LETTERS PUBLISHED ONLINE: 23 MARCH 2014 | DOI: 10.1038/NNANO.2014.39
The effect of spin transport on spin lifetime in nanoscale systems Jeremy Cardellino†, Nicolas Scozzaro†, Michael Herman, Andrew J. Berger, Chi Zhang, Kin Chung Fong, Ciriyam Jayaprakash, Denis V. Pelekhov and P. Chris Hammel* Spintronics use the electron spin as a state variable for information processing and storage1. This requires manipulation of spin ensembles for data encoding, and spin transport for information transfer. Because of the central importance of lifetime for understanding and controlling spins, mechanisms that determine this lifetime in bulk systems have been extensively studied. However, a clear understanding of few-spin systems remains challenging. Here, we report spatially resolved magnetic resonance studies of electron spin ensembles confined to a ‘spin nanowire’ formed by nitrogen ion implantation in diamond. We measure the spin lifetime of the ensemble—that is, its spin autocorrelation time—by monitoring the statistical fluctuations2 of its net moment, which is in thermal equilibrium and has no imposed polarization gradient. We find that the lifetime of the ensemble is dominated by spin transport from the ensemble into the adjacent spin reservoir that is provided by the remainder of the nanowire. This is in striking contrast to conventional spin-lattice relaxation measurements of isolated spin ensembles. Electron spin resonance spectroscopy performed on nanoscale spin ensembles by means of a novel spin manipulation protocol corroborates spin transport in strong field gradients. Our experiments, supported by microscopic Monte Carlo modelling, provide a unique insight into the intrinsic dynamics of pure spin currents needed for nanoscale devices that seek to control spins. We have studied the relationship between the lifetime and transport dynamics of electron spin ensembles confined to a nanoscale spin wire. The transport proceeds via intrinsic spin flip-flop transitions3–5 in which a pair of anti-aligned neighbouring spins exchange their spin orientations due to their dipolar interaction. Successive flipflops lead to pure, diffusive spin transport3,4,6,7 in the absence of charge transport. We detect these flip-flops by monitoring the spin noise, that is, the statistical fluctuations of the ensemble magnetization2,8–10. In contrast to non-equilibrium experiments11,12, which measure thermal polarization recovery, we continuously measure the intrinsic spin dynamics of the ensemble13 and extract the spin correlation time (see Methods), which is the time over which the spin ensemble maintains its statistical polarization. This approach is thus free from the need to impose a non-equilibrium initial state. In macroscopic systems containing a large number of particles, Fick’s law of diffusion describes the time evolution of populations moving from regions of high to low concentration, resulting in a smooth evolution towards equilibrium. By contrast, in few-spin ensembles, fluctuations can lead to transient polarizations much larger than the Boltzmann (thermal) polarization2,9,14,15, and even flow against the polarization gradient16. Our measurements focus not on the polarization itself, but on the autocorrelation time of the polarization in a nanoscale spin detection volume whose size and
position we determine. In our experiments, these correlations evolve as a consequence of spin transport out of our measurement volume. We model this process using numerical simulations that incorporate this inherent randomness of few-spin, statistically polarized ensembles, and our model corroborates our measurements. To study flip-flop-dominated spin dynamics, we fabricated a nanoscale channel of implanted nitrogen (P1) centres in diamond at a density of 6 ppm. At this density, the strength of the dipolar coupling between spins is strong, leading to a flip-flop time of Tff ≈ 1 × 1024 s, several orders of magnitude shorter than the spin-lattice relaxation time, T1 ≈ 1 s (ref. 17). The spin density outside the spin wire (0.3 ppm) is sufficiently small that flip-flop transitions out of the wire are very rare, thus confining the spin transport to the wire. We used magnetic resonance force microscopy (MRFM)11,14,15 to probe the spins within the wire (Fig. 1a). The force detector was an ultra-soft cantilever with a SmCo5 magnetic particle glued onto the tip18 (see Methods). To resonantly detect spins, we implemented the iOSCAR (interrupted oscillating cantilever-driven adiabatic reversal) timing protocol2,14. A superconducting niobium coil provided a microwave field at a frequency of vrf/2p ¼ 2.18 GHz. This frequency defines a ‘resonant slice’ where the total magnetic field (tip field plus external field) experienced by sample spins is equal to vrf/g ¼ 77.8 mT, where g/2p ¼ 28 GHz T21 is the electron gyromagnetic ratio. By driving the cantilever at its self-determined resonant frequency to a peak-to-peak amplitude of 150 nm, the resonant slice sweeps out a volume of this width, hereafter termed the detection volume. We scanned this detection volume into the spin wire (Fig. 1a–e and Supplementary Movie) and measured the force exerted on the cantilever by the selected spins. From this measurement we could extract two quantities: spin signal magnitude (Fig. 1f ) and force correlation time tm (Fig. 1g). The spin signal was obtained from the magnitude of the variance in the time record of the force signal and is directly related to the number of measured spins14. The correlation time tm describes the characteristic time for the net moment of the detected spins to decorrelate. This can arise from conventional spin relaxation processes, spurious mechanisms arising from the measurement protocol8, or spin transport out of the detection volume via random-walk diffusion processes, with each step taking an average time Tff. We rule out spurious effects, as discussed in detail in ref. 8, and conventional relaxation is much too slow to discernibly alter the measured autocorrelation time. As we discuss below, the lifetime we observe is determined by spin transport out of the volume. As the detection volume begins to enter the wire, the spin signal grows with the overlap of the detected volume with the nanowire, eventually reaching a plateau when the volume completely enters
Department of Physics, Ohio State University, Columbus, Ohio 43210, USA, † These authors contributed equally to this work. * e-mail: [email protected] NATURE NANOTECHNOLOGY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturenanotechnology
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Figure 1 | MRFM spin wire set-up, schematic of spin dynamics in wire, and spin noise measurements. a, MRFM set-up for measuring spin noise in the 200 nm × 250 nm × 4 mm spin wire of P1 centres in diamond. The coil excites resonance along a contour of the magnetic particle’s field (plus an external field), called the resonant slice. As the cantilever oscillates, the field due to the magnetic particle varies and the slice sweeps throughout the spin detection volume. Spins toward the centre of the detection volume contribute to the spin signal more strongly (solid red) than those near the edge. b–e, Schematic showing the detection volume as it is walked into the spin wire, the rectangular stripe region. The thick double arrows represent flip-flop-mediated spin diffusion in and out of the detection volume. f, The spin signal (purple circles) increases as a function of detection volume position until the volume is fully inside the spin wire as in e. g, Correlation time of the measured spins (green diamonds). At first (as in b) spins are able to quickly diffuse out of the volume, but as the detection volume is walked in further (as in c), spins must diffuse a longer distance to exit the volume, causing the correlation time to increase. Once the detection volume is fully inside the stripe (d,e), spins can diffuse out on both sides, decreasing the correlation time to roughly half the peak value. In f and g, each data point is determined by fitting a Lorentzian to the spin noise power spectrum, and error bars are the standard deviations of the fit coefficients (see Methods). Solid black lines show simulation results (Supplementary Section 2).
the wire. The correlation time shows a complex behaviour as the detection volume moves into the wire that can be understood within the framework of flip-flop-mediated spin transport: ensemble correlation is lost as spins diffuse into or out of the detection volume. When the detection volume first enters (position ¼ 0 nm in Fig. 1b), the spins inside the volume easily and rapidly interact with nearby outside neighbours, resulting in a relatively short correlation time. With increasing overlap of the detection volume and the spin wire, spins must diffuse further to exit the detection volume and change the overall magnetization of the ensemble, thus increasing tm. The correlation time peaks when 66% of the detection volume has entered the channel. This can be understood in detail from the force sensitivity profile of the detection volume: spins in the middle of the volume are most sensitively detected (solid red indicates highest sensitivity in Fig. 1a–e), while those on the edges are less so (Supplementary Section 1). As the detection volume 2
approaches 100% overlap (at 150 nm), spins become able to diffuse out either side of the most sensitive region, reducing the correlation time by a factor of 2. Scanning deeper into the channel results in no further change because the measurement geometry becomes translationally invariant. The relative impact of spin transport on lifetime depends on the relationship between the diffusion length and the characteristic length of the measurement volume. Spin transport affects spin lifetime in our case because two conditions are satisfied: there is an adjacent spin reservoir (the remainder of the nanowire), and the diffusion length is larger than the measurement volume, enabling spin transport into the reservoir. If, instead, the diffusion length is smaller than the measurement volume, still in the presence of an adjacent reservoir, the lifetime is not affected by spin transport. We have observed this behaviour in our measurements of lifetime in the low-spin-density bulk outside the nanowire, in which the
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diffusion length is significantly smaller than the measurement volume, and we find a much larger tm of 1.5 s. This is consistent with conventional electron paramagnetic resonance (EPR) measurements of T1 of P1 centres in diamond17, conducted in the regime where the measurement volume is larger than the volume of the sample. In this case, spin transport effects are not observed regardless of the spin diffusion length because flip-flops within the measurement volume preserve the spin magnetization, thus the measured lifetime is T1. By systematically varying the size of the overlap of the detection volume with the spin wire, our measurements of tm reveal the effects of spin transport on the lifetime of the measured ensemble. The average polarization gradient vanishes at thermal equilibrium, so the conventional model of diffusion driven by polarization gradient predicts that tm will be independent of detection volume position within our thermal equilibrium wire. Such a model neglects the dominant role spin fluctuations play in nanoscale ensembles. To fit the measured data in Fig. 1g we use a Monte Carlo simulation that models the flip-flops between individual spins as a Markov process (Supplementary Section 2). From fits of this simulation to our data we can extract the flip-flop time, Tff ¼ 0.21 ms, which is in good agreement with the theoretical flipflop time, Tff,th ¼ 0.13 ms (Supplementary Section 3). This corresponds to a diffusion constant of D ¼ a 2/Tff ¼ 4.8 × 1029 cm2 s21, where a ¼ n 21/3 ≈ 10 nm is the average nearest-neighbour separ23 ation of spins with a density n ¼ 1 × 1018 cm√ (6 ppm). We furthermore find a spin diffusion length of L = DT 1 ≈ 700 nm, which is significantly larger than the lateral dimensions of the wire (depth 250 nm, width 200 nm, length 4 mm). This spin diffusion length is competitive with those of metallic spin transport devices19. The above expression for Tff,th is strictly correct only in zero magnetic field gradient, but our experiments are performed in a field gradient (G ≈ 1.3 × 105 T m21) that causes neighbouring spins to experience a field difference DB, thus making the Zeeman energy splittings unequal. This can suppress flip-flops because they no longer conserve energy (Supplementary Section 4 and refs 20–22). Because our measured flip-flop time Tff agrees closely with the expected zero-gradient value Tff,th , this implies an inhomogeneous line broadening of dB ≥ Ga/2 ¼ 0.64 mT, which makes up this energy difference, allowing flip-flops to proceed. This broadening was confirmed by spectroscopic measurements, as described in the following. To measure the EPR spectrum of the spin wire, we implemented a modified iOSCAR protocol, which improves spatial resolution. A high magnetic field gradient results in multiple spatially separated resonant slices, corresponding to individual spectral features that can be revealed by probing a spatial discontinuity of spin density, so this enables spectral measurements on nanoscale spin ensembles. A conventional iOSCAR measurement utilizes the entire detection volume swept out by the cantilever oscillation, as this couples to the largest number of spins and creates the largest force signal. By interrupting the microwave power to the coil for less than half a cantilever period, we effectively truncate the detection volume (Fig. 2). This enables us to maintain higher sensitivity while coupling to spins in a smaller spatial region. Utilizing this technique (Supplementary Section 1), which we call partially interrupted OSCAR (piOSCAR), we again scanned the detection volume into the spin wire. The improved resolution enabled us to resolve a staircase structure with three steps (Fig. 3c), corresponding to the three peaks of the P1 centre’s hyperfine spectrum, as discussed below. The P1 centre is a substitutional nitrogen defect with an unpaired spin-1/2 electron hyperfine-coupled to the spin-1 nitrogen nucleus. The hyperfine interaction results in a triplet of peaks in the EPR spectrum (Fig. 3a), corresponding to the three nuclear spin states, mI ¼ 21, 0, 1. The hyperfine splitting for our diamond crystal orientation relative to the external field is 3.3 mT (Supplementary
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Figure 2 | Comparison of piOSCAR and iOSCAR resonance protocols. a, Timing diagram comparing the iOSCAR and piOSCAR spin manipulation protocols. In both protocols the microwave power is periodically interrupted to modulate the spins. The iOSCAR protocol utilizes the full volume swept out by the resonant slice as the cantilever oscillates, whereas the piOSCAR protocol modulates a definable subset of spins by means of partial interrupts. b, Schematic illustrating the spin detection volume of the iOSCAR (blue dashed lines) and piOSCAR (green dashed lines) protocols. Spins in the middle of the volume provide a stronger spin signal than those near the edge, as indicated by red shading. The piOSCAR protocol increases spatial resolution by controllably reducing the size of the measurement volume while optimizing spin sensitivity.
Section 5 and ref. 23). In the magnetic field gradient of our probe magnet this splitting results in three spatially separated spin detection volumes. Each volume is defined by a surface of constant magnetic field that satisfies the resonance condition for a particular hyperfine transition (Fig. 3b). We observe a step-like increase in the spin signal as each volume enters the wire and the cantilever becomes coupled to an additional hyperfine transition (Fig. 3c). The spacing between steps (s ¼ 25 nm) provides an accurate means of measuring the probe field gradient because the spacing is set by the ratio of the known hyperfine splitting (3.3 mT) to the gradient, which we find to be 1.3 × 105 T m21. This is consistent with our estimations of G obtained using a standard technique24. The staircase structure in Fig. 3c is a convolution of the implanted spin density (approximately a step function), our force sensitivity profile and the EPR spectrum of spins in the sample. Deconvolution reveals the EPR spectrum shown in Fig. 3d (Supplementary Section 6). From the spectrum we find an average linewidth of 0.76 mT, which agrees with the expected linewidth of 0.65 mT and gives a measured flip-flop time Tff in agreement with the expected zero-gradient value Tff,th discussed above, and substantiates the proposed spin transport mechanism. We attribute the inhomogeneous broadening to crystal lattice damage or clustering caused by nitrogen ion implantation25–27. In conclusion, we have measured the effects of flip-flop-mediated spin transport on the lifetime of nanoscale spin ensembles in an insulating diamond spin wire, in the complete absence of charge motion. This pure spin transport arises from the intrinsic spin dynamics of dipole-coupled P1 centres at thermal equilibrium, which we observe by measuring spin noise without imposing a polarization gradient. By spatially resolving the three electron spin populations (corresponding to the three spin states of the hyperfine-coupled nitrogen nucleus) we obtain the EPR spectrum of less than 100 net spins in the wire, finding a linewidth that corroborates flip-flop-mediated spin diffusion as the mechanism responsible for the observed spin transport. The spectrum also enables accurate measurement of the applied magnetic field gradient. These measurements provide insight into the mechanisms of spin transport and lifetime relevant for the development of nanoscale spin device elements.
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Figure 3 | Hyperfine spectrum measurement for spin wire. a, Energy-level diagrams for three hyperfine-split spin populations, which are simultaneously on resonance, yet spatially separated by the field gradient. b, Schematic of the three volumes that are walked into the spin wire. c, Spin signal (purple circles) increases in a step-like fashion as each volume enters the wire. Error bars are the standard deviation of the Lorentzian fit coefficients (see Methods). d, Nanovolume EPR spectrum, obtained by deconvolving the fit to the spin signal (solid black line in c) with known force sensitivity (Supplementary Section 5), resolving the three hyperfine peaks of the P1 centre.
Methods The ultrasoft silicon cantilever had a spring constant of 100 mN m21, resonance frequency of 3 kHz and approximate dimensions of 90 mm (length), 1 mm (width) and 100 nm (thickness). Displacement of the cantilever was measured using a 1,550 nm laser interferometer. In Fig. 1f,g, the peak-to-peak cantilever oscillation amplitude was 150 nm, and in Fig. 3b it was 170 nm, with an interrupt that defined a volume with a width of 15 nm. The MRFM measurements were taken at a temperature of 4.2 K. The measured moment of the SmCo5 magnetic particle was 3.6 × 10212 J T21. The niobium resonator coil had 2.5 turns and a 300 mm diameter, and the resonance bowl had a diameter of 3 mm. The diamond sample studied in this experiment had a background nitrogen concentration of 0.3 ppm (5.27 × 1016 cm23). To create the high-spin-density wire, the sample was first prepared using electron-beam patterning and then exposed to nitrogen ion implantation at a variety of energies to create a uniform spin density. Finally, the sample was annealed to yield an approximately uniform channel (6 ppm or 1.05 × 1018 cm23) with width, depth and length of 200 nm, 250 nm and 4 mm, respectively. The sample was purchased commercially from Element Six, grown via chemical vapour deposition, and the nitrogen ion implantation was performed by Leonard Kroko. Following the work of Rugar et al.14, we used their iOSCAR spin manipulation and data analysis protocol to measure the fluctuating spin magnetization with high sensitivity. This protocol imposes a modulation on the cantilever frequency, so its power spectrum peaks at this modulation frequency. However, fluctuations of the spin magnetization broaden this peak, and the inverse of this linewidth is tm. In practice, this is obtained by a lock-in measurement of the cantilever frequency using multiple digital lock-ins, each with a different bandwidth (time constant), using the imposed modulation frequency as the reference. The variance s(dv) of the lock-in output increases with increasing demodulation bandwidth dv, because the output includes an increasingly broad spectrum of fluctuations. The way in which the variance increases with bandwidth reveals the parameters of the Lorentzian lineshape: tm and the spin signal magnitude15. s(dv) was measured repeatedly for 2 h in 60 s time records for each data point in Fig. 1f,g. The experimental error for each value of s(dv) at a 4
particular bandwidth was calculated as the standard error based on this averaging. The error bars of tm and the spin signal are the errors of the fit to s(dv), which takes into account the experimental error and quality of fit. One can think of tm as the time constant over which the instantaneous statistical polarization of the spin system becomes decorrelated due to spin flips. This is described by the Wiener–Khinchin theorem, which relates the power spectrum of a timedependent signal to the Fourier transform of its autocorrelation.
Received 18 October 2013; accepted 6 February 2014; published online 23 March 2014
References 1. Wolf, S. A. et al. Spintronics: a spin-based electronics vision for the future. Science 294, 1488–1495 (2001). 2. Mamin, H. J. et al. Detection and manipulation of statistical polarization in small spin ensembles. Phys. Rev. Lett. 91, 207604 (2003). 3. Bloembergen, N. On the interaction of nuclear spins in a crystalline lattice. Physica 15, 386–426 (1949). 4. Abragam, A. The Principles of Nuclear Magnetism: The International Series of Monographs on Physics (Oxford Univ. Press, 1961). 5. Bloembergen, N., Purcell, E. M. & Pound, R. V. Relaxation effects in nuclear magnetic resonance absorption. Phys. Rev. 73, 679–712 (1948). 6. Vugmeister, B. E. Spin diffusion and spin-lattice relaxation in paramagnetic crystals. Phys. Status Solidi B 90, 711–718 (1978). 7. Zhang, S., Meier, B. H. & Ernst, R. R. Polarization echoes in NMR. Phys. Rev. Lett. 69, 2149–2151 (1992). 8. Fong, K. C. et al. Spin lifetime in small ensembles of electron spins measured by magnetic resonance force microscopy. Phys. Rev. B 84, 220405 (2011). 9. Bloch, F. Nuclear induction. Phys. Rev. 70, 460–474 (1946). 10. Sleator, T., Hahn, E. L., Hilbert, C. & Clarke, J. Nuclear-spin noise. Phys. Rev. Lett. 55, 1742–1745 (1985). 11. Vinante, A. et al. Magnetic resonance force microscopy of paramagnetic electron spins at millikelvin temperatures. Nature Commun. 2, 572 (2011).
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12. Eberhardt, K. W. et al. Direct observation of nuclear spin diffusion in real space. Phys. Rev. Lett. 99, 227603 (2007). 13. Mu¨ller, G. M. et al. Spin noise spectroscopy in GaAs (110) quantum wells: access to intrinsic spin lifetimes and equilibrium electron dynamics. Phys. Rev. Lett. 101, 206601 (2008). 14. Rugar, D. et al. Single spin detection by magnetic resonance force microscopy. Nature 430, 329–332 (2004). 15. Peddibhotla, P. et al. Harnessing nuclear spin polarization fluctuations in a semiconductor nanowire. Nature Phys. 9, 631–635 (2013). 16. Seitaridou, E. et al. Measuring flux distributions for diffusion in the smallnumbers limit. J. Phys. Chem. B 111, 2288–2292 (2007). 17. Takahashi, S. et al. Quenching spin decoherence in diamond through spin bath polarization. Phys. Rev. Lett. 101, 047601 (2008). 18. Chui, B. W. et al. In Proc. 12th Int. Conf. Transducers, Solid-State Sensors, Actuators and Microsystems 2, 1120–1123 (IEEE, 2003). 19. Jedema, F. J. et al. Electrical detection of spin precession in a metallic mesoscopic spin valve. Nature 416, 713–716 (2002). 20. Budakian, R., Mamin, H. J. & Rugar, D. Suppression of spin diffusion near a micron-size ferromagnet. Phys. Rev. Lett. 92, 037205 (2004). 21. Genack, A. Z. & Redfield, A. G. Theory of nuclear spin diffusion in a spatially varying magnetic field. Phys. Rev. B 12, 78–87 (1975). 22. Tyryshkin, A. M. et al. Electron spin coherence exceeding seconds in highpurity silicon. Nature Mater. 11, 143–147 (2011). 23. van Oort, E., Stroomer, P. & Glasbeek, M. Low-field optically detected magnetic resonance of a coupled triplet–doublet defect pair in diamond. Phys. Rev. B 42, 8605–8608 (1990). 24. Stipe, B. C. et al. Electron spin relaxation near a micron-size ferromagnet. Phys. Rev. Lett. 87, 277602 (2001).
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25. Brosious, P., Lee, Y., Corbett, J. & Cheng, L. Electron paramagnetic resonance in diamond implanted at various energies and temperatures Phys. Status Solidi A 25, 541–549 (1974). 26. Mohrange, J., Beserman, R. & Bourgoin, J. C. Study of defects introduced by ion implantation in diamond. Jpn J. Appl. Phys. 14, 544–548 (1975). 27. Warren, R., Feldman, D. & Castle, J. Spin-lattice relaxation of F centers in KCl: interacting F centers. Phys. Rev. 136, A1347 (1964).
Acknowledgements The research presented in this Letter was supported by the Army Research Office (grant no. W911NF-09-1-0147), the National Science Foundation (NSF, grant no. DMR-0807093), the Center for Emergent Materials (CEM), an NSF-funded MRSEC grant (no. DMR0820414) and the NanoSystems Laboratory at Ohio State University. The authors thank R. J. Furnstahl for valuable discussions related to this research.
Author contributions M.R.H. and K.C.F. designed and built the experimental probe. M.R.H. conceived the experiments. M.R.H. and J.D.C. performed the experiments. J.D.C. and N.S. performed the simulations. J.D.C., N.S., M.R.H. and D.V.P. analysed the data. C.Z. contributed theoretical analysis. A.J.B., J.D.C., N.S., D.V.P., C.J. and P.C.H. wrote the paper. P.C.H. supervised the project.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to P.C.H.
Competing financial interests The authors declare no competing financial interests.
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The effect of spin transport on spin lifetime in nanoscale systems Jeremy Cardellino†, Nicolas Scozzaro†, Michael Herman, Andrew J. Berger, Chi Zhang, Kin Chung Fong, Ciriyam Jayaprakash, Denis V. Pelekhov and P. Chris Hammel* Spintronics use the electron spin as a state variable for information processing and storage1. This requires manipulation of spin ensembles for data encoding, and spin transport for information transfer. Because of the central importance of lifetime for understanding and controlling spins, mechanisms that determine this lifetime in bulk systems have been extensively studied. However, a clear understanding of few-spin systems remains challenging. Here, we report spatially resolved magnetic resonance studies of electron spin ensembles confined to a ‘spin nanowire’ formed by nitrogen ion implantation in diamond. We measure the spin lifetime of the ensemble—that is, its spin autocorrelation time—by monitoring the statistical fluctuations2 of its net moment, which is in thermal equilibrium and has no imposed polarization gradient. We find that the lifetime of the ensemble is dominated by spin transport from the ensemble into the adjacent spin reservoir that is provided by the remainder of the nanowire. This is in striking contrast to conventional spin-lattice relaxation measurements of isolated spin ensembles. Electron spin resonance spectroscopy performed on nanoscale spin ensembles by means of a novel spin manipulation protocol corroborates spin transport in strong field gradients. Our experiments, supported by microscopic Monte Carlo modelling, provide a unique insight into the intrinsic dynamics of pure spin currents needed for nanoscale devices that seek to control spins. We have studied the relationship between the lifetime and transport dynamics of electron spin ensembles confined to a nanoscale spin wire. The transport proceeds via intrinsic spin flip-flop transitions3–5 in which a pair of anti-aligned neighbouring spins exchange their spin orientations due to their dipolar interaction. Successive flipflops lead to pure, diffusive spin transport3,4,6,7 in the absence of charge transport. We detect these flip-flops by monitoring the spin noise, that is, the statistical fluctuations of the ensemble magnetization2,8–10. In contrast to non-equilibrium experiments11,12, which measure thermal polarization recovery, we continuously measure the intrinsic spin dynamics of the ensemble13 and extract the spin correlation time (see Methods), which is the time over which the spin ensemble maintains its statistical polarization. This approach is thus free from the need to impose a non-equilibrium initial state. In macroscopic systems containing a large number of particles, Fick’s law of diffusion describes the time evolution of populations moving from regions of high to low concentration, resulting in a smooth evolution towards equilibrium. By contrast, in few-spin ensembles, fluctuations can lead to transient polarizations much larger than the Boltzmann (thermal) polarization2,9,14,15, and even flow against the polarization gradient16. Our measurements focus not on the polarization itself, but on the autocorrelation time of the polarization in a nanoscale spin detection volume whose size and
position we determine. In our experiments, these correlations evolve as a consequence of spin transport out of our measurement volume. We model this process using numerical simulations that incorporate this inherent randomness of few-spin, statistically polarized ensembles, and our model corroborates our measurements. To study flip-flop-dominated spin dynamics, we fabricated a nanoscale channel of implanted nitrogen (P1) centres in diamond at a density of 6 ppm. At this density, the strength of the dipolar coupling between spins is strong, leading to a flip-flop time of Tff ≈ 1 × 1024 s, several orders of magnitude shorter than the spin-lattice relaxation time, T1 ≈ 1 s (ref. 17). The spin density outside the spin wire (0.3 ppm) is sufficiently small that flip-flop transitions out of the wire are very rare, thus confining the spin transport to the wire. We used magnetic resonance force microscopy (MRFM)11,14,15 to probe the spins within the wire (Fig. 1a). The force detector was an ultra-soft cantilever with a SmCo5 magnetic particle glued onto the tip18 (see Methods). To resonantly detect spins, we implemented the iOSCAR (interrupted oscillating cantilever-driven adiabatic reversal) timing protocol2,14. A superconducting niobium coil provided a microwave field at a frequency of vrf/2p ¼ 2.18 GHz. This frequency defines a ‘resonant slice’ where the total magnetic field (tip field plus external field) experienced by sample spins is equal to vrf/g ¼ 77.8 mT, where g/2p ¼ 28 GHz T21 is the electron gyromagnetic ratio. By driving the cantilever at its self-determined resonant frequency to a peak-to-peak amplitude of 150 nm, the resonant slice sweeps out a volume of this width, hereafter termed the detection volume. We scanned this detection volume into the spin wire (Fig. 1a–e and Supplementary Movie) and measured the force exerted on the cantilever by the selected spins. From this measurement we could extract two quantities: spin signal magnitude (Fig. 1f ) and force correlation time tm (Fig. 1g). The spin signal was obtained from the magnitude of the variance in the time record of the force signal and is directly related to the number of measured spins14. The correlation time tm describes the characteristic time for the net moment of the detected spins to decorrelate. This can arise from conventional spin relaxation processes, spurious mechanisms arising from the measurement protocol8, or spin transport out of the detection volume via random-walk diffusion processes, with each step taking an average time Tff. We rule out spurious effects, as discussed in detail in ref. 8, and conventional relaxation is much too slow to discernibly alter the measured autocorrelation time. As we discuss below, the lifetime we observe is determined by spin transport out of the volume. As the detection volume begins to enter the wire, the spin signal grows with the overlap of the detected volume with the nanowire, eventually reaching a plateau when the volume completely enters
Department of Physics, Ohio State University, Columbus, Ohio 43210, USA, † These authors contributed equally to this work. * e-mail: [email protected] NATURE NANOTECHNOLOGY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturenanotechnology
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Figure 1 | MRFM spin wire set-up, schematic of spin dynamics in wire, and spin noise measurements. a, MRFM set-up for measuring spin noise in the 200 nm × 250 nm × 4 mm spin wire of P1 centres in diamond. The coil excites resonance along a contour of the magnetic particle’s field (plus an external field), called the resonant slice. As the cantilever oscillates, the field due to the magnetic particle varies and the slice sweeps throughout the spin detection volume. Spins toward the centre of the detection volume contribute to the spin signal more strongly (solid red) than those near the edge. b–e, Schematic showing the detection volume as it is walked into the spin wire, the rectangular stripe region. The thick double arrows represent flip-flop-mediated spin diffusion in and out of the detection volume. f, The spin signal (purple circles) increases as a function of detection volume position until the volume is fully inside the spin wire as in e. g, Correlation time of the measured spins (green diamonds). At first (as in b) spins are able to quickly diffuse out of the volume, but as the detection volume is walked in further (as in c), spins must diffuse a longer distance to exit the volume, causing the correlation time to increase. Once the detection volume is fully inside the stripe (d,e), spins can diffuse out on both sides, decreasing the correlation time to roughly half the peak value. In f and g, each data point is determined by fitting a Lorentzian to the spin noise power spectrum, and error bars are the standard deviations of the fit coefficients (see Methods). Solid black lines show simulation results (Supplementary Section 2).
the wire. The correlation time shows a complex behaviour as the detection volume moves into the wire that can be understood within the framework of flip-flop-mediated spin transport: ensemble correlation is lost as spins diffuse into or out of the detection volume. When the detection volume first enters (position ¼ 0 nm in Fig. 1b), the spins inside the volume easily and rapidly interact with nearby outside neighbours, resulting in a relatively short correlation time. With increasing overlap of the detection volume and the spin wire, spins must diffuse further to exit the detection volume and change the overall magnetization of the ensemble, thus increasing tm. The correlation time peaks when 66% of the detection volume has entered the channel. This can be understood in detail from the force sensitivity profile of the detection volume: spins in the middle of the volume are most sensitively detected (solid red indicates highest sensitivity in Fig. 1a–e), while those on the edges are less so (Supplementary Section 1). As the detection volume 2
approaches 100% overlap (at 150 nm), spins become able to diffuse out either side of the most sensitive region, reducing the correlation time by a factor of 2. Scanning deeper into the channel results in no further change because the measurement geometry becomes translationally invariant. The relative impact of spin transport on lifetime depends on the relationship between the diffusion length and the characteristic length of the measurement volume. Spin transport affects spin lifetime in our case because two conditions are satisfied: there is an adjacent spin reservoir (the remainder of the nanowire), and the diffusion length is larger than the measurement volume, enabling spin transport into the reservoir. If, instead, the diffusion length is smaller than the measurement volume, still in the presence of an adjacent reservoir, the lifetime is not affected by spin transport. We have observed this behaviour in our measurements of lifetime in the low-spin-density bulk outside the nanowire, in which the
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DOI: 10.1038/NNANO.2014.39
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diffusion length is significantly smaller than the measurement volume, and we find a much larger tm of 1.5 s. This is consistent with conventional electron paramagnetic resonance (EPR) measurements of T1 of P1 centres in diamond17, conducted in the regime where the measurement volume is larger than the volume of the sample. In this case, spin transport effects are not observed regardless of the spin diffusion length because flip-flops within the measurement volume preserve the spin magnetization, thus the measured lifetime is T1. By systematically varying the size of the overlap of the detection volume with the spin wire, our measurements of tm reveal the effects of spin transport on the lifetime of the measured ensemble. The average polarization gradient vanishes at thermal equilibrium, so the conventional model of diffusion driven by polarization gradient predicts that tm will be independent of detection volume position within our thermal equilibrium wire. Such a model neglects the dominant role spin fluctuations play in nanoscale ensembles. To fit the measured data in Fig. 1g we use a Monte Carlo simulation that models the flip-flops between individual spins as a Markov process (Supplementary Section 2). From fits of this simulation to our data we can extract the flip-flop time, Tff ¼ 0.21 ms, which is in good agreement with the theoretical flipflop time, Tff,th ¼ 0.13 ms (Supplementary Section 3). This corresponds to a diffusion constant of D ¼ a 2/Tff ¼ 4.8 × 1029 cm2 s21, where a ¼ n 21/3 ≈ 10 nm is the average nearest-neighbour separ23 ation of spins with a density n ¼ 1 × 1018 cm√ (6 ppm). We furthermore find a spin diffusion length of L = DT 1 ≈ 700 nm, which is significantly larger than the lateral dimensions of the wire (depth 250 nm, width 200 nm, length 4 mm). This spin diffusion length is competitive with those of metallic spin transport devices19. The above expression for Tff,th is strictly correct only in zero magnetic field gradient, but our experiments are performed in a field gradient (G ≈ 1.3 × 105 T m21) that causes neighbouring spins to experience a field difference DB, thus making the Zeeman energy splittings unequal. This can suppress flip-flops because they no longer conserve energy (Supplementary Section 4 and refs 20–22). Because our measured flip-flop time Tff agrees closely with the expected zero-gradient value Tff,th , this implies an inhomogeneous line broadening of dB ≥ Ga/2 ¼ 0.64 mT, which makes up this energy difference, allowing flip-flops to proceed. This broadening was confirmed by spectroscopic measurements, as described in the following. To measure the EPR spectrum of the spin wire, we implemented a modified iOSCAR protocol, which improves spatial resolution. A high magnetic field gradient results in multiple spatially separated resonant slices, corresponding to individual spectral features that can be revealed by probing a spatial discontinuity of spin density, so this enables spectral measurements on nanoscale spin ensembles. A conventional iOSCAR measurement utilizes the entire detection volume swept out by the cantilever oscillation, as this couples to the largest number of spins and creates the largest force signal. By interrupting the microwave power to the coil for less than half a cantilever period, we effectively truncate the detection volume (Fig. 2). This enables us to maintain higher sensitivity while coupling to spins in a smaller spatial region. Utilizing this technique (Supplementary Section 1), which we call partially interrupted OSCAR (piOSCAR), we again scanned the detection volume into the spin wire. The improved resolution enabled us to resolve a staircase structure with three steps (Fig. 3c), corresponding to the three peaks of the P1 centre’s hyperfine spectrum, as discussed below. The P1 centre is a substitutional nitrogen defect with an unpaired spin-1/2 electron hyperfine-coupled to the spin-1 nitrogen nucleus. The hyperfine interaction results in a triplet of peaks in the EPR spectrum (Fig. 3a), corresponding to the three nuclear spin states, mI ¼ 21, 0, 1. The hyperfine splitting for our diamond crystal orientation relative to the external field is 3.3 mT (Supplementary
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Figure 2 | Comparison of piOSCAR and iOSCAR resonance protocols. a, Timing diagram comparing the iOSCAR and piOSCAR spin manipulation protocols. In both protocols the microwave power is periodically interrupted to modulate the spins. The iOSCAR protocol utilizes the full volume swept out by the resonant slice as the cantilever oscillates, whereas the piOSCAR protocol modulates a definable subset of spins by means of partial interrupts. b, Schematic illustrating the spin detection volume of the iOSCAR (blue dashed lines) and piOSCAR (green dashed lines) protocols. Spins in the middle of the volume provide a stronger spin signal than those near the edge, as indicated by red shading. The piOSCAR protocol increases spatial resolution by controllably reducing the size of the measurement volume while optimizing spin sensitivity.
Section 5 and ref. 23). In the magnetic field gradient of our probe magnet this splitting results in three spatially separated spin detection volumes. Each volume is defined by a surface of constant magnetic field that satisfies the resonance condition for a particular hyperfine transition (Fig. 3b). We observe a step-like increase in the spin signal as each volume enters the wire and the cantilever becomes coupled to an additional hyperfine transition (Fig. 3c). The spacing between steps (s ¼ 25 nm) provides an accurate means of measuring the probe field gradient because the spacing is set by the ratio of the known hyperfine splitting (3.3 mT) to the gradient, which we find to be 1.3 × 105 T m21. This is consistent with our estimations of G obtained using a standard technique24. The staircase structure in Fig. 3c is a convolution of the implanted spin density (approximately a step function), our force sensitivity profile and the EPR spectrum of spins in the sample. Deconvolution reveals the EPR spectrum shown in Fig. 3d (Supplementary Section 6). From the spectrum we find an average linewidth of 0.76 mT, which agrees with the expected linewidth of 0.65 mT and gives a measured flip-flop time Tff in agreement with the expected zero-gradient value Tff,th discussed above, and substantiates the proposed spin transport mechanism. We attribute the inhomogeneous broadening to crystal lattice damage or clustering caused by nitrogen ion implantation25–27. In conclusion, we have measured the effects of flip-flop-mediated spin transport on the lifetime of nanoscale spin ensembles in an insulating diamond spin wire, in the complete absence of charge motion. This pure spin transport arises from the intrinsic spin dynamics of dipole-coupled P1 centres at thermal equilibrium, which we observe by measuring spin noise without imposing a polarization gradient. By spatially resolving the three electron spin populations (corresponding to the three spin states of the hyperfine-coupled nitrogen nucleus) we obtain the EPR spectrum of less than 100 net spins in the wire, finding a linewidth that corroborates flip-flop-mediated spin diffusion as the mechanism responsible for the observed spin transport. The spectrum also enables accurate measurement of the applied magnetic field gradient. These measurements provide insight into the mechanisms of spin transport and lifetime relevant for the development of nanoscale spin device elements.
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Nuclear spin quantum number mI = +1 mI = 0 mI = −1
Resonance frequency ωrf/2π = 2.2 GHz Magnetic field gradient
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Figure 3 | Hyperfine spectrum measurement for spin wire. a, Energy-level diagrams for three hyperfine-split spin populations, which are simultaneously on resonance, yet spatially separated by the field gradient. b, Schematic of the three volumes that are walked into the spin wire. c, Spin signal (purple circles) increases in a step-like fashion as each volume enters the wire. Error bars are the standard deviation of the Lorentzian fit coefficients (see Methods). d, Nanovolume EPR spectrum, obtained by deconvolving the fit to the spin signal (solid black line in c) with known force sensitivity (Supplementary Section 5), resolving the three hyperfine peaks of the P1 centre.
Methods The ultrasoft silicon cantilever had a spring constant of 100 mN m21, resonance frequency of 3 kHz and approximate dimensions of 90 mm (length), 1 mm (width) and 100 nm (thickness). Displacement of the cantilever was measured using a 1,550 nm laser interferometer. In Fig. 1f,g, the peak-to-peak cantilever oscillation amplitude was 150 nm, and in Fig. 3b it was 170 nm, with an interrupt that defined a volume with a width of 15 nm. The MRFM measurements were taken at a temperature of 4.2 K. The measured moment of the SmCo5 magnetic particle was 3.6 × 10212 J T21. The niobium resonator coil had 2.5 turns and a 300 mm diameter, and the resonance bowl had a diameter of 3 mm. The diamond sample studied in this experiment had a background nitrogen concentration of 0.3 ppm (5.27 × 1016 cm23). To create the high-spin-density wire, the sample was first prepared using electron-beam patterning and then exposed to nitrogen ion implantation at a variety of energies to create a uniform spin density. Finally, the sample was annealed to yield an approximately uniform channel (6 ppm or 1.05 × 1018 cm23) with width, depth and length of 200 nm, 250 nm and 4 mm, respectively. The sample was purchased commercially from Element Six, grown via chemical vapour deposition, and the nitrogen ion implantation was performed by Leonard Kroko. Following the work of Rugar et al.14, we used their iOSCAR spin manipulation and data analysis protocol to measure the fluctuating spin magnetization with high sensitivity. This protocol imposes a modulation on the cantilever frequency, so its power spectrum peaks at this modulation frequency. However, fluctuations of the spin magnetization broaden this peak, and the inverse of this linewidth is tm. In practice, this is obtained by a lock-in measurement of the cantilever frequency using multiple digital lock-ins, each with a different bandwidth (time constant), using the imposed modulation frequency as the reference. The variance s(dv) of the lock-in output increases with increasing demodulation bandwidth dv, because the output includes an increasingly broad spectrum of fluctuations. The way in which the variance increases with bandwidth reveals the parameters of the Lorentzian lineshape: tm and the spin signal magnitude15. s(dv) was measured repeatedly for 2 h in 60 s time records for each data point in Fig. 1f,g. The experimental error for each value of s(dv) at a 4
particular bandwidth was calculated as the standard error based on this averaging. The error bars of tm and the spin signal are the errors of the fit to s(dv), which takes into account the experimental error and quality of fit. One can think of tm as the time constant over which the instantaneous statistical polarization of the spin system becomes decorrelated due to spin flips. This is described by the Wiener–Khinchin theorem, which relates the power spectrum of a timedependent signal to the Fourier transform of its autocorrelation.
Received 18 October 2013; accepted 6 February 2014; published online 23 March 2014
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12. Eberhardt, K. W. et al. Direct observation of nuclear spin diffusion in real space. Phys. Rev. Lett. 99, 227603 (2007). 13. Mu¨ller, G. M. et al. Spin noise spectroscopy in GaAs (110) quantum wells: access to intrinsic spin lifetimes and equilibrium electron dynamics. Phys. Rev. Lett. 101, 206601 (2008). 14. Rugar, D. et al. Single spin detection by magnetic resonance force microscopy. Nature 430, 329–332 (2004). 15. Peddibhotla, P. et al. Harnessing nuclear spin polarization fluctuations in a semiconductor nanowire. Nature Phys. 9, 631–635 (2013). 16. Seitaridou, E. et al. Measuring flux distributions for diffusion in the smallnumbers limit. J. Phys. Chem. B 111, 2288–2292 (2007). 17. Takahashi, S. et al. Quenching spin decoherence in diamond through spin bath polarization. Phys. Rev. Lett. 101, 047601 (2008). 18. Chui, B. W. et al. In Proc. 12th Int. Conf. Transducers, Solid-State Sensors, Actuators and Microsystems 2, 1120–1123 (IEEE, 2003). 19. Jedema, F. J. et al. Electrical detection of spin precession in a metallic mesoscopic spin valve. Nature 416, 713–716 (2002). 20. Budakian, R., Mamin, H. J. & Rugar, D. Suppression of spin diffusion near a micron-size ferromagnet. Phys. Rev. Lett. 92, 037205 (2004). 21. Genack, A. Z. & Redfield, A. G. Theory of nuclear spin diffusion in a spatially varying magnetic field. Phys. Rev. B 12, 78–87 (1975). 22. Tyryshkin, A. M. et al. Electron spin coherence exceeding seconds in highpurity silicon. Nature Mater. 11, 143–147 (2011). 23. van Oort, E., Stroomer, P. & Glasbeek, M. Low-field optically detected magnetic resonance of a coupled triplet–doublet defect pair in diamond. Phys. Rev. B 42, 8605–8608 (1990). 24. Stipe, B. C. et al. Electron spin relaxation near a micron-size ferromagnet. Phys. Rev. Lett. 87, 277602 (2001).
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25. Brosious, P., Lee, Y., Corbett, J. & Cheng, L. Electron paramagnetic resonance in diamond implanted at various energies and temperatures Phys. Status Solidi A 25, 541–549 (1974). 26. Mohrange, J., Beserman, R. & Bourgoin, J. C. Study of defects introduced by ion implantation in diamond. Jpn J. Appl. Phys. 14, 544–548 (1975). 27. Warren, R., Feldman, D. & Castle, J. Spin-lattice relaxation of F centers in KCl: interacting F centers. Phys. Rev. 136, A1347 (1964).
Acknowledgements The research presented in this Letter was supported by the Army Research Office (grant no. W911NF-09-1-0147), the National Science Foundation (NSF, grant no. DMR-0807093), the Center for Emergent Materials (CEM), an NSF-funded MRSEC grant (no. DMR0820414) and the NanoSystems Laboratory at Ohio State University. The authors thank R. J. Furnstahl for valuable discussions related to this research.
Author contributions M.R.H. and K.C.F. designed and built the experimental probe. M.R.H. conceived the experiments. M.R.H. and J.D.C. performed the experiments. J.D.C. and N.S. performed the simulations. J.D.C., N.S., M.R.H. and D.V.P. analysed the data. C.Z. contributed theoretical analysis. A.J.B., J.D.C., N.S., D.V.P., C.J. and P.C.H. wrote the paper. P.C.H. supervised the project.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to P.C.H.
Competing financial interests The authors declare no competing financial interests.
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