Abstract
Spin–orbit coupling (SOC) is fundamental to a wide range of phenomena in condensed matter, spanning from a renormalisation of the freeelectron gfactor, to the formation of topological insulators, and Majorana Fermions. SOC has also profound implications in spinbased quantum information, where it is known to limit spin lifetimes (T_{1}) in the inversion asymmetric semiconductors such as GaAs. However, for electrons in silicon—and in particular those bound to phosphorus donor qubits—SOC is usually regarded weak, allowing for spin lifetimes of minutes in the bulk. Surprisingly, however, in a nanoelectronic device donor spin lifetimes have only reached values of seconds. Here, we reconcile this difference by demonstrating that electric field induced SOC can dominate spin relaxation of donorbound electrons. Eliminating this lifetimelimiting effect by careful alignment of an external vector magnetic field in an atomically engineered device, allows us to reach the bulklimit of spinrelaxation times. Given the unexpected strength of SOC in the technologically relevant silicon platform, we anticipate that our results will stimulate future theoretical and experimental investigation of phenomena that rely on strong magnetoelectric coupling of atomically confined spins.
Similar content being viewed by others
Introduction
Individual spins confined to solids are attracting considerable interest due to their potential as quantum bits (qubits) in future quantum information processors.^{1,2} In particular, electron spins bound to phosphorus donors have demonstrated exceptionally long spin life (T_{1}) and coherence (T_{2}) times, both of order seconds in isotopically purified ^{28}Si.^{3} Recent demonstrations of highfidelity^{4} spin readout,^{5} manipulation,^{6} and controllable exchange,^{7,8} promise the realisation of scalable donorbased quantum computing.^{2,9}
One reason for the extremely long spin lifetimes of electrons in silicon is their inherently weak spin–orbit coupling (SOC), which has been found to dominate spin relaxation in inversion asymmetric polar semiconductors, such as GaAs.^{10,11,12} In the absence of surfaces and interfaces,^{13,14} SOC for electrons in silicon is usually considered negligible^{15} due to its small atomic number and large band gap. In fact, the presence of bulkinversion symmetry in silicon fundamentally excludes Dresselhaus SOC, whilst structural inversion symmetry precludes Rashba SOC for the spherically symmetric ground state wave function of bulk donors. As a consequence, for donorbound electrons SOC enters spinlattice relaxation only as a weak timedependent perturbation of spin–valley states via deformation potential phonons.^{16,17,18} Typically, this has resulted in extended spin lifetimes, exceeding those in GaAs by more than an order of magnitude.^{10,11,12}
However, unlike donors in bulk silicon, individual donors as quantum bits in nanoelectronic devices^{4,5,19} experience electric fields (several MV/m^{7,15,20}), that are known to have a profound effect on the donor wave function, modifying charging energies,^{21,22,23} valley–orbit splitting,^{22,24,25} and Stark shift of the electron’s gfactor,^{26} as well as ultimately the nuclear hyperfine and exchange coupling.^{2,9} The strength and vectorial nature of such electric fields can be expected to reduce crystal symmetry,^{27} introducing higherorder coupling of spins with their electromagnetic environment, thus impacting spin lifetimes and coherence.
In this paper, we report the presence of such electric field induced magnetoelectric coupling for the first time, using measurements of electron spinrelaxation rates of an individual Si:P donor qubit within an atomically engineered nanoelectronic device. Comparing to wellestablished theoretical frameworks^{16,17} and measurements^{28} of bulk donors, we find a profoundly different magnetic field anisotropy of spinrelaxation rates in both periodicity and phase, with a difference in spinrelaxation rates by more than one order of magnitude for certain magnetic field orientations.
We explain this unusual spinrelaxation anisotropy by a coupling,^{27,29}
of spin and external electric (E) and magnetic (B) fields, where σ is the electron spin and C is a tensor representing the strength of this SOC interaction along the Cartesian directions arising from directional differences in single valleys as well as the valley weights of the wave function (see Supplementary Information Section 4). In line with conventional SOC, the above Hamiltonian can be derived as a higherorder perturbation from k·p theory,^{29} and induces an effective magnetic field B_{eff}. However, unlike conventional SOC, the particular structure of H_{EB} demands that for external magnetic fields applied along the main symmetry axes, this effective field is oriented perpendicular to the spin quantisation axis, allowing effective mixing of spin states facilitating fast spin relaxation. Whilst a related magnetoelectric coupling has been theoretically predicted for acceptors^{27} with known strong SOC^{30} more than 5 decades ago, it has never been observed experimentally, nor considered in donor spinrelaxation theory. Demonstrating its impact on spin relaxation in the controlled electromagnetic environment of an atomically engineered nanoelectronic device, we thus solve a longstanding mystery as per why spin lifetimes in silicon devices (~sec) have been found substantially shorter,^{5,15} compared to those in bulk (~min).^{28} Indeed, we show that careful design of the local electromagnetic environment in nanoelectronic devices allows us to eliminate these lifetimelimiting effects—an important factor in designing future qubit architectures—and allow us to demonstrate the bulklimit of donor spin lifetimes within a nanoelectronic device for the first time.
Results
An overview scanning tunneling microscope (STM) image of the nanoelectronic device (see Methods) is shown in Fig. 1a. A single Si:P donor qubit was placed at the dashed white circle,^{4} d ~20 nm from the island of a single electron transistor (SET). Four inplane gates (G_{1,2}, G_{SET} and G_{T}) tune the donor and SET electrochemical potentials with a dominant electric field E_{y} ≃ 5 MV/m along the G_{T} − G_{SET} axis to facilitate spin readout (green arrow, see electrostatic model in Fig. 3). The diagram in Fig. 1b shows the respective orientations of E_{y} (green), the external magnetic field B (blue), as well as the effective H_{EB} magnetic field B_{eff} (red). The angles θ and ϕ enclose outofplane (θ) and inplane (φ) alignments of B.
Spinlattice relaxation in silicon is strongly influenced by the structure of its conduction band with six degenerate valleys. At the donor site (Fig. 1c), the sixfold degeneracy is broken into a spindegenerate 1s(A_{1}) ground state, with valley–orbit doublet 1s(E) and triplet 1s(T_{2}) excited states. Spinlattice relaxation within the 1s(A_{1}) ground state occurs as acoustic deformation potential phonons modulate the electron gtensor by inducing a timedependent lattice strain.^{16,17} SOC mixes the donor ground state with the higher 1s excited states separated from the 1s(A_{1}) ground state by the valley–orbit splitting E_{vo} (Hasegawa’s valley repopulation mechanism)^{16} or with continuum states inside the same conduction band valley (Roth’s singlevalley mechanism).^{17} Both valleyrepopulation and singlevalley theories predict a dependency of the spinrelaxation rate on the magnitude of the external magnetic field, 1/T_{1} ∝ B^{5} (lowtemperature limit^{5}), with a characteristic anisotropy depending on the alignment of the magnetic field vector with the crystal symmetry axes.^{16,17} For bulk donors, Wilson and Feher^{28} confirmed in the early 1960s that spinrelaxation rates could be described by a combination of valleyrepopulation and singlevalley contributions, with measured spin lifetimes of up to 18 min at 1.25 K and at a weak magnetic field, B = 0.8 T. To compare with our own measurements, performed at higher magnetic fields, we replot their experimental data at 1 K (black circles), rescaled to B = 3.5 T in the lowtemperature limit^{5} (Fig. 1d), showing maximum spin lifetimes T_{1} ≃ 1.25 s for B[001].
Our measurement of spinrelaxation rates for a single donor in a nanoelectronic device at the same magnetic field strength is shown in Fig. 1e. Here, we observe a profoundly different magnetic field anisotropy in magnitude, periodicity and phase. Along B[001] and at B = 3.5 T, we find T_{1} ≃ 40 ms—a factor ~30 lower than bulk donors at the same magnetic field and temperature, consistent with previous measurements of donors in a nanoelectronic device at the same magnetic field.^{4,5} This large discrepancy in T_{1} obtained, compared to bulk donors, and combined with the observation of a fundamentally different spinrelaxation anisotropy, underpins the presence of a previously unreported spinrelaxation pathway. We note that T_{1} ∝ B^{5}, as shown in Fig. 1f, so that spin lifetimes are of order seconds at B = 1.5 T for all magnetic field orientations, in agreement with previous single dopant devices.^{5} Consistency across two separate measurement cooldowns of the device and two different planes of rotation (outofplane (θ) and inplane (φ)), is demonstrated in Fig. 2a–c. Indeed, here we recover T_{1} = 1.25 s at B[010] and B = 3.5 T, comparable to that in bulk samples at the same magnetic field and temperature (Fig. 1d).
Discussion
To explain the observed spinrelaxation anisotropy, we turn to the full effective mass Hamiltonian for donor electron spins in an electric field, expanded up to fourth order in perturbation theory^{29}
Here, \(H_{\mathrm{Z}} = {\textstyle{{\mu _{\mathrm{B}}} \over 2}}\sigma \cdot {\bf{g}} \cdot B\) is the Zeeman Hamiltonian (second order perturbation) with the gtensor according to Roth,^{17,18} H_{R} = σ · R · (E × k) is the bulk Rashba Hamiltonian (third order), and H_{EB} the Hamiltonian introduced in Eq. (1) (fourth order). This terms arises in an inversion asymmetric system in the presence of both electric and magnetic fields, and in our specific case, is influenced by the anisotropic multivalley nature of the conduction band. Different from bulk donors, both H_{R} and H_{EB} can be shown to enter valleyrepopulation theory for donors in nanoelectronic devices (see Supplementary Information), dominating spinrelaxation rates and anisotropy in the presence of external electric fields.
Donor qubits in electric fields are thus different from other, inversion asymmetric, electronic systems—such as semiconductor quantum dots^{11,31}—in which electronic wave functions are confined by interfaces, and in which the effect of any higherorder magnetoelectric coupling is dwarfed by lower order Rashba and Dresselhaus SOC terms. Atomically confined spins in lowSOC materials—such as P donors in silicon—typically have spherically symmetric ground state wave functions, and hence negligible Rashba and Dresselhaus SOC due to inversion symmetry. Although electric fields from nearby gate electrodes can introduce a nonnegligible bulk Rashba contribution in donors, its effects are much weaker compared to quantum dots as the donor wave function remains strongly bound to the Coulomb potential over a wide range of realistic electric field strengths. Indeed, our calculation shows that the ratio of the coupling constants is given by (see Supplementary Information).
Here, (ħ/m_{0})p_{μν} = 1.5 × 10^{−10} eV is the momentum matrix element between the silicon valence bands, evaluated at the conduction band minimum, and E_{αμ} ~ 4 eV is the energy difference between the conduction band minimum and the valence band, both at k_{0}. μ and ν are the indices of the two highest valence bands, α is the index of the lowest conduction band, and k_{0} is the momentum vector where the conduction band minimum occurs.
Singleshot spin readout of donorbound electrons requires the application of considerable gate electric fields^{20} to bring the Zeemansplit donor ground state into resonance with the electrochemical potential of a singleelectron transistor charge sensor. Such electric fields, together with the external magnetic field thus induce an effective magnetic field, B_{eff}, that is oriented normal to the external magnetic field orientation when aligned along one of the main symmetry axes (compare Fig.1b). In this case, the magnitude of this field maximises whenever B⊥E and is reduced to zero when they are aligned. From an electrostatic model of our device (Fig. 3a, b, see Methods) we find that E_{y} ≃ 5 MV/m along the G_{T} − G_{SET} axis (green arrow in Fig. 1a) at the readout point (Supplementary Information), confirming that we expect rapid spin relaxation for B[001] and B[100] where B⊥E (Fig. 2a–c).
Following the same argument, we show that the lifetimelimiting effects of H_{EB} can be eliminated by aligning EB, such that B_{eff} vanishes. Indeed, at both B[010] and \(B{\mathrm{}}[0\bar 10]\) (Fig. 2a–c), we measure T_{1} = 1.25 s (1/T_{1} ≃ 0.8 1/s) at B = 3.5 T. This value is an enhancement by a factor ~6–7 compared to previous measurements in nanoelectronic devices^{5} and identical to spin lifetimes measured in bulk donors along the equivalent B[001] axis.^{28} Such consistency in spinrelaxation times for equivalent crystal directions thus give a strong indication that we have reached the bulk limit of donor spin lifetimes within a nanoelectronic device. Given that spin lifetimes as long as 1111 s (18 min) at 1.25 K and B = 0.8 T have been measured in bulk,^{28} such timescales should be attainable in nanoelectronic devices by careful design of the electromagnetic environment and at low enough magnetic fields.^{12} A further enhancement of spinrelaxation times beyond the bulk limit may be achieved using single spins confined to donor clusters instead of single donors.^{19}
For an estimate of the respective bulk Rashba SOC and H_{EB} coupling strengths, we can fit the spinrelaxation anisotropy (solid red and blue lines in Figs. 1e and 2a, b, see Supplementary Information for detail), from which we extract R = 9.05 × 10^{−19}e m^{2} and C = 5.86 × 10^{−14}e m/T (outofplane), as well as R = 9.53 × 10^{−19}e m^{2} and C = 6.01 × 10^{−14}e m/T (inplane). Not only are the extracted coupling coefficients in remarkable agreement across both measurement cooldowns, we also confirm their theoretically predicted ratio as expressed in Eq. (3). The extracted H_{EB} coupling strength now allows us to to calculate a maximum effective magnetic field strength B_{eff} = CE_{y}B_{z}/μ_{B} ~ 18.9 mT for E_{y} = 5 MV/m and B = 3.5 T—more than two orders of magnitude larger than the vanishing bulk Rashba SOC field B_{R} = Rk_{0m}E_{y}/μ_{B} ~ 0.1 mT.
The evolution of the spinrelaxation rates with increasing electric field strength E_{y} from bulk donors (circles) to donors in electric fields (squares) is shown in Fig. 3c. For simplicity, here we have assumed E_{x} = E_{z} = 0, consistent with our COMSOL calculations (<1 MV/m), accounting for a small phase shift in θ. As evident from Fig. 3c, our theory describes both donors in zero and finite electric fields, providing a consistent theoretical framework for spin relaxation of donorbound electrons.
To conclude, we have demonstrated a previously unreported spin–orbit induced magnetoelectric coupling for donorbound electron spins in silicon. In contrast to bulk donors, we find that this higherorder coupling provides a spinrelaxation pathway, that can dominate spin relaxation for certain orientations of external electric and magnetic fields. By careful alignment of an external vectormagnetic field, we can eliminate the lifetimelimiting effect, thereby recovering the bulkregime of donor spin relaxation. Thus, we resolve the longstanding puzzle as to the significant differences of spin lifetimes in bulk donors to those in nanoelectronic devices. Given the strength of the observed magnetoelectric coupling—two orders of magnitude larger than bulk Rashba SOC—it should allow for electrically driven spin resonance,^{32,33,34} not previously thought possible in donors.
Finally, we note that this higherorder coupling could be observable for atomically confined spins in inversion asymmetric semiconductors, particularly when multiple valleys are present. A similar prediction of such magnetoelectric coupling has recently been made for bilayer graphene.^{35} We thus expect that our discovery will stimulate future theoretical and experimental investigations of spin control via higherorder electromagnetic coupling.
Methods
Device fabrication
The qubit device was patterned^{7} on the hydrogenterminated Si(001)–2 × 1 surface of pdoped substrates (N_{A} ≃ 5 × 10^{15}–5 × 10^{16} cm^{−3}). Following STMlithography, the template was exposed to PH_{3} gas followed by annealing (350 °C, 1 min), which dopes the device electrodes to N_{2D} ≃ 2 × 10^{14} cm^{−2}, allowing quasimetallic conduction.^{36} The device was subsequently encapsulated with ~45 nm of epitaxial silicon at low temperature (250 °C), followed by the alignment of Ohmic contacts.^{7}
Electrical measurements
Electron spinrelaxation rates were measured over two separate thermal cycles (cooldowns) in a toploading Leiden cryofree threeaxis vector field fridge (base temperature ≤25 mK, electron temperature (T_{el} ≃ 100 mK) and B_{z} ≤ 9 T, B_{x} ≤ 5 T, B_{y} ≤ 1 T). In a first cooldown, the device was mounted such that B_{z}[001], parallel to the surface normal (crystal growth direction) and B_{x}[110]. In a second cooldown, the device was mounted such that B_{z}[110] and \(B_x{\mathrm{}}[1\bar 10]\), allowing a rotation of the field vector within the sample plane. Any possible misalignment in either cooldown was estimated no greater than ±5°. Electron spin readout was performed by spin selective tunnelling from the donor to the SET, using a threelevel voltage pulse sequence on the G_{1} and G_{2} gates, upon monitoring the SET drain current (bandwidth: 3 kHz).
Electrostatic modelling
Electric fields in the device were calculated using the electrostatics module of the finite element solver COMSOL. We assume metallic electrodes with a thickness, t = 2 nm, as previously shown to represent STMpatterned quantum dots.^{7} The surface of the device was terminated with 2 nm of SiO_{2}, assuming Neumann boundary conditions. Both cooldowns were modelled separately with slightly different gate voltages (outofplane: V_{G1} = V_{G2} = 400 mV, inplane: V_{G1} = V_{G2} = 150 mV), consistent with the experiment. In both cooldowns V_{GT} = V_{SET} = 900 mV.
Data availability
Supplementary Information is available at npj Quantum Information website. The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
References
Loss, D. & DiVincenzo, D. P. Quantum computation with quantum dots. Phys. Rev. A 57, 120 (1998).
Kane, B. E. A siliconbased nuclear spin quantum computer. Nature 393, 133–137 (1998).
Tyryshkin, A. M. et al. Electron spin coherence exceeding seconds in highpurity silicon. Nat. Mater. 11, 143–147 (2012).
Watson, T. F., Weber, B., House, M. G., Büch, H. & Simmons, M. Y. Highfidelity rapid initialization and readout of an electron spin via the single donor D ^{−} charge state. Phys. Rev. Lett. 115, 166806 (2015).
Morello, A. et al. Singleshot readout of an electron spin in silicon. Nature 467, 687–691 (2010).
Pla, J. J. et al. A singleatom electron spin qubit in silicon. Nature 489, 541–545 (2012).
Weber, B. et al. Spin blockade and exchange in Coulombconfined silicon double quantum dots. Nat. Nanotechnol. 9, 430–435 (2014).
Dehollain, J. P. et al. Singleshot readout and relaxation of singlet and triplet states in exchangecoupled ^{31}P electron spins in silicon. Phys. Rev. Lett. 112, 236801 (2014).
Hollenberg, L. C. L., Greentree, A. D., Fowler, A. G. & Wellard, C. J. Twodimensional architectures for donorbased quantum computing. Phys. Rev. B 74, 045311 (2006).
Elzerman, J. M. et al. Singleshot readout of an individual electron spin in a quantum dot. Nature 430, 431–435 (2004).
Scarlino, P. et al. Spinrelaxation anisotropy in a GaAs quantum dot. Phys. Rev. Lett. 113, 256802 (2014).
Camenzind, L. C. et al. Hyperfinephonon spin relaxation in a singleelectron gaas quantum dot. Nat. Commun. 9, 3454 (2018).
Ferdous, R. et al. Valley dependent anisotropic spin splitting in silicon quantum dots. npj Quantum Inf. 4, 26 (2018).
Ferdous, R. et al. Interfaceinduced spin–orbit interaction in silicon quantum dots and prospects for scalability. Phys. Rev. B 97, 241401 (2018).
Zwanenburg, F. A. et al. Silicon quantum electronics. Rev. Mod. Phys. 85, 961–1019 (2013).
Hasegawa, H. Spinlattice relaxation of shallow donor states in Ge and Si through a direct phonon process. Phys. Rev. 118, 1523–1534 (1960).
Roth, L. Massachusetts Institute of Technology Lincoln Laboratory Reports (1960).
Roth, L. M. gfactor and donor spinlattice relaxation for electrons in germanium and silicon. Phys. Rev. 118, 1534–1540 (1960).
Watson, T. F. et al. Atomically engineered electron spin lifetimes of 30s in silicon. Sci. Adv 3, 1–6 (2017).
Mohiyaddin, F. A. et al. Noninvasive spatial metrology of singleatom devices. Nano Lett. 13, 1903–1909 (2013).
Sellier, H. et al. Transport spectroscopy of a single dopant in a gated silicon nanowire. Phys. Rev. Lett. 97, 206805 (2006).
Lansbergen, G. P. et al. Gateinduced quantumconfinement transition of a single dopant atom in a silicon finfet. Nat. Phys. 4, 656–661 (2008).
Rahman, R. et al. Electric field reduced charging energies and twoelectron bound excited states of single donors in silicon. Phys. Rev. B 84, 115428 (2011).
Rahman, R. et al. Engineered valley–orbit splittings in quantumconfined nanostructures in silicon. Phys. Rev. B 83, 195323 (2011).
Lansbergen, G. P. et al. Lifetimeenhanced transport in silicon due to spin and valley blockade. Phys. Rev. Lett. 107, 136602 (2011).
Rahman, R. et al. Gateinduced gfactor control and dimensional transition for donors in multivalley semiconductors. Phys. Rev. B 80, 155301 (2009).
Bir, G., Butekov, E. & Pikus, G. Spin and combined resonance on acceptor centres in Ge and Si type crystals—I: paramagnetic resonance in strained and unstrained crystals. J. Phys. Chem. Solids 24, 1467–1474 (1963).
Wilson, D. K. & Feher, G. Electron spin resonance experiments on donors in silicon. III. Investigation of excited states by the application of uniaxial stress and their importance in relaxation processes. Phys. Rev. 124, 1068–1083 (1961).
Winkler, R. Spin–Orbit Coupling Effects in TwoDimensional Electron and Hole Systems. Springer Verlag, Berlin Heidelberg (2003).
Salfi, J., Mol, J. A., Culcer, D. & Rogge, S. Chargeinsensitive singleatom spin–orbit qubit in silicon. Phys. Rev. Lett. 116, 246801 (2016).
Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in fewelectron quantum dots. Rev. Mod. Phys. 79, 1217 (2007).
Nowack, K. C., Koppens, F. H. L., Nazarov, Y. V. & Vandersypen, L. M. K. Coherent control of a single electron spin with electric fields. Science 318, 1430 (2007).
Corna, A. et al. Electrically driven electron spin resonance mediated by spin–valley–orbit coupling in a silicon quantum dot. npj Quantum Inf. 4, 6 (2018).
Jock, R. M. et al. A silicon metal–oxide–semiconductor electron spin–orbit qubit. Nat. Commun. 9, 1768 (2018).
Winkler, R. & Zülicke, U. Electromagnetic coupling of spins and pseudospins in bilayer graphene. Phys. Rev. B 91, 205312 (2015).
Weber, B. & Simmons, M. Y. Determining the quantumcoherent to semiclassical transition in atomicscale quasionedimensional metals. Phys. Rev. B 94, 081412 (2016).
Acknowledgements
We thank S. Mahapatra for useful discussions. The device was fabricated in part at the New South Wales node of the Australian National Fabrication Facility. This research was conducted by the Australian Research Council (ARC) Centre of Excellence for Quantum Computation and Communication Technology (project number CE110001027), the US National Security Agency and the US Army Research Office under contract number W911NF17110202, and the ARC Discovery Project scheme. BW acknowledges an ARC DECRA fellowship (project number DE160101334) and a Singapore National Research Foundation (NRF) Fellowship (NRFNRFF201711). M.Y.S. acknowledges an ARC Laureate Fellowship.
Author information
Authors and Affiliations
Contributions
B.W. and T.F.W. prepared the sample, performed the measurements, and analysed the data. R.L. and A.R.H. assisted with the vector fridge measurement. Y.H., R.R., and L.C.L.H. developed the theory. B.W. planned the experiment. M.Y.S. supervised the project. B.W. and M.Y.S. wrote the paper with input from all authors.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Weber, B., Hsueh, YL., Watson, T.F. et al. Spin–orbit coupling in silicon for electrons bound to donors. npj Quantum Inf 4, 61 (2018). https://doi.org/10.1038/s4153401801111
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4153401801111
This article is cited by

Optimisation of electron spin qubits in electrically driven multidonor quantum dots
npj Quantum Information (2022)

Review of performance metrics of spin qubits in gated semiconducting nanostructures
Nature Reviews Physics (2022)

Topological phases of a dimerized Fermi–Hubbard model for semiconductor nanolattices
npj Quantum Information (2020)

The subband structure of atomically sharp dopant profiles in silicon
npj Quantum Materials (2020)

Gatereflectometry dispersive readout and coherent control of a spin qubit in silicon
Nature Communications (2019)