The initialization and manipulation of quantum information stored in silicon by bismuth dopants

The initialization and manipulation of quantum information stored in silicon by...

(Parte 1 de 2)

The initialization and manipulation of quantum information stored in silicon by bismuth dopants

A prerequisite for exploiting spins for quantum data storage and processing is long spin coherence times. Phosphorus dopants in silicon (Si:P) have been favoured1–10 as hosts for such spins because of measured electron spin coherence times

(T2)longerthananyotherelectronspininthesolidstate:14ms at 7K with isotopically purified silicon11. Heavier impurities such as bismuth in silicon (Si:Bi) could be used in conjunction with Si:P for quantum information proposals that require two separatelyaddressablespinspecies12–15.However,thequestion of whether the incorporation of the much less soluble Bi into Si leads to defect species that destroy coherence has not been addressed. Here we show that schemes involving Si:Bi are indeed feasible as the electron spin coherence time T2 is at least as long as for Si:P with non-isotopically purified silicon. We polarized the Si:Bi electrons and hyperpolarized the I = 9/2 nuclear spin of 209Bi, manipulating both with pulsed magnetic resonance. The larger nuclear spin means that a Si:Bi dopantprovidesa20-dimensionalHilbertspaceratherthanthe four-dimensionalHilbertspaceofanI=1/2Si:Pdopant.

Kane’s suggestion for a Si:P quantum computer1, where the electronandnuclearspinsofPimpuritiesareregulatedandreadout using electrical gates, has inspired many researchers. Two particular challenges in building the Kane quantum computer are placing phosphorus dopants with atomic precision2 below the surface, and depositing metallic contacts between them.

Alternative schemes12–15, which are conceptually more complex but impose less stringent requirements on fabrication, take advantage of other group v elements that also substitute for silicon. With qualitatively similar behaviour to Si:P dopants, these other dopants exhibit electron spin resonance (ESR) at field/frequency combinations distinct16 from Si:P, allowing selective excitation and detection with microwave pulses. As a result of their lower solubility and higher binding energies, they have been of far less relevance to microelectronics, and so in contrast to Si:P, their spin relaxation has remained relatively unexplored. This is particularly true for the heaviest element, bismuth, which has the highest binding energy and also the largest nuclear spin (I =9/2), both of which should be advantageous for quantum computing because they would permit higher temperature operation and a larger auxiliary state space for quantum data storage, respectively. Although Bi seems attractive in principle, a potential problem is that Bi is the largest and least soluble of the group v elements17—if the site of incorporation is too distorted or should Bi entrain other substitutional or interstitial impurities (including vacancies), extra decoherence could ensue. Accordingly, we have used pulsed ESR to measure the spin-lattice relaxation time T1 and the decoherence time T2, as well as to

1London Centre for Nanotechnology, University College London, London WC1H 0AH, UK, 2Department of Physics and Astronomy, University College London, London WC1E 6BT, UK, 3National High Magnetic Field Laboratory and Florida State University, Tallahassee, Florida 32310, USA, 4Institute of Structural and Molecular Biology, University College London, London WC1E 6BT, UK. *

demonstrate the controlled preparation of quantum states (using Rabi oscillations) and nuclear spin manipulations using pulsed electron–nuclear double resonance (ENDOR).

Figure 1 shows the ESR spectra obtained with both standard- (9.7GHz) and very high- (240GHz) frequency microwave radiation. In each case Si:Bi yields the ten resonances expected for an electron (spin 1/2) coupled to a nuclear spin of 9/2. The resonant fields of all transitions are well simulated as shown in the figure and described in the Supplementary Information. The Gaussian linewidth is shown in the Supplementary Information to be 0.41±0.002mT with 9.7GHz radiation, which agrees with the value reported previously16 and attributed to interactions with the natural (4.7%) concentration of 29Si nuclear spins.

Figure 1a shows a spin-echo-detected field-swept spectrum recorded with a pulsed ESR spectrometer (Bruker E580) operating at 9.7GHz. The Supplementary Information describes in more detail all pulse sequences used in our experiments. The spectrum in Fig. 1b was recorded in continuous-wave (CW) mode at 240GHz with a quasi-optic spectrometer18,19 at the National High Magnetic Field Laboratory in Tallahassee, Florida.

For the 240GHz experiments, a magnetic field above 8.3T together with a temperature of 3K ensures that the electron spin polarization is above 95%, providing a good initial state for a quantum computation using the electron spin. This almost pure state avoids the problems encountered by liquid-state NMR quantumcomputerswiththeuseofpseudo-purestartingstates.

In addition, we have initialized the electron–nuclear spin system by transferring some of the large electron polarization to the bismuth nuclei. Figure 1b shows that this was achieved with abovebandgap white light, without the resonant excitation used in some previous experiments10,20. This dynamic nuclear polarization is due to the Overhauser effect whereby the electron spin of the photoelectrons relaxes by ‘flip-flopping’ with the 209Bi nuclear spin. The photoelectrons are initially unpolarized and to move towards thermal equilibrium it is necessary for ∼45% of them to ‘flip’ spins. These flips can conserve angular momentum if a 209Bi nuclear spin ‘flops’ in the other direction. The energy required to flop a nuclear spin is negligible compared with the thermal energy in our experiment. Very similar effects have been seen with 29Si nuclear spins in silicon21, but anti-polarization occurs with Si:P (refs 6,7) owing to the trapping of conduction electrons by the P donors4. The application of light with energy only slightly larger than the bandgap can favour the formation of bound excitons in Si:Bi and a different mechanism for dynamic nuclear polarization producing anti-polarization22. The results obtained in ref. 2 came from photoluminescence measurements that are not


Energ y (

G H z)

Magnetic field (T)


Si:Bi CW ESR signal ( arb. units )

Light off Light on

3 K 5 K 4 K


Energy (GHz)

Spin echo intensity (arb. units)

Figure 1 | Qubit initialization. a,b, ESR spectra of Si:Bi with a frequency of 9.7GHz (a) and a frequency of 240GHz (b). At the high magnetic field required for the higher frequency experiment, the ten Si:Bi resonances are evenly spaced (b). This is not the case for the lower-frequency experiment as the correspondingly lower magnetic field is not strong enough to define the axis of quantization: the large hyperfine interaction of a=1.4754GHz (for definition, see Supplementary Information) is comparable in size to the Zeeman term in the Hamiltonian. The red lines are simulations of the Si:Bi resonances and the blue line is a simulation of the Si:P resonances that account for the sharp but weak features visible in the high-field data. In all spectra we attribute the small broad signal around g=2 (0.34T for 9.7GHz and 8.55T for 240GHz) to dangling-bond defects in the silicon. 209Bi nuclear polarization manifests itself in the form of a larger signal for the low-field resonance lines at temperatures below 5K when above-bandgap light is applied. The spectra have been offset for clarity and the arbitrary units are the same for each spectrum. c,d, The simulated energy levels as a function of low and high magnetic field respectively, with the arrows indicating transitions that flip the electron spin state but leave the nuclear spin state unchanged.

Pulse πlength Echo t t Time Integrated echo intensity (arb. units)

Length of first pulse,(μs)τ

τ π/2π/2π/2 RFπEcho t t’

Time Pulsed ENDOR signal (arb. units)

RF frequency (MHz)

Figure 2 | Qubit manipulation. a, Pulsed ENDOR manipulates the Bi nuclear spin as well as the electron spin. A microwave frequency of 240GHz (black rectangles in inset that control the electron spin) was used at a temperature of 3K and the length of the radiofrequency (RF) pulse (green rectangle that controls the nuclear spin) was 150µs. b, Rabi oscillations of the electron spin at 25K with 9.7GHz radiation. The spin is flipped using a pulse of τ =13ns duration.

able to coherently manipulate spin qubits. The magnitude of the 209Bi nuclear polarization near the surface may be much larger than inthebulkbecausethelightdoesnotpenetratetheentiresample6.

To use the full twenty-dimensional Hilbert space available from a Si:Bi donor for quantum computing, it is necessary not only to initialize the spin system and manipulate the electron spin, but also tomanipulatethenuclearspin.Todemonstratethefeasibilityofthis we carried out pulsed ENDOR (ref. 23) at 240GHz, as illustrated in Fig. 2a. We fit the ENDOR resonance with a 0.24MHz Gaussian line but this linewidth is inhomogeneous: it is not due to the nuclear relaxation times of the 209Bi.

To characterize the quality of our electron spin manipulations we recorded Rabi oscillations as shown in Fig. 2b, obtaining a spin flip time of 13 ns. The characteristic timescale for the decay of these oscillations is around 100ns, but this provides only a lower bound on the time for the decay of spin coherence.

To measure the spin coherence times of Si:Bi we recorded the electron spin echo size as a function of the separation (t) of the pi refocusing pulse and the initial pi/2 pulse (the inset of Fig. 3a shows the pulse sequence). For temperatures above ∼18K the decay is exponential: e−2t/T . The exponential decay constant for spins in a solid, Tm, is referred to here as T2 because the electron spin density is low enough that these spins interact only weakly. At lower temperatures the coherence decay is clearly non-exponential as shown in Fig. 3a. We fitted this decay with the same function8 that has been used for similar experiments with


Si:Bi Si:P Monoexponential fits

Echo t’’ t Time

Integrated echo intensity (arb. units)

Time, t’’ (ms)

Echo Time

Si:Bi Si:P Fit with exp(¬2t/T ¬(2t)/T)

Integrated echo intensity (arb. units) Time, 2t (ms)

Transient microwave intensity (arb. units)

Time (ms)

Si:Bi peaks of CPMG echoes Si:P peaks of CPMG echoes

Monoexponential fitsπEchoπEcho… Repeating unit ttTimeTransient microwave intensity (arb. units)

Zoom bd t

Figure 3 | Storage of quantum information. Electron spin relaxation of Si:Bi at 9.7GHz, with measurements of Si:P for comparison. a, Spin echo decays measured at 10K. b, Inversion-recovery measurements of T1 at 10K. c, CPMG experiment: decay of spin echoes produced by 1,020 pi pulses at 8K. The signal acquisition time for each sample was 20min. d, Expansion of the indicated region showing the resolved echoes.

Si:P: e−2t/T −(2t) /T . TS characterizes the coherence decay owing to the presence of 29Si nuclear spins. The sample was oriented with the [1] direction perpendicular to the applied magnetic field; aligning the crystal with the magnetic field along the [100] direction produces longer TS times in Si:P experiments8,1. The best-fit value of the exponent n was between 2.5 and 2.6 for temperatures of

8–14K. All relaxation time measurements are accompanied by directly comparable measurements of a sample of Si:P with a lower concentration of 0.3–1×1015 cm−3.

Figure 4 shows the temperature (T)-dependent T2 and TS times. For T >14K the spin decoherence is dominated by the spin-lattice relaxation time, T1, which we measured with the inversion-recovery pulse sequence23. The Si:Bi inversion recoveries were well fitted by monoexponential decays such as in Fig. 3b. T1 is the typical time taken for the electron spins to polarize, and can be thought of as the timescale for storing classical information, in contrast to the quantum information storage time, T2. As expected, T1 is dominated by phonons, but most of these have energies that are much larger than the energy gap between spin up and spin down. As a result, two-phonon processes (such as the absorption of a high-energy phonon and the emission of an even higher-energy phonon) occur more frequently than single-phonon processes. The dependence of the spin-lattice relaxation rate, 1/T1, on T is well described by an e−1E/T Orbach term (two-phonon process using an excited state) added to a T7 Raman term (two-phonon process using a virtual excited state). The 1E used in Fig. 4 was 500K, but values from 450 to 800K provide an acceptable fit. A value of 1E = 400K has previously been measured with ESR of a more concentrated sample24, and identified25 with the energy gap to the

1s(T2) orbital excited state. Si:Bi has a larger ionization energy than the other group v donors, which reduces the number of phonons with enough energy

(Parte 1 de 2)