# Nitrogen-vacancy center

The nitrogen-vacancy center (N-V center) is one of numerous point defects in diamond. Its most explored and useful property is photoluminescence, which can be easily detected from an individual N-V center, especially those in the negative charge state (N-V). Electron spins at N-V centers, localized at atomic scales, can be manipulated at room temperature by applying a magnetic field, electric field, microwave radiation or light, or a combination, resulting in sharp resonances in the intensity and wavelength of the photoluminescence. These resonances can be explained in terms of electron spin related phenomena such as quantum entanglement, spin-orbit interaction and Rabi oscillations, and analysed using advanced quantum optics theory. An individual N-V center can be viewed as a basic unit of a quantum computer, and it has potential applications in novel, more efficient fields of electronics and computational science including quantum cryptography and spintronics.

## Structure

Primitive picture of the N-V center

The nitrogen-vacancy center is a point defect in the diamond lattice. It consists of a nearest-neighbor pair of a nitrogen atom, which substitutes for a carbon atom, and a lattice vacancy.

Two charge states of this defect, neutral N-V0 and negative N-V, are known from spectroscopic studies using optical absorption,[1][2] photoluminescence (PL),[3] electron paramagnetic resonance (EPR)[4][5][6] and optically detected magnetic resonance (ODMR),[7] which can be viewed as a hybrid of PL and EPR; most details of the structure originate from EPR. A nitrogen atom has five valence electrons. Three of them covalently bond to the carbon atoms and two remain non-bonded and are called a lone pair. The vacancy has three unpaired electrons. Two of them make a quasi covalent bond and one remains unpaired. The overall symmetry, however, is axial (trigonal C3V); one can visualize this by imagining the three unpaired vacancy electrons continuously exchanging their roles.

The N-V0 thus has one unpaired electron and is paramagnetic. However, despite extensive efforts, electron paramagnetic resonance signals from N-V0 avoided detection for decades until 2008. Optical excitation is required to bring the N-V0 defect into the EPR-detectable excited state; the signals from the ground state are presumably too broad for EPR detection.[8]

In the negative charge state N-V, an extra electron is located at the vacancy site forming a spin S=1 pair with one of the vacancy electrons. As in N-V0, the vacancy electrons are "exchanging roles" preserving the overall trigonal symmetry. This N-V state is what is commonly, and somewhat incorrectly, called "the nitrogen-vacancy center". The neutral state has not yet been explored for spin manipulations.

The N-V centers are randomly oriented within a diamond crystal. Ion implantation techniques can enable their artificial creation in predetermined positions.[9]

## Production

Nitrogen-vacancy centers are typically produced from single substitutional nitrogen centers (called C or P1 centers in diamond literature) by irradiation followed by annealing at temperatures above 700 °C.[1] A wide range of high-energy particles are suitable for such irradiation, including electrons, protons, neutrons, ions, and gamma photons. Irradiation produces lattice vacancies, which are a part of N-V centers. Those vacancies are immobile at room temperature, and annealing is required to move them. Single substitutional nitrogen produces strain in the diamond lattice;[10] it therefore efficiently captures moving vacancies,[11] producing the N-V centers.

During chemical vapor deposition of diamond, a small fraction of single substitutional nitrogen impurity (typically <0.5%) traps vacancies generated as a result of the plasma synthesis. Such nitrogen-vacancy centers are preferentially aligned to the growth direction.[12]

Diamonds are notorious by having relatively large lattice strain. Strain splits and shifts optical transitions from individual centers resulting in broad lines in the ensembles of centers.[1] Special care is taken to produce extremely sharp N-V lines (line width ~10 MHz)[13] required for most experiments: high-quality, pure natural or better synthetic diamonds (type IIa) are selected. Many of them already have sufficient concentrations of grown-in N-V centers and are suitable for applications. If not, they are irradiated by high-energy particles and annealed. Selection of a certain irradiation dose allows tuning the concentration of produced N-V centers such that individual N-V centers are separated by micrometre-large distances. Then, individual N-V centers can be studied with standard optical microscopes or, better, near-field scanning optical microscopes having sub-micrometre resolution.[7][14]

## Basic optical properties

Optical absorption and emission of the N-V center at room temperature

N-V centers emit bright red light which can be conveniently excited by visible light sources, such as argon or krypton lasers, frequency doubled Nd:YAG lasers, dye lasers, or He-Ne lasers. Excitation can also be achieved at energies below that of the zero phonon emission.[15] Laser illumination, however, also converts some N-V into N-V0 centers.[3] Emission is very quick (relaxation time ~10 ns).[16][17] At room temperature, no sharp peaks are observed because of the thermal broadening. However, cooling the N-V centers with liquid nitrogen or liquid helium dramatically narrows the lines down to few megahertz width.

An important property of the luminescence from individual N-V centers is its high temporal stability. Whereas many single-molecular emitters bleach after emission of 106–108 photons, no bleaching is observed for the N-V centers at room temperature.[7][14]

Because of these properties, the ideal technique to address the N-V centers is confocal microscopy, both at room temperature and at low temperature. In particular, low temperature operation is required to specifically address only the zero-phonon line (ZPL).

## Energy level structure and its manipulation by external fields

Schematic energy level structure of the N-V center. Electron transitions between the ground 3A and excited 3E states, separated by 1.945 eV (637 nm), produce absorption and luminescence. The 3A state is split by 1027 gauss[4][5] (~12 µeV) and the 3E state by 508 gauss[18] (~5.9 µeV). Numbers 0, ±1 indicate spin value; splitting due to the orbital degeneracy is not shown.

The energy level structure of the N-V center was established by combining optical, electron paramagnetic resonance and theoretical results, as shown in the figure. In particular, several theoretical works have been done, using the Linear Combination of Atomic Orbitals (LCAO) approach, to build the electronic orbitals to describe the possible quantum states, looking at the NV center as a molecule. Moreover, group theory results are used, to take into account the symmetry of the diamond crystal, and so the symmetry of the NV itself. The energy levels are labeled according to the group theory, and in particular are labelled after the irreducible representations of the C3V symmetry group of the defect center, A1, A2 and E. The numbers 3 in 3A and 1 in 1A represent the number of allowable ms spin states, or the spin multiplicity, which range from –S to S for a total of 2S+1 possible states. If S = 1, ms can be −1, 0, or 1. The 1A level is predicted by theory but not directly observed in experiment[citation needed], and it is believed to play an important role in the quenching of photoluminescence.

In the absence of an external magnetic field, the ground and excited states are split by the magnetic interaction between the two unpaired electrons at the N-V center (see microscopic model): when two electrons have parallel spins (ms=±1), their energy is higher than when spins are antiparallel (ms=0). The farther apart the electrons are, the weaker their interaction energy D (roughly D ~1/r3).[5] Thus the smaller splitting in the excited state can be viewed in terms of larger electron-electron separation in the excited state. When an external magnetic field is applied to the N-V center, it does not affect the ms=0 states nor the 1A state (because it has S = 0), but it splits the ms = ±1 levels. If a magnetic field is oriented along the defect axis and reaches about 1027 G (or 508 G) then the ms = –1 and ms = 0 states in the ground (or excited) state become equal in energy; they strongly interact resulting in so-called spin polarization, which strongly affects the intensity of optical absorption and luminescence transitions involving those states.[18]

In order to understand why this happens, we have to keep in mind that transitions between electronic states are mediated by a photon which cannot change overall spin. Thus optical transitions must preserve the total spin and occur between levels of the same total spin. For this reason, transitions 3E↔1A and 1A ↔ 3A are non-radiative and quench the luminescence. Whereas ms = −1 (excited state) ↔ ms = 0 (ground state) transition was forbidden in the absence of an external magnetic field, it becomes allowed when a magnetic field mixes the ms = −1 and ms = 0 levels in the ground state. As a measurable outcome of this phenomenon, luminescence intensity can be strongly modulated by magnetic field.

An important property of the non-radiative transition between 3E and 1A is that it is stronger for ms = ±1 and weaker for ms = 0. This property results in a very useful manipulation of N-V center, which is called optical spin-polarization. First, an off-resonance excitation which has a higher frequency (typically 2.32 eV (532 nm)) than the frequencies of all transitions and thus lays in the vibronic bands for all transitions. By using a pulse of this wavelength, people can excite all spin states and create phonons as well. For a spin state with ms = 0, due to conservation of spin in transition, it will be excited to the corresponding ms = 0 state in 3E and then go back to original state. However, for a spin state with ms = ±1 in 3A, after the excitation, it has a relatively high probability to jump to the intermediate state 1A by non-radiative transition and go to the ground state with ms = 0. After sufficient cycles, the state of the NV center can be regarded as in the ms = 0 state. Such a process can be used in the initialization of quantum state in quantum information processing.

There is an additional level splitting in the excited 3E state due to the orbital degeneracy and spin-orbit interaction. Importantly, this splitting can be modulated by applying a static electric field,[13][19] in a similar fashion to the magnetic field mechanism outlined above, though the physics of the splitting is somewhat more complex. Nevertheless, an important practical outcome is that the intensity and position of the luminescence lines can be modulated by applying electric or/and magnetic fields.

The energy difference between the ms = 0 and ms = ±1 states corresponds to the microwave region. Thus by irradiating the N-V centers with microwave radiation, one can change the relative population of those levels, thereby again modulating the luminescence intensity.

There is an additional splitting of the ms = ±1 energy levels, which originates from the "hyperfine" interaction between the nuclear and electron spins. Thus finally, the optical absorption and luminescence from the N-V center consists of roughly a dozen sharp lines with a separation in the MHz-GHz range, and all those lines can be resolved, given proper sample preparation. The intensity and position of those lines can be modulated using the following tools:

1. Amplitude and orientation of magnetic field, which splits the ms = ±1 levels in the ground and excited states.
2. Amplitude and orientation of elastic field (strain), which can be applied by, e.g., squeezing the diamond. Similar effects can be induced by applying electric field,[13][19] and the electric field can be controlled with much higher precision.
3. Continuous-wave microwave radiation, which changes the population of the sublevels within the ground and excited state.[19]
4. Tunable laser, which can selectively excite certain sublevels of the ground and excited state.[19][20]
5. In addition to those static perturbations, numerous dynamic effects (spin echo, Rabi oscillations, etc.) can be exploited by applying a carefully designed sequence of microwave pulses.[21][22][23][24][25] The first pulse coherently excites the electron spins, and this coherence is then manipulated and probed by the subsequent pulses. Those dynamic effects are rather important for practical realization of quantum computers, which ought to work at high frequency.

As a final remark, it should be noted that the above-described energy structure is by no means exceptional for a defect in diamond or other semiconductor.[26] It was not this structure alone, but a combination of several favorable factors (previous knowledge, easy production and excitation, etc.) which suggested the use of the N-V center.

## Spin dynamics

Diagram for spin dynamics in the NV center on diamond. The primary transition between triplet ground and excited states is predominantly spin conserving. Decay via the intermediate singlets gives rise to spin polarization by preferentially switching spin from ms = ±1 to ms = 0. Both absorption and emission wavelengths are indicated,[27] since they differ due to Stokes and anti-Stokes shift. The diagram was mainly inspired in the one present on Rogers et al.[28]

Thinking of the NV center as a multielectronic system, we can draw the diagram in the figure at right, where the states are labeled according to their symmetry and with a left superscript that indicates with a 3 if it is a triplet (S=1) and with a 0 if it is a singlet (S=0). It is well accepted today that we have two triplet states and two intermediate singlet states.[29]

The optical excitations conserve the spin state, but there is a high probability of the states ${\textstyle \left|^{3}{\text{E}},\pm 1\right\rangle }$ decaying non-radiatively to the singlet state ${\textstyle \left|^{1}{\text{A}}_{1}\right\rangle }$, a phenomenon called intersystem crossing (ISC). This happens at an appreciable rate because the energy curve in function of the position of the atoms for the ${\textstyle \left|^{3}{\text{E}},\pm 1\right\rangle }$ state intersects the curve for the ${\textstyle \left|^{1}{\text{A}}_{1}\right\rangle }$ state. Therefore, for some instant during the vibrational relaxation that the ions undergo after the excitement, it is possible for the spin to flip with little or no energy required in the transition.[30] It is important to note that this mechanism also leads to a transition from ${\textstyle \left|^{3}{\text{E}},0\right\rangle }$ to ${\textstyle \left|^{1}{\text{A}}_{1}\right\rangle }$, but the rate of this ISC is much lower than the ${\textstyle \left|^{3}{\text{E}},\pm 1\right\rangle }$ states rate, therefore this transition is indicated with a thin line. The diagram also shows the non-radiative and infrared competing decay paths between the two singlet states, and the fine splitting in the triplet states, whose differences in energy correspond to microwave frequencies.

Some authors explain the dynamics of the NV center by admitting that the transition from ${\textstyle \left|^{1}{\text{E}}\right\rangle }$ to ${\textstyle \left|^{3}{\text{A}}_{2},\pm 1\right\rangle }$ is small, but as Robledo et al. shows,[31] only the fact that the probability of decaying to ${\textstyle \left|^{1}{\text{A}}_{1}\right\rangle }$ is smaller for ${\textstyle \left|^{3}{\text{E}},0\right\rangle }$ than for ${\textstyle \left|^{3}{\text{E}},\pm 1\right\rangle }$ is enough to polarize the spin to ms = 0.

## Potential applications

The spectral shape and intensity of the optical signals from the N-V centers are sensitive to external perturbation. This phenomenon has been explored in prototype ultrasensitive sensors of strain, electric and magnetic field. Such sensors can detect magnetic fields of a few nanotesla[32] or electric fields of about 10 V/cm[33] at kilohertz frequencies after 100 seconds of averaging. This sensitivity allows detecting a magnetic or electric field produced by a single electron located tens of nanometers away from an N-V center. Another possible use is as a detector to measure the full mechanical stress tensor in the bulk of the crystal. For this application, the stress-induced splitting of the zero-phonon-line is exploited, and its polarization properties.[34]

In addition to the quantum optical applications, luminescence from the N-V centers can be applied for imaging biological processes, such as fluid flow in living cells.[35] This application relies on good compatibility of diamond nano-particles with the living cells and on favorable properties of photoluminescence from the N-V centers (strong intensity, easy excitation and detection, temporal stability, etc.). Compared with large single-crystal diamonds, nanodiamonds are cheap (about 1 USD per gram) and available from various suppliers. N-V centers are produced in diamond powders with sub-micrometre particle size using the standard process of irradiation and annealing described above. Those nanodiamonds are introduced in a cell, and their luminescence is monitored using a standard fluorescence microscope.[36]

Further N-V center has been hypothesized to be a potential bio-mimetic system for emulating radical pair spin dynamics of the avian compass.[37]

## Historical remarks

The microscopic model and most optical properties of ensembles of the N-V centers have been firmly established in the 1970s based on the optical measurements combined with uniaxial stress[1] and on the electron paramagnetic resonance.[4][5] However, a minor error in EPR results (it was assumed that illumination is required to observe N-V EPR signals) resulted in the incorrect multiplicity assignments in the energy level structure. In 1991 it was shown that EPR can be observed without illumination,[6] which established the energy level scheme shown above. The magnetic splitting in the excited state has been measured only recently.[18]

The characterization of single N-V centers has become a very competitive field nowadays, with many dozens of papers published in the most prestigious scientific journals. One of the first results was reported back in 1997.[7] In that paper, it was demonstrated that the fluorescence of single N-V centers can be detected by room-temperature fluorescence microscopy and that the defect shows perfect photostability. Also one of the outstanding properties of the NV center was demonstrated, namely room-temperature optically detected magnetic resonance.

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