Longitudinal spin relaxation in nitrogen-vacancy ensembles in diamond
- Mariusz Mrózek^{1}Email author,
- Daniel Rudnicki^{1},
- Pauli Kehayias^{2},
- Andrey Jarmola^{2},
- Dmitry Budker^{2, 3} and
- Wojciech Gawlik^{1}Email author
Received: 14 May 2015
Accepted: 13 October 2015
Published: 29 October 2015
Abstract
We present an experimental study of the longitudinal electron-spin relaxation of ensembles of negatively charged nitrogen-vacancy (NV^{−}) centers in diamond. The measurements were performed with samples having different NV^{−} concentrations and at different temperatures and magnetic fields. We found that the relaxation rate \(T_{1}^{-1}\) increases when transition frequencies in NV^{−} centers with different orientations become degenerate and interpret this as cross-relaxation caused by dipole-dipole interaction.
Keywords
PACS Codes
1 Introduction
Nitrogen-vacancy (NV^{−}) color centers are point defects in the diamond lattice, which consist of a substitutional-nitrogen atom adjacent to a lattice vacancy. They possess nonzero electron spin (\(S=1\)) and can be optically initialized and read out, which allows numerous applications including electric-field, magnetic-field, pressure, and temperature sensing, as well as nanoscale NMR [1–10]. Nanodiamonds containing NV^{−} centers are non-toxic, photostable, and can be easily functionalized, meaning they can be used as fluorescent markers or sensors in biological materials [11–17].
Successful applications require sound knowledge of interaction of the NV^{−} centers with environment. This motivates recent efforts to study the relaxation dynamics of the NV^{−} electron-spin polarization. Particularly, measurements of the longitudinal relaxation time \(T_{1}\) (the decay lifetime for NVs population initialized to a ground-state magnetic sublevel) might result in development of new techniques to use NV^{−} \(T_{1}\) as a spin probe, for example, in biological systems [18–24].
The temperature and magnetic dependences of the longitudinal electron-spin relaxation time \(T_{1}\) have been experimentally studied previously [25, 26]. In particular, in Ref. [25] the \(T_{1}\) relaxation was studied through the temperature dependence of the decay of the optically enhanced EPR signal, hinting that two-phonon Raman- and Orbach-type processes are the main relaxation mechanisms. The results of Ref. [26] revealed temperature independent longitudinal relaxation at low temperatures where the \(T_{1}\) relaxation is magnetic field dependent and strongly affected by the cross-relaxation between differently aligned NV^{−} centers as well as between NV^{−} centers and substitutional nitrogen impurities (known as the P1 centers).
In this paper we build upon previous work to improve our understanding of NV^{−} longitudinal relaxation in ensembles as a function of the strength and direction of the magnetic field. Following Ref. [26] we propose a qualitative description of the mechanisms responsible for the \(T_{1}\) relaxation at zero magnetic field and near 595 G in terms of the dipole-dipole interactions between the neighboring centers. The dipole-dipole interactions were also considered responsible for narrow luminescence resonances detected as a function of magnetic field in Ref. [27, 28].
In Section 2 we present the experimental setup and the method of measuring the longitudinal relaxation rates; in Section 3 we describe our results of studying the magnetic-field dependence of the relaxation rates. We concentrate on the effects of varying the direction of the magnetic field and on the zero-field relaxation resonance. Section 4 is devoted to a discussion of the cross-relaxation and analysis of the zero-field resonance in terms of magnetically enhanced dipole-dipole interaction. The paper is concluded in Section 5 and supplemented by appendix which presents results of temperature measurements obtained for several samples.
2 Experimental
Details on sample preparation and characteristics
Sample | Synthesis | [N] (ppm) | [NV ^{ − } ] (ppm) | Radiation dose (cm ^{ −2 } ) | Annealing (°C) |
---|---|---|---|---|---|
E1 | HPHT | <200 | 18 | 10^{18} | 650 |
E2 | HPHT | <200 | 20 | 10^{18} | 650 |
S5 | HPHT | <200 | 40 | 8 × 10^{17} | 700 |
W5 | HPHT | <200 | 2 | 1.5 × 10^{17} | 750 |
E8 | CVD | <1 | 0.02 | 10^{16} | 650 |
The measurements were conducted in a range of temperatures between 10 and 400 K. For driving transitions between the spin states of the NV^{−} ground state, a MW field was applied to the NV^{−} ensemble by either a copper wire 70 μm in diameter, placed on the top of diamond surface (samples E2, S5, W5) or by a stripline printed circuit placed inside the cryostat directly under the sample (samples E1, E8) [30].
Our study was performed with magnetic fields of 0 to 400 G. The field direction was controlled with a system of three pairs of Helmholtz coils or with a permanent magnet aligned along a chosen direction and mounted at an adjustable distance from the sample.
3 Results
3.1 Magnetic-field dependence
The axis of NV^{−} center is parallel to one of the [111] crystallographic directions in a diamond crystal so there are four alignments of NVs that constitute four spatial subensembles. For the field directed along [111], one spatial subensemble of NVs is parallel to the field and three subensembles are at the angle of ≈109° to B. Considering their electronic structure, the three subensembles are degenerate. In a field along [100] all NVs have the same angle about 55° with B.
At fields of 0 and ∼595 G, the transition frequencies become degenerate for various orientations of the crystallographic axes, i.e. at line crossings of appropriate transitions between magnetic sublevels. As described below, the relaxation rates exhibit resonant increase at these crossing points (similar resonances have been observed near 514 G due to P1 centers [24, 26]).
(a) Direction of the magnetic field
As shown in Figure 5(a), for some directions of the magnetic field between the [100] and [110] and between the [110] and [111] axes some components of the ODMR spectrum become degenerate. When such degeneracy occurs, i.e. when NVs of different orientations (other spatial subensembles) are degenerate, \(T_{1}\) becomes shorter [Figure 5(b)]. Quantitatively: (i) when a fourfold-degenerate resonance I (at [100]) becomes doubly-degenerate then \(T_{1}\) lengthens by a factor of about two [corresponding to halving the relaxation rate as seen in Figure 5(b)]; (ii) when the non-degenerate resonance IIb becomes triply-degenerate (at [111]), \(T_{1}\) shortens by a factor of about four; (iii) when a non-degenerate resonance overlaps with non-degenerate resonances, IIa and IIb (at [110]), \(T_{1}\) time shortens by a factor of about two.
The effect appears analogous to the relaxation resonances that occur when electronic transitions have the same energies at \(\mathrm{B}=0\) and 595 G. The later resonance was studied in Ref. [26] and the former is described in more detail below and discussed in Section 4.
(b) Zero-field resonance
The range of intermediate magnetic fields, i.e. such that the ODMR components are partly overlapping, is associated with an interesting transformation of the shape of the corresponding zero-field relaxation resonances. These shapes are different for the internal and external components of the ODMR spectrum. Particularly, the resonances corresponding to the internal ODMR lines exhibit W-like shapes resembling second derivative of a Lorentzian: the relaxation rate first drops as one approaches the resonance and then peaks around the center at \(\mathrm{B}=0\). We found that these unusual shapes are artefacts caused by our procedure of assigning single relaxation rates to the decay curves discussed in Section 2, corresponding to individual components of the ODMR spectrum. As long as the components are well resolved, we can assign them well-defined relaxation rates. In the range of weak magnetic fields, however, the ODMR components overlap and cannot be assigned with single rates. The width of the overlap of the ODMR components depends on MW power and is visibly broadened at the power level corresponding to reliable measurements of \(T_{1}^{-1}\) (amplifier output +48 dBm). The discussed nonstandard shape of the overlapping zero-field resonances is particularly well seen in the \(\mathrm{T}=77\mbox{ K}\) data in Figure 6(c) where shading marks the region of overlapping ODMR components. Open symbols mark the relaxation rates associated with three overlapped subensembles.
4 Discussion on cross-relaxation
The NV^{−} cross-relaxation contributes to \(T_{1}\) relaxation, and the effect is enhanced when different NV sub-ensembles are degenerate. It is thus a collective effect characteristic of ensembles, rather than a degeneracy of energies occurring within a single center. In that way it is analogous to the “line crossing” [32] rather than the “level-crossing” effect [33] known in atomic spectroscopy.
The cross relaxation effect is not limited to the crossing at 595 G. Whenever a spatial subensemble of NVs (which can be seen as a single resonance in the ODMR spectra) becomes degenerate with other subensemble, the cross relaxation increases the \(T_{1}^{-1}\) rate. We have demonstrated this effect in Figure 5(b) by varying the angle of the magnetic field (\(\mathrm{B}=50\mbox{ G}\)) relative to the crystallographic directions. Another special case of the transition frequency degeneracy is the line crossing at zero magnetic field. Since we never achieve 100% NV^{−} polarization, the described mechanism can also play important role even in \(\mathrm{B}=0\) field (green arrows in Figure 8).
5 Conclusions
The present study of electron-spin relaxation in NV^{−} diamond performed with various strengths and orientations of a magnetic field reveals the important role of cross-relaxation processes in NV^{−} ensembles. We find distinct resonances of the relaxation rate \(T_{1}^{-1}\) when transition frequencies between the ground-state spin states of NV^{−} centers with different orientations become degenerate. Such degeneracy has been described previously in 595 G [26–28] and attributed to cross-relaxation due to dipole-dipole interaction. In this work we have demonstrated that the effect is general: the cross-relaxation shortens \(T_{1}\) whenever the transition frequencies in NVs with different orientations overlap. In particular, such degeneracy takes place in zero magnetic field and is responsible for the zero-field relaxation resonance
By studying several NV^{−} samples in the [100] direction at 77 K, we have found that measurements of \(T_{1}^{-1}\) in non-zero magnetic field can be used for estimation of the NV^{−} density (Appendix). While the present accuracy of our estimation of the NV^{−} concentration was accurate only to about an order of magnitude, we expect that future work with samples with calibrated P1 and NV^{−} concentrations will enable a practical technique for determining local concentrations from the \(T_{1}^{-1}\) dependences.
The presented interpretation still leaves some open questions and is part of the ongoing investigation aimed at a more complete picture of the \(T_{1}\) relaxation. On the other hand, the results presented in Figure 7, provide confirmation that the zero-field resonance, even if overwhelmed by the phonon background for some samples becomes visible at low temperature and for high NV^{−} density.
By using more samples prepared under different conditions, we aim at getting a more quantitative picture of the longitudinal relaxation in NV^{−} ensembles. The recently developed hole-burning methodology [34] should be useful here. The ability to select a well-defined frequency class in the inhomogeneously broadened ODMR profile with controlled application of strain and magnetic field should provide more information on various contributions to the \(T_{1}\) relaxation.
Declarations
Acknowledgements
We thanks S. D. Chemerisov, K. Aulenbacher and V. Tyukin for irradiating the samples. This work was conducted as part of the Joint Krakow-Berkeley Atomic Physics and Photonics Laboratory and was supported by the NATO Science for Peace Programme (CBP.MD.SFP 983932), and NCN (2012/07/B/ST2/00251), MNSW (7150/E-338/M/2013), POIG (02.01.00-12-023/08 and 02.02.00-00-003/0), by the AFOSR/DARPA QuASAR program, and by the DFG DIP project Ref. FO 703/2-1 1 SCHM 1049/7-1.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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