Addressable electron spin resonance using donors and donor molecules in silicon
Phosphorus donor impurities in silicon are a promising candidate for solid-state quantum computing due to their exceptionally long coherence times and high fidelities. However, individual addressability of exchange coupled donor qubits with separations is challenging. Here we show that by using atomic-precision lithography we can place a single P donor next to a 2P molecule apart and use their distinctive hyperfine coupling strengths to address qubits at vastly different resonance frequencies. In particular the single donor yields two hyperfine peaks separated by , in contrast to the donor molecule which exhibits three peaks separated by . Atomistic tight-binding simulations confirm the large hyperfine interaction strength in the 2P molecule with an inter-donor separation of , consistent with lithographic STM images of the 2P site during device fabrication. We discuss the viability of using donor molecules for built-in addressability of electron spin qubits in silicon.
Phosphorus donor atoms in silicon are very attractive as the basis of a solid-state quantum computer, because they combine the long-lived quantum memory of a nuclear spin with the rapid control and strong interactions possible with an electron spin Kane (1998). With weak coupling to their environment, phosphorus donors in isotopically purified Si have demonstrated minutes-long nuclear Steger et al. (2012) and millisecond-long electron Tyryshkin et al. (2012) spin coherence times. Additionally, high fidelity single-qubit quantum gate operations using resonant magnetic fields Muhonen et al. (2015) and high fidelity state readout Watson et al. (2017); Pla et al. (2013) have recently been demonstrated for single P donor qubits.
Due to the strong Coulomb potential well, donors provide a means of producing uniform electron spin qubits with reproducible tightly confined wavefunctions, and thus a non-degenerate, low-lying valley and orbital ground-state. To achieve an accurate entangling two-qubit quantum gate, via the exchange Koiller et al. (2001); Song and Das Sarma (2016) or dipole de Sousa et al. (2004) interaction, donors must be placed with high precision and on the order of tens of nanometres apart. We achieve this fine positional control through scanning tunnelling microscope (STM) hydrogen resist lithography Schofield et al. (2003) which has an effective resolution less than the silicon lattice constant. One of the challenges therefore, is to individually address nominally identical qubits when they are very close together. In gate-defined silicon quantum dot single-spin qubits, typically placed apart, the nesseccary addressability in electron spin resonance has been achieved via a slanting Zeeman field generated by a surface micromagnet Noiri et al. (2016, 2016), or via a Stark shift in the electron spin resonances Veldhorst et al. (2014). Similar tunability of qubit resonance frequencies has been observed for a single donor qubit Laucht et al. (2015). However, in moving to multi-qubit systems, with inter-donor separations, a higher B- or E-field gradient will be needed to avoid overlapping qubit resonance frequencies Büch et al. (2013).
In this paper we demonstrate the successful implementation of an alternate strategy for addressing donor-bound electron spin qubits. Here we use the differentiated hyperfine coupling of an electron confined by the potential well formed by a single P donor and by a pair of donors in a donor molecule Büch et al. (2013); Wang et al. (2016a). We present electron spin resonance (ESR) measurements of the hyperfine spectrum of both a single donor (1P) and a donor molecule (2P) within a single double quantum dot device.
The spin states of each individual quantum dot may be described by the generalised Hamiltonian containing an electron Zeeman term and nuclear Zeeman terms as well as hyperfine interaction terms for each donor nucleus, .
Here () is the electron (nuclear) g-factor, () the Bohr (nuclear) magneton, and () the electronic (nuclear) spin angular momentum. The static magnetic field is oriented parallel to the patterned surface, and aligned with the crystal axis, and the hyperfine coupling tensor for each nuclei is decomposed into a scalar Fermi contact interaction part and an anisotropic dipolar component , which is often treated as negligible.
ii.1 Alignment of ESR antenna
The device presented is a planar donor-defined nanostructure, in which 2D regions of a silicon crystal are heavily phosphorus-doped beyond the metal-insulator transition by STM hydrogen resist lithography, to produce gates, reservoirs and also localised charge islands Schofield et al. (2003). Figs. 1A,B show the lithographic mask at the two quantum dot sites, relative to the dimer reconstruction on the silicon surface during fabrication. To incorporate a single P atom, three adjacent exposed dimers are required. This permits the phosphine gas () molecule to fully dissociate at the surface (Wilson et al., 2006). From the images (both showing 5 consecutive exposed dimers), we expect at least one P atom in each quantum dot, with a possibility for two at the left site due to the presence of multiple additional exposed dimers nearby. The actual number of donors incorporated at each site was subsequently verified by measuring each quantum dot’s charging energy. Here we determined the single electron addition energies to be meV and meV, consistent with NEMO3d tight-binding simulations reflecting 2P and 1P for the left and right quantum dot respectively Weber et al. (2014). These assignments are further confirmed by the ESR results following. The two sites are separated by nm, and are both tunnel-coupled (at a distance of 19nm) to a larger charge-sensing single electron transistor (SET) as shown in Fig. 1C, for energy selective spin readout Elzerman et al. (2004). The SET additionally functions as an electron reservoir for the two donor sites. The full layout including electrostatic gates is shown in Fig. 1D, where red regions represent metallic conductive structures of delta-doped epitaxial silicon, with a carrier density of McKibbin et al. (2010). The conducting phosphorus structures are buried below a 55nm thick encapsulation layer, and contacted by etching vias and depositing aluminium surface contacts.
Following initial characterisation of the device including independent spin readout and spin correlation measurements Broome et al. (2017a), a broadband microwave antenna was post-fabricated on the chip. This is a remarkable feature of donor-based all-epitaxial devices. Since the dopant layer is protected by the crystalline silicon environment, which is conductive () at room temperature due to background doping, electrostatic discharge is unlikely. Hence these devices can be measured at cryogenic temperatures (where background dopants freeze out) multiple times, and be re-processed to add additional surface gates, waveguides or antennas, before being measured again.
The antenna geometry is impedance matched Dehollain et al. (2012) to minimise radiative and reflective loss of microwave power, whilst maximising the oscillating magnetic field, . The post-fabrication process requires additional electron beam lithography, achieved with positional uncertainty of nm relative to the buried atomic scale device, by reference to pre-etched alignment markers Ruess et al. (2005). Physical vapour deposition of 100nm of aluminium onto the naturally oxidised silicon surface produces an antenna capable of withstanding up to 2V DC bias relative to the buried phosphorus layer with minimal current leakage ().
The inner region of the completed antenna is seen in Fig. 1E (coloured blue), positioned with the donors inside the loop of the antenna where the simulated ratio of oscillating magnetic field, , to in-plane oscillating electric fields is maximised (see Supplementary material I). A vertical cross-section of the device structure is shown in Fig. 1F.
ii.2 Addressable resonance spectra
We operate in the high magnetic field regime (T) such that , where the eigenstates are to first order separable into electron and nuclear subspaces, and we perform our measurements at in a dilution fridge. The spin resonance experiment proceeds by applying voltages proportionally to the left and right gates, in order to detune the donor potential relative to a fixed SET Fermi energy (see Supplementary material II). Since the electron spin relaxation time is much longer than the characteristic tunneling time between SET and donor, we initialise the state by ionising the donor and deterministically loading an electron in the spin down state . The donor-bound electron is in Coulomb blockade whilst a microwave pulse is applied. This is followed by a single-shot readout sequence in which spin dependent tunneling Elzerman et al. (2004); Büch et al. (2013) converts the projected electron spin state to a charge state, observable via the SET current signal.
The sequence used for the 2P molecule is equivalent, but instead of conditionally ionising the molecule by removing the single electron, it utilises a spin-dependent transition into the 2 electron spin singlet state Watson et al. (2015) for readout. This technique has the additional benefit of faster tunnelling rates, which permits faster operation relative to the single donor (1P). Our microwave ESR pulses are applied with a nominal power at the signal generator of (we estimate attenuation at the device) for , and modulated with a linear frequency chirp of . This adiabatic passage pulse Garwood and DelaBarre (2001) inverts the electron spin eigenstates irrespective of the exact pulse duration or precise instantaneous resonance frequency, enhancing our spin resonance signal Laucht et al. (2014).
The spin resonance spectrum of the single donor (1P) is shown in Fig. 2A for . This data shows the fraction of outcomes, , over 640 single-shot measurements at each frequency, . We observe two peaks, corresponding to the resonance conditions for driving transitions between electron and states when the single nuclear spin state is either (left peak) or (right peak). The transition frequencies here are: and , separated by the single donor hyperfine coupling strength . The solid curve in Fig. 2A is a fit to the sum of two Gaussian peaks sharing a common full width at half maximum and with amplitudes and .
The hyperfine coupling strength for the single P atom is comparable to other values in the literature for ion-implanted P donor devices ranging from to Muhonen et al. (2015); Laucht et al. (2015). The difference between this and the value of the hyperfine reported for bulk ensembles of P donors Feher (1959) can be attributed to a Stark shift, since within our device we have an electric field at the 1P site of (see Supplementary Material III), perturbing the wavefunction. At this E-field, the reduced the electron density over the nucleus leads to a quadratic Stark effect in P donors Pica et al. (2014); Rahman et al. (2007) and thus is reduced by a factor , consistent with our measurement.
We note that in Fig. 2A the peak has more than twice the amplitude of the resonance, indicating that there is some polarisation of the nuclear spin, with the state more likely to be occupied than . This polarisation reflects nuclear spin dynamics likely arising from an inelastic electron-nuclear flip-flop process, pumped by spin resonant excitation at the frequency Pines et al. (1957); McCamey et al. (2009). Here an electron spin ‘flips’ from to , and the nuclear spin simultaneously ‘flops’ from to . The total spin is thus conserved, and energy conservation is satisfied by the emission of a phonon. Since the energy difference between the states is larger than the thermal energy , the reverse transition involving absorption of a phonon is suppressed. Any alternate cross-process involving the and states would require a change in total spin of and so is forbidden by spin conservation. Interestingly, we infer from fluctuations in our recorded spin-up signal over time (see Supplementary material IV for further analysis on the nuclear dynamics) that the timescale for the flip-flop process may be as short as , orders of magnitude faster than expected for P donors at our magnetic field and temperaturePines et al. (1957); McCamey et al. (2009). We may attribute this increased rate to an enhancement in the electron-phonon interaction due to the non-trivial valley structure of the electric field-perturbed wavefunction inside our nanostructureBoross et al. (2016). Partial repopulation of the state can be explained by an ‘ionisation shock’ Pla et al. (2013), where mis-alignment of the nuclear spin eigenstates in the neutral and ionised donor charge states provides a small non-zero probability of flipping the nuclear spin on each electron ionisation event.
In Fig. 2B we present the resonance spectrum measured for the electron bound to the 2P molecule. This data is based on 2000 single shots (compared to 640 for the 1P single donor due to the faster tunnel rate between the SET and 2P molecule), with a microwave pulse power of and the same chirp parameters as above. Here we observe three resonant frequencies. The solid curve is a fit to three Gaussian peaks of width and amplitudes: and . The three peaks reflect the transition frequencies, shown in the inset to Fig. 2B, separated by and , the contact hyperfine interaction coefficients representing the electron wavefunction density at the location of the two donor sites (labelled and ) of the 2P molecule. At zero electric field the electron wavefunction is symmetric and the hyperfine interaction at the two donor sites is expected to be equal , producing two degenerate transition frequencies . However, at the operating point of the 2P molecule in our device, we calculate an electric field of around (see Supplementary material III, and VII), which serves to break this degeneracy . Since we resolve only one central peak in the resonance spectrum, the hyperfine asymmetry must be less than the width of the observed peak . The average peak separation, representing the donor molecule hyperfine interaction energy is more than twice the single donor value, consistent with the anticipated range for a pair of donors with small () spatial separation Wang et al. (2016a). We can calculate the Stark shift expected for the 2P molecule for the same electric fields (see Supplementary Material VII). The value ranges from (with the electric field perpendicular to the molecular axis of the 2P molecule) to (with the electric field parallel to the molecular axis of the 2P molecule).
Interestingly, we note that the asymmetry in the peak amplitudes seen in the 1P case is absent in the 2P molecule’s ESR spectrum. In Fig. 2B we see an equal probability for each of the 4 nuclear spin state resonances. The nuclear and states are approximately degenerate and indistinguishable, producing a peak with approximately twice the amplitude of the and resonances.
To understand the nuclear spin dynamics in the 2P molecule, we consider the full tensor form of the hyperfine interaction , which for each of the donors consists of the Fermi-contact hyperfine scalar , proportional to the electron wavefunction density at the position of a donor atom and the traceless dipole-dipole interaction tensor (see Supplementary material V). Since donor atoms have a strong Coulomb confinement, the electron wavefunction is highly concentrated over the donor nuclei, and dominates by several orders of magnitude over for a single donor, even with significant perturbation by an electric field Pica et al. (2014); Park et al. (2009). The dipolar tensor is expected to be more anisotropic in the case of the molecular 2P wavefunction Saraiva et al. (2015), since it is inherently non-spherical. Indeed we find that there is no correlation in the nuclear spin state between successive electron readout events, confirming a significant enhancement of the ‘ionisation shock’ mechanism relative to that observed for the single donor. In short, the nuclear spin state in the 2P molecule is randomised faster than it is polarised by any inelastic relaxation.
Next we turn to consider the disparity in resonance linewidths in the 1P and 2P cases, important as this reflects the coherence properties of the bound electron spin. The 1P resonance peaks have a linewidth of 8MHz as a result of being artificially broadened by the linear frequency chirp we apply to our microwave pulse to adiabatically invert the spin state. The spin dephasing time for a single P donor in natural silicon was recently measured as 55ns Pla et al. (2012) corresponding to a natural linewidth (FWHM) around . This is limited by random fluctuations in the local magnetic field due to presence of nuclear spins. We use the adiabatic pulse strategy [24,25] to combat the fluctuations and selected the chirp span to cover the expected natural linewidth. The 2P resonances are markedly wider at , which we attribute to a stronger interaction between the bound electron and local nuclear spins within the 2P wavefunction envelope, as compared to the single donor. Whilst this suggests a dephasing time for the current 2P bound electron, using isotopically purified silicon substrate promises to completely suppress nuclear-spin limited decoherence. Thus we expect that dephasing times approaching milliseconds are possible for 2P electron qubits, just as observed already for single donorsMuhonen et al. (2015).
ii.3 Hyperfine metrology of donor position
Given the atomic scale of our device, it is possible to model the full electron wavefunctions, accounting for the silicon lattice and bandstructure, donor potentials and the potential profile across the nanostructure. Hence we compare our measured 2P hyperfine coupling strength, , with atomistic tight-binding simulations. From the size of the lithographic patches (Fig 1A,B), we restrict ourselves to consider pairs of lattice sites within a distance of . Fig. 3A shows the calculated hyperfine energy within a 2P molecule hosting a single electron, where we vary the location and thus the separation between the two P atoms. The datapoints in Fig. 3A indicate the values for configurations where both donors lie in the  crystallographic plane, for 4 different inter-atomic distances (additional out-of-plane donor configurations are discussed for comparison in Supplementary material VI). The general trend is a reduction in the hyperfine interaction with increasing donor separation, as expected. However the interplay between the tetrahedral symmetry of the silicon crystal lattice around the donor atom, and the cubic symmetry of the 6 conduction band minima in the silicon bulk gives rise to a highly structured 2P wavefunction with deviations from a smooth exponential decay curve that is dependant on the orientation of the donor pair with respect to the crystal latticeMartins et al. (2005); Wang et al. (2016a). The dominant uncertainty in the calculated hyperfine coupling A2P is due to the potential Stark shift that could be observed in such a system for the electric field strength and orientation used in the device (see Supplementary Materials VII). The magnitude of the uncertainty () is reflected by the size of the markers themselves.
Over the past decade, research on the Si:P system has established that during an anneal at , a phosphorus atom incorporates into the surface layer of silicon, forming a P-Si heterodimer with a strong phosphorus-silicon bond Wilson et al. (2006). Consequently, when encapsulated with silicon in a low temperature () epitaxial growth process Goh et al. (2004), calculations indicate that the P-Si bond is resilient to segregation or diffusion Bennett et al. (2010). Experimental evidence confirms that the donor atom remains localised Oberbeck et al. (2004); Usman et al. (2016) in the crystal at the lithographically defined site, to an uncertainty on the order of one lattice constant. The orange band represents the hyperfine energy measured experimentally for our 2P molecule, . The geometrical layout of the in-plane configurations of donors are displayed in the inset, colour-coded to the hyperfine energy plot. Due to lattice symmetry there are several equivalent sites for the second P atom, at any given distance from the central reference site (coloured black) which represents the location of the first P atom of the molecule. Considering our experimental value of , only the eight geometries coloured orange and yellow are likely representations of the relative configuration of donors in our device. These are consistent with the size of the lithographic patch that was fabricated, as can be seen with reference to the surface dimer reconstruction marked by dashed lines.
An interesting feature observed in our resonance measurements is a relative offset in the central frequencies, evident when the 1P and 2P spectra are displayed on a single axis as in Fig. 4A. To gain an understanding of this offset, we examine the magnetic field dependence of the resonant frequencies. Fig. 4B (C) plots the relative offset in resonance frequencies for the 1P electron (2P electron) as a function of magnetic field. The values plotted are obtained from fits to the recorded spectra at each magnetic field setting, with a linear Zeeman term subtracted for clarity. The electron g-factor used in each figure is obtained by a linear fit to each dataset, producing values of (1P) and (2P). These are both consistent with the bulk g-factor of for donor-bound electrons Feher (1959); Stegner et al. (2006), with a uncertainty limited by the current-to-field calibration of our superconducting magnet.
The offset between the central frequencies of the 1P and 2P spectra remains constant across the field range. In magnetic field terms this corresponds to , consistent with the difference in g-factor we observe between the 1P and 2P cases. We attribute this variation to a Stark shift of the g-factor Bradbury et al. (2006); Friesen (2005), since the electric field is different in both magnitude and direction for our two quantum dots (see Supplementary material III). Importantly, the difference in peak splitting measured for our 1P and 2P sites , as shown in Fig. 4A, is considerably larger than any variability expected from a Stark shift of the hyperfine coupling or g-factor, for either the single P atom or 2P molecule.
The range of hyperfine values available for closely separated 2P molecules means that our addressing scheme can be extended to a larger number of qubits. For instance, utilising the four different 2P hyperfine interaction strengths shown in Fig. 3A along with a single donor may produce five individually addressable qubits with unique resonance frequencies, and with an offset greater than inhomogeneous broadening due to nearby nuclear spins, or electric field shifts. An area requiring further investigation however, is to determine the impact of multiple host nuclear spins on coherence times and the overall electron spin dynamics.
We note that hosting single electrons in donor molecules carries a number of additional benefits beyond the intrinsic frequency detuning, including extended relaxation times Hsueh et al. (2014) and deeper confinement of the two electron charge state. The latter is particularly relevant for implementing singlet-triplet based qubits using donor-bound electrons and to realise SWAP-type two-qubit gates Kalra et al. (2014); Wang et al. (2016b). The use of larger donor molecules, patterned by STM hydrogen lithography provides additional scope for wavefunction engineering, and permits strong confinement of multi-electron states, as required to achieve Pauli spin blockade for high fidelity state readout Broome et al. (2017b).
These results represent an important step toward achieving full control over multiple donor spin qubits in silicon. The addressability demonstrated, with frequency detunings an order of magnitude larger than the inhomogeneous linewidth in natural silicon will facilitate selective control over individual qubits with low cross-talk. Combined with isotopically purified Si and NMR control over the nuclear spin states Morton et al. (2008), donor molecules provide individual addressability for electron spin qubits, and are attractive for quantum simulation and multi-qubit architectures.
Iv Materials and Methods
The STM hydrogen lithography was performed in ultrahigh vacuum with an Omicron Variable Temperature instrument. A chemically cleaned Si(001) wafer was annealed at and passivated in a beam of atomic hydrogen. The hydrogen mask was removed in the required areas by scanning with a tip voltage of around and current setpoint of . Following lithography, the wafer was dosed with phosphine gas, then heated to to incorporate the P donors (Wilson et al., 2006) before a encapsulation layer of epitaxial silicon was grown at a rate of . The donor layer was electrically contacted by depositing aluminium onto contact vias formed by reactive ion etching. The contact structures, as well as the microwave antenna were all defined by electron beam lithography using a PMMA mask.
Measurements were performed in a dilution refrigerator with a base temperature of . A superconducting solenoid magnet provided the external magnetic field. DC voltage signals applied to the gates were generated by Yokogawa 7651 and Stanford Research Systems SIM928 voltage sources. Voltage pulses were generated by a National Instruments USB6363 DAC/ADC device and added to the DC signals with simple resistive voltage dividers. The combined gate control signals were then filtered by two-stage lumped element RC filters inside the dilution fridge with a low-pass cutoff of , and additional high frequency (GHz) noise was suppressed by distributed ’Eccosorb LS’ RF absorber material within the filter enclosure. The microwave signals were supplied to the on-chip antenna from a Keysight E8267D vector signal generator (with phase and pulse modulation signals supplied by a Tektronix 5014C arbitrary waveform generator) via a lossy stainless steel coaxial cable (and additional 1dB attenuator at 4K). The readout signal was collected from the SET by a low noise Femto DLPCA200 transimpedance amplifier and then electrically decoupled and filtered by a Stanford Research Systems SIM910 JFET isolation amplifier and SIM965 Bessel filter before being digitised by the National Instruments USB6363 DAC/ADC.
Our tight-binding method uses the NEMO-3D atomistic solver. The model applies an adjustable cut-off potential at the donor site (a central-cell correction) while elsewhere each donor potential is Coulombic Wang et al. (2016a). This model of the central cell correction has been successful in reproducing the full single donor spectrum, the Stark shifts of the hyperfine coupling , as well as the system g-factor, the orbital energies of the bound electrons, and replicating STM based imaging of the donor wavefunction. The Schrodinger and Poisson equations are then self-consistently solved within a 30nm cubic domain to produce the ground-state wavefunction from which the hyperfine interaction strength is computed for each donor.
This research was supported by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (Project No. CE110001027), the U.S. National Security Agency and the U.S. Army Research Office under Contract No. W911NF-13-1-0024. M.Y.S. acknowledges an Australian Research Council Laureate Fellowship. This work was performed in part at the NSW Node of the Australian National Fabrication Facility. S.J.H., M.G.H., E.P. and M.Y.S. designed the experiment; S.J.H., S.K.G., M.B. and J.G.K. fabricated the device; S.J.H. and L.F. carried out spin resonance measurements and analysed the data; C.C.Y., Y.W. and R.R. performed tight-binding simulations. S.J.H. and M.Y.S. wrote the article with input from all authors. M.Y.S. is Editor-in-Chief of Nature Partner Journals (NPJ) Quantum Information and receives financial benefits from the position. The other authors declare that they have no competing interests. All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors. The authors thank Yuling Hsueh, Arne Laucht, Lloyd Hollenburg and Sven Rogge for helpful discussions.
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