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. 2025 Mar 27;25(14):5762–5769. doi: 10.1021/acs.nanolett.5c00365

Stable Unpaired Electron States in the Lu–Lu Bond Leading to the Absence of Odd–Even Parity in the Kondo Effect of Lu2@C82 Transistors

Jun Chen †,‡, Yuan Shui §, Wangqiang Shen ∥,⊥, Feng Wang †,‡, Yifu Mao †,‡, Haoran Qi †,‡, Xinrong Liu †,‡, Yu Du †,‡, Fucong Fei †,#, Yuyuan Qin †, Jianguo Wan †,‡, Lipiao Bao ∥, Minhao Zhang †,‡,*, Tao Yang §,*, Xing Lu ∥,*, Fengqi Song †,‡,*
PMCID: PMC12005644  PMID: 40145565

Abstract

graphic file with name nl5c00365_0005.jpg

Spin qubits constructed in endohedral fullerenes benefit from the protective shielding of the carbon cage, which effectively mitigates external decoherence and enables ultralong coherence times. However, endohedral fullerene spin qubits face the challenge of charge transfer in complex electrical environments, such as during qubit readout or large-scale integration, which can induce spin state modifications. In this study, we developed transistors based on the endohedral fullerene Lu2@C82 and observed the absence of parity dependence in the Kondo effect; this result was contradictory to the typical behavior of the Kondo effect observed in C60. Density functional theory calculations revealed that upon electron loss, a spin-1/2 electron predominantly from the s-orbitals formed in the Lu–Lu bond and its orbital energy was significantly lower than that of the highest occupied molecular orbital. Based on these results, Lu2@C82 held stable unpaired electron states across multiple charge states and has potential applications in spin quantum devices.

Keywords: spin qubits, endohedral fullerenes, transistors, Kondo effect, density functional theory

An endohedral fullerene (EF) consists of a carbon cage that encloses an embedded entity, where the cage protects the internal spin from decoherence. As a result, an EF exhibits extended coherence times, making it an ideal candidate for spin-based quantum computing.1−6 Additionally, the EF system can be conveniently assembled into complex structures using chemical engineering techniques,2,7,8 for example, embedding EFs into single-walled carbon nanotubes to arrange them as a one-dimensional linear qubit register.9 Alternatively, embedding EFs into three-dimensional metal–organic frameworks creates a three-dimensional qubit array.2,10

The spin-1/2 state of an electron is one of the simplest two-level quantum systems.1,11,12 The spin-1/2 unpaired electron state in the dimetallofullerene (di-EMF) with a single-electron metal–metal bond exhibits the potential for electron spin qubits.7,13−16 Recent studies report that CaY@Cs(6)-C82, which possesses a single-electron Ca–Y bond and implements a single-qubit logic gate, exhibits a decoherence time of up to 7.74 μs and maintains quantum coherence at 170 K.16 In addition to the protection provided by the carbon cage, the unpaired electron in this bond is associated primarily with s-orbitals, which lack orbital angular momentum and do not exhibit spin–orbit coupling. These factors contribute to the prolongation of the spin–lattice relaxation time.16,17 However, spin qubit devices based on EFs experience charge transfer in complex electrical environments, such as during multiqubit integration and electrode contact, which may result in spin state perturbations.18−22 To ensure reliable performance, the spin qubits must exhibit sufficient robustness, remaining stable across multiple charge states.

Lu2@C82 is a notable di-EMF in which the formation of a two-electron Lu–Lu bond thermodynamically stabilizes the cluster structure,23−27 and a covalent bond also exists between the Lu2 unit and the carbon cage.28 Upon the loss of an electron through oxidation, the Lu–Lu bond transitions from a two-electron covalent bond to a one-electron covalent bond.13,29 The Lu–Lu bond is primarily composed of s-orbitals,28 which endow the electron spin with advantages of long coherence time and high operation temperature.3,16,30 Furthermore, the phase coherence time can be further enhanced through the clock transition mechanism.17,31 This indicates that Lu2@C82 is highly suitable for quantum applications that require long spin coherence times.

In this study, we synthesized and purified the Lu2@C82 cluster and fabricated single-cluster transistors (SCTs); here, we observed a distinctive zero-bias conductance peak (ZBP) across multiple charge states. Temperature and magnetic field measurements indicated that this ZBP originated from the Kondo effect. We performed orbital and spin density functional theory (DFT) calculations on Lu2@C82 in various charge states and proposed that Lu2@C82 harbored stable unpaired electrons in multiple cationic charge states; this accounted for the absence of parity dependence in the Kondo effect. Furthermore, the spin-1/2 electron in Lu2@C82 has significant potential as a robust electron spin qubit with long coherence times; thus, it is a promising candidate for quantum computing applications.

We synthesized lutetium endohedral metallofullerenes in soot via the direct current arc discharge technique23,32 and subsequently isolated the di-EMF Lu2@C3v(8)-C82 using a multistep high-performance liquid chromatography (HPLC) strategy (see Supplementary Note 1 for details). The high purity of isolated Lu2@C3v(8)-C82 was confirmed by analytical HPLC (Figure 1a) and laser desorption ionization time-of-flight (LDI-TOF) mass spectrometry. Figure 1b presents the LDI-TOF mass spectrum, confirming that the molecular weight of Lu2@C3v(8)-C82 is 1334 Da, consistent with its theoretical molecular formula. The inset shows that the measured isotopic distribution of Lu2@C3v(8)-C82 matches well with the calculated results. Figure 1c displays the visible–near-infrared (vis–NIR) absorption spectrum of Lu2@C3v(8)-C82, which presents two distinct peaks at 705 and 895 nm with an onset at 1024 nm, corresponding to a relatively large optical band gap (1.21 eV). Single-crystal X-ray diffraction was used to determine the structure of Lu2@C3v(8)-C82.23 The above characterization results are consistent with the high-purity Lu2@C3v(8)-C82 reported in the literature.23

Figure 1.

Figure 1

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(a) Analytical HPLC chromatograms of Lu2@C3v(8)-C82. (b) LDI-TOF mass spectrum of Lu2@C3v(8)-C82. Insets: measured and calculated isotopic distributions of Lu2@C3v(8)-C82. (c) Vis–NIR absorption spectrum of Lu2@C3v(8)-C82 in CS2 at room temperature, with the inset showing a CS2 solution of Lu2@C3v(8)-C82. (d) Schematic of the Lu2@C82 SCT structure. For clarity in the illustration, the carbon atoms blocking Lu2 are omitted.

We then fabricated Lu2@C82 SCTs, with the architecture shown in Figure 1d. A single cluster was placed in the gap between the gold electrodes, which functioned as the source and drain. The gap was created by breaking a gold nanowire using the feedback-controlled electromigration break junction (FCEBJ) technique. Figure 2a shows the changes in the current–voltage (I–V) curves during the FCEBJ process; here, the conductance of the gold nanowire gradually decreased as the break junction progressed, ultimately forming a nanometer-scale gap; the specific break junction process is detailed in the literature.33−36 The SCT was fabricated when the cluster was placed into the gap, establishing an appropriate coupling with the source and drain electrodes. Figure 2b also shows a comparison of the I–V curves for the SCT and the case where no cluster is present in the gap. The I–V curve without a cluster exhibits an open circuit. The success rate of detecting cluster signals using the FCEBJ method is approximately 10–20%.37,38

Figure 2.

Figure 2

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(a) Source–drain I–V traces of SCT during the break junction process using the FCEBJ method. The first I–V trace at the onset of electromigration is marked in black, while the final I–V trace at the termination is marked in red. The process of FCEBJ is indicated with a black dashed arrow. (b) Comparison of the I–V curves for the SCT formed after the break junction process and the open circuit (without SCT). (c) Distribution of the differential conductance of Device A as a function of the gate voltage and source–drain bias. The dashed lines indicate the boundaries of the Coulomb blockade region. (d) Distribution of the differential conductance for Device B. The differential conductance is calculated numerically from the current vs bias voltage data.

The bottom substrate of the device is p-type heavily doped silicon, which serves as the gate electrode, and the middle layer is a thin layer of silicon dioxide, which acts as the gate dielectric layer; the detailed device fabrication process can be found in the Supplementary Note 2. The gate electric field modulates the energy levels of the cluster, thus altering the transport characteristics of the SCT. The I–V characteristics of the transistor show different conductance behaviors under varying gate voltages (Figure S1).

In SCTs, electrons undergo single-electron tunneling through the discrete energy levels of the cluster. Under the control of the gate and bias voltages, Coulomb blockade effects are observed (Figures 2c,d).39,40 The dashed lines indicate the boundaries of the Coulomb blockade region with the intersections forming two degenerate points (Figure 2c). These points define three different charge states: (N+1)+, N+, and (N–1)+.40−42 Based on the Coulomb diamond pattern in Figure 2c, the charging energy of the device is estimated to be relatively large (exceeding 100 meV), indicating that the transport signal originates from the target cluster rather than from gold particles formed due to electromigration.34,43,44 The clear Coulomb blockade pattern indicates that only a single cluster is properly coupled to the electrodes on both sides.36,45 The subtle variations in the coupling between the cluster and the electrodes result in distinct patterns in Device A and Device B.19,46

In Figures 2c and 2d, ZBPs are observed, extending across multiple charge states with three in Device A and two in Device B. This phenomenon occurs in a subset of Lu2@C82 SCTs. ZBPs in SCTs are generally attributed to the Kondo effect; this effect describes a many-body interaction between the conduction electrons and localized spins.36,39,47 In SCTs, the Kondo effect enhances the forward scattering, leading to the formation of a virtual localized state near the Fermi level; this causes the ZBP.40,47,48 Although other potential sources of ZBPs have been identified in systems such as quantum wires and quantum dots, their conductance values are typically low.49,50 Moreover, the logarithmic dependence of the ZBP peak on temperature serves as a key indicator that it originates from the Kondo effect.40,48

We performed temperature-dependent measurements of the ZBP for Device A (Figure 3a) and observed a significant decrease in the peak height as the temperature increased. The ZBP peak height as a function of temperature was fitted to the formula predicted by numerical renormalization group theory for the single-channel Kondo effect for different spin quantum numbers (Figure 3b, fitting function shown in Supplementary Note 5).51,52 The fit based on the spin S = 1/2 Kondo effect formula showed excellent agreement, and a Kondo temperature of TK = 18 K was attained. This Kondo temperature was in close agreement with those reported in other SCTs.39,52,53

Figure 3.

Figure 3

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(a) Differential conductance as a function of bias voltage at different temperatures. (b) Temperature dependence of the ZBP peak height. The curves are fitted using Kondo effect formulas for different spin values. (c) Distribution of differential conductance with respect to the bias voltage and magnetic field. (d) Differential conductance as a function of bias voltage at various magnetic fields. The inset shows the change in the height of the ZBP as a function of the magnetic field. All measurements were taken from Device A, with a gate voltage of zero. (e–i) Schematic diagram of the sources of parity and nonparity Kondo effects. (g) Normal parity Kondo effect. (e) Triplet ground state. (f) Orbital Kondo effect. (h) Kondo effect in high-spin centers. (i) Protected unpaired electron.

We also conducted measurements of the ZBP under varying magnetic fields (Figure 3c,d) and observed a clear reduction in peak height with an increasing magnetic field; the direction of the magnetic field was perpendicular to the direction of the current. The ZBP associated with the Kondo effect split when the magnetic field exceeded a critical value, given by the formula54,55Bc = 0.75kBTK/gμB, where kB is the Boltzmann constant, μB is the Bohr magneton, and g ≈ 2 is the electron Landé factor.36,39,47 The device’s Bc is approximately 10 T. Since the critical field exceeded the experimental measurement range, we did not observe splitting of the ZBP. In the case of the spin-1 Kondo effect, splitting would occur at a lower magnetic field.52 The ZBPs observed in the three charge states of Device A decrease in peak height under a 5 T magnetic field, but no splitting is observed (Figure S2).

The typical ZBP phenomenon associated with the spin-1/2 Kondo effect is parity dependent;40,47,48 thus, it only appears in the charge states with an odd number of electrons.56 This occurs because the Kondo effect requires the presence of a local net spin, and in orbitals with spin degeneracy, electrons fill the states in an antiparallel manner due to the Pauli exclusion principle.47,57 In charge states with an even number of electrons, the net spin is 0 (as depicted in Figure 3g). However, in our experimental observations, Device A exhibits a Kondo effect without the expected parity dependence and ZBPs are observed in three consecutive charge states (Figure 2c). In some previous studies on quantum dot system devices, ZBPs have been observed in two consecutive charge states.52,58−61 Some of these occurrences have been explained by the fact that the net spin of adjacent states in high-spin magnetic molecules is nonzero58 (as shown in Figure 3h). Other sources are explained by magnetic field or electron spin exchange interactions, which lead to the triplet state being the ground state in the even charge state, giving rise to the spin-1 underscreened Kondo effect52,59,60 (Figure 3e). In single-walled carbon nanotubes, the double degeneracy of spin and orbital states also leads to the non-odd–even Kondo effect61 (Figure 3f). Here, we propose a new explanation by combining experimental observations and theoretical calculations described below: the carbon cage protects the unpaired electron, allowing it to maintain Kondo interactions with the conduction electrons across several charge states (Figure 3i).

We calculated the energies of different spin states near the neutral charge state (see Table 1), where the spin states marked in bold have energies lower than those of the others; thus, these spin states were the spin ground states for each charge state. At the same time, we also observe that in the even 2+ charge state, a specific situation arises where the net spin is nonzero, resulting in multiple consecutive cationic states exhibiting nonzero net spins. Additionally, due to the charge transfer phenomena that occur between the cluster and the gold electrode, the cluster may be in a charged state under zero gate voltage.18−21 To investigate this, we calculated the differential charge density of the Lu2@C82 single-cluster device and found electron transfer to the gold electrodes (see Figure S5), leaving Lu2@C82 in a cationic state at zero gate voltage. Therefore, the several consecutive charge states exhibiting the Kondo effect in Figure 2c for Device A may correspond to the cationic states of Lu2@C82.

Table 1. DFT Calculations of the Energies for the Different Spin States near the Neutral Charge State.

  Charge State
  5+ 4+ 3+
Spin 1/2, 3/2, 5/2 0, 1, 2 1/2, 3/2, 5/2
Energy Difference (eV)
graphic file with name nl5c00365_m001.jpg
 Es=0 – Es=1 = 0.20
graphic file with name nl5c00365_m002.jpg
  Charge State
  2+ 1+ 0
Spin 0, 1, 2 1/2, 3/2 0, 1
Energy Difference (eV) Es=0 – Es=1 = 0.18
graphic file with name nl5c00365_m003.jpg
 Es=1 – Es=0 = 0.93
  Charge State
  1– 2– 3–
Spin 1/2, 3/2 0, 1, 2 1/2, 3/2, 5/2
Energy Difference (eV)
graphic file with name nl5c00365_m004.jpg
 Es=1 – Es=0 = 0.23
graphic file with name nl5c00365_m005.jpg
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To investigate the unique spin filling in the cationic state of Lu2@C82, we performed further DFT calculations to examine the orbital characteristics of Lu2@C82. We calculated the orbital distribution near the Fermi level of neutral Lu2@C82 (see Figure 4a). The results indicated that the highest occupied molecular orbital (HOMO) of neutral Lu2@C82 was 97% localized on the Lu–Lu bond and the remainder was primarily distributed on the carbon cage. This result suggests that near the Fermi level, the Lu2 orbitals and the carbon cage orbitals exhibit minimal hybridization and remain largely independent.

Figure 4.

Figure 4

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(a) DFT calculation of the orbital distribution near the Fermi level. (b) DFT calculation of the orbital arrangements for different charge states. The gray horizontal lines represent the carbon cage orbitals, and the blue horizontal lines represent the Lu2 orbitals. The blue spheres and arrows represent the electrons and their spins residing in the carbon cage, while the red spheres and arrows represent the electrons and their spins residing on Lu2. (c) DFT-calculated spin density distributions for various charge states. Additional cationic state distributions are shown in Figure S4.

We further calculated the orbital configuration in different charge states (Figure 4b). In the neutral state of Lu2@C82, the Lu2 orbital occupied the HOMO, which was fully occupied by two paired electrons. In [Lu2@C82]+, one electron was removed from the Lu2 orbital, forming a single-electron Lu–Lu bond. This orbital experienced a significant energy shift, moving to the HOMO–7. Upon further electron loss, the Lu2 orbital remained at the HOMO–7 or HOMO–6. This phenomenon was also observed in certain di-EMFs13,62,63 and was attributed mainly to the interaction between the metal–metal bond and the carbon cage. Based on this energy shift, for the cationic variants of Lu2@C82, the primary mode of electron–environment interaction was mediated by the carbon cage. Furthermore, the C82 cage, which is an effective electron donor, played a crucial role in maintaining the stability of the spin-1/2 orbital of Lu2. We also calculated the effect of the electric field on the energy shifts of the orbitals64 (Supplementary Table 1). The shielding effect of the carbon cage on the embedded Lu2 results in the orbital energy of Lu2 being far less influenced by the electric field compared to that of the carbon cage.

We also conducted DFT calculations to analyze the spin density distributions of neutral Lu2@C82 and its cationic states [Lu2@C82]+, [Lu2@C82]2+, and [Lu2@C82]3+ (Figure 4c). The three cationic states all exhibited localized spin density on the Lu–Lu bond, indicating that an unpaired electron with spin-1/2 remains in the Lu–Lu bond. For the even-electron state, [Lu2@C82]2+, both the Lu2 and the carbon cage exhibited localized spin, resulting in a “triplet state”.

Based on these theoretical calculations and the structure of Lu2@C82, we propose a specific transport mechanism to explain the nonparity Kondo effect in Lu2@C82 SCTs65 (schematic diagram shown in Figure S3). The charge states measured in Device A correspond to the cationic states of Lu2@C82, where a stable unpaired electron resides in the Lu–Lu bond. Conducting electrons interact with the localized spin-1/2 on the Lu–Lu bond; this produces the Kondo effect and results in the ZBP. The additional electron in the intermediate N+ state compared to the (N+1)+ state does not occupy the Lu2 orbital but instead fills the orbital of the carbon cage. Owing to the independent nature of the Lu2 and carbon cage orbitals, the Lu2 orbital retains its localized spin-1/2. Therefore, the N+ state still has a Kondo effect and ZBP. Similarly, in the (N–1)+ state, the newly added electron preferentially fills the carbon cage orbital; thus, the localized spin-1/2 on the Lu2 orbital continues to interact with the conducting electrons, the Kondo effect is maintained, and the ZBP phenomenon is preserved in the (N–1)+ state.

In summary, we synthesized and purified the Lu2@C82 cluster and fabricated Lu2@C82 SCTs using the FCEBJ method. Electrical transport measurements of the SCTs revealed the presence of Kondo ZBPs, which appeared to be independent of the electron number parity. Based on DFT results and the unique structure of Lu2@C82, we propose that the stable unpaired electron states in its cationic states lead to the absence of odd–even parity in the Kondo effect. Furthermore, our calculations indicate that the Lu–Lu bond is primarily composed of s-orbitals in both neutral and cationic states (Supplementary Table 2). These characteristics endow Lu2@C82 with the potential to serve as a robust electron spin qubit with a long coherence time. Notably, the transistor structure enables efficient initialization and manipulation of the electron spin qubit via gate-applied pulsed voltages,66−68 while allowing direct electrical readout of the spin state.18

Acknowledgments

We acknowledge the financial support of the National Key R&D Program of China (Grant No. 2022YFA1402404), the National Natural Science Foundation of China (Grant Nos. 92161201, 12025404, T2221003, 12422410, 12104221, 12274337, 22431005, 92261204, 21925104, 22001084, 22201086, 12474272, 12374043, 12274208, T2394473, T2394470, and 62274085), and the Natural Science Foundation of Jiangsu Province (Grant Nos. BK20243013, BK20233001, BK20240166, and BK20230079).

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.5c00365.

  • Synthesis and isolation of Lu2@C3v(8)-C82; device fabrication; theoretical calculations; data processing method; fitting formula for Kondo ZBPs; the current–voltage (I–V) curves of Device A under different gate voltages; differential conductance response of Device A to magnetic field; schematic of the transport process for the different charge states; DFT calculation of the spin density distribution in the 4+ and 5+ charge states; differential charge density of the Lu2@C82 single-cluster device; drain–source current as a function of gate voltage at a fixed drain–source bias of 2 mV for Device A; DFT calculations of the effect of the electric field on the energy shifts of the different molecular orbitals; amount of s spin density population in different charge states (PDF)

Author Contributions

Jun Chen, Yuan Shui, and Wangqiang Shen contributed equally.

The authors declare no competing financial interest.

Supplementary Material

nl5c00365_si_001.pdf (634.8KB, pdf)

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