Two-electron conduction band of a graphene quantum dot and coherent spin manipulation

To get a carbon-based qubit, we pay attention to the two-electron conduction band of a graphene quantum dot (GQD) in the presence of an external magnetic field and an extrinsic Rashba spin-orbit interaction (SOI). To help understand the formation of the two-electron spectra, we first calculate the tight-binding (TB) spectra. There exist the sensitivity of the conduction band to magnetic fields and the mixing of spin states induced by a Rashba SOI. The two factors inspire the study of the magnetic-field modulation of the conduction band for realizing a spin qubit. We present the method for calculating the electronic structure of a few-electron GQD. The roles of the Coulomb interaction and the Rashba SOI in the two-electron conduction band are investigated. The Coulomb interaction contributes to a singlet-triplet level crossing and the Rashba SOI leads to a singlet-triplet mixing. The fast initialization and coherent manipulation of spin states are demonstrated by the magnetic control of singlet-triplet splitting.


Introduction
Electron spin is one of the most significant candidates for solid-state qubits [1] and some progresses have been made using gate-defined semiconductor quantum dots [2,3].To suppress hyperfine-induced dephasing, it is desirable to develop qubits in carbon-based materials, where hyperfine interaction is absent [4].GQDs exhibit very long spin-decoherence times and thus are expected to become an ideal platform for fabricating spin qubits [5][6][7][8][9].For extending spin decoherence times, it is helpful to use two-electron spins with a different degree of freedom from nuclear spins [10][11][12].The electronic properties and magnetism of GQDs have been widely explored both theoretically and experimentally [13][14][15][16][17][18], which provides an essential basis for realizing a graphene qubit.
As the GQD sizes become smaller, the shapes and edge types of GQDs together play a dominant role in determining the electronic and magnetic properties.The most investigated shapes of GQDs include triangle, rectangle and hexagon [15] and the most stable edge types are armchair and zigzag edges [19].Examples of GQDs with different shapes and edges are schematically shown in figure 1. Single-layer GQDs have been mostly realized by top-down lithography [20,21].While many of the basic quantum transport properties such as Coulomb blockade [20,22,23] and charge detection [24] have been experimentally demonstrated, the limitation of top-down lithography leads the sample edges to remain rough on the atomic scale and thus the orbital and spin properties of specific states remained elusive.Recent synthesis of zigzag triangular GQDs with the atomically precise edges overcame the issue of rough edge states [25][26][27][28].The electronic structures of the zigzag triangular GQDs were unraveled by measuring scanning tunneling spectroscopy [26,28].The single-electron transport based on scanning tunneling microscopy is proposed [29] and the few-electron regime in such synthetic GQDs is believed to be within reach.In bilayer graphene, GQDs can be created via electrostatically induced quantum confinement and have a tunable energy gap controlled by out-of-plane electric fields [30].The time-reversal symmetry of bilayer graphene can be controllably broken by electric fields, providing an additional valley degree of freedom to realize qubits [31].The demonstration of spin and valley blockade enables qubit manipulation and readout [32,33].The experimental synthesis of atomically precise bilayer GQDs remains a great challenge while the electronic properties of such GQDs with well-defined edges have been extensively investigated [34][35][36][37].
To manipulate the spins of single-layer GQDs with well-defined edges, the effects of external electric and magnetic fields on them have been extensively theoretically investigated [38][39][40][41][42][43][44][45][46].The magnetic-field manipulation with Zeeman spin splitting has its own advantage over the electric-field manipulation [10,47].In theoretical studies of manipulating the GQD spins by magnetic fields, zigzag triangular GQDs have gained a lot of attention due to the spin polarization induced by their zero-energy degenerate shells [44][45][46].The two-electron spin polarization in zigzag triangular GQDs has been both theoretically predicted [48] and experimentally observed [49].Unfortunately, the orbital energies of the degenerate shells are rather insensitive to external magnetic fields and thus the magnetic-field manipulation can only rely on the Zeeman effect [44][45][46].For two-electron spin states of zigzag triangular GQDs, given a larger singlet-triplet energy difference, it is difficult to compensate the energy difference by the Zeeman energy and thus achieving ground-state spin switches solely through magnetic fields is not feasible [46].
In this paper, we explore the possibility of realizing a spin qubit based on the conduction band of a few-electron zigzag triangular GQD.To achieve this goal, it is necessary to understand the complex Coulomb interactions and to realize the coupling of spin states.Three specific things are performed.First, we present the method for calculating the electronic structure of a few-electron GQD, including the effects of an external magnetic field and an extrinsic Rashba SOI.Second, for a triangular zigzag GQD, we analyze the effects of the Coulomb interaction and the Rashba SOI on the two-electron conduction band.Third, we demonstrate the Rabi oscillations of two-electron spin states dominated by the competition between a singlet-triplet splitting and a Rashba SOI splitting.

TB energy spectra
As a starting point, we calculate the electronic states of a charge-neutral GQD within the TB model, including the effects of an external magnetic field and an extrinsic Rashba SOI.The magnetic field B is perpendicular to the plane of the GQD.The Rashba SOI is characterized by the interaction strength t R , which can be tuned within the range 0-0.225 eV by an external electric field [50,51] or a Ni(111) substrate [52].The TB model cannot capture all edge effects, for example, the deviation of the electron density from the bulk value [53].Nevertheless, the TB model provides the most fundamental understanding of the roles of the magnetic field and the Rashba SOI in the energy spectra.
The TB Hamiltonian takes the form [35,44]: The operator c † iσ (c iσ ) creates (annihilates) an electron with spin σ =↓, ↑ at site i = 1, . .., N, where N is equal to the total number of atoms.The hopping integral t i,l between sites i and l is taken to be −2.5 eV for nearest neighbors and −0.1 eV for next-nearest neighbors [54].The first part of the Hamiltonian describes the effect of B on the orbital energy by the Peierls substitution [44,55].The phase ϕ il is accumulated by an electron going from site i to l due to the magnetic field [46].The second part is the Zeeman energy with the g factor g = 2 for graphene [56].The last part describes the Rashba SOI, where ⃗ τ = (τ x , τ y , τ z ) is the Pauli matrix vector and ⃗ d ⟨il⟩ is the unit vector connecting the nearest-neighbor sites i and l.
For a triangular zigzag GQD consisting of N = 61 carbon atoms (see figure 1(c)), the TB spectra contains 2N = 122 eigenstates, taking into account spin states.The TB spectra near the Fermi level and their evolution with B are shown in figures 2 and 3.Although the range of B under investigation is too large to be achieved in laboratories, the range of B allows to physically identify the roles of the magnetic field and the Rashba SOI.Thus, it is interesting to discuss results for the theoretical B range.Moreover, within the more realistic range of B ⩽ 20 T, these roles are significant in the two-electron spectra, as will be shown in section 4.
To focus on the effect of the magnetic field, we ignore the Rashba SOI for the moment.For t R = 0, the TB energy spectra near the Fermi level are shown in figure 2(a).For a triangular zigzag GQD, a characteristic feature of the TB spectra is that the zero-energy shell is completely degenerate.According to equation (1), for t R = 0, the z component of the spin s z is a conserved quantity and thus the eigenstates keep either spin-up or spin-down.The orbital energy of the degenerate shell is immune to the magnetic field (see figure 2(b)) and the Zeeman energy lifts the spin degeneracy of the degenerate shell (see figure 2(c)).In contrast, the orbital energies of the conduction and valence band significantly change with B, which even closes the energy gap.
Next, we investigate the effect of the Rashba SOI on the TB spectra by taking two different interaction strengths and the evolutions of the spectra with B are shown in figure 3.For the smaller strength t R = 0.02,   the eigenstates generally preserve spin-up or spin-down, indicating that the influence of the Rashba SOI is negligible (see figure 3(a)).However, by observing the details of the spectra, one may see level anticrossings when the spin-up and spin-down levels will cross (see figure 3(c)), which means the mixing of spin states.For the larger strength t R = 0.2, the spin s z is significantly not conserved and the level anticrossings become more apparent (see figure 3(b)), suggesting that the mixing of spin states has been driven by the Rashba SOI to a large extent.

Method for a few-electron GQD
In the frame of Coulomb blockade, the degenerate shell or the conduction band of a GQD can be filled with only few electrons and thus a few-electron GQD can be obtained [21].Few-electron systems usually deviate from charge neutrality, thus leading to a significant enhancement of intraband interactions.We use a combination of the mean-field Hartree-Fock (HF) approximation and the configuration interaction method to calculate the many-body correlation effects.Next, the methodology is described in detail through four steps.
The first step is to calculate the single-electron states with only one electron occupying the considered energy band.Here, due to the lack of intraband interactions, Coulomb interactions, namely interband interactions, can be described using the mean-field HF approximation [57].In the studies of the electronic properties of zigzag triangular GQDs, the mean-field model can provide numerical simulation results consistent with those obtained from density functional theory [58].Moreover, the consistency between the mean-field calculations and the results of scanning tunneling spectroscopy experiments can be found [26,49,59].Based on the TB model, the HF Hamiltonian can be expressed as [46] where ρ jkσ ′ is the density matrix element for the GQD, ρ o jkσ ′ is for the bulk graphene and ⟨ij|V|kl⟩ is the two-body Coulomb matrix element.In this step, the Zeeman energy and the Rashba SOI are not taken into account, which will be considered below.
The second step is to construct the basis set of many-body configurations for calculating the considered energy band.In the occupation number representation, the basis set of configurations can be written as where n α is the number of electrons occupying the αth HF state.For a given number of electrons n el and a given number of HF states n st , the total number of all the configurations is N conf = C n el nst .The third step is to construct the full many-body Hamiltonian for few interacting electrons occupying the considered energy band.The Hamiltonian may be written as where The operator a † sσ (a sσ ) creates (annihilates) a HF quasi-particle occupying HF eigenstate s with spin σ and energy ϵ sσ .The term H orb describes the energy of non-interacting electrons occupying HF orbitals.The term H int describes the two-body Coulomb interaction with the coefficient C scaling the interaction strength and with the interaction matrix element ⟨sd|V|fp⟩.The term H Z describes the Zeeman effect.The term H R describes the Rashba SOI, with the operator R taking the form of the last part in equation (1).
The last step is to calculate the intraband interactions by the configuration interaction method.Using the constructed many-body basis set of configurations (equation( 3)), we can obtain the matrix form of the many-body Hamiltonian (equation ( 4)).By diagonalizing the matrix, the energy spectra of a few-electron GQD can be calculated with the correlations among electrons occupying the band correctly captured.

Two-electron conduction band
For the triangular zigzag GQD (N = 61), the HF conduction band filled with only one electron is obtained by solving equation (2).As mentioned previously, the purpose of obtaining these HF eigenstates is to construct the basis set for calculating the few-electron conduction band.In view of this purpose, the Zeeman energy and the Rashba SOI are not included in the HF calculation.Due to the lack of intraband interactions and the Zeeman energy, the HF conduction band is spin degenerate.Figure 4(a) shows the low-lying HF conduction band labeled by eigenstate index from 67 to 80. Figure 4(b) shows the evolution of the HF conduction band with B. Compared to the case of charge neutrality, namely the TB conduction band, the HF conduction band is shifted upward as a whole due to the increase of the electron filling.Among the HF conduction band, the states with index 67-70 are closely involved in the following discussion on the two-electron states.Noted that the spin-down 67th and spin-up 68th HF state share the same orbital wavefunction, and so are the 69th and 70th HF states.The orbital energies of the conduction band change with B, which is an important magnetic behavior and will later help to cross the spin singlet and triplet levels.
For the GQD (N = 61), the magnetic-field evolution of the two-electron conduction band is obtained by solving equation ( 4) and shown in figure 5. We use the realistic strength of t R = 0.02 for the Rashba SOI, which induced an energy change of ∼1 µeV in the investigated levels.This means that the effect of the Rashba SOI is invisible with the energy resolution of ∼1 meV, as shown in figures 5(a)-(c).In the calculation, the two-electron configurations in the form of equation ( 3) involve only the lowest 8 HF eigenstates from the conduction band.Thus, one gets N conf = 28 possible configurations: The validity of the approximation can be assessed by checking the convergence between the approximate computation and the computation including more HF eigenstates in the two-electron configurations.The approximation significantly decreases the dimension of the Hilbert space and thus effectively reduces the computational cost.Now, we analyze the role of the Coulomb interaction in the spectra.To highlight the role, the spectra are calculated for three Coulomb strengths [C = 0, 0.5 and 1].Here, it is good approximation to ignore the Rashba SOI.With this approximation, for C = 0, only diagonal matrix elements are nonzero in the Hamiltonian matrix given by equation ( 4); that is, the basis set of configurations (equation ( 9)) is exactly eigenstates.More specifically, the ground eigenstate is |11000000⟩, which corresponds to two electrons occupying the 67th and 68th HF state.Therefore, the ground state can also be characterized by the symmetrized HF state |67, 68⟩.As the symmetrized HF representation can facilitate the discussion on interactions, the five lowest two-electron states are characterized by the occupation number representation as well as by the symmetrized HF representation, as listed in table 1.The degenerate eigenstates |67, 70⟩ and |68, 69⟩ span a 2D eigenspace.The another basis set of the subspace, |67, 70⟩ ± |68, 69⟩ √ 2 , are the excited singlet and the triplet T 0 .For C = 0 there is no crossing between the ground singlet S and the ground triplet T -, as shown figure 5(a).
As C deviates from zero, the eigenstates are no longer these configurations but their mixing.Nevertheless, the main configuration component in each two-electron state remains unchanged, which allows that these configurations can still approximately indicate the eigenstates.Based on the stability, we can explain the changes of these lowest energy levels with C. All the five lowest levels are shifted upward as C increases, as shown in figures 5(a)-(c).In comparison, the ground level |67, 68⟩ moves faster.This can be explained by the fact that the 67th and 68th HF states share the same orbital wavefunction, while the two HF states forming each of the other four two-electron states have different orbital wavefunctions.In addition, the degeneracy between the excited singlet and the triplet T 0 is not lifted, implying the lack of exchange interaction.For C = 1, there are the level crossings between |67, 68⟩ and the other four two-electron states.Among these crossings, the S-T -crossing between |67, 68⟩ and |67, 69⟩ will receive further attention below.Noted that the ground singlet-triplet crossing is based on tuning the orbital energy instead of the Zeeman energy.Additionally, we mention that for the S-T -crossing, the high magnetic field (∼ 16 T) is not a necessary   condition.Here, the magnetic field is mainly used to adjust the energy difference between singlet and triplet states, which can also be adjusted in advance to a small value by an in-plane electric field, so that the required magnetic field strength can be arbitrarily small [46].Next, we analyze the role of the Rashba SOI in the spectra.If t R = 0, the singlet |67, 68⟩ and the ground triplet |67, 69⟩ belong to different spin subspaces, thereby keeping unmixed at the crossing point.For t R ̸ = 0, the term H R of equation ( 4) does not conserve the projection of the spin onto z-axis, S z , leading to the mixing between the singlet |67, 68⟩ and triplet |67, 69⟩ when the two states are close to each other.The spin mixing leads to an S-T -level anticrossing and for t R = 0.02 the anticrossing energy is ∼1 µeV, as shown figure 5(d).The singlet-triplet mixing will later play a key role in the spin manipulation.
Different GQD sizes are also used for calculating the spectra and the S-T -mixing is not universal.For example, for the GQD (N = 118), there is an S-T -level crossing but not a level anticrossing, suggesting no singlet-triplet mixing.The reason is that the specific wavefunctions lead the related matrix elements ⟨sσ|R|pσ ′ ⟩ in equation ( 8) to be trivial.For the GQD (N = 33), the ground singlet level cannot even approach the ground triplet level.

Coherent spin manipulation
Two-level system.The S-T -mixing in the two-electron conduction band allows to define a two-level system (or qubit) using the two states, as schematically shown in figure 6(a).The two-level system can be expressed with the Hamiltonian where ϵ 0 (B) is the S-T -energy difference for t R = 0 and ϵ R (B) is the Rashba SOI energy.At ϵ 0 (B) = 0, with B 0 denoting the B value, the Hamiltonian becomes and the eigenstates S and T − are split by the energy ϵ 0 (B).To facilitate demonstrating the following spin manipulation, we define a Bloch sphere for the two-level system that has S and T -at the north and south poles (z axis) and the eigenstates of H = 1 2 ϵ R (B 0 )τ x , |−⟩ and |+⟩, as the poles along the x axis, as shown in figure 6(b).
Rabi oscillations.Two modes of Rabi oscillations are shown by tuning the magnetic field, as shown in figure 6(b).One mode is the oscillation between S and T -.Tuning B ≫ B 0 , the system favors a singlet ground state.Then, the field B is swept by an adiabatic passage to B 0 , namely, quickly relative to S-T -mixing.At this point, the system is initialized into the S state while the Hamiltonian takes the form of equation (11).So, the Rashba SOI drives the system to rotate around the x axis: New J. Phys.26 (2024) 053044 W-J Xu et al where θ = ϵ R (B 0 )t/h.Another mode is the oscillation between |−⟩ and |+⟩.Tuning B = B 0 , the system favors a |−⟩ ground state.Then, the field B is swept to a certain value (quickly relative to mixing between |−⟩ and |+⟩) and a certain ϵ 0 (B) is obtained.The system is initialized into the |−⟩ state while the Hamiltonian takes the form of equation (12).So, the system begins to rotate around the z axis: where φ = π + ϵ 0 (B)t/h.Both of the two energy splittings ϵ R (B 0 ) and ϵ 0 (B) are not less than ∼ 1 µeV.Thus, the intrinsic decoherence mechanism does not pose an obstacle for the proposed coherent spin manipulations [43,60].

Summary
In the paper, we have studied the scheme of realizing a spin qubit based on the conduction band of a few-electron GQD.The method is developed to calculate the electronic structure of a few-electron GQD with the effects of an external magnetic field and an extrinsic Rashba SOI.In the magnetic-field evolution of the two-electron conduction band, the Coulomb interaction and the variation of orbital energies together lead to the level crossing between the ground singlet and triplet states.The singlet-triplet spin mixing can be obtained by introducing the Rashba SOI.In the demonstrated Rabi oscillations, the spin states can be prepared and coherently manipulated using the fast magnetic control of the singlet-triplet splitting.Our work is inspired by successful synthesis of the atomically precise GQDs with zigzag edges.The spin states involved in the study are closely related to the shape, sizes and edges of the quantum dots, thus the proposed qubit scheme has strong design tunability.Moreover, our work would motivate further studies, for example, demonstrating Coulomb blockade effect in synthetic GQDs and designing the readout of two-electron spin states for the system.

Figure 1 .
Figure 1.Examples of GQDs with different shapes and edges.(a) Rectangular dot with armchair and zigzag edges.(b) Hexagonal armchair dot.(c) Triangular zigzag dot consisting of 61 carbon atoms.Hydrogen passivation is assumed for all the edges.

Figure 2 .
Figure 2. TB spectra of the GQD shown in figure 1(c) without the Rashba SOI.(a) Levels vs. eigenstate index.The zero-energy shell is completely degenerate.Magnetic-field evolution (b) excluding and (c) including Zeeman energy.In (c), red curves are for spin-up states and blue curves for spin-down states.The orbital energy of the degenerate shell is unchanged with the magnetic field and the Zeeman energy lifts the spin degeneracy.

Figure 3 .
Figure 3. TB spectra of the GQD (N = 61) as functions of B. (a) For tR = 0.02, the eigenstates generally preserve spin-up or spin-down.(b) For tR = 0.2, the spin is significantly not conserved and the level anticrossings is apparent.(c) Zoom-in of the black box in (a).Level anticrossings can be seen in the details.

Figure 4 .
Figure 4. Low-lying HF conduction band filled with only one electron.The Zeeman energy and the Rashba SOI are not included.(a) Levels vs. eigenstate index.Spin-down (spin-up) states are indicated by open (solid) circles.(b) Levels vs. B. Compared to the TB conduction band, the HF conduction band is shifted upward due to the increase of the electron filling.Blue color is for HF spectra and black color for TB spectra.

Figure 5 .
Figure 5. Low-lying two-electron conduction band of the GQD (N = 61) with tR = 0.02, for (a) C = 0, (b) C = 0.5 and (c) C = 1.In (a), each level is labeled by the corresponding symmetrized HF state.As C increases, the five lowest levels are shifted upward and the ground level |67, 68⟩ moves faster.For C = 1, there are the level crossings between |67, 68⟩ and the other four two-electron states.(d) Zoom-in of the black box in (c).For tR = 0.02, the S-T-anticrossing energy is ∼1 µeV.

Figure 6 .
Figure 6.(a) Two-level diagram near the S-T-anticrossing.At B = B0, ϵ0(B) = 0 and eigenstates |+⟩ and |−⟩ are split by ϵR(B0).When B deviates from B0, effect of ϵR(B) becomes negligible and eigenstates S and T-are split by ϵ0(B).(b) Two modes of Rabi oscillations.Upper Bloch sphere corresponds to oscillation between S and T-.Tuning B ≫ B0, the system favors a singlet ground state.Then the field B is swept to B0 and the system begins to rotate around the x axis.Lower Bloch sphere to oscillation between |−⟩ and |+⟩.Tuning B = B0, the system favors a |−⟩ ground state.Then the field B is swept to a certain value and the system begins to rotate around the z axis.Green (red) arrows depict the initial (evolving) states.

Table 1 .
Five lowest two-electron states in the occupation number representation and the symmetrized HF representation.The corresponding spin states are labeled.