Spectroscopy of photoionization from the ${}^{1}E$ singlet state in nitrogen–vacancy centers in diamond

The nitrogen vacancy (N$V$) center in diamond has emerged as an important platform for quantum technology.[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] The negative charged state (N$V^{-}$) garners the most interest due to its $S=1$ triplet ground state. These ground state sublevels have long spin coherence times and can be polarized and read out optically at room temperature.[13, 14].

Initialization and readout of the N$V^{-}$ spin state involves transitions between four electronic states within the diamond band gap: the S = 1 spin triplet ground (${}^{3}A_{2}$) and excited (${}^{3}E$) electronic states, and the S = 0 spin singlet metastable (${}^{1}E$) and excited (${}^{1}A_{1}$) electronic states (Fig. 1(a)), which we refer to as triplet and singlet states respectively. For illumination wavelengths between 470 nm and 637 nm, N$V^{-}$ centers will absorb photons and transition from ground to the excited ${}^{3}E$ states while preserving $m_{z}$. Relaxation back to the ground state can occur directly by emitting a photon, or indirectly via the inter-system crossings (ISCs) and a transition between singlet states, typically without emitting a photon (Fig. 1(a)). Importantly, the ISC path is more probable for $m_{z}=\pm 1$ than for $m_{z}=0$.[6] The spin-dependence of the upper ISC makes it possible to prepare the $m_{z}=0$ spin state with good fidelity and to read it out via photoluminescence (PL). The lower ISC is a slow process that gives the metastable ${}^{1}E$ state a lifetime that is an order of magnitude longer than any other excited state.[6]

The ${}^{3}A_{2}$—${}^{3}E$ and ${}^{1}E$—${}^{1}A_{1}$ transition energies in the N$V^{-}$ are determined by the photon energies of the zero phonon lines (ZPL) present in the absorption and PL spectra near 1.95 eV and 1.190 eV respectively.[15, 6, 16, 17, 18] The single-photon ionization and recombination thresholds place the N$V^{-}$ ground state 2.60 eV below the conduction band and 2.94 eV above the valence band.[19]

Using ionization thresholds to place the ${}^{1}A_{1}$ and ${}^{1}E$ singlet states within the bandgap has proven more difficult. Previous studies have focused on measuring the ${}^{1}A_{1}$—${}^{3}E$ and ${}^{3}A_{2}$—${}^{1}E$ energy gaps, labeled $\Delta$ and $\Sigma$ respectively in Fig. 1 (a). Estimates of $\Delta$ at cryogenic temperatures and temperatures above 300 K have been calculated by relating ISC rates to these energy gaps, and values of $\Sigma$ have been inferred using these calculations for $\Delta$ and the ZPL energy of the ${}^{1}E$—${}^{1}A_{1}$ transition without investigating $\Sigma$ directly. At cryogenic temperatures, $\Delta$ is estimated to lie between 0.321 eV and 0.414 eV,[20, 21, 22] and earlier work done at high temperature is at variance with these results, yielding an estimate of $\Delta\approx 0.8$ eV.[23] These differing results suggest a temperature dependent phenomena inherent to the ${}^{1}A_{1}$—${}^{1}E$ singlet manifold. This is the subject of investigation in this work.

Often these values of $\Delta$ are used in conjunction with the ZPL energy of the ${}^{1}E$—${}^{1}A_{1}$ transition and the ionization energy of the ${}^{3}A_{2}$ to locate the ${}^{1}E$ with respect to the diamond conduction band edge, however a recent publication indicates that this common technique is inappropriate for accurately determining the ${}^{1}E$ ionization energy as it erroneously treats the inherently multi-particle energy levels of the N$V^{-}$as single-particle energy levels.[7] In light of this new information, the ${}^{1}E$ ionization energy must be directly measured, not inferred from existing data. To accomplish this, the ${}^{1}E$ state can be selectively populated using a magnetic field to mix the $m_{z}=0$ and $m_{z}=-1$ triplet sublevels while under laser excitation, and then photoionized with wavelength tuned laser excitation, which would allow the ionization energy of the ${}^{1}E$ to be measured. This technique can be applied in a temperature controlled cryostat in order to determine the ${}^{1}E$ ionization energy as a function of temperature.

Ionization of N$V^{-}$ results in an increase of N$V^{0}$ concentration and a corresponding increase in N$V^{0}$ PL. The N$V^{-}$ system can be polarized into the ${}^{1}E$ by exploiting a magnetic field[24] or microwave resonance to increase the population of the ${}^{3}E$ $m_{z}=\pm 1$ states, which can then relax to of ${}^{1}E$. The ${}^{1}E$ can then be selectively ionized by applying laser excitation to the system and reading out the change in N$V^{0}$ PL.[25, 24, 26, 27]

In the present work, we use field modulation to investigate ionization dynamics as a function of temperature, wavelength, and optical power. The diamond has a high concentration of N$V$ centers suitable for ensemble measurements of interest to this work. As shown in Fig. 1(b), population transfers from the triplet ground to the ${}^{1}E$ singlet state when laser excitation is applied in the presence of a magnetic field. We exploit the fact that the PL spectra of N$V^{-}$ and N$V^{0}$ behave differently under the influence of a magnetic field to track their relative defect populations and identify ionization thresholds. We find that in the absence of a magnetic field, the fraction of N$V$ centers in the N$V^{-}$ charge state is approximately constant for photon energies lower than a threshold between 2.54 eV and 2.60 eV, which we identify as the ionization energy of the ${}^{3}A_{2}$ ground state.[19] At energies below that threshold, the concentration of N$V^{0}$ increases by a few percent when a spin-mixing magnetic field is applied, due to increased ionization from the newly populated ${}^{1}E$ state. A threshold for the N$V^{0}$ contrast between 532 nm and 550 nm brackets the singlet state ionization energy between 2.25 eV and 2.33 eV for temperatures between 1.6 K and 300 K.

The relative PL amplitude of the N$V^{-}$ and N$V^{0}$ yields information about the charge state of N$V$ centers. The fraction of the integrated PL amplitude contributed by N$V^{-}$ in zero applied field is plotted in Fig. 4. Figs. 4(a) and (b) both show a dropoff in N$V^{-}$PL amplitude between 476 nm and 488 nm (2.605 eV and 2.55 eV respectively). This threshold is consistent with ionization of N$V^{-}$ from its ground state to N$V^{0}$ with an optically measured threshold near 2.6 eV (477 nm)[19], and a photoconductivity threshold between 2.6 eV and 2.7 eV[40]. Calculated thresholds include 2.67 eV,[41], 2.7 eV.[40], and 2.76 eV[42]. The threshold for ionization of the ${}^{3}A_{2}$ appears to shift to lower photon energies and/or broaden at the higher temperatures. A slight decreasing trend with power at all excitation wavelengths in Fig. 44(b) shows that the N$V^{-}$ PL fraction is only weakly power dependent. We therefore consider only the error-weighted average over all powers when determining ionization thresholds from PL amplitudes as a function of temperature and wavelength.

We next compare the measured PL fraction with several models of charge dynamics, all of which assume that the integrated PL amplitude for N$V^{-}$ and N$V^{0}$ are proportional to the species concentrations $c_{NV^{-}}$, $c_{NV^{0}}$ and the excited state populations $n_{ex}^{-}(\lambda_{p})$, and $n_{ex}^{0}(\lambda_{p})$ respectively.

A simple rate-balance photodynamic model of the steady state assumes that the rates of ionization and recombination must be equal,

where $\Gamma_{NV^{-}}$ and $\Gamma_{NV^{0}}$ are the ionization rate of N$V^{-}$ and recombination rate of N$V^{0}$, respectively. For excitation wavelengths greater than 490 nm and powers on the order of 10 $\mu$W, the transition rates are quadratic in laser power and on the order of 10³ s^-1.[19]. The transition rates are modeled as

for photon flux $J$ and cross sections, $\sigma$. Superscripts indicate the charge state of the N$V$ center and subscripts $ex$, $s$ and $g$ refer to excited, singlet and ground states. Because the populations of states other than the ground state are linear in power, the net ionization and recombination rates are quadratic. Ionization from the ground state is linear in power for excitation wavelengths shorter than 485 nm.

We also consider a fixed-concentration model, where the concentrations are only slightly perturbed by quadratic ionization and recombination processes, which agrees empirically with our observations.

Fig. 5 shows the behavior of the N$V^{-}$ PL fraction using rate-balance and fixed-concentration models as a function of excitation wavelength. Measured results are provided for 1.6 K (black) and 295 K (grey fill) data.

The blue dashed line in Fig. 5 is the balanced rate model with ionization allowed only from the excited state. Under this assumption the N$V^{-}$ PL fraction is nearly constant. The green dot-dash curve also includes ionization from the singlet state for wavelengths below 550 nm, and from the ground state below 485 nm. The solid red curve corresponds to the fixed-concentration model with $c_{NV^{-}}=0.66$ and $c_{NV^{0}}=0.34$, and the resulting N$V^{-}$ PL fraction increases smoothly across the pump wavelength range. In view of the poorly determined parameter values, firm conclusions should not be drawn from these models. However, the results are in good agreement with previous measurements of two-step ionization/recombination rates on individual centers.[19]

Qualitative agreement between the fixed-concentration model results and measurement data suggests the N$V^{-}$ and N$V^{0}$ concentrations are approximately unchanged for excitation wavelengths longer than 490 nm in the absence of an external magnetic field. Figs. 4, 5 both suggest an ionization threshold from the ${}^{3}A_{2}$ ground state between 2.54 eV and 2.60 eV (488 nm and 476 nm resp.).

The effects of the applied magnetic field provide additional information about changes in charge state. The off axis field is known to decrease the population of the ${}^{3}E$ excited states, quenching the PL from N$V^{-}$. The field also increases the population in the ${}^{1}E$ singlet state.[6, 38, 36] By changing the populations, the applied field also affects net ionization rates of these states. The decrease in ${}^{3}E$ population and increase in ${}^{1}E$ population induced by the field could increase or decrease the total ionization rate, depending on their respective cross sections.

Fig. 6 shows the contrast parameter $\epsilon$ which describes a reduction in N$V^{-}$ PL with applied field as defined in (Eqns. 1, 2). The contrast $\epsilon$ shows a broad plateau between 100 K and 300 K bounded at shorter wavelengths by the ground state ionization threshold near 480 nm, and by a valley of suppressed contrast between 100 K and 1.6 K. Focusing first on the plateau, $\epsilon$ is expected to be independent of excitation wavelength in the low-power, linear-response regime as long as concentrations are not wavelength-dependent. The decrease below 480 nm coincides with depressed N$V^{-}$ PL due to photoionization of the ${}^{3}A_{2}$ triplet state (shown in Fig. 4).

The suppressed $\epsilon$ contrast shown in Fig. 7 around 50 K and across all wavelengths also appears in earlier results by Rogers et al.[43] and others by Ernst et al. and Happacher et al. contemporaneous with this work.[44, 45, 46] We propose an explanation for this effect related to thermal averaging of the excited state orbitals. At room temperature, the excited state behaves like a single spin triplet. However, at low temperature the ${}^{3}E$ excited state of N$V^{-}$ is an orbital doublet which is split by transverse strain into ${}^{3}E_{x}$ and ${}^{3}E_{y}$ electronic states separated by tens of GHz.[47, 48] At temperatures above 100 K, these spin triplets are thermally averaged, and the excited state appears as one triplet state.[43, 49, 46]. To explain the suppressed contrast, we propose that the thermal averaging process produces an effective field noise that causes fast relaxation of the spin states at intermediate temperatures. The spin relaxation rate will have a maximum when the characteristic switching time is at the GHz frequencies corresponding to transitions between the spin triplet sublevels. This spin-disordering mechanism will reduce polarization into the $m_{z}=0$ state in zero field, so further mixing due to an applied field will produce a suppressed PL quenching effect. Further mixing and PL quenching due to an applied field will have a suppressed effect. In support of this mechanism, we note that shallow minima in the zero-field N$V^{-}$ PL fraction are visible in the same 50 K region as the suppressed contrast $\epsilon$. See Fig. 4(a). A thorough study of the contrast temperature dependence has recently been made public.[44].

The N$V^{0}$ has no known spin-dependent relaxation mechanisms, so its optical properties are considered immune to modest magnetic field. The contrast parameter $\delta$ is therefore an indicator of changes in N$V^{0}$ concentration. The field-immunity of N$V^{0}$ also implies field-independent recombination mechanisms, so we attribute $\delta$ to changes in N$V^{0}$ concentration induced by a field-dependent change in ionization rate of the N$V^{-}$. The plots of $\delta$ in Figs. 7(a) and (c) show a region of negative contrast between approximately 480 nm and 540 nm, strongest below 100 K. Negative contrast here corresponds to an increase in N$V^{0}$ PL induced by the applied magnetic field.

As shown in Fig. 1(b), the population of the excited state decreases with applied field and the population of the singlet state sees a corresponding increase, strongly implying that the negative values of $\delta$ are due to ionization from the singlet state. This result adds to a growing body of experimental evidence [26, 27, 24] indicating a wavelength dependent increase in N$V^{0}$ population is a direct result of resonant CW and pulsed microwave excitation[24, 26, 27] or DC field quenching (this work) inducing an increase in $\ket{\pm 1}$ spin state population.

Regions of negative $\delta$ in Figs. 7(a) and (c) constitute the main results of this paper. The low wavelength boundary of the negative=$\delta$ region near 480 nm coincides with the abrupt dropoff in N$V^{-}$ PL, which we interpret as the threshold for ionization from the ground state. We interpret the high wavelength boundary of the negative-$\delta$ region near 540 nm to be a threshold for ionization from the ${}^{1}E$ long-lived singlet state. The threshold appears between excitation wavelengths of 532 nm and 550 nm, bracketing the singlet ionization threshold between 2.25 eV and 2.33 eV. At 200 K and above, the negative contrast weakens significantly or changes sign, indicating a change in ionization rate as a function of temperature that is likely a result of multiple competing processes, including decreased lifetime of the singlet state,[18, 37, 16], thermal occupation of vibronic states of the ${}^{1}A_{1}$ and ${}^{1}E$,[50] temperature dependent shifts in valence and conduction band energies,[7] and small shifts in the singlet state energies within the band gap.[7]

The increase in N$V^{0}$ concentration indicated by negative values of $\delta$ suggests reactions that reduce the number of charged centers in the diamond. Ionization, ${\rm N}V^{-}\rightarrow{\rm N}V^{0}+e^{-}$, neutralizes an N$V$ center, and if the freed electron then goes on to neutralize a N${}_{\rm s}^{+}$ ion, the total number of charged centers in the sample is reduced by 2. The accompanying reduction in random electric fields would be observable as a narrowing of ZPL peaks as Stark shift broadening is reduced.[51]

Figs. 7(b) and (d) show the fractional change in the width of the N$V^{0}$ ZPL with applied field; negative values correspond to narrowing. The results shown in Figs. 7(b) and (d)) include a wavelength region above 488 nm for temperatures below 150 K where the width of the N$V^{0}$ ZPL line decreases by $\approx$2 % with applied field, in agreement with the $\delta$ contrast for that region. This line width change and the $\delta$ contrast both point to significant ionization from the singlet state.

We do not attempt to explain the anomalous behavior at 100 K in Figs. 7(c) and (d), which includes one sharp peak at certain wavelengths below 490 nm and adjacent pit above 490 nm. These anomalies appear consistently over different laser powers and the different wavelengths were measured on different days.

Surface plots of the results presented in Fig. 7 are included in the supplemental material for a synoptic overview of these results.

The N$V$ center in diamond has become a canonical quantum system, featured in work on fundamental quantum effects, quantum sensing and physics education. An energy level diagram similar to Fig. 1(a) appears in countless publications. So, it may be surprising that there are parts of the N$V^{-}$ energy level diagrams that are not well-known. In particular, the energies of the singlet states relative to the triplet states have proven difficult to measure. In this work, we probed the ionization dynamics of N$V^{-}$ centers using applied magnetic field modulation. We have identified ionization thresholds for both the ${}^{3}A_{2}$ ground state and the long-lived ${}^{1}E$ singlet state, and discovered a temperature and wavelength dependence of the ionization rate from the ${}^{1}E$ singlet, which has not been measured previously. Knowledge of ionization pathways will be important as new research in electrical readout and spin-charge conversion strives to circumvent the poor optical readout characteristics of N$V$ centers.

For temperatures between 2 K and 200 K, the ionization threshold of ${}^{1}E$ was found between 2.33 eV and 2.25 eV (excitation wavelengths 532 nm and 550 nm resp.), which is in reasonable agreement with 2.22 eV predicted by ab initio calculations.[42] For temperatures above 200 K the ionization rate appears to change as a function of temperature. Ionization from the singlet state is identified by increases in N$V^{0}$ PL with applied field, supported by narrowing of the ZPL linewidth due to the Stark effect. Combined with our observation of a ground state ionization threshold between 2.54 eV and 2.60 eV, we bracket the energy difference 0.21 eV $<\Sigma<$ 0.35 eV, which is in reasonable agreement with prior results inferring this value from values of $\Delta$. [20, 21, 22, 16].