WO2023115107A1

WO2023115107A1 - Exchange-based magnetic field sensors and sensing methods

Info

Publication number: WO2023115107A1
Application number: PCT/AU2022/051520
Authority: WO
Inventors: Michelle Yvonne Simmons; Samuel Keith Gorman; Joris KEIZER; Ludwik Kranz; Daniel Keith
Original assignee: Silicon Quantum Computing Pty Limited
Priority date: 2021-12-22
Filing date: 2022-12-16
Publication date: 2023-06-29
Also published as: AU2022421945A1

Abstract

A method of determining the atomic configuration of nuclear spin atoms in a semiconductor processing element is disclosed. The semiconductor processing element includes at least one semiconductor layer and at least two electrons/holes confined within the at least one semiconductor layer. The at least two electrons/holes exchange are coupled with each other. The method includes the steps of: performing exchange oscillations between the at least two electrons/holes by applying timed exchange pulses to the semiconductor processing element; measuring the spins of the at least two electrons/holes after the exchange oscillations; calculating differences in energy splitting between the at least two electrons/holes; determining hyperfine coupling values between the at least two electrons/holes and the nuclear spin atoms; and determining the location of the nuclear spin atoms based on the determined hyperfine coupling values.

Description

EXCHANGE-BASED MAGNETIC FIELD SENSORS AND SENSING METHODS

TECHNICAL FIELD

[0001] Aspects of the present disclosure are related to advanced processing systems and methods for operating the same and more particularly, to ultra-sensitive exchange-based magnetic field sensors.

BACKGROUND

[0002] The developments described in this section are known to the inventors. However, unless otherwise indicated, it should not be assumed that any of the developments described in this section qualify as prior art merely by virtue of their inclusion in this section, or that those developments are known to a person of ordinary skill in the art.

[0003] Large-scale quantum processing systems hold the promise of a technological revolution, with the prospect of solving problems which are out of reach with classical machines. To date, a number of different structures, materials, and architectures have been proposed to implement quantum processing systems and fabricate their basic information units (quantum bits or qubits). Qubits can be understood as quantum-mechanical systems encoded into two discrete energy levels.

[0004] One way of fabricating qubits, for example, is to use the nuclear or the electron spin of phosphorus donor atoms in silicon such that the nuclear/electron spin of each phosphorus donor atom acts as a qubit. This fabrication technique offers near perfect qubit state encoding due to the addressability and long coherence of the phosphorus spins. Further, qubits fabricated in this manner have demonstrated second-long lifetimes and benefit from a semiconducting host enabling electrical addressing and high fidelities. In particular, there has been substantial progress made recently towards a donor-based quantum computer, with both single-qubit and two-qubit gates exceeding the fault-tolerant threshold (> 99%).

[0005] Many important operational characteristics of donor-based qubits, such as the hyperfine strengths and relaxation times, depend directly on the exact locations of donors within the silicon crystal lattice. For the development of a full-scale quantum computer it is essential to be able to understand the interplay between the atomic qubit composition and the corresponding operational characteristics, and leverage this understanding to optimise qubit behaviour at the atomic level. Therefore, it is critical to establish robust methods for mapping out the atomic arrangement of qubits.

SUMMARY

[0006] According to a first aspect of the present disclosure there is a provided a method of determining the atomic configuration of nuclear spin atoms in a semiconductor processing element, the semiconductor processing element comprising: at least one semiconductor layer, at least two electrons/holes confined within the at least one semiconductor layer, the at least two electrons/holes exchange coupled with each other; the method comprising the steps of: performing exchange oscillations between the at least two electrons/holes by applying timed exchange pulses to the semiconductor processing element; measuring the spins of the at least two electrons/holes after the exchange oscillations; calculating differences in energy splitting between the at least two electrons/holes; determining hyperfine coupling values between the at least two electrons/holes and the nuclear spin atoms; and determining the location of the nuclear spin atoms based on the determined hyperfine coupling values.

[0007] In some embodiments, the semiconductor processing element is a quantum processing element.

[0008] Further, the nuclear spin atoms may be dopant atoms and the semiconductor processing element may include a plurality of dopant dots embedded in the at least one semiconductor layer, each dopant dot comprising one or more dopant atoms and the at least two electron/holes are confined within adjacent dopant dots.

[0009] In some embodiments, determining the location of the nuclear spin atoms further comprises: comparing the calculated differences in energy splitting with predetermined energy splitting values for different atomic configurations of a test processing element having similar number of dopant dots and dopant atoms as the semiconductor processing element to identify matching differences in energy splitting; and determining the atomic configuration of the semiconductor processing element based on the atomic configuration of the predetermined energy splitting value that matches the calculated energy splitting value.

[0010] The method may further include simulating or modelling the predetermined energy splitting values for the different atomic configurations of the test processing system. [0011] Further still, the method may include detecting discrete exchange oscillation frequencies in response to performing the exchange oscillations between the at least two electrons/holes. In such cases, the method also includes simulating or modelling predetermined exchange oscillation values for different atomic configurations of the test processing element. Yet still, the method in such cases includes: comparing the detected exchange oscillation frequencies with the predetermined exchange oscillation frequencies to identify matching exchange oscillation frequencies; and determining the atomic configuration of the semiconductor processing element based on the atomic configuration of the predetermined exchange oscillation frequency values that matches the detected exchange oscillation frequencies.

[0012] In some embodiments, the method of the first aspect includes initializing the at least two electrons or holes in a mixture

and |IT) states. This initialization includes loading an electron or hole with a random spin from a reservoir on a first dopant dot by plunging the energy levels of the dopant dot below the Fermi energy of the reservoir; and loading an electron or hole in a spin-down state on the second dopant dot by aligning the reservoir’s Fermi energy in between the energy levels on the second dopant dot.

[0013] In some examples, at least one of the dopant dots in the pair of the dopant dots includes multiple dopant atoms. Further at least one of the dopant dots in the pair of the dopant dots includes multiple electrons/holes.

[0014] In some embodiments, the nuclear spin atoms are phosphorus atoms. In other embodiments, the semiconductor processing element comprises a plurality of gate-based quantum dots formed in the semiconductor, the nuclear spin atoms are one or more silicon-29 atoms present in the gate-based quantum dots and the at least two electron/holes are confined within adjacent quantum dots.

[0015] In such systems, the method further includes comparing the determined hyperfine coupling values with predetermined distribution of hyperfine couplings for different positions of the one or more silicon-29 atoms in the adjacent quantum dots; and determining the position of the one or more silicon-29 atoms based on the positions of the one or more silicon-29 atoms with the predetermined hyperfine coupling values that matches the determined hyperfine coupling values.

[0016] According to another aspect of the present invention there is provided a method for determining the nuclear spin environment in a semiconductor device, by an exchange-based sensor, the exchange based sensor comprising at least two exchange -coupled electrons/holes, the method comprising: performing exchange oscillations between the at least two electrons/holes by applying timed exchange pulses to the semiconductor device; measuring the at least two electrons/holes after the exchange oscillations; calculating differences in energy splitting between the at least two electrons/holes; determining hyperfine coupling values between the at least two electrons/holes and nuclear spin atoms present within the electron wavefunctions of the at least two electrons/holes; and determining the location of the nuclear spin atoms based on the determined hyperfine coupling values.

[0017] In some embodiments in the method of the second aspect determining the location of the nuclear spin atoms further comprises: comparing the calculated differences in energy splitting with predetermined energy splitting values for different atomic positions of nuclear spins atoms within the electron wavefunction to identify matching differences in energy splitting; and determining the position of the nuclear spin atoms based on the atomic positions of the predetermined energy splitting value that matches the calculated energy splitting value.

[0018] The method may further include detecting discrete exchange oscillation frequencies in response to performing the exchange oscillations between the at least two electrons/holes.

[0019] In some embodiments, the method further comprising initializing the at least two electrons or holes.

[0020] The method further includes calculating or modelling predetermined exchange oscillation values for different atomic positions of nuclear atoms in the electron wavefunction. In such cases, the method further includes comparing the detected exchange oscillation frequencies with the predetermined exchange oscillation frequencies to identify matching exchange oscillation frequencies; and determining the atomic position of the nuclear atoms based on the atomic positions of the predetermined exchange oscillation frequency values that matches the detected exchange oscillation frequencies.

BRIEF DESCRIPTION OF DRAWINGS

[0021] While the invention is amenable to various modifications and alternative forms, specific embodiments are shown by way of example in the drawings and are described in detail. It should be understood, however, that the drawings and detailed description are not intended to limit the invention to the particular form disclosed. The intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the present invention as defined by the appended claims.

[0022] Fig. 1A is a Scanning Tunnelling Microscopy (STM) micrograph of a two-qubit quantum device according to aspects of the present disclosure.

[0023] Fig. IB is an STM image of the qubit sites of the quantum device of Fig. 1A.

[0024] Fig. 1C is an STM image of the left qubit of Fig. 1A.

[0025] Fig. ID is an STM image of the right qubit of Fig. 1 A.

[0026] Fig. IE is a graphical representation of the left quantum dot of Fig. 1A.

[0027] Fig. IF is a graphical representation of the right quantum dot of Fig. 1A.

[0028] Figs. 2A and 2B are schematics illustrating the energy splitting of the left and right electron spin qubits as a function of nuclear spin configurations of the three and two atoms that host the left and right qubits, respectively.

[0029] Fig. 2C is a schematic representation of some of the different combinations of nuclear spin configurations in a two-qubit 3P-2P system.

[0030] Fig. 3 A is a schematic Bloch sphere diagram and simulated exchange two-spin probabilities for a fixed energy difference between qubits.

[0031] Fig. 3B is a schematic Bloch sphere diagram and simulated exchange two-spin probabilities for a system that includes two discrete values of energy difference between qubits.

[0032] Fig. 3C is a schematic Bloch sphere diagram and simulated exchange two-spin probabilities for a system that includes five discrete values of energy difference between qubits.

[0033] Fig. 4A is stability diagram of the device of Fig. 1A showing SET current as a function of left and right gate voltages.

[0034] Fig. 4B is a zoomed in view of a portion of the stability diagram of Fig. 4A.

[0035] Fig. 4C voltage map illustrates an example method for measuring exchange oscillations.

[0036] Fig. 4D is a schematic chart showing the example method of Fig. 4C.

[0037] Fig. 5 A is a chart displaying exchange oscillations measured for the device of Fig.

1A using the method of Fig. 4C. [0038] Fig. 5B is a schematic showing the frequency spectrum of the |Tl) exchange oscillations depicted in Fig. 5A.

[0039] Fig. 5C is a schematic showing the two most probably atomic configurations of the donor atoms in the left and right quantum dots of the device shown in Fig. 1A.

[0040] Fig. 6A is a chart showing measured exchange oscillations and modelled exchange oscillations for the two atomic configurations shown in Fig. 5C.

[0041] Fig. 6B is a schematic showing a simulated frequency spectrum of the modelled exchange oscillations for the two atomic configurations shown in Fig. 5C.

DETAILED DESCRIPTION

[0042] As described above, it is important to establish robust methods for mapping out the atomic environment/placement of qubits.

[0043] Conventionally, to determine the atomic-scale characterization of qubits, Electron Spin Resonance (ESR) spectroscopy is used. This technique can be applied to electron spins of qubits to extract hyperfine couplings (i.e., interactions) between the given electron spin and each of the nuclear spins present within the electron’s wave-function. The extracted hyperfine couplings can be then combined with atomistic simulations to determine the spatial locations of donor atoms.

[0044] While ESR provides a direct measure of hyperfine couplings, it can introduce a number of challenges. For example, to perform ESR, a high-frequency AC signal of around 30 - 40 GHz (which is in the microwave regime) is required. The generation, processing, and transmission of microwave signals is known to be extremely demanding due to the short signal wavelength. Therefore, ESR needs expensive specialised equipment and wiring. Further ESR relies on an antenna placed in close proximity to the qubits such that an AC magnetic field can be produced at the qubit site. These antennas require additional nano-fabrication steps, and can introduce additional unwanted noise to the qubits. Further still, ESR experiments are typically performed at a narrow range of static magnetic fields due to the limited frequency range of commercially available microwave generators.

[0045] Aspects of the present disclosure employ a novel exchange-based spectroscopy technique and a novel exchange-based magnetic field sensor to accurately determine the position of nuclear spins in solid-state devices. For instance, aspects of the present disclosure can be used to determine the position of dopant atoms in semiconductor processing elements (and in particular in dopant-based quantum dots). Similarly, aspects of the present disclosure can be used to determine the position of nuclear spins (such as ²⁹Si) in gate-controlled quantum dots. In particular, the exchange-based spectroscopy of the claimed invention utilizes a pair of exchange-coupled electrons/holes to determine the atomic-scale magnetic fields produced by nuclear spins in the vicinity of/in the wavefunctions the exchange-coupled electrons/holes.

[0046] The exchange-based spectroscopy technique includes calibrating the solid-state device such that exchange coupling can be applied to two adjacent electrons/holes in a device. Next, the electron/holes can be initialized and timed exchange pulse may be applied to the electrons/holes to perform exchange oscillations. By measuring the spins of the individual electrons/holes, the two-spin probabilities can be measured. The anti-parallel two-spin states (down-up and up-down) are exchange-coupled and produce so-called exchange oscillations when plotted as a function of the duration of the applied exchange pulse. From the measured data, the frequency spectrum of the exchange oscillations, Q, can be measured and the corresponding differences in energy splitting between the electron spin qubits can be determined. Alternative or in addition, the hyperfine coupling values between the at least two electrons/holes and the nuclear spin atoms can also be determined based on the exchange oscillations. The measured data can then be compared with simulation data of a similar device to determine the spatial locations of nuclear spins.

[0047] For instance, in the case of dopant-based qubits, the differences in energy splitting between the electron spin qubits (AE) can be compared with simulated AE values for every atomic configuration in which nuclear spins can be present in the dopant dots. The atomic configurations that result in the best matches between the measured and simulated AE values are determined to be probable locations of the dopant atoms in that system. In another example, a set of discrete exchange frequencies can be detected in the exchange oscillations that correspond to different orientations of nuclear spins. By comparing these discrete exchange frequencies with atomistic tight-binding simulations of electron wave-functions, the exact locations of donor atoms can be determined.

[0048] In the case of gate-based quantum dots, the determined hyperfine coupling values data can be compared with simulated hyperfine coupling values determined from electrostatic simulations of electron wavefunctions to find the spatial locations of nuclear spins. [0049] In comparison to traditional spectroscopic methods such as ESR, the exchangebased spectroscopy does not require AC magnetic fields, microwave generators, special wiring, or on-chip antennas. This makes the exchanged-based sensors much easier to implement and with less specialised equipment. Further still, the technique described herein can be executed at a much wider range of static magnetic field strengths than conventional ESR techniques.

[0050] Although the exchange-spectroscopy method described herein can be used in any solid-state quantum processing system to map-out the exact nuclear spin environment in the wave function of a pair of exchange-coupled electrons/holes, it will be described in the remainder of this disclosure with reference to a dopant-based quantum dot processing system and in particular a donor quantum dot based system.

***

[0051] The single electron spin qubit (spin- 1/2 qubit) is a natural and paradigmatic quantum mechanical two-level system, where quantum information is encoded using the spin degree of freedom (|l) and |T) basis). Such a qubit can be realised in a straightforward way using a single quantum dot, a single donor, or multiple donors in close proximity to confine the electron.

[0052] Regardless of the implementation, electron spin- 1/2 qubits are typically operated at magnetic fields of Bo ~1- 1.5 T, corresponding to a Zeeman energy splitting of E ~ 28 - 42 GHz. Single-qubit gates can be executed by applying a microwave drive at the frequency corresponding to E. The difference in energy splitting between two adjacent qubits, AE, plays an important role during two-qubit gate operations. To date, the most explored and successful method of coupling single electron spin qubits is via the exchange interaction (J), where the magnitude of the exchange interaction can be precisely controlled using voltage pulses. When J coupling is applied, the effective coupling strength Q between the anti -parallel electron spin states | IT) and |Tl) depends not only on J but also on AE, and can be expressed as fl = V/² + AE².

[0053] In donor-based qubits, AE is dominated by the hyperfine interactions A between the electron spins and the nuclear spins of the phosphorus atoms (P) that confine the electrons. As a result, each electron spin qubit experiences a local hyperfine field that depends on the spatial arrangement of the P atoms within a quantum dot and the temporal orientation of their nuclear spins. Consequently, nuclear spin flips of the qubit-hosting P atoms can change the value of AE in a quantised fashion, modulating the effective coupling strength. [0054] This quantisation leads to beating in the coherent exchange SWAP oscillations between |lT)and |Tl) states. A detailed analysis of the nuclear spin beating observed allows use of the exchange coupled electron spins as precise local probes for sensing the nuclear spin environment of the qubit, and provides a unique insight into the exact atomic configurations of the donors within each qubit.

[0055] This effect is demonstrated in the present disclosure using a multi-qubit device having 3 and 2 phosphorus donors in left and right quantum dots, respectively. One such quantum processing device 100 is displayed in Fig. 1A. In particular, Fig. 1A shows a Scanning Tunnelling Microscope (STM) image of the two-qubit device 100.

[0056] The device 100 may be fabricated on a p-type Si substrate (1-10 Q cm). The substrate may be subjected to a series of high-temperature annealing processes up to -1,100 °C followed by a controlled cool-down to -330 °C, at which point the surface is terminated with mono-atomic hydrogen via thermal cracking. The result is a fully terminated H:Si (2 x 1) reconstructed surface from which hydrogen can be selectively removed with an STM tip. Using the STM tip a lithographic mask representing the device and donor qubits is created on the Si surface. Subsequent adsorption and incorporation (at 350 °C) of gaseous PH? precursor metallizes the exposed area with -1/4 monolayer of phosphorus. Then, a layer of Si is grown epitaxially to encapsulate the device 100. The typical thickness of encapsulation layer is between 20 nm and 100 nm.

[0057] The whole device 100 may be epitaxial - i.e., the donor dots 102, 104 may be fabricated within a substrate (such as a p-type Si substrate (1-10 cm)). Positioning the donor dots epitaxially can significantly reduce impact of noise on the qubits. In some examples, the qubits are formed in the quantum dots 102, 104 about 20-50 nm from the surface and separated by approximately 10-15 nm.

[0058] The qubits are tunnel coupled to a single-electron transistor (SET) 106 that acts as a charge sensor and electron reservoir to load electrons onto the donor dots 102, 104. Further, the qubits may be controlled by one or more gates. Fig. 1A illustrates three gates - left gate 108, middle gate 112, and right gate 110, which can be used to control the electrochemical potentials of the donor dots 102, 104, whereas the SET gate 114 is predominately used to control the electrochemical potential of the SET 106. In one implementation, the gates may be metal contacts on the surface. In another implementation, the gates may be phosphorus-doped silicon (SiP) gates fabricated epitaxially within the semiconductor substrate. In either case, the gates allow full electrostatic control of the qubits.

[0059] Although an SET 106 is depicted in Figs. 1A, qubit readout can be performed using other mechanisms. For instance, it may be performed dispersively using the left, right, or middle gates mentioned previously.

[0060] In the most basic implementation, a global or local nuclear magnetic resonance (NMR) antenna allows control of the nuclear spins of the donors via radio frequency (RF) magnetic fields in the range of between 1 MHz and 100 MHz. The NMR antenna (not shown) can be manufactured on chip, or off chip (cavity or coil). The control of the nuclear spins is necessary for initialization and gate operations.

[0061] Electronic structures for readout and control can be placed on chip, or on the printed circuit board (PCB) which holds the silicon chip. They include waveguides, resonators, bias tees, amplifiers, filters, mixers circulators, etc. Any of these structures can be implemented using on chip lithographic structures or on the PCB using commercially available surface mount devices (SMD).

[0062] Fig. IB show a zoomed in view of the portion 116 in Fig. 1A taken after hydrogen lithography. Figs. 1C and ID show zoomed in views of portions 120 and 130 of Fig. IB taken after hydrogen lithography. In particular, Fig. IB shows the top part of the SET transistor 106 and the two donor dots 102, 104 separated by 12 ± 0.5 nm. The left donor dot may have 3 donor atoms and the right donor dot may have 2 donor atoms. The separation in distance between the two dots has been engineered at the atomic scale to provide exchange coupling between the qubits that is sufficiently large for a two-qubit lSWAP gate . It will be appreciated that this is an example and that the distance between the qubits can be between 10-15nm without departing from the scope of the present disclosure.

[0063] Fig. 1C shows a close-up of a lithographic patch 110 of the left donor dot 102. The diagonal lines 116 represent the dimer rows on the hydrogen-terminated silicon surface. The squares 118 represent the sites on the surface of the silicon lattice, for which the hydrogen mask was removed. Fig. ID shows a close-up of a lithographic patch 120 of the right donor dot 104, which includes potential donor sites 118 for two donor atoms. As with Fig. 1C, the white diagonal lines 116 in Fig. ID represent the dimer rows on the hydrogen-terminated silicon surface. The squares 118 represent the sites on the surface of the silicon lattice, for which the hydrogen mask was removed. [0064] The fabrication of the multi-donor qubits shown in Figs. 1A-1D relies on patterning lithographic patches of a specific size within the hydrogen mask. If the size of the lithographic patch is exactly 3 dimers along a dimer row (6 black squares in the image above), then most likely no more than 1 donor will be incorporated. The exact dependence between the patch size and donor number is to some extent probabilistic in nature due to the different chemical pathways that can take place. However, generally, the bigger the patch the more donors can be incorporated. In the example shown in Figs, 1B-D, 18 hydrogen atoms (black squares) were desorbed in the left patch and 15 hydrogen atoms were desorbed in the right patch, which resulted in 3 and 2 donors in the left and right donor dots 102, 104, respectively. To create donor dots with other numbers of donor atoms, different sized patches may be desorbed. For instance, to create 2P-1P donor dots, 15 hydrogen atoms may be desorbed from the left donor dot 102 (to incorporate 2 P atoms) and 6 hydrogen atoms are desorbed from the right donor dot 104 (to incorporate 1 P atom). The number of donors incorporated within a given lithographic patch can be regulated not only by controlling the size of the lithographic patch, but also with other methods such as tip-assisted incorporation, control of the phosphine dosing parameters, and control of the incorporation parameters.

[0065] In this way, to achieve a desired hyperfine interaction and hence a desired qubit energy difference AE, an optimal number of donor atoms in each qubit can be incorporated during the fabrication stage.

[0066] Figs. IE- IF schematically show the left and right (L and R donor dots 102, 104, where the circles 132 represent the donors incorporated within the silicon crystal lattice, and the ovoids 134 represent the approximate extent of the donor-bound electron wave-functions. It is important to realise that each P donor nucleus (L1-L3 in the left donor dot and R1-R2 in the right donor dot) within the electron wavefunction contributes to the overall energy splitting of the electron spin. For an electron spin qubit hosted on N phosphorus atoms, the qubit energy can be written as,

E = gg_BB_Q +5 o z)i (1) where g is Lande g-factor, g.B is the Bohr magneton, Bo is the global magnetic field,

= ±1/2 is the expectation value of the nuclear spin operator of the i-th donor, and Ai is the hyperfine coupling between the electron spin and the nuclear spin of an i-th donor nuclear spin.

[0067] Fig. 2A and 2B are energy diagrams 200, 210 for 3P and 2P qubits respectively. In particular energy diagram 200 shows the energy splitting of the left electron spin qubit, EL, as a function of nuclear spin configurations of the three P atoms that host the left qubit. This figure shows how the orientation of each nuclear spin in the qubit impacts the overall qubit energy splitting EL. The hyperfine couplings A between electron spin and the individual P nuclear spins are denoted as ALI, AL2, and Au. For a 3P qubit, there are 8 discrete values of the energy splitting EL.

[0068] The energy diagram 210 shows the energy splitting of the right electron spin qubit, ER, as a function of nuclear spin configurations of the two P atoms that host the right qubit. This figure shows how the orientation of each nuclear spin in the qubit impacts the overall qubit energy splitting ER. The hyperfine couplings between electron spin and the individual P nuclear spins are denoted as ARI and AR2. For a 2P qubit there are 4 discrete values of energy splitting ER.

[0069] When considering a system of two adjacent multi -donor electron spin qubits, AE can have several contributions. Small (~ 10 MHz) qubit-to-qubit variations in the Zeeman term gpnBo are expected due to the non-homogeneities in the global magnetic field Bo, as well as variations in g-factor due to the local electric fields. These variations in the Zeeman energies, however, are much smaller than variations in the hyperfine couplings, A, which are typically on the order of tens or hundreds of MHz depending on the exact atomic qubit arrangement. Therefore, in donor qubits AE arises mainly from unequal hyperfine interactions on each qubit and can be approximated as,

where NL and NR represent the number of donors in the left and right qubit, respectively and ALL and AR.L are the hyperfine energies of the k-th donor on the left and right qubit, respectively. For device 100, there are 2³ ■ 2² = 32 possible configurations of P nuclear spin configurations, and hence 32 discrete AE values.

[0070] Fig. 2C depicts some of these combinations schematically. The energy difference (AEi, AE2 ... AE32) between neighbouring qubits is dominated by the local hyperfine fields. Consequently, the temporal orientation of nuclear spins within both qubits ultimately determines AE between them.

[0071] Importantly, each time a nuclear spin of the k-th donor within a qubit is flipped, the operator (I_z)_k in equation 1 above reverses its polarity (from 'A to - A or vice versa) and consequently the AE value changes by Ak. This means that when the nuclear spin of one of the qubit-hosting P atoms flips, for instance due to relaxation processes or ionisation shock, the AE value can change suddenly (by tens or hundreds of MHz). Typical nuclear spin lifetimes are on the order of < few seconds. Hence, during a long-running measurement (typically several minutes/hours), the switching value of AE produces several discrete exchange oscillation frequencies fl = ^/j² + AE², which causes a beating in the exchange oscillations between |lT)and |Tl) states.

[0072] The details of how this beating arises from the discrete oscillation frequencies is illustratively explained below.

[0073] Figs. 3A-3C show the two-spin probabilities |H), |IT), |Tl), and |TT), simulated for the three different sets of AE values, respectively. In particular, Fig. 3A-3C illustrate schematic Bloch sphere diagrams 302, 304, 306 and simulated exchange two-spin probabilities 308, 310, 312 for three different sets of AE values. All cases assume that a |l) state is initialised on the left qubit and a random spin state on the right qubit. The exchange interaction acts on the | IT) and |T4) states during the ./ pulse, while the |tt)and |TT) states remain unaffected. For all cases, an exchange energy of J = 150 MHz, detuning noise of <J_E = 4.5 peV and magnetic noise of (Jnuc = 3.2 MHz are assumed.

[0074] Fig. 3 A depicts the Bloch sphere diagram 302 and simulated exchange two-spin probabilities 308 for a simple case of exchange oscillations with a fixed value of AE = 30 MHz. The | IT) and |T4) states oscillate with a single oscillation frequency of fl = - J² + AE². In this case, dephasing due to charge noise causes decay of the oscillating probabilities overtime.

[0075] Fig. 3B depicts the Bloch sphere diagram 304 and simulated exchange two-spin probabilities 310 for two discrete values of Zeeman energy difference AEi = 30 MHz and AE2 = 70 MHz. The resulting |4T) and |T4) probabilities are averaged over two distinct oscillation frequencies Qi and Q2 giving rise to a beating effect. In such a scenario, a beating is observed with the envelope frequency of,

where h is the reduced Planck's constant. In this particular case the beating frequency, fbeat = 6.2 MHz, corresponds to the envelope period of Tbeat ⁼ 1/fbeat = 159.25 ns. Thus, as seen in Fig. 3B, the first node of the beating envelope is observed at Tbeat/4 = 39.8 ns. [0076] Fig. 3C depicts the Bloch sphere diagram 306 and simulated exchange two-spin probabilities 312 for a scenario in which AE has five equally probable values; AEi = 30 MHz, AE₂ = 70 MHz, AEs = 120 MHz, AE₄ = 160 MHz and AEs = 170 MHz. Consequently, the |4T) and |Tl) probabilities oscillate in a complex fashion, where five discrete oscillation frequencies (Qi, O2, O3, Qi Os) can be distinguished. Importantly, the visibility of coherent oscillations is reduced when AE switches between different values, due to the overlapping beating envelopes. This effect is particularly visible for the case depicted in Fig. 3C, where the |IT) and |T4) populations barely cross each other.

[0077] For illustrative purposes, the AE values were chosen in an arbitrary way. In practice, the number of AE values and their magnitude is determined by hyperfine interactions.

[0078] To observe the predicted beating due to the nuclear spin dynamics described above, the exchange oscillations of the donor device 100 are measured as a function of the exchange pulse duration. Fig. 4A shows the charge stability map 400 of the device 100 taken in a dilution refrigerator with a base temperature of ~ 50 mK. The charge stability map was obtained by measuring the SET current, ISET, as a function of right and left gate voltages. The dashed lines 402 and the solid lines 404 correspond to the charge transitions of the L and R dots, respectively, and the numbers in brackets indicate the electron occupation on both donor dots (L, R). The diagonal lines 406 correspond to the peaks in the SET conductance. The breaks in the diagonal SET lines 406 indicate the charge transitions of the two dots. The exchange oscillations are measured near the (1,1) to (2,0) inter-dot transition, highlighted with the box 408. This inter-dot transition area 408 is zoomed in and depicted in Fig. 4B.

[0079] Figs. 4C and 4D show the gate pulse sequences used to measure the exchange oscillations. As shown in these figures, the pulse sequence includes seven steps. The sequence commence s in the ( 1 , 1 ) charge configuration where the two qubits are initialized in the classical mixture of |ll) and | IT) states. This is achieved by:

1. Loading a random spin on the right (R) quantum dot 104 by plunging the R energy levels below the Fermi energy of the SET, and

2. Loading a spin-down state on the left (L) quantum dot 102 by aligning the SET Fermi energy in-between the energy levels on the left quantum dot 102.

[0080] Next, over steps 3-5, a fast exchange pulse is performed towards the (2,0) charge region that maps |4T) and |T4) eigenstates to singlet and triplet eigenstates, S and To. The detuning energy 6 = a_eAV_e during the pulse is proportional to the voltage difference AV_e between the point (4) and the charge anti-crossing, at which G = 0. The proportionality constant a_e is the lever arm along the (l,l)-(2,0) direction (see grey arrow 410 in Fig. 4C). (Xe = 0.083 eV/V in device 100.

[0081] Finally, at steps six and seven, single-shot measurements are performed of the electron spins on both R and L quantum dots.

[0082] The resulting experimental exchange oscillations 500 are shown in Fig. 5A. In particular, this figure depicts the resulting two-spin probabilities as a function of the wait-time spent near the anti -crossing, at detuning G = -0.167 meV . Coherent oscillations of the two-spin probabilities can be observed for measuring | IT) 504 and |T4) 506 states, while |4T) 502 and |TT) 508 states are unaffected by the exchange interaction. The |4T) 504 and |T4) 506 probabilities cannot be described by a simple decaying cosine function, suggesting the presence of multiple oscillation frequencies.

[0083] Fig. 5B shows the experimental exchange spectrum 510 or Fast Fourier Transform (FFT) of the oscillating |T4) probability, where discrete spectral components can be clearly distinguished. These quantised oscillation frequencies Q can be attributed to the varying hyperfine fields that arise from the dynamics of the donor nuclear spins.

[0084] To investigate the impact of individual nuclear spin flips on the measured exchange oscillations (shown in Fig. 5A-5B), a theoretical model is constructed which allows simulation of exchange oscillations for different spatial locations of the P atoms within each donor dot. The simulation is commenced by finding all the geometric arrangements of the donors in the left and right quantum dots. In the example of device 100, this means finding the geometric arrangements of the three donors within the L quantum dot 102, and the two donors within the R quantum dot 104 that are feasible. The considerations can be constrained to in-plane atomic arrangements where the donors within each dot are contained within the area, slightly larger than the size of the lithographic openings 110, 120.

[0085] For each 2P and 3P configuration, numerical tight binding simulations are then performed of the donor-bound electron wavefunction, and the corresponding hyperfine couplings are extracted. In one example, the simulations can be performed using NEM0-3D, a semi-empirical tight-binding modelling tool. During simulation, each P donor in a substitutional lattice site is modeled as a positive charge screened by the silicon dielectric constant with a cut-off potential Uo at the center representing the central-cell correction. The electronic wavefunction at each lattice site is expressed in a spin-resolved sp³d⁵s* atomic orbital basis. The hyperfine constant can be obtained by evaluation of the electron density at the donor site (|y(Rdonor)|²).

[0086] In one example, the 2P and 3P systems of Fig. 1 can be simulated separately, each in a 30.4 x 30.4 x 30.4 nm3 block of silicon crystal - the domain is so chosen in order to avoid any distortion of the wavefunction due to confinement of the block.

[0087] Using the calculated hyperfine couplings, the full 11940 combinations of 3P-2P qubit pairs can then be considered and the corresponding AE values can be determined. Table 1 shows a portion of the lookup table that may store the hyperfine couplings and AE values for the 11940 combinations of 3P-2P qubit pairs. In particular, table 1 shows the atomic qubit configurations labelled A-J. For each atomic qubit configuration A-J, there are five nuclear spins across both qubits. This means that there are 2⁵ = 32 possible configurations of nuclear spins (excluding superposition of states). It is important to note that AE is defined by the absolute value of the hyperfine energy difference (see Eq. 2), i.e., negative difference in hyperfine values between the qubits have the same effect on the two-spin probabilities as a positive difference in hyperfine values. Hence, the two opposite configurations such as | ftftft) | ft ft) and |ftftft) | ft ft) yield the same magnitude of AEz. Additionally, the same value of AE is shared between some configurations, for example |ftftft) |ftft) and |ftftft) |ftft), due to the degeneracy of the |ftft) and |ftft) states. In total, amongst 32 configurations there are 12 distinct AE values. These 12 discrete AE values are listed in ascending order and labelled with Roman numerals in Table 1. For each AE value the corresponding discrete oscillation frequency value is also provided, where J = 80 MHz.

Atomic Qubit Configuration

Table 1 : Example portion of lookup table showing energy splitting and hyperfine values for 10 different combinations of a 3P-2P quantum processing system.

[0088] In this manner, the AE and fl values for each of the different atomic configurations possible for a 3P-2P device 100 can be determined. These values may be stored in a lookup table or database and compared with the AE and fl values measured for a given device to determine the atomic configuration of the given device.

[0089] In one example, when the measured energy splitting and hyperfine values of the 3P-2P device 100 are compared with the modelled values stored in the lookup table, it was determined that two atomic configurations, labelled as configurations A and B in Figs. 5C and 5D, provided the best agreement with the AE and fl values measured for that device. Accordingly, in this example, these two configurations are the most likely atomic arrangements of the qubits.

[0090] Next, numerical simulation is performed of the exchange oscillations of these two configurations to assess if they match the measured exchange oscillations. The simulation is based on the time -evolution of the two-electron-spin Hamiltonian in the |S) and |T₀) basis,

" - (AE “)

where the exchange energy J depends on the detuning energy f between qubits of the L and R quantum dots 102, 104 and is well approximated with the Hubbard model using equation,

where t_c is the inter-dot tunnel coupling, defined as t_c = J(^e = 0). For device 100, tc was measured to be 1.8 ± 0.1 GHz.

[0091] Using the time-evolution Hamiltonian simulations, the exchange oscillations 600 and the corresponding FFT traces 610 for the atomic configurations A and B are modelled, as presented in Fig. 6A and 6B. Amongst the 32 possible configurations of nuclear spins, there are 12 distinct AE values possible in the considered 3P-2P system. For each of the twelve AE values, the oscillation frequency can be written a

+ AE². This means that there are 12 discrete frequency components in the FFT signal, labelled with Roman numerals from I to XII, with FFT peaks occurring at frequencies corresponding to Qi - xn values. For each FFT peak, in Fig. 6B the underlying nuclear spin configurations are also provided. The FFT peakwidths are determined by the charge noise, o_e, acting on J(e), and magnetic noise from the ²⁹Si nuclear spin bath, o_nuc, resulting in fluctuations in AE.

[0092] For the modelling, it was assumed that de = 4.5 peV and (Jnuc = 3.2 MHz which corresponds to magnetic noise in natural silicon. It is important to note that as the AE values increase from AEi to AExn, the heights of the FFT peaks become smaller, and so does the visibility of the exchange oscillations. The visibility a of the exchange oscillations is given by,

[0093] This means that the visibility of the exchange oscillations is large (a ~ 1) when the two electron spins are in the singlet-triplet basis (J » AE). and small (a ~ 0) when the two electron spins are isolated (J «AE). For a fixed magnitude of the exchange energy (J = 80 MHz in this case), the oscillation frequency varies dramatically for different nuclear spin configurations, from Qi = 80.1 MHz to xn = 458.5 MHz. At the same time, the visibility of exchange oscillations ranges from ai = 0.996 to axn = 0.03.

[0094] The theoretically modelled FFT spectra for both atomic configurations, A (dashed line) and B (dotted line) are very similar. This is due to the fact that the hyperfine couplings for both A and B are identical for the left dot (AL = {49.0, 265.2, 304.4} MHz) and vary only slightly for the right dot (AR = { 142.2, 142.2} MHz for configuration A and AR = { 141.8, 141.8} MHz for configuration B). The simulated FFT spectra are in a good agreement with the measured data shown in Fig. 5B, both in terms of frequency components and visibilities. In particular, the measured spectrum confirms the presence of multiple discrete AE values arising from the nuclear spin dynamics. The exchange oscillations are dominated by the spectral components I and II, for which AE < J, hence the highest visibility (see Eq. 5) and the strongest FFT signal. While the peaks I-V are observed in the measured data, the higher- frequency peaks are characterised by low visibility and are not resolved within the noise floor of the measurement.

[0095] In Fig. 6A the theory model is compared with the measured data in the time domain. Here, the open-circle markers with error bars correspond to the measured data, while dashed and dotted lines correspond to the theory traces for atomic configurations A and B, respectively. From the time domain data one can clearly observe the beating effect arising from discrete switching of oscillation frequency. Specifically, the dominating spectral components I and II correspond to beating frequency of f_b |fl_; — fl | = 5.6 MHz, creating a beating envelope with period Tb = 1/ft = 179 ns. The first node of this beating envelope occurs at Tb/4 causing the oscillations to temporarily flatten out around the 40-50 ns mark. At the same time, the low visibility (~ 5%) of the observed oscillations can be explained by the fact that most of the AE values are larger than J, in which case the two-spin states remain in the single-spin basis (|4T), | IT), |T4), |TT)), resulting in a CPHASE-like operation rather than a SWAP operation. The oscillation visibility is expected to be larger in the high-J regime (J » max(AE) = 458 MHz), where two-spin states are always in the singlet-triplet basis during the exchange pulse. However, the high J regime suffers from larger charge noise and, consequently, a short decay time of < 3 ns which causes difficulties in measuring the fullamplitude exchange oscillations. [0096] To emphasize, the analysis of the exchange oscillations has allowed determination of the hyperfine couplings of both qubits, thereby representing a novel and precise metrology tool. By modelling the frequency components of exchange oscillations, the exact location of nuclear spins in a material can be determined.

[0097] In Figs 1C and ID, the size of the lithographic patches 110, 120 are approximately 1.9 nm X 1.9 nm for the left quantum dot 102 and 1.5 nm X 1.5 nm for the right quantum dot 104. When searching for the best-matching atomic configurations, it was assumed that the donor atoms spread across a 2.7 nm X 2.7 nm area within each quantum dot to account for ±1 nm lateral diffusion. Nevertheless, it was found in this case that the best matching atomic donor configurations (A and B) are compact enough such that they are consistent with the size of the lithographic patches 110, 120.

[0098] This is in agreement with a previous atomic metrology studies where the atomic configuration deduced from the ESR measurements was consistent with the size of the lithographic patch. This confirms that the level of diffiision/segregation of P atoms after encapsulation at 250°C remains small, and that the STM images of the lithographic patches provide a good estimate of the final position of the P atoms within the silicon crystal lattice.

[0099] The methods and systems for determining the atomic configuration of donor atoms are described with reference to a 3P-2P system. It will be appreciated that this is merely an example device and that the methods and systems can be used with other multi-qubit quantum processing systems such as 2P-1P systems without departing from the scope of the present disclosure. Further, instead of donor atoms, the methods and system may be used to determine the atomic configuration of acceptor atom system where the spin qubits are formed from unpaired holes in the quantum dots.

[0100] Further still, the exchange-based sensor and sensing methods described herein can be used to determine the nuclear spin environment of electrons in any semiconductor system such as a two-dimensional electron gas system, nanowires etc.), to sense the nuclear spin environment of electrons. More generally, the above described exchanged-coupled sensor and sensing method can be applied in any solid-state system to sense any nuclear spins present within the electron/hole wavefunctions of the exchange-coupled sensor electrons/spins.

[0101] The term "comprising" (and its grammatical variations) as used herein are used in the inclusive sense of "having" or "including" and not in the sense of "consisting only of. [0102] It will be appreciated by persons skilled in the art that numerous variations and/or modifications may be made to the invention as shown in the specific embodiments without departing from the spirit or scope of the invention as broadly described. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive.

Claims

1. A method of determining atomic configuration of nuclear spin atoms in a semiconductor processing element, the semiconductor processing element comprising: at least one semiconductor layer, at least two electrons/holes confined within the at least one semiconductor layer, the at least two electrons/holes exchange coupled with each other; the method comprising the steps of: performing exchange oscillations between the at least two electrons/holes by applying timed exchange pulses to the semiconductor processing element; measuring the spins of the at least two electrons/holes after the exchange oscillations; calculating differences in energy splitting between the at least two electrons/holes; determining hyperfine coupling values between the at least two electrons/holes and the nuclear spin atoms; and determining the location of the nuclear spin atoms based on the determined hyperfine coupling values.

2. The method of claim 1, wherein the semiconductor processing element is a quantum processing element.

3. The method of claim 1 or 2, wherein the nuclear spin atoms are dopant atoms and the semiconductor processing element comprises a plurality of dopant dots embedded in the at least one semiconductor layer, each dopant dot comprising one or more dopant atoms and the at least two electron/holes are confined within adjacent dopant dots.

4. The method of any one of claims 1-3, wherein determining the location of the nuclear spin atoms further comprises: comparing the calculated differences in energy splitting with predetermined energy splitting values for different atomic configurations of a test processing element having

22 similar number of dopant dots and dopant atoms as the semiconductor processing element to identify matching differences in energy splitting; and determining the atomic configuration of the semiconductor processing element based on the atomic configuration of the predetermined energy splitting value that matches the calculated energy splitting value.

5. The method of claim 4, further comprising simulating or modelling the predetermined energy splitting values for the different atomic configurations of the test processing system.

6. The method of any one of the preceding claims, further comprising detecting discrete exchange oscillation frequencies in response to performing the exchange oscillations between the at least two electrons/holes.

7. The method of claim 6, further comprising simulating or modelling predetermined exchange oscillation values for different atomic configurations of the test processing element.

8. The method of claim 7, further comprises: comparing the detected exchange oscillation frequencies with the predetermined exchange oscillation frequencies to identify matching exchange oscillation frequencies; and determining the atomic configuration of the semiconductor processing element based on the atomic configuration of the predetermined exchange oscillation frequency values that matches the detected exchange oscillation frequencies.

9. The method of any one of the preceding claims, further comprising initializing the at least two electrons/holes in a mixture of 144) and |4T) states.

10. The method of claim 9, wherein initializing the at least two electrons/holes in the mixture of 144) and |4T) states comprises: loading an electron/hole with a random spin from a reservoir on a first dopant dot by plunging the energy levels of the dopant dot below the Fermi energy of the reservoir; and loading an electron/hole in a spin-down state on the second dopant dot by aligning the reservoir’s Fermi energy in between the energy levels on the second dopant dot.

11. The method of any one of claims 3-10, wherein at least one of the dopant dots in the pair of the dopant dots includes multiple dopant atoms.

12. The method of any one of claims 3-11, wherein at least one of the dopant dots in the pair of the dopant dots includes multiple electrons/holes.

13. The method of any one of the preceding claims where the nuclear spin atoms are phosphorus atoms.

14. The method of claim 1, wherein the semiconductor processing element comprises a plurality of gate-based quantum dots formed in the semiconductor, the nuclear spin atoms are one or more silicon-29 atoms present in the gate-based quantum dots and the at least two electron/holes are confined within adjacent quantum dots.

15. The method of claim 14, further comprising: comparing the determined hyperfine coupling values with predetermined distribution of hyperfine couplings for different positions of the one or more silicon-29 atoms in the adjacent quantum dots; and determining the position of the one or more silicon-29 atoms based on the positions of the one or more silicon-29 atoms with the predetermined hyperfine coupling values that matches the determined hyperfine coupling values.

16. A method for determining the nuclear spin environment in a semiconductor device, by an exchange -based sensor, the exchange based sensor comprising at least two exchange-coupled electrons/holes, the method comprising: performing exchange oscillations between the at least two electrons/holes by applying timed exchange pulses to the semiconductor device; measuring the at least two electrons/holes after the exchange oscillations; calculating differences in energy splitting between the at least two electrons/holes; determining hyperfine coupling values between the at least two electrons/holes and nuclear spin atoms present within the electron wavefunctions of the at least two electrons/holes; and determining the location of the nuclear spin atoms based on the determined hyperfine coupling values.

17. The method of claim 16, wherein determining the location of the nuclear spin atoms further comprises: comparing the calculated differences in energy splitting with predetermined energy splitting values for different atomic positions of nuclear spins atoms within the electron wavefunction to identify matching differences in energy splitting; and determining the position of the nuclear spin atoms based on the atomic positions of the predetermined energy splitting value that matches the calculated energy splitting value.

18. The method of claim 16 or 17, further comprising detecting discrete exchange oscillation frequencies in response to performing the exchange oscillations between the at least two electrons/holes.

19. The method of any one of claims 16-18, further comprising initializing the at least two electrons or holes.

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20. The method of any one of claims 16-19, further comprising calculating or modelling predetermined exchange oscillation values for different atomic positions of nuclear atoms in the electron wavefunction.

21. The method of claim 20, further comprises: comparing the detected exchange oscillation frequencies with the predetermined exchange oscillation frequencies to identify matching exchange oscillation frequencies; and determining the atomic position of the nuclear atoms based on the atomic positions of the predetermined exchange oscillation frequency values that matches the detected exchange oscillation frequencies.

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