The ability to measure weak signals such as pressure, force, electric field and temperature with nanoscale devices and high spatial resolution offers a wide range of applications in fundamental and applied sciences. Here we present a proposal for a hybrid device composed of thin film layers of diamond with colour centres and piezoactive elements for the transduction and measurement of physical signals. The magnetic response of a piezomagnetic layer to an external stress or a stress induced by a signal is shown to affect significantly the spin properties of nitrogen-vacancy centres in diamond. Under ambient conditions, realistic environmental noise and material imperfections, we show that this hybrid device can achieve significant improvements in sensitivity over the pure diamond-based approach in combination with nanometre-scale spatial resolution. Furthermore, the proposed hybrid architecture offers novel possibilities for engineering strong coherent couplings between nanomechanical oscillator and solid state spin qubits.
The sensitive measurement of signals, such as force, pressure, electric field, temperature, with high spatial resolution possesses a wide range of applications in physics, nano science, engineering and the life sciences. Conventional methods—for example, for pressure detection—can usually operate at length scales down to the micron scale. Going beyond this regime while retaining highest sensitivity represents a significant challenge.
Recently, the negatively charged nitrogen-vacancy (NV) centre in diamond1 has been established as a highly sensitive room-temperature nanoscale probe for magnetic2,3 and electric fields4 as well as temperature5,6,7 building on the strength of the excellent coherence properties of the NV-centre electron spin. The operational principle of such kind of pure diamond quantum sensor is based on the fact that a magnetic/electric field or temperature change will directly affect the spin properties of the NV centre, which can then be detected by optical means. The effect of pressure acting on a diamond lattice, for example, leads to the coupling between the orbital and spin dynamics of the NV centre. This offers a method to measure very high pressure with a sensitivity better than existing techniques8. At room temperature, however, the coupling between the strain of the diamond lattice and the ground state manifold of the NV-centre spin is insufficient for measuring a sub-MPa pressure.
In this work, we propose a hybrid device that combines an NV-centre quantum sensor in diamond with a surface layer of piezomagnetic material as a signal transducer. The growth of ultrapure isotopically modified diamond and subsequent nitrogen implantation make it possible to engineer NV centres close to the diamond surface9,10,11 (see also Müller, C. et al., manuscript in preparation) and therefore in close proximity to the piezomagnetic material layer. This allows for the detection of changes in the stray magnetic field3,12,13,14,15,16 generated by the piezomagnetic material because of an external stress. We find that the effect of pressure on the NV-centre spin can be amplified by three orders of magnitude when compared with its direct effect on the diamond8. However, as the NV centre lies close to the magnetic material, the magnetic noise is also expected to become more significant and deteriorate the measurement sensitivity. We analyse the sensitivity of such a hybrid diamond–piezomagnetic device for the measurement of stress (force) with detailed numerical studies, and show that a sensitivity of (tens of ) can be achieved for the measurement of pressure (force) under ambient conditions with state-of-the-art experimental capabilities by choosing appropriate control parameters. In combination with piezoelectric and thermally sensitive elements, our proposed hybrid piezomagentic-NV approach also provides a new method for the measurement of electric field and temperature with an improved sensitivity as compared with standard diamond NV sensors4,5,6,7.
We consider a system (see Fig. 1a) made up of a diamond layer grown on the surface of a thin piezomagnetic film such as Tb0.27Dy0.73Fe2 (Terfenol-D) that exhibits a high magnetostrictive effect. The surface area of the piezomagnetic film can be reduced to a designed size by subsequent selective etching. The NV centre is implanted in the diamond layer with a depth denoted as d from the interface between the magnetic film and diamond. The ground 3A2 level of the NV-centre spin exhibits a zero field splitting of D=2.87 GHz between the ms=0 and ms=±1 spin sublevels. The spin Hamiltonian, including the strain effect of the diamond lattice and the Zeeman interaction with a magnetic field, is given by
where γ=28 MHz mT−1 is the electron gyromagnetic ratio, is the combination of the stray magnetic field Bp generated by the piezomagnetic film and the applied external magnetic field along the NV axis (that connects the nitrogen and vacancy sites). The effect of strain on the NV centre, as quantified by E, induces ground-state spin sublevel mixing and is usually much smaller (on the order of MHz) than the energy splitting of the ground-state triplet. The zero-field splitting D is slightly affected by the stress because of pressure σ, with a gradient of about dD(σ)/dσ~\n15 kHz/MPa at room temperature8. In this work, we will be mostly interested in the case of a weak (sub MPa) pressure, and therefore such a direct effect of stress in the diamond lattice on the NV spin is negligible. The coupling strength between an electric field and the NV ground-state spin is also negligible in the present model18. The spin-dependent fluorescence of an NV centre (see Fig. 1b) provides an efficient mechanism to optically initialize and readout the ground-state spin, and to perform optically detected magnetic resonance (ODMR) measurements in the ground state. The resonance frequencies ω±1 in the ODMR spectra corresponding to the electronic transitions from the spin sublevel ms=0 to ±1, respectively, depend on the magnetic field acting on the NV centre.
The piezomagnetic material consists of magnetic domains, in which the magnetization is uniform Mk=MSmk, where MS is the saturation magnetization of the material, and with being the direction cosines with respect to the crystal axes. The response of the piezomagnetic materials that have large magnetostriction, such as Terfenol-D, leads to a change of the magnetization directions of magnetic domains, and in turn of the stray magnetic field19,20 that affects the spin properties of an NV centre.
ODMR response to a uniaxial stress
To estimate the stress-induced change of the stray magnetic field and the corresponding magnetic noise acting on the NV-centre spin, we perform micromagnetic simulation using Landau–Lifshitz–Gilbert equation. We first calculate the steady state of Terfenol-D film (characterized by the parameters in Supplementary Table 1) under varying stress by using the Landau–Lifshitz–Gilbert micromagnetic dynamic equation (Supplementary Equation 1 in Supplementary Note 1). To account for the thermal fluctuation at room temperature, we add a random thermal fluctuation field Hth to the effective magnetic field Hk in the micromagnetic dynamic equation, and describe the thermal field Hth as a Gaussian random process with the strength determined by the fluctuation-dissipation theorem21,22,23, see Supplementary Note 1. We apply an external magnetic field along the NV axis, which is assumed to be in the ‹001› direction and set it as the z axis, and consider a stress along the ‹111› direction with respect to the crystal axes in the piezomagnetic material (see Supplementary Note 2), along which it has the largest magnetostriction coefficient, see Supplementary Table 1. By solving the Langevin equation using the stochastic Heun scheme24, we calculate the effective magnetic field acting on the NV centre and observe a prominent resonance frequency shift in the ODMR spectra, see Fig. 2. The shift of Δ=ω+1−ω−1 is approximately linear with the stress, and the gradient is about dΔ/dσ~\n17 MHz/MPa for the NV centre at a depth of d=15 nm, which is about three orders of magnitude larger than the shift of the zero-field splitting parameter D under a direct pressure8. We remark that the stress-induced frequency shift of the NV centre can be further improved by optimizing the dimension of the piezomagnetic film and the depth of the NV centre. For example, we find that dΔ/dσ~\n36 MHz/MPa for d=10 nm, although the magnetic noise may also increase (see Supplementary Note 3). Besides the frequency gradient dΔ/dσ, the shot-noise-limited sensitivity also depends on the coherence time of the NV centre, which we will determine in the following section.
Noise power spectrum and measurement sensitivity
At a finite temperature, the thermal fluctuation of the piezomagnetic film induces a fluctuating magnetic field acting on the NV-centre spin, which gives rise to the ODMR linewidth broadening and thereby the decay of spin coherence. Using a pulsed sensing scheme, the NV spin is first prepared into a coherent superposition state by applying a microwave driving field Hd=Ω[cos(ω+1t)|+1›‹0|+cos(ω−1t)|−1›‹0|] +h.c. for a pulse duration tπ/2=π/Ω. We remark that by choosing the spin sublevels ms=−1 and +1 to encode quantum coherence, the temperature instability5,6,7 and electric noise will shift the energies of these two levels equally (Supplementary Equation 10 in Supplementary Note 4), and thus would not cause dephasing. The ideal evolution of the NV spin is . The electronic spin coherence time of an NV centre in isotopically engineered diamond can exceed 100 μs even at room temperature25,26; thus, the dominant magnetic noise in the present model arises from the fluctuation in the magnetic material (see Supplementary Note 4 for the analysis of other noises). By solving the stochastic Langevin equation including the thermal random field to the simulation of micromagnetic dynamics, we obtain the time correlation function Rζ(t)=‹Bζ(t)Bζ(0)›−‹Bζ(t)›‹Bζ(0)› of the magnetic noise from the Terfenol-D film, where ζ=x, y, z represents the field component along the x, y, z directions. In Fig. 3a,b, we plot the noise correlation functions, which show a fast decay on a timescale on the order of nanoseconds. The noise correlation functions can be fitted by Rζ(t)=Rζ(0)e−tξcos (ω0t), where ξ−1 characterizes the the correlation time of the magnetic noise, and it depends on the Gilbert damping constant αd of the piezomagnetic material (for Terfenol-D, the value of αd~\n0.1 (ref. 27), see Supplementary Note 5 for detailed discussion). The oscillatory feature of the noise correlation function is related to the precession of magnetic domains, whose frequency ω0 is dependent on the applied external magnetic field B0 and the demagnetization field among magnetic domains. The latter is determined by the parameters of the piezomagnetic material. The noise power spectrum is obtained from the Fourier transformation of the correlation function as
As the correlation time of the magnetic noise is much shorter than the coherence time of the NV spin, we can describe the NV spin dynamics under the environmental noise by a quantum master equation as derived in Supplementary Note 6. The fluorescence measurement of the state ms=0 population after an interrogation (that is, free evolution) time ta and a subsequent (π/2) pulse is
where the dephasing and relaxation coefficients χ||(t)=exp (−4t Sz(0)), χ⊥(t)=exp(−t S⊥(ω−1)/2) are determined by the magnetic noise power spectrum, where S⊥=(Sx+Sy)/2.
To obtain the optimal measurement sensitivity, we choose ta such that the signal has the maximum absolute slope with respect to the stress, namely |sin(taΔ)|=1. The shot-noise-limit sensitivity for the measurement of stress within a total experiment time T is obtained as follows
where tp is the preparation and readout time in each experiment run, accounts for the NV state initialization, readout efficiency and the signal contrast. The value of can exceed 0.3 by improving the collection efficiency using a plasmonic or optical waveguide17,28. The quantity ησ,,T characterizes the smallest change in the stress that can be identified within the total experiment time T. By choosing a larger interrogation time of ta, it is possible to improve the slope of the signal (see equation (4)) while in the mean time the NV-centre spin would suffer more from the magnetic noise. This implies an optimal interrogation time to achieve the best sensitivity. In Fig. 3c,d, we plot the simulated fluorescence signal at two different interrogation times, and the shot-noise-limit sensitivity for the measurement of stress (force), which can achieve and , respectively. Such a sensitivity is close to the one achieved by atomic force spectroscopy (see, for example, ref. 29), while requiring less stringent operating environment and being more robust. We remark that the sensitivity may be further improved by optimizing the dimension of the hybrid system and using an array of NV centres.
Electric field and temperature measurement
Although the NV centre has been used to measure electric field4 and temperature5,6,7, the sensitivity is largely limited by the efficiency of the direct coupling between the NV-centre spin and electric field (temperature). Here we propose to combine the hybrid diamond and piezomagnetic device with a piezoelectric element island on a substrate to measure electric field (see Supplementary Note 7). An electric field E induces a strain ε=εe·E of the piezoelectric element island, which generates a stress σ=ε·Y acting on the attached piezomagnetic layer where Y denotes the Young’s modulus of the piezoelectric material. For a piezoelectric island that has large piezoelectric constants, such as Pb[ZrxTi1−x]O3 and simultaneously low optical absorption30, the electric-field-induced strain can be as large as εe~\n0.0002 (MV/m)−1, the corresponding Young’s modulus is Y~\n105 MPa (ref. 31). The mediated coupling coefficient between the NV spin (at a depth of 15 nm) and the electric field is thus dΔ/dE=(εe·Y)(dΔ/dσ)=34 kHz (V/cm)−1, as compared with the direct coupling coefficient of ~\n10 Hz (V/cm)−1 (ref. 18). The sensitivity for the measurement of electric field thus reaches ηe=ησ/(εe·Y). As the coupling between the NV centre and the electric field is rather weak compared with the magnetic noise, and because the charge noise decreases rapidly with the distance from the NV centre to the piezoactive layer32, at a distance of, for example, 10–15 nm, it is at least three orders of magnitude smaller than the magnetic noise effect considered above (see Supplementary Note 4). In the mean time, the charge noise in an isotopically engineered diamond25 is also negligible. The sensitivity for the pressure measurement therefore implies the sensitivity for the measurement of electric field as high as , which significantly exceeds the sensitivity of by a pure diamond NV sensor4. This sensitivity would allow for the detection of the electric field produced by a single elementary charge at a distance from the NV-spin sensor of ~\n8 μm in around 1s, going beyond the single electron spin detection capacity of magnetic resonance force microscopy33, and thus opens the possibility of remote sensing of a single charge under ambient conditions. We remark that a similar sensitivity can be achieved in an NV-diamond electrically induced transparency device, which however requires an ensemble of NVs and liquid–nitrogen temperature34.
In a similar way, it is possible to combine the hybrid diamond and piezomagnetic device with a thermally sensitive element island on a substrate to measure temperature (see Supplementary Note 7). A change of temperature induces the expansion of the thermally sensitive element island as characterized by its thermal expansion constant εT, which can be as high as 2.3 × 10−5K−1 (aluminium). The thermal expansion produces a stress acting on the attached piezomagnetic layer as σT=(εT·T)Y, where Y denotes the Young’s modulus of the thermally sensitive material, for example, Y=7 × 104MPa (aluminium). Thus, the response of the resonance frequency of an NV centre at a depth of 15 nm is dΔ/dT=(εT·Y)(dΔ/dσ)=27 MHz K−1, which represents significant improvement over the value dΔ/dT~\n74 kHz K−1 of a pure diamond NV sensor35. The sensitivity for the measurement of temperature can reach and does not require an interrogation time of millisecond5,6,7.
The proposed that diamond–piezomagnetic hybrid sensor combines high sensitivity under ambient conditions with nanometre-scale dimensions improving the sensitivity of standard diamond sensing devices by up to three orders of magnitude. The high spatial resolution is achieved, thanks to the atomic size of the NV centre and the ability for positioning it with high accuracy5,9,10,11,36,37,38 in nanoscale diamond. Special attentions to, for example, surface preparation and heating control would be necessary for an even better sensitivity. Such a device can find a wide range of applications for measuring force and pressure, electric field and temperature changes under ambient conditions, such as protein folding and DNA stretching (see Supplementary Note 8) in life science39. Furthermore, it offers a precise way to determine the short-range forces between two plates (or spheres) and may have implications for the study of fundamental quantum physics phenomena such as an accurate description of the Casimir effect40 or non-Newtonian forces (for example, Yukawa-type forces) where forces have to be measured at smallest distances (below 100 nm)41. It also provides a possible way to detect an elementary charge (electron) over a distance of micrometres. Besides its application in signal transduction and precision measurement, it may also find applications in spin and mechanical oscillator hybrid systems. A hybrid diamond–piezomagnetic (film) mechanical nanoresonator can significantly enhance the coherent coupling between its vibrational mode and NV electronic spins in diamond42,43, and thereby boost the phonon-mediated effective spin–spin interactions. This would allow fast mechanical control of spin qubits and efficient phonon cooling by an NV centre42,43,44,45,46,47 and would facilitate the generation of spin squeezing at room temperature, which can be used as a resource for magnetometry and phonon-mediated quantum information processing. The extension of the present model to arrays may find applications in tactile imaging48 and instantaneous pressure visualization by interactive electronic skin49. The proposed device may also find applications in nanoscale surface characterization, for example, in read/write heads of hard disks where the enhanced sensitivity to magnetic fields in combination with nanoscale spatial resolution may significantly increase the characterization speed (by decreasing the number of repetitive interrogation times while sustaining a readout precision criteria) which is a crucial constraint in industrial applications.
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We thank Liam McGuinness and Boris Naydenov for discussion about experiments. This work was supported by the Alexander von Humboldt Foundation, the DFG (FOR1482, FOR1493, SPP1601 and SFB TR21), the EU Integrating Projects SIQS and DIADEMS, the EU STREPs EQUAM and DIAMANT, DARPA via QUASAR, and the ERC Synergy grant BioQ. Computations were performed on the bwGRiD.
The authors declare no competing financial interests.
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Cai, J., Jelezko, F. & Plenio, M. Hybrid sensors based on colour centres in diamond and piezoactive layers. Nat Commun 5, 4065 (2014). https://doi.org/10.1038/ncomms5065
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