Introduction

Deep-level defects in wide-bandgap semiconductors exhibit localized electronic energy levels with paramagnetic spin states, and hence they constitute promising platforms for quantum bits (qubits).1 Defect-based solid-state spin qubits have been widely studied, including the nitrogen-vacancy (NV) center in diamond,2,3 vacancies in SiC,4,5,6,7 donor spins in silicon,8 and rare-earth ions,9 targeting quantum computing, and sensing applications.10,11 The spin coherence time T2, usually measured by Hahn-echo experiments,12 is a critical property for qubit applications.13 In quantum computers, in order to achieve quantum error correction, at least 104 quantum gate operations need to be performed within T2, yielding a desired coherence time larger than 100 μs.13,14 The sensitivity of quantum sensors, which is proportional to \(\frac{1}{{\sqrt {T_2} }}\),15 benefits as well from qubits with long coherence times.

The decoherence of spin qubits can arise from different sources, including the magnetic noise from high-density paramagnetic defects and dangling bonds on the material surface.16,17 In highly purified bulk solids, the phase decoherence is mainly caused by the entanglement between the electron spin qubit and the nuclear spin bath of the host material.18 Thus, isotopic purification is an effective technique to enhance T2.19,20 For example, the ensemble T2 of the NV center in diamond has been extended from 0.63 ms21 to 1.8 ms19 by reducing the concentration of 13C nuclear spins from 1.1 to 0.3%. In general, solids with a low concentration of nuclear spins are desirable spin qubits hosts.1

Two-dimensional (2D) materials exhibit a variety of promising properties for the realization of hosts of defect-based spin qubits, including their facile transfer and integration into hybrid quantum systems.22,23,24,25 In 2D systems, point defects can be created and precisely located using a scanning transmission electron microscope or an annular dark field microscopy.26 Furthermore, compared with thin films of three-dimensional (3D) solids, 2D layers are free from unsaturated dangling bonds at the surface that are detrimental to spin coherence. However, the nuclear spin environments in many 2D materials are much more complex than, for example, in diamond.

In this work, we investigate the coherence of spin qubits in 2D materials and propose strategies to search for ideal host materials. We first explore the effect of dimensionality, crystal geometry, and nuclear spin moment on the spin coherence. Especially, we discuss the interplay of dimensionality and nuclear spin concentration in determining T2. We then predict T2 in single layer and multi-layer MoS2 and h-BN, which are among the most studied 2D host candidates,27,28,29,30,31,32,33,34,35 and we propose design rules to further improve the spin coherence and obtain spin qubits with a performance superior to that, for example, of 3D materials such as diamond and SiC.

Results and discussion

Theoretical framework

We describe an electron spin S immersed in a nuclear spin bath using a central spin Hamiltonian H = HS + HB + HS−B, where HS and HB represent the electron spin and the nuclear spin bath, respectively, and HS−B describes the qubit–bath interaction:36,37,38,39

$$H_{\mathrm{S}} = - \gamma ^{\mathrm{e}}{\boldsymbol{B}} \cdot {\boldsymbol{S}} + {\boldsymbol{S}} \cdot {\boldsymbol{D}} \cdot {\boldsymbol{S}},$$
(1)
$$H_{\mathrm{B}} = - {\boldsymbol{B}}\cdot \mathop {\sum}\limits_i {\gamma _i^{\mathrm{n}}} {\boldsymbol{I}}_i + H_{{\mathrm{nn}}},$$
(2)
$$H_{{\mathrm{nn}}} = \frac{{\mu _0}}{{4\pi }}\mathop {\sum}\limits_{i < j} {\gamma _i^{\mathrm{n}}} \gamma _j^{\mathrm{n}}\frac{{{\boldsymbol{I}}_i \cdot {\boldsymbol{I}}_j - 3({\boldsymbol{I}}_i \cdot \widehat {\boldsymbol{r}}_{ij})({\boldsymbol{I}}_j \cdot \widehat {\boldsymbol{r}}_{ij})}}{{r_{ij}^3}},$$
(3)
$$H_{{\mathrm{S}} - {\mathrm{B}}} = {\boldsymbol{S}} \cdot \mathop {\sum}\limits_i {{\boldsymbol{A}}_i} \cdot {\boldsymbol{I}}_i,$$
(4)

where γe and γn are the gyromagnetic ratios of electron and nuclear spins, and rij is the distance between the nuclear spins Ii and Ij. HS describes the Zeeman splitting of the spin sublevels due to an applied magnetic field \(\left( {{\boldsymbol{B}} = B\widehat {\boldsymbol{z}}} \right)\) and D is the zero-field splitting, different from zero for \(S > \frac{1}{2}\). In the case of a large level splitting arising from HS, the electron spin Sz is a good quantum number. HB is the sum of the Zeeman energy and the dipolar interaction of nuclear spins. The latter describes a fluctuating magnetic noise present in the system, which is entangled with the electron spin through the hyperfine interaction entering HS−B.40

The hyperfine interaction between a nuclear and an electron spin consists of two parts, the Fermi contact and the dipolar interaction. Because of the localized nature of the spin density of deep center defects, only a small number of nuclear spins near the spin qubit contributes to the Fermi contact, which is thus considered negligible and not included in our calculations. In addition, as Sz is a good quantum number in the presence of a strong magnetic field or a large zero-field splitting, the Sx and Sy terms are neglected in HS−B.

Effect of reduced dimensionality

We first focused on the effect of reduced dimensionality (from 3D to 2D materials) on the T2 value, and we considered host materials with only one nuclear spin species. The NV center in diamond, where the ground state spin |0〉 and |1〉 states of the defect are used as a two-level system, is an ideal example as only 13C contributes to the decoherence. We investigated the variation of T2 as a function of the thickness of a diamond slab. In practice, a diamond thin slab with tunable 13C concentration could be obtained through delta-doping a thin layer of 13C nuclear spins embedded in a 3D nuclear-spin-free diamond matrix.41 In order to approach the 2D limit, we also considered graphite as a model host material with a NV-like spin state localized at a C-vacancy site. Although a NV-like spin qubit in graphite is hypothetical, we note that spin qubits have recently been realized as edge states of graphene nanoribbons.42

We placed a spin qubit in the center of a thin diamond slab of width d or in the central layer of a n-layer graphite (see Supplementary Information) and we considered various nuclear spin concentration. We applied a magnetic field B = 0.05 T along the three-fold axis of the defect; in this direction T2 attains its maximum value.21,38 The chosen value of B corresponds to a strong field under which T2 reaches saturation and the dominant decoherence mechanism arises from nuclear spin flip-flop transitions.43,44

Our predicted coherence times for diamond thin slabs and for multi-layer graphites are plotted in Fig. 1 as a function of the 13C nuclear spin concentration η. The values α and β obtained by fitting the log–log plot in Fig. 1 with T2 = αη−β are summarized in Table 1. The figure and the table show that reducing the slab thickness can effectively enhance T2 (α2D > αslab > α3D). This finding is not surprising since the nuclear spin density is reduced in 2D. Most importantly, our results show that the enhancement of T2 relative to the bulk is more prominent at lower nuclear spin concentrations (β2D > βslab > β3D). We have also clearly identified the thickness of the slab at which T2 is significantly enhanced, which turns out to be few nanometers. The interplay between the nuclear spin concentration and the slab thickness can be understood as follows. The major contributions to decoherence originate from nuclear spins belonging to a region of radius R centered on the spin qubit and R decreases as the concentration increases. For example, in bulk diamond, we found that R ≈ 5 nm for 1.1% 13C, and it reduces to 3 nm for 10% 13C, and to 1.5 nm for 100% 13C. A major increase of T2 due to a decrease of the slab thickness d is expected only if d ≪ R; hence, for thin slabs the enhancement of T2 is larger when the nuclear spin concentration is lower.

Fig. 1
figure 1

Computed coherence time T2 as a function of 13C concentration for bulk and thin films diamond (left panel) and for bulk and thin layers graphite (right panel). T2 is predicted to be 0.88 ms for nitrogen-vacancy (NV) center in bulk diamond at the nuclear spin natural abundance (1.1% 13C), in agreement with previous reports38,44,48,49

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Table 1 Computed coherence time T2 = αη−β of diamond and graphite, as obtained by fitting the curves shown in Fig. 1, where η is the 13C concentration
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Effect of crystal geometry

Crystal geometry, in particular the interatomic distance, has a significant impact in determining T2, since the energy scale of nuclear spin flip-flop transitions is proportional to \(\frac{1}{{r^3}}\) (see Eq. 3).44,45 An example is shown in Fig. 1: the T2 of a bulk graphite is always larger than that of a bulk diamond, due to the larger average inter-atomic distance in graphite. Silicon has the same lattice structure as carbon diamond, but with a much larger lattice constant. In addition, deep-level defects have been observed in Si as potential spin qubits.46,47 For a localized defect in Si with the same spin states as a NV center, for example, a tin vacancy,47 we predict T2 in the bulk and in a 4-nm slab to be 1.15 and 1.25 ms, respectively, for 4.7% of 29Si, values that are about 35–30% larger than the corresponding ones in C diamond with 1.1% of 13C. This points at thin Si slabs as potentially excellent host materials for qubits, since quantum confinement is expected to increase the band gap of Si, possibly to desired values to create stable spin-defects within the gap.

Effect of large nuclear spins

So far, we only considered a nuclear spin \(I = \frac{1}{2}\), such as 13C and 29Si. As most nuclei of 2D materials have \(I > \frac{1}{2}\), we qualitatively investigated the dependence of T2 on I, by arbitrarily varying I of 13C from \(\frac{1}{2}\) to \(\frac{9}{2}\) in our bulk diamond calculations. We found that the coherence time is approximately inversely proportional to I (in particular \(T_2(I)/T_2\left( {I = \frac{1}{2}} \right) = - 0.045 + 0.522/I\)) and the result can be rationalized as follows. The decoherence is determined by the strength of nuclear spin flip-flop transitions and the number of transition channels. In a volume V with N nuclear spins, the number of nuclear spin pairs is \(N_{{\mathrm{pair}}} = \frac{{N(N - 1)}}{2}\). In bulk diamond, T2 is approximately inversely proportional to the 13C nuclear spin density \(\eta = \frac{N}{V}\) as observed in Table 1; therefore, \(T_2 \sim \frac{1}{N} \sim \frac{1}{{\sqrt {2N_{{\mathrm{pair}}}} }}\). For an arbitrary nuclear spin I, the total number of flip-flop channels is Ncn = (2I)2 and \(T_2 \sim \frac{1}{{\sqrt {2N_{{\mathrm{pair}}}N_{{\mathrm{cn}}}} }}\) giving \(T_2(I)/T_2\left( {I = \frac{1}{2}} \right) \sim \frac{1}{{2I}}\). This scaling relation is also useful to understand the role of quadrupole interactions (see Supplementary Information). For symmetrical electric field gradients, the quadrupole Hamiltonian shifts the spin sublevels with different Iz; thus, only flip-flop transitions that swap the magnetic quantum numbers are allowed. In this case, Ncn = 2I and \(T_2(I)/T_2\left( {I = \frac{1}{2}} \right) \sim \frac{1}{{\sqrt {2I} }}\). Therefore, in the calculations on 2D systems presented below, neglecting the quadrupole term in the spin Hamiltonian leads to an underestimate of the calculated T2.

Spin coherence in MoS2 and h-BN

We finally consider specific 2D materials as qubit hosts, namely MoS2 and h-BN. Both materials have large nuclear spin concentrations (>25% in MoS2 and 100% in h-BN), multiple nuclear species, and large nuclear spins. Suitable defects have not yet been identified in MoS2 and h-BN and here we consider mostly hypothetical defects with the same electron spin state S = 1 and \(\hat z\) direction of the magnetic axis as in the diamond NV center. The magnetic field is applied along the magnetic easy axis to maximize T2.21,38 Below, we present results for h-BN, with a nitrogen–substitution–vacancy complex NBVN where the qubit is at the vacancy site, and for MoS2, where the qubit is located at a molybdenum site. We also considered qubits located at the boron site in h-BN and at the sulfur site in MoS2, and we found minor variations of T2 (~3%). The similarity of T2 for different defects in the same host material is not surprising, as T2 is determined by hundreds of nuclear spins and the effect of the local structure is negligible as long as the defect is a deep center with localized wavefunctions. We report below results for spin S = 1, but we also considered the case of spin \(S = \frac{1}{2}\), where we used the \(\left| {\frac{1}{2}} \right\rangle\) and \(\left| { - \frac{1}{2}} \right\rangle\) as the two-level system. Despite a minor change (~10%) in the numerical value of T2, we found the same trends for spin \(S = \frac{1}{2}\) and S = 1.

As multiple nuclear spin species are present in both h-BN and MoS2, we first investigated the coupling of the various spin baths in the material. The spin coherence functions from each nuclear spin bath are shown in Fig. 2a, c. For non-interacting nuclear spin species, the total spin coherence function \({\cal L}\) can be written as a product \({\cal L} = {\mathrm{{\Pi}}}_\alpha {\cal L}_\alpha\), where \({\cal L}_\alpha\) is the coherence function from the αth nuclear spin species. In Fig. 2b, d, we show that the computed \({\cal L}\) coincides with the one obtained from the product of the functions of each species, proving that different nuclear spin baths are decoupled. The reason for this decoupling is that the hetero-nuclear-spin flip-flop transitions are greatly suppressed in the presence of large magnetic fields because of the different gyromagnetic ratios from different nuclear spin species. A similar bath decoupling effect has also been observed in SiC.44,45

Fig. 2
figure 2

Spin coherence functions computed for single-layer MoS2 (left panels) and h-BN (right panels). The upper panels show results for different spin baths separately. The lower panels compare results obtained for the total coherence function computed directly (red curve) and as a product (gray curve) of the coherence functions of the various spin baths, showing that the baths are decoupled

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Second, we considered the layer dependence of central spin decoherence in multi-layer MoS2 and h-BN in their natural abundance nuclear bath (see Table 2). The magnetic field used in the simulation is 0.1 T for MoS2 and 0.5 T for h-BN, which have been tested to be in the strong field limit (see Supplementary Information). The T2 increases with decreasing layer numbers, but there is only a two-fold enhancement for a single layer compared to the bulk. This insensitivity of T2 to the number of layers is due to the large concentration of nuclear spins in both materials (see Fig. 1), and we would expect a more prominent layer dependence of T2 in isotopic purified MoS2.

Table 2 Computed coherence time T2 (ms) for multi-layer and bulk h-BN and MoS2 for natural abundance nuclear concentrations
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In MoS2, the concentration of 33S is negligible compared to that of Mo isotopes, and the decoherence of the electron spin mainly originates from the presence of 95Mo and 97Mo nuclear spins, as shown in Fig. 2a. We note that nuclear-spin-free Mo isotopes are commercially available. Therefore, we considered isotopically purified Mo and computed T2 in single-layer, multi-layer, and bulk MoS2. The results at different nuclear spin concentrations are summarized in Fig. 3 (more details in the Supplementary Information). As the nuclear spin concentration decreases, the enhancement of T2 is very large, especially for a single layer: at 1% 95Mo, 1% 97Mo, and 0.76% 33S, the single layer T2 is 16 times larger than that computed for natural abundance concentration, while the T2 of the bulk shows a six-fold increase. In other words, isotopic purification is a more effective tool in 2D than in 3D materials to engineer an enhancement of T2. By further isotopic purification, the T2 of a single layer MoS2 can exceed 30 ms, an exceptionally long coherence time. In addition, we notice that the nuclear spin bath with 5% 95Mo and 5% 97Mo has a longer T2 than the one with 10% 95Mo (see Supplementary Information), despite the similarity in the total nuclear spin concentrations. This is because the gyromagnetic ratios of 95Mo and 97Mo are similar (they differ by only 2%), but at large magnetic field, their mutual flip-flops are suppressed. Therefore, the presence of multiple nuclear spin species offers a larger parameter space for isotopic engineering, and in turn this allows for the realization of longer T2 without the need of eliminating all nuclear spins, which can then be used as additional quantum resources.

Fig. 3
figure 3

Computed coherence time T2 (ms) for multi-layer MoS2 as a function of Mo concentration. Here both 95Mo and 97Mo have the same concentrations represented by the Mo concentration axis and the concentration of 33S is 0.76%, corresponding to its natural abundance

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Figure 2c shows that in h-BN the decoherence is dominated by the 11B nuclear spin bath. This is due to the high concentration of 11B compared to 10B, and, most importantly, to the much larger nuclear gyromagnetic ratio of 11B relative to 10B and 14N. Hence, isotope engineering of T2 in h-BN should be focused on reducing 11B nuclear spin concentration. The total nuclear spin concentration from B nuclei cannot be changed, as η(10B) + η(11B) = 1; therefore, we varied the relative concentration between η(10B) and η(11B). We found that T2 increases monotonically as the 10B concentration increases, from 0.036 ms at 20% to 0.086 ms, when all 11B nuclei are replaced by 10B nuclei.

Effect of gyromagnetic ratio

For spin natural abundance (see Table 2), the coherence time of MoS2 is almost two orders of magnitude longer than that of h-BN. The difference in nuclear spin concentration is not the only reason determining the large difference in T2: indeed at 100% 95Mo, the difference between the T2 of the two materials is still about one order of magnitude (see Supplementary Information). The difference in gyromagnetic ratios between 11B and 95Mo is another important factor. As γ(11B)/γ(95Mo) ≈ 5, and the nuclear spin dipolar interaction Hnn (see Eq. 3) is proportional to γ2, the difference in gyromagnetic ratios accounts for roughly 25-fold enhancement of T2 in MoS2 compared to h-BN. The small gyromagnetic ratios of 95Mo and 97Mo is also responsible for T2 of bulk MoS2 with more than 20% nuclear spins being longer than that of diamond with only 1.1% nuclear spins. Many nuclei have a smaller gyromagnetic ratio than 13C and compounds containing such nuclei may potentially be host materials with long coherence times.

In summary, we revealed, through simulations, the role of dimensionality and crystal geometry in controlling the decoherence of spin qubits in solids, which is as important as nuclear spin concentration. A key result of our study is that isotopic purification is much more effective in 2D than in conventional 3D materials such as Si and C. We also demonstrated that in 2D, long spin-coherence times may be engineered in solids with multi-nuclear-spin environments, without sacrificing nuclear spins. In addition, our study provided a quantitative understanding of the influence of high nuclear spin baths (I > 1/2) on the spin coherence time T2 of a defect-based spin qubit, and we identified the role of multiple physical parameters in controlling the qubit decoherence; such identification is critical in searching for promising host materials for spin qubits. Overall, we predicted that single-layer MoS2 layers, delta-doped diamond slabs, and Si thin slabs can be promising hosts for spin qubits, while the T2 of h-BN appears to be much more challenging to engineer, due to the large gyromagnetic ratio of the boron nuclei. In particular, we showed that T2 of an isotopically purified layer of MoS2 is of the order of 30 ms.

Methods

Using the spin Hamiltonian H (Eqs. 1–4) and the cluster-correlation expansion method,39 we computed the coherence function \({\cal L} = \frac{{{\mathrm{Tr}}[\rho (t){\boldsymbol{S}}_ + ]}}{{{\mathrm{Tr}}[\rho (0){\boldsymbol{S}}_ + ]}}\) for an ensemble of electron qubits with randomly distributed nuclear spins, where ρ is the density operator and S+ is the electron spin raising operator. The time T2 was obtained from the profile of the ensemble averaged coherence function using \({\cal L} = e^{ - (t/T_2)^n}\), where n is the stretching exponent. Numerical convergence tests are reported in the Supplementary Information.