Abstract
The Kondo effect is a cornerstone in the study of strongly correlated fermions. The coherent exchange coupling of conduction electrons to local magnetic moments gives rise to a Kondo cloud that screens the impurity spin. Here we report on the interplay between spin–orbit interaction and the Kondo effect, that can lead to a underscreened Kondo effects in quantum dots in bilayer graphene. More generally, we introduce a different experimental platform for studying Kondo physics. In contrast to carbon nanotubes, where nanotube chirality determines spin–orbit coupling breaking the SU(4) symmetry of the electronic states relevant for the Kondo effect, we study a planar carbon material where a small spin–orbit coupling of nominally flat graphene is enhanced by zeropoint outofplane phonons. The resulting twoelectron triplet ground state in bilayer graphene dots provides a route to exploring the Kondo effect with a small spin–orbit interaction.
Introduction
Electronic conduction at low temperatures can be affected by a small amount of magnetic impurities, a phenomenon known as the Kondo effect^{1}. The spin of a localized electron coherently couples to the spins of delocalized electrons in the host material, resulting in a net spin of zero and the formation of the Kondo screening cloud^{2}. Due to phasespace constraints, mainly electrons near the Fermi surface are affected, leading to the characteristic signature of the Kondo effect: a narrow resonance at the Fermi energy. The effect has been observed in a variety of materials, including graphene, with impurities acting as the host for the localized spin. Kondo temperatures in graphene were shown to reach up to 90 K^{3}.
The Kondo effect was discovered experimentally in semiconductor quantum dots in 1998^{4,5}, confirming theoretical predictions^{6,7,8,9,10}. Quantum dots with their net spins act as magnetic artificial atoms, and nearby leads take the role of the surrounding Fermi sea. The high tunablity of quantum dots is an important aspect of studies of the Kondo effect^{11}. Measurements on electrostatically defined quantum dots in GaAs enabled the observation of the unitary limit of the Kondo effect^{11}. In studies of quantum dots in carbon nanotubes the singlet triplet^{12} and the SU(2) and SU(4) Kondo effect were explored^{13}. Moreover, spin–orbit interaction in carbon nanotubes^{14} were found to change significantly the lowenergy Kondo physics, which often complicates studies of strongly correlated effects.
Improvements in the fabrication of nanostructures^{15,16,17} in 2D materials pave the way to reveal Kondo physics in quantum dots electrostatically defined in a flat bilayer graphene sheet with small spin–orbit coupling. In addition an unusual twohole triplet ground state^{15}, and an exceptional tunability of tunnel rates, dot size and valley magnetic moment^{18}are present. Measuring the Kondo resonance in different magnetic fields, allows to identify a clear level scheme for the first two charge carriers loaded into the dot as well as a spin orbit splitting of 80 μeV. The interplay between a small spin–orbit coupling and the Kondo effect is further known to lead to an underscreened spin1 Kondo effect, in which the net spin of the conduction electrons only partially compensates the localized spin. This effect has been observed in mechanically controlled individual cobalt complexes^{19}, but not in quantumdot systems, which have the advantage of being electrically tunable.
Results and discussion
Our gatedefined quantum dots in bilayer graphene (shown in Fig. 1a, b) are investigated through twoterminal AC measurements (see methods for details). The lowtemperature differential conductance G of a strongly coupled dot is presented in Fig. 1d as a function of DC source–drain bias V_{SD} and plunger gate voltage V_{P}. The regions of low conductance (white) are caused by Coulomb blockade, whereas the lines of higher conductance (light blue) are edges of the Coulomb blockade diamonds. Within each diamond, the quantum dot hosts a fixed integer number N of holes. When filled with many holes, a fourhole shellfilling periodicity emerges^{16,20,21}, which reflects the fourfold degeneracy of the graphene spectrum: one factor of two from the spin and one from the isospin (K, K′) that stems from the valley degree of freedom.
Most importantly, the differential conductance exhibits lines of high conductance centred at V_{SD} = 0 within the N = 1, 2, and 3 Coulomb diamonds, which is a signature of Kondoassisted tunneling through the quantum dot. The zerobias resonance is absent for N = 4, where all spin and valley states pair up (see Fig. 1c). For odd filling (N = 1, 3), the QD is charged with spin S = 1/2, leading to a mixing of the spin and valley Kondo effect. For halffilling (N = 2), a weaker Kondo resonance is observed. The QD is charged with two holes with the same spin and the spin–1 Kondo effect is expected. The presence of the Kondo effect is confirmed by the temperature dependence of the zerobias conductance as a function of gate voltage (Fig.1e). The individual traces taken at different temperatures show an increasing conductance with decreasing temperature for N = 1, 2, 3.
In Fig. 2a, we show how for N = 1 the single Kondo peak at B = 0 splits into four peaks at finite magnetic field, when measured in the centre of the N = 1 diamond at constant V_{P}. This is made visible in Fig. 2b by measuring line cuts at B = 0 and 0.4 T. The observed split peaks are labeled α (red arrows), and β, γ (green arrows) in Fig. 2a, b. We use a peakfinder algorithm to identify the conductance peaks α and β from Fig 2a and plot them as blue and red data points in Fig. 2c, respectively. The slope of the conductance resonance α is described by the spin gfactor of two. Extrapolating the conductance peak belonging to this resonance with a straight line to B = 0 (red dashed lines in Fig. 2a) gives a zerofield splitting Δ_{SO} = 80 μeV, which we interpret as a spin–orbit splitting; note that this splitting is not directly resolved at B = 0 in Fig. 2a, b. The observed splitting is of the same order of magnitude as the spin–orbit gap in previous measurements in quantum point contacts in bilayer graphene^{22} and more than a factor of three smaller than in carbon nanotubes^{23,24}. This difference is due to the different origin of the spin–orbit coupling: in nanotubes it originates from the nanotube chirality, whereas in graphene spin–orbit coupling is of Kane–Mele type^{25} and, according to ref. ^{26}, its value is determined by π − σ bands mixing promoted by the zeropoint outofplane phonons to the value of 0.1 meV, in good agreement with our measurements. Fig. 2
d shows the onehole energylevel spectrum. The respective alignment of the magnetic moments is shown with four small icons, where the red arrow represents the spin magnetic moment and the blue arrow the valley magnetic moment. In the presence of spin–orbit coupling, it is composed of two Kramer pairs split by the spin–orbit gap. The ground state exhibits a preferred parallel alignment of the valley and the spin magnetic moments. Applying a perpendicular magnetic field separates each energy doublet into two states with slopes proportional to either g_{v} ± g_{s} or − g_{v} ± g_{s}, with valley and spin gfactors g_{v} and g_{s}, respectively. This onehole level scheme results in the three magnetic fielddependent excitations α, β, and γ (see Fig. 2).
The conductance peak shown in blue in Fig. 2c includes the two expected excitations β and γ, which are difficult to separate experimentally (blue and green lines in Fig. 2c). This is due to the strong valley splitting with ∣g_{v}∣ = 38 and the finite V_{SD} bias window, which limits the observation of these transitions to small B_{⊥}. However, the valley splitting depends only on the perpendicular component of the magnetic field and is shifted to lower V_{SD} when the sample is rotated by 80^{∘}. The spin splitting remains constant upon rotation as it depends on the total magnetic field. Therefore both excitations β and γ can be observed separately at higher magnetic fields above 2 T as shown in Fig. 2e. At large enough field the spins now align with the almost parallel magnetic field leading to a transition \(\epsilon ^{\prime}\) similar to ϵ. This measurement confirms the presence of the energy level spectrum shown in Fig. 2d.
In a magnetic field applied parallel to the sample (Fig. 2f), the spin degeneracy of the zerofield spectrum is lifted while the valley states are not affected. For N = 1 the Kondo resonance splits into three peaks (Fig. 2f). The outer two resonances (marked with red dashed lines) split off as described by g_{s} = 2. Within our model, the resonance at V_{SD} = 0 and finite parallel field is a pure valley Kondo resonance with fluctuations of the state in the dot between K↑ and \(K^{\prime} \uparrow\).
In the absence of spin–orbit interaction an SU(4) Kondo model will describe the zerobias resonance for N = 1 at zero magnetic field, with a characteristic temperature dependence of the Kondo peak conductance^{27}. Deviations from this prediction due to the presence of the spin–orbit gap can therefore be studied via the temperature dependence of the maximum conductance of the Kondo resonance at V_{SD} = 0 and B = 0. The corresponding data (blue data points in Fig. 2g) can be fitted with a model for the temperature dependence of the SU(2) Kondo effect with spin S = 1/2, given approximately by^{28}
with \({T}_{{{{{{{{\rm{K}}}}}}}}}^{\prime}={T}_{{{{{{{{\rm{K}}}}}}}}}/{({2}^{1/s}1)}^{1/n}\), where s = 0.22 and n = 2 for a spin1/2 system, G_{0} is the amplitude of the peak, T_{K} the Kondo temperature and T the electron temperature. We compare this fit (blue solid line in Fig. 2g) with the corresponding fitted temperature dependence of the SU(4) Kondo model with s = 0.2 and n = 3 (ref. ^{27}; blue dashed line). The data agree better with the SU(2) Kondo model. Numerical renormalization calculations (see Supplementary Note 1) confirm that a small spin–orbit energy splitting will lead to a better fit to the peak by the SU(2) form. Yet it is not enough to split the zerobias peak (calculations shown in the Supplementary Fig. 1). (Another reason for the apparent transition from SU(4) to SU(2) are different tunnel rates of different channels^{29}.) In summary, the measurements of N = 1 h can be completely described by a SU(2) Kondo effect due to spin–orbit splitting of the 4fold degeneracy.
Figure 3a shows the magnetic field dependent energy spectrum for N = 2^{15}. The six twohole states can be constructed from linear combinations of the four degenerate spin and valley singleparticle states. Exchange interaction splits these states into a spintriplet ground state (labeled \(\left{S}_{V}\right\rangle \left{T}_{S}^{}\right\rangle\), \(\left{S}_{V}\right\rangle \left{T}_{S}^{0}\right\rangle\) and \(\left{S}_{V}\right\rangle \left{T}_{S}^{+}\right\rangle\) in Fig. 3a) that is threefold degenerate at zero magnetic field, a spinsinglet state with a twofold valley degeneracy (labeled \(\left{T}_{V}^{}\right\rangle \left{S}_{S}\right\rangle\), \(\left{T}_{V}^{+}\right\rangle \left{S}_{S}\right\rangle\)) at zero magnetic field, and a single spinsinglet valleytriplet state at the highest energy^{15}. Applying a parallel magnetic field, the triplet ground state splits into its three spin components, while all the other excited states remain unaffected. Applying a perpendicular magnetic field splits the valleytriplet spinsinglet excited state, leading to a strong energy reduction of \(\left{T}_{V}^{}\right\rangle \left{S}_{S}\right\rangle\) with increasing magnetic field and a strong energy increase of \(\left{T}_{V}^{+}\right\rangle \left{S}_{S}\right\rangle\). The \(\left{S}_{V}\right\rangle \left{T}_{S}^{}\right\rangle\) sets the origin for the cotunneling excitations in parallel magnetic field and for small perpendicular fields. The excitations (shown as vertical arrow and labeled α, β, γ, and ϵ) belong to the transitions from the ground state to the excited states shown in Fig. 3a.
Figure 3b shows the splitting of the Kondo resonances as a function of B_{∥}. The resonances α and γ introduced in Fig. 3a are observed and marked with red and green dashed lines. Slope and offset are given by the spin Zeeman effect (g_{s} = 2) and the exchange energy ΔE_{Exch} = 0.37 meV. Resonance α appears as a splitting of the B = 0 Kondo resonance, whereas resonance γ is seen as a cotunneling resonance already at B = 0 but finite V_{SD} (see also Fig. 1c). Because the excited state energy is independent of B_{∥}, this resonance runs in parallel to the split Kondo resonance. The expected transition labeled β is not seen experimentally in Fig. 3b and c as it would require spin flips of both holes.
Corresponding data for B_{⊥} are depicted in Fig. 3c. The Kondo resonance α (red dashed line) is observed at \({B}_{\perp } \; < \;{B}_{\perp }^{\star }=0.2\ {{{{{{{\rm{T}}}}}}}}\), but its Zeeman splitting remains unobservable at these low fields. The cotunneling transition γ (green dashed line), splitoff from V_{SD} by the exchange energy at B_{⊥} = 0, moves down linearly with increasing B_{⊥} according to the valley Zeeman effect, hitting V_{SD} = 0 at \({B}_{\perp }^{\star }\), as expected from Fig. 3a. The extracted ∣g_{v}∣ = 38 is in excellent agreement with the valley splitting determined for N = 1. At \({B}_{\perp }^{\star }\), the excitations α and γ converge, resulting in an enhanced zerobias peak. The state \(\left{T}_{V}^{}\right\rangle \left{S}_{S}\right\rangle\) is the system ground state for \({B}_{\perp } \; > \;{B}_{\perp }^{\star }\) the transition to the \(\left{S}_{V}\right\rangle \left{T}_{S}^{}\right\rangle\) state is the experimentally dominating cotunneling transition involving a simultaneous valley and spin flip.
Figure 3d shows the differential conductance at B = 0 for temperatures between 0.08 K (blue) and 0.8 K (red). The finitebias resonances at V_{SD} = ± 0.37 mV are marked by red and green arrows and the Kondo resonance by a blue arrow. We study the temperaturedependent conductance of the three resonances in detail in Fig. 3e, where the blue data points belong to the Kondo resonance, while the red and green data points belong to the finitebias resonances. The nonequilibrium data at finitebias are in surprisingly good agreement with an equilibrium SU(2) Kondo scaling.
Due to the spin triplet ground state in bilayer graphene quantum dots, we expect a spin1 Kondo effect. Localized spin1 magnetic impurities connected to two conduction channels each with s = 1/2 screening capacity can be screened by two (fully screened), one (underscreened) or none (nonKondo regime) of the channels. In Fig. 3e we show fits to the data points using the underscreened^{19} (blue solid line) and fully screened (blue dashed line) spin1 Kondo models. The underscreened Kondo model is in better agreement with the data, in particular at the lowest temperatures (for details of the fitting parameters see the NRG calculation results in the Supplementary Note 1). Note that changing G_{0} will also change the slope of the fullyscreened spin1 model, hence a better agreement with the measurement is not possible (see also Supplementary Fig. 3). The fully screened spin1 model required exact symmetry of the two channels. Thus any splitting, e.g., due to spin–orbit interaction, will lead to a difference between the Kondo temperatures of each conduction channel, and to a twostage Kondo effect^{30}. The data agree with such a scenario where the temperature T lies around the firststage Kondo temperature, such that only one channel participates in the screening and the secondstage Kondo temperature lies below the measured temperature T.
We have studied the breaking of the SU(4) symmetry of the Kondo effect for N = 1 due to spin–orbit coupling with a magnitude of 80 μeV. This spin–orbit coupling strength is in agreement with the theoretical estimate^{26} of the enhancement of the KaneMele gap by the outofplane zeropoint vibrations of graphene. Furthermore, the spin triplet ground state for N = 2 allows us to study the dependence of the spin1 Kondo resonance on magnetic field as well as temperature together with a small spin–orbit interaction in the QD. This spin–orbit interaction in bilayer graphene can lead to an underscreening of the spin in the quantum dot. The spintriplet ground state, together with the wide range of tunability, makes quantum dots in bilayer graphene an interesting experimental platform for studying the underscreened spin1 Kondo effect as well as the interplay between spin–orbit coupling and the Kondo effect in graphene nanostructures.
Methods
The investigated bilayer graphene flake is embedded between two hBN flakes (20 nm and 39 nm thick) and stacked on top of a graphite back gate using the dry transfer technique. Source and drain contacts are fabricated by etch contacting^{31}. The gate structure fabricated on top of the stack is shown in the atomic force microscopy (AFM) image in Fig. 1b. The split gates (shown in grey) can be used to form a 800 nm long and 100 nm wide channel in the bilayer graphene flake. As a second layer of gates, five finger gates (20 nm wide) are deposited on top of an Al_{2}O_{3} layer. Back and top gates can be used to (i) open a band gap below the gates and (ii) tune the Fermi energy into the band gap, rendering these regions insulating. An ntype channel is formed between the split gates by applying a positive voltage to the graphite back gate (V_{BG} = 3.7 V) and a negative voltage to the split gates (V_{SG} = −2.92 V). An inplane sourcedrain bias voltage V_{SD} is applied to the channel using the pair of Ohmic contacts.
A fully tunable quantum dot can be formed in the channel using the three finger gates (colored red and blue) in Fig. 1b. The finger gate on top of the quantum dot (colored blue) is used to form an ntype quantum dot and controls the number of charge carriers in the quantum dot. The outer two finger gates (colored red) are used to tune the tunnel coupling over wide ranges by gate voltages. We can deplete the dot down to the last electron as seen from Coulomb blockade resonances.
The electrical properties are investigated through twoterminal AC measurements, in which a variable DC voltage V_{SD} and a AC component V_{AC} = 0.020 mV are applied between source and drain contacts, where the differential conductance G = ∂I/∂V_{SD} is measured by standard lockin techniques. The hole occupancy of the dot is controlled by the centre finger gate (shown in blue in Fig. 1b) by a voltage V_{P}. We measure the device in a ^{3}He/^{4}He dilution refrigerator with a base temperature of 80 mK, fitted with a rotatable sample stick for outofplane rotations of the sample in magnetic fields of up to 8 T.
Data availability
All data generated in this study have been deposited in the ETH database under accession code https://doi.org/10.3929/ethzb000504566 [http://hdl.handle.net/20.500.11850/504566].
Code availability
The code used for the NRG calculations is publicly available at http://www.phy.bme.hu/d̃mnrg/.
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Acknowledgements
We thank Peter Märki, Thomas Bähler as well as the staff of the ETH cleanroom facility FIRST for their technical support. We also acknowledge financial support by the European Graphene Flagship and the ERC Syngery Grant Quantropy. Growth of hexagonal boron nitride crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan, Grant Numbers JPMXP0112101001, JSPS KAKENHI Grant Number JP20H00354 and the CREST(JPMJCR15F3), J.S.T. We acknowledge funding from the European Union’s Horizon 2020 research and innovation program under the Marie SkłodowskaCurie Grant Agreement No. 766025. YM acknowledges the support of the Israel Science Foundation (grant no. 359/20). F.K.d.V acknowledges support from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 862660/QUANTUM E LEAPS.
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A.Ku. performed the experiments and fabricated the sample with help of M.E., C.T. and R.G. A.Ku. analyzed the data with the assistance of C.G., M.E., C.T., R.G., C.M. and F.V. K.W. and T.T. synthesized the hBN crystals. Y.K. and Y.M. developed the theoretical understanding and Y.K. performed NRG calculations. A.Kn. and V.F. developed the theoretical understanding of the spin–orbit coupling. Y.M., T.I. and K.E. supervised the project. All authors discussed the results.
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Kurzmann, A., Kleeorin, Y., Tong, C. et al. Kondo effect and spin–orbit coupling in graphene quantum dots. Nat Commun 12, 6004 (2021). https://doi.org/10.1038/s41467021261493
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DOI: https://doi.org/10.1038/s41467021261493
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