling of conduction electrons to local magnetic moments^{5,6}, a central problem of condensedmatter physics, has not been realized in graphene, and, given carbon’s lack of d or f electrons, magnetism in graphene would seem unlikely. Nonetheless, magnetism in graphitic carbon in the absence of transitionmetal elements has been reported^{7,8,9}, with explanations ranging from lattice defects^{10} to edge structures^{11} to negative curvature regions of the graphene sheet^{12}. Recent experiments suggest that correlated defects in highlyordered pyrolytic graphite (HOPG), induced by proton irradiation^{8} or native to grain boundaries^{7}, can give rise to ferromagnetism. Here we show that point defects (vacancies) in graphene^{13} are local moments which interact strongly with the conduction electrons through the Kondo effect^{6,14,15,16}, providing strong evidence that defects in graphene are indeed magnetic. The Kondo temperature T_{K} is tunable with carrier density from 30 to 90 K; the high T_{K} is a direct consequence of strong coupling of defects to conduction electrons in a Dirac material^{16}.
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Main
We previously reported the resistivity of graphene with vacancies induced by ion irradiation in ultrahigh vacuum (UHV; ref. 13). Here we present a detailed study of the gate voltage (V_{g}) and temperature (T) dependence of the resistivity ρ(V_{g},T) in similar graphene with vacancies over a wider temperature range 300 mK<T<290 K. Apart from weaklocalization (WL) corrections^{17,18}, we find that ρ(V_{g},T) is explained by a temperatureindependent contribution ρ_{c}(V_{g}) due to nonmagnetic disorder plus a temperaturedependent contribution ρ_{K}(V_{g},T), not present in asprepared graphene^{13}, which follows the universal temperature dependence expected for Kondo scattering from a localized 1/2spin with a single scaling parameter T_{K}.
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 ^{3}He sampleinvacuum cryostat. Figure 1a shows σ(V_{g}) 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 ^{3}He cryostat.V_{g} is applied to the Si substrate to tune the carrier density n=c_{g}V_{g}/e, where c_{g}=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 cm^{2} V^{−1} s^{−1}, respectively, for these three measurements; the conductivity and mobility recover significantly after annealing and air exposure, consistent with our previous study^{13}. From the postannealing mobility we estimate that this device has a defect density, n_{imp}, of approximately 3×10^{11} 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(V_{g}) curve is also observed in the irradiated sample, which could indicate a nonzero onsite energy for the defects in graphene^{19}.
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 defects^{17,18}. Figure 1c shows a detail of the magnetoresistance at small B, at 300 mK and at V_{g}−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 V_{g}−V _{g,min}<5 V, SdH corrections may affect the data slightly at 1 T. However, as shown below, the ρ(T) behaviour for V_{g}−V _{g,min}>5 V and V_{g}−V _{g,min}<5 V show no qualitative differences.
Figure 2a shows the temperaturedependent 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 V_{g} not too near V _{g,min}, indicating phonon contributions^{20}; between ∼10 and ∼100 K, we find dρ/dT<0 and the resistivity increases logarithmically with decreasing temperature at all V_{g}. At low temperature the resistivity at all V_{g} saturates (dρ/dT→0), indicating that there is no disorderinduced metal to insulator transition (MIT; ref. 16) or opening of a bandgap.
In metallic systems where localized magnetic moments couple antiferromagnetically to the conduction electrons, spinflip scattering gives rise to an anomalous component of the resistivity ρ_{K}(T) which is characterized by a Kondo temperature T_{K} (refs 6, 21). For T≈T_{K}, ρ_{K}(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≪T_{K}, the conduction electrons screen the spins of the local moments and the resistivity saturates, with a negative correction proportional to T^{2} (ref. 22). To compare the observed data in graphene with vacancies to theories of the Kondo effect, we model the temperaturedependent 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 ρ_{K,0} is the Kondo resistivity at zero temperature, ρ_{c1} and ρ_{c2} the nontemperaturedependent part of the resistivity, presumably from impurity scattering that does not involve the spin degree of freedom^{23,24}. The numerical factors in equations (1) and (2) are from the theory of the spin1/2 Kondo effect^{21}. As ρ(V_{g},T=0) is known, there are three degrees of freedom in these equations at each V_{g} : ρ_{c1}, ρ_{c2} and T_{K}; if ρ_{K}(T) follows the universal Kondo form then ρ_{c1}=ρ_{c2}. We keep ρ_{c1} and ρ_{c2} 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(V_{g},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 V_{g} and compare them to the universal Kondo behaviour^{21,25}. Figure 2b shows the normalized Kondo resistivity (ρ−ρ_{c1})/ρ_{K,0} versus T/T_{K} 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<∼3T_{K} and (2) the functional form matches well the universal Kondo behaviour from NRG calculations. At higher temperature (T>200 K), phonon contributions become important^{20} 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, ρ_{c1}, ρ_{c2}, ρ_{K,0} and T_{K}. Figure 3a shows ρ_{c1} and ρ_{c2} versus V_{g}, which peak around the actual minimum conductivity gate voltage V _{g,min}≈5.3 V. We find that ρ_{c1} and ρ_{c2} are practically identical, which indicates that the logarithmic divergence and T^{2} saturation of the resistivity indeed arise from the same effect (the Kondo effect). From now on we use ρ_{c1} for the nonKondo resistivity and label it as ρ_{c}. Figure 3b shows the nonKondo conductivity G_{c}=1/ρ_{c} as a function of V_{g}, which has a similar gate voltage dependence as an asprepared graphene sample. That is linear G(V_{g}) at high V_{g}, and a minimum G of a few e^{2}/h where h is Planck’s constant. It is worth noting that the minimum nonKondo conductivity G_{c,min}=6.9e^{2}/h is the same as the minimum conductivity of the pristine sample before irradiation (see Fig. 1a).
Figure 3c shows the Kondo resistivity ρ_{K,0}=ρ(T=0)−ρ_{c} versus V_{g}, which also peaks around the V _{g,min}, and decreases rapidly with increasing V_{g}−V _{g,min}, and Fig. 3d shows G_{K,0}=1/ρ_{K,0} versus V_{g}. In the lowtemperature limit (saturated resistivity), we expect G_{K,0}≈(πe^{2}/h)(n/n_{imp}) (ref. 15). However, G_{K,0} is 3–10 times larger than expected for n_{imp}=3×10^{11} cm^{−2}, and varies more rapidly; the red solid line is a power law fit to G_{K,0}(V_{g}) that yields G_{K,0}∼A+B V_{g}^{α}, 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 T_{K}. T_{K} is of order 50 K, which indicates strong coupling between the localized magnetic moment and the conduction electrons. Moreover, T_{K} is tunable by gate voltage, with a minimum of about 30 K near V _{g,min} and maxima close to 90 K at V_{g}−V _{g,min}≈−20 V (hole conduction) and close to 70 K at V_{g}−V _{g,min}≈25 V (electron conduction). At higher gate voltages (V_{g}−V _{g,min}>25 V), T_{K} decreases slightly with gate voltage, although the experimental error becomes large at large V_{g}, as the Kondo resistivity become very small. T_{K} versus V_{g}−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 T_{K}, and might result in a systematically smaller T_{K} obtained from the fitting. It is also possible that T_{K} depends on the initial disorder in the sample, as electron–hole puddling may play a role in determining T_{K}.
In a conventional (nonrelativistic) Fermi liquid, we can roughly estimate the Kondo temperature:
where k_{B} 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 T_{K} vanishes exponentially as E_{F}→0. This is in contrast to the observed variation of only ∼3× in T_{K}. The Kondo effect in a relativistic electron system is expected to differ qualitatively from the conventional nonrelativistic Kondo effect^{14,26,27}. In particular, for systems with ρ(E_{F})∝E_{F}^{r}, the case of r=1 (expected for graphene) is critical, and T_{K} vanishes as E_{F}→0 either linearly or as a power law, depending on the sign of E_{F}. For r>1,T_{K} remains finite, and for r<1,T_{K} is exponentially suppressed, as E_{F}→0. A recent theoretical study^{27} of (AB sublattice symmetric) Co on graphene found large T_{K}, of order 100 K, and strong asymmetry in T_{K} about E_{F}=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 E_{F}→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(E_{F}) 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 unity^{16} as E_{F}→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 E_{F}. However, we suppose that a state located modestly below the Dirac point could be filled at all observable E_{F} (for V_{g}<50 V,E_{F}<240 meV) and still produce a strongly resonant scattering around E_{F}=0 (ref. 19). It is also probable that in disordered graphene with potential fluctuations^{28}, the resistivity probes only the few defects with the largest T_{K}; this is consistent with G_{K,0} larger than expected in the lowtemperature 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 T_{K} in graphene with its small density of states is a unique consequence of defect scattering in a Dirac system^{16}, 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 range^{29}; as a small native concentration of defects could be present in these (and perhaps all) graphene devices^{30}.
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Acknowledgements
This work has been supported by NSFUMDMRSEC grant DMR 0520471 (JH.C., W.G.C., E.D.W., M.S.F.) and the US ONR grant N000140610882 (W.G.C., E.D.W., M.S.F.). The MRSEC SEFs were used in this work. Infrastructure support has also been provided by the UMD NanoCenter and CNAM. We would also like to thank D. GoldhaberGordon, S. D. Sarma, E. Rossi, E. Hwang and J. Zhu for useful discussions.
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JH.C. and M.S.F. conceived the experiments, JH.C. and L.L. fabricated devices, JH.C. performed the experiments and analyzed the data, JH.C., W.G.C., E.D.W. and M.S.F. cowrote the paper. All authors discussed the results and commented on the manuscript.
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Chen, JH., Li, L., Cullen, W. et al. Tunable Kondo effect in graphene with defects. Nature Phys 7, 535–538 (2011). https://doi.org/10.1038/nphys1962
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DOI: https://doi.org/10.1038/nphys1962
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