The development of magnetic sensors simultaneously satisfying high magnetic sensitivity and high spatial resolution becomes more important in a wide range of fields including solid-state physics and life science. The nitrogen-vacancy (NV) center in diamond is a promising candidate to realize nanometer-scale magnetometry due to its excellent spin coherence properties, magnetic field sensitivity, atomic-scale size and versatile operation condition. Recent experiments successfully demonstrate the use of NV center in various sensing and imaging applications. In this chapter, we review the basic sensing mechanisms of the NV center and introduce imaging applications based on scanning magnetometry and wide field-of-view optics.
- magnetic sensor
- diamond NV center
- quantum sensing
- scanning magnetometry
- wide field-of-view optics
Understanding magnetic properties at the microscopic level plays an important role in the development of modern science and technology [1, 2]. For instance, writing and reading information using nanometer size magnetic bits is the heart of massive data storage indispensable in the modern information technology . Magnetic resonance imaging (MRI) which is an important medical tool of imaging the inner structures of human body is also based on sensing the magnetic response of minuscule protons with respect to radio frequency (RF) electromagnetic waves . For fundamental research, on the other hand, studying magnetic phases and spin textures at the nanometer scale are one of the hottest topics in solid-state physics due to the recent discovery of exotic materials and topological phases [3, 4, 5]. Therefore, it is not too much to say that the continuous advances in modern science and technology strongly reply on the precise sensing and control of magnetism at the atomic level.
The paradigm of modern science and technology seems to shift from charge-based devices to spin-based systems. Nonetheless studying spins is a lot more difficult than electric charges mainly due to the lack of sensitive measurement techniques of magnetic field. For instance, the size of magnetic bits used in MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor) and STT-MRAM (Spin-Transfer Torque Magnetic Random-Access Memory) are less than 10 nanometers and eventually reaches at the level of single spins requiring sensitive detection of individual spins with high spatial resolution [6, 7]. However, detecting single electron spin takes more than 13 hours even with the best magnetometer , while sensing single electron charge takes only 1 picosecond [9, 10]. This motivates to develop new magnetic sensors with high magnetic field sensitivity and high spatial resolution.
Existing magnetometers are insufficient to satisfy both requirements especially when trying to measure minute magnetic fields at the length scale of 100 nm or below. For instance, SQUID (Superconducting Quantum Interference Devices), atomic vapor cell and Hall bar are very sensitive magnetometers but their spatial resolutions are typically limited to tens of micrometers [11, 12]. On the other hand, scanning probe type tools such as SP-STM (Spin-Polarized Scanning Tunneling Microscope) and MFM (Magnetic Force Microscope) exhibit very high spatial resolution but their sensitivity is relatively low and not quantitatively defined [13, 14]. Moreover, magnetic films coated at the tips may produce unwanted stray field affecting the magnetic samples to be measured.
Here, we introduce a novel magnetometer enabling non-invasive, extremely sensitive magnetic sensing and imaging at the nanometer scale. It is based on diamond NV (nitrogen-vacancy) center which is an atomic size point defect in the diamond crystal providing high spatial resolution. It is also a spin qubit (i.e. quantum bit) possessing remarkable magnetic and quantum properties satisfying high field sensitivity [15, 16]. Since it can also operate over a wide range of temperature from room temperature down to cryogenic temperatures and is chemically inert and non-toxic, the NV center already has been applied in various experiments including magnetic imaging of solid-state materials and biomedical samples [17, 18]. In this chapter, we will discuss basic working principles of diamond NV centers (Section 2) and their sensing mechanisms (Section 3). Furthermore, we will provide two examples of imaging applications; scanning probe type imaging (Section 4.1) and wide field-of-view optical imaging (Section 4.2) ( Figure 1 ).
2. Background of diamond NV center
The diamond NV center is a hetero-molecular defect in a diamond crystal consisting of a substitutional nitrogen defect combined with an adjacent carbon vacancy [15, 16] ( Figure 2a ). It is a color center as it absorbs photons in the visible range of wavelength (e.g. 532 nm) and emits photons of a broad range of wavelength (e.g. 632–800 nm). The NV center can exist in a natural diamond, but it can be created in more controllable fashion, for instance, by implanting nitrogen ions into the diamond. Subsequent high temperature annealing (e.g. at 800°C) results in the thermal migration of carbon vacancies and NV centers are formed once the vacancies meet the implanted nitrogen impurities. The density and location of NV centers in diamond are well controlled with various techniques. Figure 2b shows an example of precise positioning of NV centers less than a few hundred of nanometers uncertainty .
When negatively charged, the NV center has total six electrons (i.e. three electrons from three carbons, two electrons from the substitutional nitrogen and one electron from the diamond lattice). Four of them form pairs and the remaining two unpaired electrons make spin triplet states (i.e. S = 1) in the ground energy level. The spin triplet states are split into ms = 0 and ms = 1 whose separation is about 2.9 GHz at room temperature due to the crystal field and spin–spin interaction [17, 18] (this is called zero-field splitting). As shown in Figure 3a , the degenerated ms = 1 states are split further if there is non-zero magnetic field along the NV crystal axis (i.e. the quantized axis of NV spin). Sensing magnetic field (i.e. magnetic field component along the NV axis) is realized by measuring the amount of Zeeman splitting [17, 18] (e.g. 5.6 MHz splitting per 1 Gauss field).
The optical response of the NV center varies depending on its spin states. When it is in the ms = 0 state, almost 100% cycling transition occurs upon optical pumping. On the other hand, for the case of ms = 1 states, 10–30% of the excited electrons undergo intersystem crossing (ISC) to the spin singlet states and relax into the ms = 0 ground state. This dark transition results in the reduction of the number of emitted photons relative to the ms = 0 state. Furthermore, the transition via the shelving states produces spin flip from the ms = 1 states to the ms = 0 state. The spin-sensitive fluorescence and spin-flip transition allow optical readout of the spin states as well as optical initialization of the qubit state [15, 16, 17, 18].
The energy levels of NV center are located well within the bandgap of diamond (i.e. 5.3 eV) making it effectively isolated from the hosting material and enabling to preserve its intrinsic quantum properties. Thus, the NV center has exceptionally long spin coherence times even at room temperature  (e.g. T2 > 1 ms). In addition, the NV spin is highly sensitive to various fields, including temperature, magnetic, electric and strain fields . For example, the magnetic field sensitivity for a single NV center is on the order of which is about 105 smaller than typical earth magnetic field [16, 17, 18]. Higher sensitivity can be also possible using either NV ensembles  (e.g. <) or advanced sensing protocols . The high magnetic field sensitivity is one of the key ingredients of realizing novel magnetometer introduced in this chapter.
3. Sensing mechanism
In this section, we will discuss various sensing methods specifically designed to study magnetism in different spectral regimes from static spin distributions to high frequency magnetic excitations. Sections 3.1 and 3.2 elucidate basic mechanisms used for sensing static and dynamic magnetic fields. The section ends with a brief outlook on advanced sensing techniques (Section 3.3).
3.1 Sensing dc magnetic field
Probing static field from magnetic textures or current flow in transport devices is an important capability to study microscopic magnetism in condensed matter physics. The diamond NV centers already have been used to study a diverse set of magnetic systems including skyrmions , superconducting vortices [32, 33], domain walls , magnetic nanowires , and steady-state current distributions in graphene . Various experimental methods have been implemented to detect static dc magnetic field and we will examine the two most common protocols; continuous wave electron spin resonance (CW-ESR) and Ramsey interferometry.
As seen in Figure 3b , the ground spin states of NV center are subject to change by external magnetic field via the Zeeman effect. Upon continuous illumination of the pumping laser and gigahertz microwave photons, ESR transitions of ms = 0 ms = −1 and ms = 0 ms = 1 occurs. The ESR signals appear as negative peaks in the photoluminescence (PL) measurement due to the dark transitions associated with the ms = 1 states. Since the ESR spectrum is obtained by optical means, this method is often called as optically detected magnetic resonance (ODMR).
The amount of dc magnetic field is extracted from the ESR splitting of , where is the frequency difference of the two transitions, is the gyromagnetic ratio (i.e. 2.8 MHz/G) and is the magnetic field along the NV axis. The CW-ESR signal can be analyzed by fitting the spectrum with a Lorentzian peak,
where is the photon count rate, is the optical contrast between the spin states, is the ESR central frequency, and is the full width half maximum of the resonance. The sensitivity of dc magnetic field based on this method is determined by the minimum resolvable frequency shift ( Figure 4 ) which is defined as , where is total number of photons during the measurement time, T, i.e. . For the shot noise limited photon measurement, and the minimum detectable magnetic field and the sensitivity become .
As seen in Eq. (2), the sensitivity is limited by the ESR linewidth and in principle it can be as narrow as the inverse of NV’s inhomogeneous dephasing time, . However, practical linewidth suffers from the power broadening due to continuous laser and microwave excitation. Therefore, typical magnetic sensitivity based on CW-ESR is limited to on the order of .
The power broadening problem can be avoided by using pulsed laser and microwave photons. Ramsey interferometry is one of the basic pulse techniques used to measure the free induction decay of a spin qubit. Figure 5a shows Ramsey pulse sequences used in the NV measurement. The two laser pulses are used for optical initialization and readout while the two microwave pulses are used to make qubit superposition state and to project it back to the initial state. The basic idea of Ramsey interferometry is very similar with Michelson interferometry. The first laser pulse polarizes the NV center to the ms = 0 state and the subsequent microwave pulse rotates the spin into the equal superposition of ms = 0 and ms = +1 (or ms = −1) state. This works as a 50:50 beam splitter used in the Michelson interferometry experiment. Under external magnetic field, the two spin states evolve together but with different phases each other and the amount of accumulated phase depends on the magnitude of dc field. If the microwave frequency is detuned from the qubit energy by , the Ramsey signal oscillates at the frequency of and is written as
where is the free induction time. Shift in the oscillation frequency gives the information about static magnetic field and the resolvable frequency shift is now limited by . The minimum detectable magnetic field and the sensitivity are written as.
where is the single shot readout time (e.g. a few hundreds of nanoseconds) and is the total number of measurement cycles . Based on this method, the dc field sensitivity from a single NV center can be as high as for .
Figure 5b and c shows an example of the Ramsey measurement. The ESR resonance exhibits three hyperfine structures due to the dipole-dipole interaction between NV electron spin and 14N nuclear spin (I = 1). Therefore, the beatings of three oscillations appear in the Ramsey signal ( Figure 5c ) which is defined as,
where are the detunings of three hyperfine levels and are the phase offsets. The fast Fourier transformation (FFT) of the signal also reveals three frequencies (subset in Figure 5c ) whose shifts are used to probe static field.
3.2 Sensing ac magnetic field
The NV center can also detect ac magnetic field up to gigahertz. The large bandwidth sensing capability is important to study spin dynamics in solid-state systems and biological samples. For example, nuclear spin precession in biomolecules occurs at 100 kHz–MHz regime while spin excitations or charge fluctuations in solids happen at higher frequency of MHz–GHz . Various sensing methods with its detection bandwidth are listed in Figure 6 and, in this section, we will explain two basic dynamical decoupling methods such as spin Hahn echo and Carr-Purcell-Meiboom-Gill (CPMG).
Spin Hahn echo measurement consists of a similar pulse sequence as Ramsey interferometry but having an extra pulse in the middle of the sequence ( Figure 7a ). This additional pulse flips the sign of accumulated phase during the free induction evolution resulting in the cancelation of dc and low frequency magnetic field. When the pulse duration matches to the period of ac magnetic field, however, the phase survives and continues to be accumulated. In this way, one can selectively probe ac magnetic field.
This can be better viewed with the Bloch sphere representation shown in Figure 7b . In the rotating frame at the qubit frequency, the spin rotates around the equator by an angle, , which is determined by the free precession time, , between and pulses (② and ⑤ in Figure 7b ). The angles from the first (②) and the second period (⑤) of the echo pulse are written as,
where is the amplitude of ac magnetic field and is a phase offset. And their difference is
This difference determines the projected probability of the qubit and becomes non-zero if the sign of flips before and after thepulse. The probabilities of the and states after a single echo sequence are obtained as,
Finally, an average probability of the qubit state after repeated Hahn echo cycles (but with random phase offset,) can be expressed as the first order Bessel function,
Figure 8 shows an example of the spin Hahn echo measurement as a function of the free precession time, , and the ac field strength, . For the experiment, external field at 500 kHz frequency is applied with a wire. Compared to a monotonic exponential decay in Figure 8a , the first order Bessel functions discussed in Eq. (9) are clearly observed for the cases of non-zero ac fields ( Figure 8b and c ). The field sensitivity can be determined from the maximum change of the PL signal with respect to . For instance, Figure 8d shows the normalized PL as a function of measured at the time, . The minimum detectable field (or sensitivity) is obtained from the ratio of the maximum slope over the noise level in the PL signal (shot noise limited in this measurement) which is at 500 kHz.
By adding more periodic microwave pulses in the sequence, one can extend the spin coherence time and realize improved sensitivity. For example, CPMG sequence utilizes n pairs of two pulses separated by ( Figure 9 ) which prolong the spin coherence by . The axis of pulse can be also alternated between x and y in the Bloch sphere which can mitigate potential pulse error. Such dynamical decoupling sequences are called XY4 or XY8.
The ac field sensitivity is similar as the dc field sensitivity in Eq. (4) but now the intrinsic spin dephasing time, , limits the sensitivity i.e. . The ac field sensitivity based on as a single NV center can be as high as for .
3.3 Current limitations and advanced sensing protocols
In general, the bandwidth of dynamical decoupling methods is limited to ∼ 10 MHz ( Figure 6 ). Higher frequency ac field (∼GHz) can be measured by relaxometry [18, 30]. Dynamic field fluctuation around the qubit frequency can directly affect the qubit relaxation time which can be measured by the relaxometry technique. For instance, Figure 10 shows that gigahertz spin fluctuations of Gd3+ ions result in much faster relaxation of the qubit .
In terms of the field sensitivity, higher sensitivity can be realized using either ensembles of NV center or advanced sensing protocols. The former is possible since the sensitivity is proportional to where is the number of NV centers. For instance, sub-picotesla sensitivity has been demonstrated based on an ensemble of NV centers . The latter relays on advanced sensing methods utilizing various quantum techniques such as sensing assisted by entanglement or auxiliary qubits . For example, nuclear spins typically exhibit 1000 times longer coherence time compared to electron spins such NV center. By using a nuclear spin as an auxiliary qubit via entanglement with the NV electron spin, one can realize extended coherence time and obtain enhanced sensitivity. This method is called quantum memory and an example is shown in Figure 11 .
4. Imaging applications
In this section, we will introduce magnetic imaging techniques based on the diamond NV center. The combined properties of atomic-scale size and high field sensitivity make the NV center as a novel imaging tool to study nanoscale magnetism in the field of condensed matter physics and biology. Multiple imaging techniques have been implemented in several experiments and we will discuss two examples of the diamond imaging techniques; scanning magnetometry and wide field-of-view optics.
4.1 Scanning magnetometry for solid-state systems
Scanning probe microscopy (SPM) is a versatile tool to image sample surface with high spatial resolution. The probe tip can scan over the surface with nanometer step size while maintaining vertical distance with feedback techniques. Scanning tunneling microscope (STM) and atomic force microscope (AFM) are the most common SPM methods used in various experiments. In NV-based imaging applications, different geometries of SPM are possible depending on the position of NV center either on tip or on surface. For example, magnetic molecules that are attached at the end of AFM tip are scanned over a bulk diamond surface where NV centers are located underneath the surface  (e.g. 5–20 nm). On the other hand, the NV center can be positioned at the apex of AFM tip and is scanned over magnetic samples [31, 32, 33, 34]. There are several benefits of the former geometry. A simple SPM design is possible and there are no needs for complicated diamond fabrication. Moreover, one can use NV centers in a bulk diamond which typically possess good coherence properties. However, this method is not suitable to image large area of sample since it is not easy to be prepared on the tip. Therefore, the geometry of NV-on-tip is more suitable to study solid-state materials.
Figure 12a shows a schematic of the NV-on-tip geometry. A fabricated diamond probe with pillar structures or a diamond nano-particle is glued at the end of an AFM tuning fork. An objective lens focuses on the NV center at the tip apex for the optical excitation and readout. A scanning stage maneuvers the sample in three dimensional directions with nanometer step size. In this way, the NV center can effectively scan over the sample and detects local magnetic fields at every scan position on the surface. Figure 12b shows examples of the scan images on various magnetic samples including superconducting vortex [32, 33], and multiferroic materials . Since the NV center can operate from room temperature down to cryogenic temperatures, this novel method provides an efficient way to study temperature-dependent evolution of magnetic orders in exotic materials.
4.2 Wide field-of-view optics for life science
The diamond SPM method provides high resolution imaging but is quite slow due to the scanning process (e.g. a few hours per image) and is not an efficient method to study dynamical features. Fast magnetic imaging can be realized by simultaneous mapping the sample with wide field-of-view optics. While confocal optics used in the SPM collects photons only from a focused spot, wide field-of-view optics records optical signals from every point within the optical field-of-view (e.g. ) with a CCD (Charge-coupled device) camera. Even though the spatial resolution is not as good as the SPM methods and is diffraction limited, the faster imaging capability gives more advantages when one studies biological samples.
In the diamond wide field-of-view optics experiment, ensembles of NV center are typically used to enhance the field sensitivity and biological samples are placed on a bulk diamond containing NV ensembles. Simultaneous excitation of a number group of NV centers within the field-of-view requires high power of pumping laser. In order to avoid potential damage to the bio-sample due to the high power laser beam, total internal reflection fluorescence (TIRF) technique is typically adopted . With the TIRF configuration, the excitation laser from the backside of the diamond is totally reflected at the interface between the diamond and the sample and is only illuminated onto the NV ensembles.
Figure 13a shows an example of the wide field-of-view magnetic imaging on biological samples e.g. bacteria called magnetotactic bacteria (MTB) . MTB contain magnetic nano-particles in the body forming one dimensional chain that and are aligned along the external magnetic field. The diamond wide field-of-view optics successfully maps the stray field produced by a single MTB and identifies orientations of the magnetic chains from the field distribution.. Motivated by this work, scientists try to identify cancer cells among normal cells by selectively dosing magnetic nano-particles into the target cells and imaging the resulting magnetic field .
Owing to the capability of bio-imaging, the NV center gets increasing attentions in the field of neural science. Since neurons communicate each other via the motions of ions, it can be viewed as current flows in a conducting wire which produces magnetic field around it. By mapping the induced magnetic field with the NV center, therefore, one can study brain activities in the neural networks. As a first step toward this goal, recent experiment successfully demonstrates detection of magnetic field due to the action potential along an axon ( Figure 13b ) . The NV center is also used to probe nuclear magnetic resonance (NMR) signal from a single protein  which opens up the possibilities of MRI at the nanometer-scale  ( Figure 13c ).
In this chapter, we introduce the diamond NV center as a novel magnetometer satisfying high magnetic field sensitivity and high spatial resolution. We review the basic working principles of sensing dc and ac magnetic field and discuss two imaging applications which are based on scanning magnetometry and wide field-of-view optics techniques. The excellent properties of NV centers such as sub-nanotesla sensitivity, nanometer-scale resolution, gigahertz range of detection bandwidth, wide range of the operation temperature, non-toxic and bio-friendly host material position the NV center as a unique tool of sensing and imaging microscopic magnetic phenomena. Improving sensing protocols and imaging techniques is on-going efforts in this field. In conclusion, the novel magnetometer introduced in this chapter has a promising potential to be used in various research fields particularly solid-state physics and life science.
The authors acknowledge the support from the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1A02937119) and the Korea University Future Research Grant (K1822781).