Spin Amplification for Magnetic Sensors Employing Crystal Defects
Marcus Schaffry, Erik M. Gauger, John J. L. Morton, Simon C. Benjamin
SSpin Amplification for Magnetic Sensors Employing Crystal Defects
Marcus Schaffry, Erik M. Gauger, John J. L. Morton,
1, 2 and Simon C. Benjamin
1, 3 Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom Clarendon Laboratory, University of Oxford, Parks Road, OX1 3PU, United Kingdom Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 (Dated: April 3, 2019)Recently there have been several theoretical and experimental studies of the prospects for magnetic fieldsensors based on crystal defects, especially nitrogen vacancy (NV) centres in diamond. Such systems couldpotentially be incorporated into an AFM-like apparatus in order to map the magnetic properties of a surface atthe single spin level. In this Letter we propose an augmented sensor consisting of an NV centre for readout andan ‘amplifier’ spin system that directly senses the local magnetic field. Our calculations show that this hybridstructure has the potential to detect magnetic moments with a sensitivity and spatial resolution far beyond thatof a simple NV centre, and indeed this may be the physical limit for sensors of this class.
A key goal for future sensor technologies is the detectionof weak magnetic fields at molecular length scales. There arenumerous potential applications in materials science, medicalscience and biology. An ambitious goal would be to detect thefield due to single nuclear spins, in order to gain direct infor-mation about the structure of a molecular complex depositedon the surface. This would require unprecedented combina-tion of sensitivity and spatial resolution.Among the most sensitive magnetic field sensing devicesare Hall sensors [1], SQUID sensors [2], force sensors [3],and potentially NV centres in diamond [4–8]. In this Letter wepropose a significant improvement of the NV centre sensor bycoupling it to an amplifying spin system (see Fig. 1). We showthat the improvement in magnetic moment sensitivity can beexpected to be of about three orders of magnitude; indeed theaugmented NV centre may have the highest in-principle per-formance of any such device. r NV r A (a) (b) FIG. 1. (Color online) (a) The sensor structures: an NV centre (redsphere) is embedded in the middle of a diamond nano-crystal thatis attached to an AFM-tip (yellow cone). The magnetic field gen-erated by local spins (green spheres) affects the dynamic of the NVcentre. The strength of the magnetic field can be inferred by mea-suring the NV centre through optical means and manipulating it withmicrowaves. (b) Amplified NV centre sensor: the surface of the nan-odiamond is decorated with an another spin system (blue sphere) thatcouples to the NV centre inside the diamond. This additional spinsystem has an amplifying effect and enormously increases the mag-netic field sensitivity of the sensor and its spatial resolution.
We begin with a general discussion of measuring an un-known magnetic field using two energy levels of a probe spin s system (2 s ∈ N ), the underlying principle of the NV cen-tre sensor. Let the probe spin be governed by the ZeemanHamiltonian H = − µ Prb ( B + B ) S z , where µ Prb is the mag-netic moment of the probe spin, S z is the usual spin operator, B is a known external field applied in the z -direction, and B corresponds to the magnetic field (also in the z -direction) thatwe wish to estimate. Consider two eigenstates | (cid:105) and | (cid:105) of H whose z-projections differ by an integer m . The field es-timation then proceeds as follows: we start by creating thesuperposition | + (cid:105) [here |±(cid:105) = / √ ( | (cid:105) ± | (cid:105) ) ], for exampleby using a suitable microwave or radio-frequency pulse se-quence. This state evolves in time to | ψ ( t ) (cid:105) = / √ ( | (cid:105) + exp ( im µ Prb ( B + B ) t / ¯ h ) | (cid:105) ) and therefore by successive mea-surements of the spin, we can infer the strength of the mag-netic field. In any real-world experiment the state | ψ ( t ) (cid:105) willalso be affected by decoherence processes, typically dephas-ing is predominant [9]. In this case, the coherence between | (cid:105) and | (cid:105) decays as exp ( − γ ( t )) for a positive non-decreasingfunction γ ( t ) . The time evolution of the two level system’sdensity matrix ρ ( t ) is then fully described by ρ ( t ) = (cid:32) e i m µ Prb ( B + B ) ¯ h t − γ ( t ) e − i m µ Prb ( B + B ) ¯ h t − γ ( t ) (cid:33) . (1)The uncertainty of estimating B is limited by the quantumCram´er-Rao bound (CRB), i.e. even for an ideal system andperfect measurements the uncertainty in δ B cannot becomesmaller than b CR . The CRB is given by the inverse squareroot of the quantum Fisher information F with respect to B [10], yielding the following inequality δ B ≥ b CR = √ F = e γ ( τ ) ¯ hm τ | µ Prb | , (2)where τ is the duration for which the spin has experienced themagnetic field. Repeating the measurement many times givesa further reduction in the uncertainty of the unknown field.Assuming the preparation of the | + (cid:105) state takes the time t p ,the lower bound for the uncertainty δ B after N = T / ( τ + t p ) repetitions within a total time T is given by b CR = √ NF = e γ ( τ ) ¯ h √ τ + t p √ T m τ | µ Prb | . (3) a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r As an example, we set γ ( t ) = tT . Minimizing the CRB withrespect to τ yields the optimal sensing time for each run of theprotocol: τ ∗ = (cid:18) T − t p + (cid:113) T + t p ( T + t p ) (cid:19) . (4)It is easily seen that τ ∗ ≈ T in the limit of weak dephasing( T (cid:29) t p ), in contrast to τ ∗ ≈ T when the dephasing time iscomparatively short, T (cid:28) t p .Since a magnetic field smaller than b CR cannot be resolved,the CRB provides a natural limit of the sensitivity of a mag-netic field sensor. Interestingly, this limit does not depend onthe actual value of B . However, a na¨ıve intuition is that theproximity of the probe spin and the sample will be an impor-tant characteristic of a practicable sensor, since it determinesthe strength of the measured magnetic field as well as the sen-sor’s spatial resolution. To quantify this intuition, one can askthe question: how long does it take until the sensor spin de-tects the magnetic moment associated with an electron or anuclear spin? In the following, we introduce a magnetic mo-ment sensitivity (in units of T / √ Hz) as a suitable figure ofmerit for this question: the magnetic moment sensitivity S of asensor is given by the number of proton magnetons that can beresolved within the time window √ T for a distance r betweenthe probe spin and the sample. The dipole field originatingfrom a proton with magnetic moment µ p at the position ofthe probe spin is of order b = µ π µ p r (where µ is the vacuumpermeability), so we obtain S = min τ ( b CR ) √ Tb . (5)For example, once again assuming γ ( t ) = tT and employingEq. (4) gives the following magnetic moment sensitivity S = π ¯ h µ µ p e τ ∗ T (cid:112) τ ∗ + t p r m τ ∗ | µ Prb | . (6)We shall now briefly discuss the example of field sensingwith an NV centre before describing the inherent limitationsof this approach and presenting our proposal for improvingthe characteristics of this class of device. An NV centre indiamond possesses spin 1 with the three levels | (cid:105) and | ± (cid:105) .The levels | ± (cid:105) are degenerate in the absence of an exter-nal field and shifted by a zero-field-splitting (ZFS) of about2 .
87 GHz. For the purpose of sensing the magnetic field, vari-ous two level manifolds can be used, for example {| (cid:105) , | ± (cid:105)} or { / √ ( | (cid:105) + | − (cid:105) ) , ( / √ ( | (cid:105) − | − (cid:105) ) } [5].Figure 2 shows the magnetic moment sensitivity of variousclasses of sensors. Clearly, NV centres hold the promise ofa high field sensitivity combined with a fairly small probe tosample separation. However, in order to further improve thespatial resolution and obtain an even better magnetic momentsensitivity (according to the definition in Eq. (5)) the NV cen-tre must be brought even closer to the sample. Due to thecubic dependence of S on r even a modest reduction in the Magnetic field sensitivity T yp i ca l ti p - s a m l e s e p a r a ti on ( m ) -12 -9 -6 -3 -9 -8 -7 -6 -5 SQUID sensors Hall sensorsForcesensors NV centresensors s i ng l e p r o t on s p i n s i ng l e e l ec t r on s p i n Amplified NVcentre sensors I m p r ov i ng m a gn e ti c m o m e n t s e n s iti v it y FIG. 2. (Color online) Performance of various state-of-the-art andproposed magnetic field sensors (figure adapted from Ref. [11]).The plot shows magnetic sensitivity (per √ Hz) (horizontal) versusa typical tip-sample separation. The data points are reported sensi-tivities from experiments: force sensors [12–14], Hall sensors [15],SQUID sensors [2, 16, 17], and NV centre sensors [4]. The diagonallines sketch the boundaries for the magnetic moment sensitivity (seeEq. (5)) required to detect 1 , , , or 10 protons (solid black)and a Bohr magneton (blue dash dotted) within one second. The po-tential of our proposed pre-amplified NV centre is illustrated by thedark green ellipse. separation leads to a large gain, and for this reason NV cen-tres have been embedded in smaller and smaller nano-crystals[4, 6]. However, the size of the diamond crystal surround-ing the NV centre cannot become arbitrarily small withoutseverely affecting the remarkable coherence time (and thusthe sensitivity) of the NV centre. In C isotopically enrichedbulk diamond the room temperature coherence time can be aslong as 1 . . . µ s [20]. Smaller crystals with a diameter of only 5 nmhave been studied in Ref. [21], however the properties of en-closed NV centres are even further degraded in this case. Aminiaturisation of the crystal hence leads to a reduced fieldsensitivity in exchange for a smaller separation between theprobe and the sample.We propose to overcome this trade-off situation by bringingthe NV centre ”effectively” closer to the sample without re-ducing the size of the nanocrystal. This is achieved by attach-ing an additional spin system on the surface of a nanocrystalwhich relays the sample magnetic field to the NV centre (seeFig. 1). In the simplest implementation, a single electron spinserves as the amplifier. Of course, further improvements willbe possible by using higher spin systems such as N@C [22]or a molecular magnet [23, 24]. It may even be possible tomake use of more sensitive entangled states with suitably en-gineered molecular systems [25, 26].For simplicity, we consider an electron spin as the ampli-fier in the following. The NV centre and the amplifier spinare dipolar coupled, and we assume the vector connecting thetwo spins is aligned with the z -axis. Both spins experience an(known) homogeneous external magnetic field B = ( , , B ) and a small magnetic field whose z -component B at the po-sition of the amplifier is to be measured. In comparison tothe amplifier spin the NV centre experiences a weaker samplefield cB where the factor c < H = − µ NV ( B + cB ) S NV , z + D NV S , z − µ A ( B + B ) S A , z + d ( S NV , z S A , z − S NV , x S A , x − S NV , y S A , y ) , (7)where d = µ π γ NV γ A ¯ h ∆ is the dipolar coupling constant; γ NV and γ A are the gyromagnetic ratios of the spins, ∆ is the distancebetween the NV centre and the amplifier spin, and D NV de-notes the NV centre’s ZFS constant. Our protocol assumesthat the influence of cB on the NV centre is much smaller thanits coupling to the amplifier spin; this will certainly be justi-fied if the amplifier is an electron spin and the sample field isdue to nuclear spins. In addition we assume that flip-flops be-tween the spins are heavily suppressed and can be neglected,for example because D NV represents the largest energy in thesystem, yielding the following effective Hamiltonian H ≈ − µ NV B S NV , z + D NV S , z − µ A ( B + B ) S A , z + dS NV , z S A , z , (8)whose level structure is schematically depicted in Fig. 3. NV centre NV + amplifier spin (b)(a) E A FIG. 3. (Color online) (a) Level structure of an NV centre. Theground state is a spin triplet with zero field splitting and the excitedstate has a manifold of levels. Following excitation with green lightthe dominant decay is spin preserving. However, spin is not alwaysconserved due to an additional decay path to | (cid:105) via a metastablestate, so that continuous excitation eventually results in a polarisedstate. (b) Eigenspectrum of H for an amplifier spin system. The field sensing protocol now proceeds as follows. Ini-tially, the system starts in a completely mixed spin state, ,so that it needs to be polarised with laser and microwavepulses. As a first step we polarise the NV centre by illumi-nating it with green light, after about 1 µ s the NV centre willhave relaxed to the state | (cid:105) with high probability [5], leavingthe combined system in the state | (cid:105)(cid:104) | ⊗ . Next, we ap-ply a selective microwave π -pulse on the transition | (cid:105)| ↓(cid:105) ↔| (cid:105)| ↓(cid:105) . For a distance ∆ =
10 nm between the NV centre and the amplifier, this transition is split from the neighbouring | (cid:105)| ↑(cid:105) ↔ | (cid:105)| ↑(cid:105) transition by 2 d = µ g NV g A µ B π ∆ ¯ h ≈ .
65 Mrad / s(i.e. 104 kHz). A highly selective π -pulse is beneficial for thepresently discussed protocol. For this reason we now estab-lish the conditions under which imperfections in the pulse se-lectivity can be considered negligible. Assuming Lorentzianline broadening of both transitions with a FWHM of 2 / T , NV ( T , NV is the coherence time of the NV centre), a detailed anal-ysis [27] shows that a linewidth overlap of up to 10% meetsthis requirement and translates into T , NV > µ s. The prod-uct of pulse duration and frequency passband T × ∆ ω is typ-ically between 2 and 5 [28]. Taking T = / ∆ ω and allowingthe passband to overlap with the area of unwanted transition’sLorentzian by at most 5% (allowing us to neglect imperfectpulse selectivity), we thus obtain τ π = ( d − cot ( . π ) / T , NV ) for the duration of the desired selective π -pulse (reducing to τ π ≈ / d in the limit of long T , NV ). Finally, the state of theNV centre is measured by exciting it once more with greenlight and detecting the spin-dependent photoluminescence.Importantly, this puts the NV centre back into the | (cid:105) state,leaving the system in one of the well-defined states | (cid:105)| ↓(cid:105) or | (cid:105)| ↑(cid:105) .Without loss of generality we assume the system is inthe state | (cid:105)| ↓(cid:105) , ready for the magnetic field estimationthrough several repetitions of the following sensing cycle: Amicrowave π -pulse creates the superposition 1 / √ ( | ↓(cid:105) + | ↑(cid:105) ) . It is safe to assume that the amplifier spin can be ro-tated fast and with high fidelity as long as the unknown fieldcorresponds to the smallest energy scale of the system, as canbe accomplished with a modest external field. After a sens-ing time τ , the amplifier spin acquires a relative phase pro-portional to B . The phase is first mapped onto a populationdifference between | ↓(cid:105) and | ↑(cid:105) with another π -pulse, andthen entangled with the NV centre spin state with a selective π -pulse in the same way as in the initialisation process. TheNV centre spin state is now read out using spin-dependentphotoluminescence detection (also initialising it for the nextsensing cycle).Having described the protocol, we now benchmark the sen-sitivity improvement of the amplified system over a conven-tional single NV centre sensor. Obviously, the performanceof both sensors depends on the decoherence model, which weassume to be fully characterised by γ ( t ) . Different forms of γ ( t ) corresponding to different predominant dephasing mech-anism occur, however, typically γ ( t ) can be written as ( t / T ) n for n = , | (cid:105) and | (cid:105) of anNV centre in the middle of a nanocrystal with radius 10 nmand a control and measurement duration of t p , NV ≈ µ s percycle. In contrast, using the electron spin on the surface ofthe augmented sensor entails an additional control overheadof about τ π , i.e. t p , A ≈ t p , NV + τ π . Figure 4 shows the ratio S NV / S A as a function of the amplifier coherence time T , A for several illustrative values of T , NV . The red regions ofthe inset show the parameter space for which our protocolis not necessarily advantageous, either due to excessive line-broadening preventing a highly selective π -pulse on the tran-sition | (cid:105)| ↓(cid:105) ↔ | (cid:105)| ↓(cid:105) , or because T , NV is so much larger than T , A that it more than compensates the benefit of the amplifierspin. Figure 4 impressively demonstrates that the magneticmoment sensitivity can be enhanced by up to three orders ofmagnitude for T , NV ≈ T , A with realistic parameters. Fur-ther improvements may be feasible by substituting the elec-tron spin amplifier with a high spin system or molecular mag-net.
20 1000 806040 500100015002000250030000 FIG. 4. (Color online) Ratio S NV / S A as a function of the amplifierspin coherence time T , A for T , NV = µ s (black), 30 µ s (red), and15 µ s (blue). This ratio gives the factor by which the amplified sen-sor outperforms a conventional NV centre sensor. Here γ ( t ) = t / T ,but we note that the curves for γ ( t ) = ( t / T ) , look almost identi-cal. Other parameters: r A = , r NV =
11 nm , ∆ =
10 nm , t p , NV = µ s , t p , A = µ s + τ π . Inset: excessive line-broadening or a signifi-cant mismatch in the coherence times prevent a sensitivity enhance-ment in the red region (see text). We now turn to another important consideration: Given asensor that can resolve magnetic fields at the level of the singleBohr or even nuclear magneton, its spatial resolution is vitallyimportant for many applications such as mapping the con-stituent molecules and nuclei of complex organic molecules.Unsurprisingly, it is highly beneficial to be able to move veryclose to the sample in order to reliably distinguish individualmagnetic moments. Figure 5 shows the z -component (i.e. themeasured component) of the magnetic field for a regular ar-ray of protons at two different heights above the surface. Ifthe probe sample separation is of order the distance betweenindividual dipoles, the nuclear spins can be unambiguouslyresolved (given the required field sensitivity). On the otherhand, the field is almost indistinguishable from that of a sin-gle stronger dipole if the sensor is scanning at z =
10 nm, evenwith the ability to sense small magnetic fields of order 1 nT.In summary, we have proposed a practical way of enhanc-ing NV centre based magnetic field sensor with a pre-amplifiersystem. Such an augmented sensor possesses a magnetic mo-ment sensitivity that is capable of resolving individual pro-tons, both in terms of their field strength as well as spatiallyon an atomic scale. The sensor consists of two essential parts:the NV centre for optical readout coupled to an entirely spin-based system for the magnetic field detection in close proxim-
FIG. 5. (Color online) The z -component of the magnetic field of aregular array of protons calculated in two planes z = z =
10 nm. The 36 dipoles with the magnetic moment of a proton areplaced in the z = z -direction as indicatedby the red arrows. The field range of the contour plots spans 43 . z =
10 nm plane and 2 . µ T for z = z = ity to the sample. Despite this partitioning of tasks, the fun-damental complexity of this improved sensor design is similarto that of existing proposals and implementations. Acknowledgements -
We thank Brendon Lovett andStephanie Simmons for fruitful discussions. This work wassupported by the National Research Foundation and Ministryof Education, Singapore, the DAAD, the Royal Society, StJohn’s College and Linacre College, Oxford. [1] E. Ramsden,